Integral Equations and Operator Theory - Volume 67

Integr. Equ. Oper. Theory 67 (2010), 1–14 DOI 10.1007/s00020-010-1759-x Published online March 24, 2010 c Birkh¨ auser...

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Integr. Equ. Oper. Theory 67 (2010), 1–14 DOI 10.1007/s00020-010-1759-x Published online March 24, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On the Negative Spectrum of One-Dimensional Schr¨ odinger Operators with Point Interactions N. Goloschapova and L. Oridoroga Abstract. We investigate negative spectra of one-dimensional (1D) Schr¨ odinger operators with δ- and δ -interactions on a discrete set in the framework of a new approach. Namely, using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of Albeverio and Nizhnik (Lett Math Phys 65:27–35, 2003; Methods Funct Anal Topol 9(4):273–286, 2003). For instance, we propose an algorithm for determining the number of negative squares of the operator with δ-interactions. We also show that the number of negative squares of the operator with δ -interactions equals the number of negative strengths. Mathematics Subject Classification (2000). Primary 47A10; Secondary 34L40. Keywords. Schr¨ odinger operator, point interactions, self-adjoint extensions, number of negative squares.

1. Introduction The object of this paper is to investigate some spectral properties of Schr¨ odinger operators with point δ- and δ -interactions in L2 (R). They have the following representations, respectively, d2 d2 αk δk (x), LX,β = − 2 + βk ·, δk δk (x), (1.1) LX,α = − 2 + dx dx k∈I

k∈I

where δk (x) := δ(x − xk ) and δ(x) is the Dirac delta-function, α = {αk }k∈I , β = {βk }k∈I ⊂ R, k ∈ I, and I equals either N or Z. We assume that X = {xk }k∈I ⊂ R is an increasing sequence such that dk := xk+1 − xk > 0, k ∈ I, and d∗ := inf dk > 0, k∈I

d∗ := sup dk < ∞. k∈I

(1.2)

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Schr¨ odinger operators with point interactions have been studied extensively in the last decades (numerous results and a comprehensive list of references may be found in [1,2], see also Appendix K by Exner [1]). In the recent publications [3,4], Albeverio and Nizhnik investigated the numbers κ− (LX,α ) and κ− (LX,β ) of negative squares of the operators LX,α and LX,β in the case |X| = n < ∞. They described κ− (LX,α ) by means of certain continued fractions (cf. [3, Theorem 3]) and proposed an elegant algorithm for determining κ− (LX,α ). In particular, they obtained necessary and sufﬁcient conditions formulated in terms of the distances dk and the strengths αk for the validity of equalities κ− (LX,α ) = n and κ− (LX,α ) = 0 (cf. [3, Theorem 5] and [3, Theorem 4], respectively). Regarding the operators with δ -interactions, it is shown in [4, Theorem 6] that the equality κ− (LX,β ) = n holds if and only if all intensities are negative. The point δ- and δ -interactions can be expressed in terms of “interior boundary conditions”. Namely, Schr¨ odinger operators LX,α and LX,β with d2 δ- and δ -interactions are deﬁned in L2 (R) by the differential expression − dx 2 on the domains, respectively, f (xk +) = f (xk −), 2 D(LX,α ) = f ∈ W2 (R\X) : ,x ∈X , f (xk +) − f (xk −) = αk f (xk ) k D(LX,β ) = f ∈ W22 (R\X) :

(1.3) f (xk +) = f (xk −), ,x ∈X . f (xk +) − f (xk −) = βk f (xk ) k (1.4)

Note that the operators LX,α and LX,β are self-adjoint if (1.2) is satisﬁed ([1], see also [6,9]). We emphasize that the expression (1.1) for LX,α is not formal. It determines LX,α directly. In this paper, we present a new approach to investigate negative spectra of the operators with δ- and δ -interactions on the discrete set X satisfying (1.2). We consider the operators LX,α and LX,β as self–adjoint extensions of the symmetric operator d2 ˚ 22 (R\X), , D(Lmin ) = W X = {xk }k∈I . (1.5) 2 dx Applying the technique of boundary triplets and the corresponding Weyl functions (see [8,13] and also Sect. 2), we establish a connection between the Hamiltonians LX,α and LX,β and certain classes of Jacobi matrices. Using this connection, we describe κ− (LX,α ) and κ− (LX,β ) by means of entries of these matrices (Theorem 3.1). The latter enables us to complete and substantially generalize previous results from [3,4] mentioned above. Namely, for δ-type interactions, we propose an algorithm for determining κ− (LX,α ) (Theorem 3.5). In the case |X| = n, our algorithm differs from the one proposed by Albeverio and Nizhnik, but it is close to that (see Remark 3.11). One of our main results is the following equality κ− (LX,β ) = κ− (β) (Theorem 4.1). It means that the number of negative squares of LX,β equals the number of negative intensities. In the particular case κ− (β) = |X| = n < ∞, this result coincides with [4, Theorem 6]. It is interesting to mention that Lmin = −

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for the operator with δ-interactions such equality does not hold (cf. [3,18]) and, moreover, the operator LX,α with an arbitrary number of negative intensities might be non-negative (see [3, Theorem 4] and also Corollary 3.6). We also obtain sufﬁcient conditions for the inequality κ− (LX,α ) ≥ m, as well as for the equality κ− (LX,α ) = m, with an arbitrary m ∈ N (cf. Theorem 3.3). Our condition differs from the conditions recently obtained by Ogurisu in [18,19], however implies sufﬁcient conditions proposed by Albeverio and Nizhnik [4, Criterion 3] in the case when all intensities are negative, i.e., κ− (α) = |X| = n. The results of the paper were partially announced (without proofs) in [11] and published as a preprint in arXiv.org [12]. Notation. Let H and H stand for the separable Hilbert spaces. [H, H] stands for the space of bounded linear operators from H to H, [H] := [H, H]. The set of closed operators in H is denoted by C(H). Let X be a discrete subset of R. |X| stands for the cardinal number of ˚22 (R\X) we denote the Sobolev spaces the set X. By W 2 (R\X) and W 2

W22 (R\X) ˚22 (R\X) W

:= {f ∈ L2 (R) : f, f ∈ ACloc (R\X), f ∈ L2 (R)}, := {f ∈ W22 (R) : f (xk ) = f (xk ) = 0 for all xk ∈ X}.

2. Preliminaries Boundary triplets and closed extensions. In this section, we recall basic notions of the theory of boundary triplets (we refer to [8,13] for a detailed exposition). Let A be a closed densely deﬁned symmetric operator in H with equal deﬁciency indices n± (A) = dim N±i ≤ ∞, Nz := ker(A∗ − z). Definition 2.1 ([13]). A triplet Π = {H, Γ0 , Γ1 } is called a boundary triplet for the adjoint operator A∗ of A if H is an auxiliary Hilbert space and Γ0 , Γ1 : D(A∗ ) → H are linear mappings such that (i)

the second Green identity, (A∗ f, g)H − (f, A∗ g)H = (Γ1 f, Γ0 g)H − (Γ0 f, Γ1 g)H ,

(ii)

holds for all f, g ∈ D(A∗ ), and the mapping Γ := (Γ0 , Γ1 ) : D(A∗ ) → H ⊕ H is surjective.

Since n+ (A) = n− (A), a boundary triplet Π = {H, Γ0 , Γ1 } for A∗ exists but is not unique [13]. Moreover, dim H = n± (A) and A = A∗ ker(Γ0 ) ∩ of A admits the representation ker(Γ1 ). Further, any proper extension A (see [7]) = D(AC,D ) := D(A∗ ) ker{DΓ1 − CΓ0 }, D(A)

where

C, D ∈ [H]. (2.1)

Note that the representation (2.1) is not unique. In what follows, we will also denote A0 = A∗ ker(Γ0 ). Note that ∗ A0 = A0 .

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Definition 2.2 ([8]). Let Π = {H, Γ0 , Γ1 } be a boundary triplet for A∗ . The Weyl function corresponding to Π is the unique mapping M (·) : ρ(A0 ) → [H] satisfying Γ1 fz = M (z)Γ0 fz ,

for all fz ∈ Nz ,

z ∈ ρ(A0 ).

(2.2)

Before formulating the next results we need the following definition. Definition 2.3 ([14]). Let T = T ∗ ∈ C(H) and let ET (λ) = ET (λ − 0) be the spectral function of T . Dimension of the subspace ET (−∞, 0)H is called a number of negative squares of T and is denoted by κ− (T ). The Weyl function M (·) enables us to describe the number of negative squares of self-adjoint extensions of a non-negative symmetric operator A. Theorem 2.4 ([8]). Let A be a closed densely defined non-negative symmetric operator in H with equal deficiency indices, and let AC,D be its self-adjoint extension. Assume that Π = {H, Γ0 , Γ1 } is a boundary triplet for A∗ such that A0 = AF , where AF is the Friedrichs extension of A. Then the following assertions hold. (i) The strong resolvent limit M (0) := s − R − lim M (x) (see [14, Chapter x↑0

(ii)

8]) exists. If M (0) ∈ [H], then κ− (AC,D ) = κ− (CD∗ − DM (0)D∗ ).

(2.3)

The Sylvester criterion. Description of κ− (LX,α ) is substantially based on the following fact (see, for instance, [17, Lemma 4]). ∗ Proposition 2.5. Let the operator T = T admit the block-matrix represenT11 T12 tation T = with respect to the orthogonal decomposition H = T21 T22 ∗ H1 ⊕ H2 , where T11 ∈ [H1 ], T12 = T12 ∈ [H2 , H1 ] and T22 ∈ C(H2 ). If 0 ∈ ρ(T11 ), then −1 T12 ). κ− (T ) = κ− (T11 ) + κ− (T22 − T21 T11

(2.4)

3. Operators with δ-type interactions 3.1. The case of infinite number of δ-type interactions Consider the following Jacobi matrix in l2 (N), ⎛ −d−1 0 α1 + d−1 1 1 −1 ⎜ −d−1 α2 + d−1 + d−1 −d 1 1 2 2 S=⎜ −1 −1 ⎝ 0 −d2 α3 + d2 + d−1 3 ... ... ...

⎞ ... ...⎟ ⎟. ...⎠ ...

(3.1)

Notice that S = S ∗ since d∗ = inf dk > 0 (cf. [5, Theorem VII.1.3]). k∈N

The main result of this Section is the following description of κ− (LX,α ). Theorem 3.1. Let the set X = {xk }∞ k=1 satisfy (1.2). Let also the operator LX,α and the matrix S be defined by (1.3) and (3.1), respectively. Then κ− (LX,α ) = κ− (S).

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Proof. Consider the minimal operator (1.5). Note that Lmin is non-negative and n± (Lmin ) = ∞. Since X satisﬁes (1.2), the triplet Π = {H, Γ0 , Γ1 }, where H = ⊕∞ k=0 Hk ,

k Γ1 = ⊕∞ k=0 Γ1 ,

(3.2)

= −f (x1 −), = f (x1 −), and (3.3) f (xk +) f (xk +) Γk0 f = , Γk1 f = , k ∈ N, −f (xk+1 −) f (xk+1 −) (3.4)

H0 = C, Hk = C2 ,

k Γ0 = ⊕∞ k=0 Γ0 ,

Γ00 f

Γ01 f

is a boundary triplet for L∗min [15, Lemma 1]. The corresponding Weyl function is √ M0 (z) = i z, (3.5) M (z) = ⊕∞ k=0 Mk (z), √ √ √ √ −√ z ctg( √zdk ) − √z/ sin(√ zdk ) Mk (z) = , k ∈ N, (3.6) − z ctg( zdk ) − z/ sin( zdk ) where √ the square root is deﬁned on C with a cut along [0, ∞) and ﬁxed by

m z > 0 for z ∈ / [0, ∞). Using (3.2)–(3.4), we obtain the representation (2.1) for D(LX,α ), where ⎛ ⎞ ⎛ ⎞ −1 1 0 0 ... 0 α1 0 0 ... ⎜ 0 ⎜ 1 0 0 0 ... ⎟ 1 0 0 ...⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 −1 1 ... ⎟ . . . 0 0 0 α , D = C=⎜ 2 ⎜ ⎟. ⎜ ⎟ ⎝ 0 ⎝ 0 0 0 0 ... ⎠ 0 1 1 ...⎠ ... ... ... ... ... ... ... ... ... ... Without loss of generality, it can be assumed that C is bounded. If there exists {αkj }∞ j=1 such that lim |αkj | = ∞, then we put j→∞

= KC, = KD, 1 − CΓ 0 }, C D and D(LX,α ) = D(L∗min ) ker{DΓ = diag(1, . . . , 1, α−1 , 1, . . . , 1, α−1 , 1, . . .). where K k1 k2 After straightforward calculations we get the matrix T := CD∗ − DM (0)D∗ , ⎛ ⎞ α1 + d−1 0 −d−1 0 0 ... 1 1 ⎜ 0 0 0 0 0 ...⎟ ⎜ ⎟ −1 −1 −1 ⎜ −d−1 0 α + d + d 0 −d ...⎟ 2 1 1 2 2 ⎜ ⎟. T =⎜ 0 0 0 0 0 ...⎟ ⎜ ⎟ −1 ⎝ 0 0 −d−1 0 α3 + d−1 ...⎠ 2 2 + d3 ... ... ... ... ... ... 1 ⊕H 2 , where H 1 = span{e2k−1 }∞ With respect to the decomposition H = H k=1 ∞ 2 = span{e2k } and H , the operator T admits the representation k=1 T = S ⊕0H2 . Hence κ− (T ) = κ− (S), and, applying Theorem 2.4, we complete the proof. Further, we obtain sufﬁcient condition for κ− (LX,α ) ≥ m, as well as for the equality κ− (LX,α ) = m, with arbitrary ﬁnite m by using the following Gerschgorin theorem.

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Theorem 3.2 ([16, Theorem 7.2.1]). All eigenvalues of a matrix A = (aij )ni,j=1 ∈ [Cn ] are contained in the union of Gerschgorin’s disks ⎧ ⎫ ⎨ ⎬ |aij | , i ∈ {1, . . . , n}. Gi = z ∈ C : |z − aii | ≤ ⎩ ⎭ i =j

Theorem 3.3. Let K = {ki }m i=1 be a subset of N such that |K| = m. Assume that −1 αki < −2(d−1 ki −1 + dki ),

ki ∈ K.

(3.7)

/ K, then κ− (LX,α ) = Then κ− (LX,α ) ≥ m. If, in addition, αk > 0 for all k ∈ m. Proof. Due to the equality κ− (LX,α ) = κ− (S), the problem is reduced to proving the inequality κ− (S) ≥ m and the equality κ− (S) = m, respectively. We divide the proof into two cases. (a) Assume that K = {1, . . . , m}. Denote by Sm ∈ [Cm ] the submatrix in the upper left corner of the matrix S deﬁned by (3.1). According to the minimax principle (see, for instance, [10]), κ− (Sm ) ≤ κ− (S) = κ− (LX,α ).

(3.8)

Applying Theorem 3.2 to Sm and using (3.7), we obtain κ− (Sm ) = m. Therefore κ− (LX,α ) ≥ m and the ﬁrst assertion of the theorem holds. Further, setting αk = 0 for k ∈ {1, . . . , m}, we obtain a non-negative X,α . It is obvious that LX,α is an m-dimensional perself-adjoint operator L X,α . Thus, from the minimax principle follows turbation of the operator L that κ− (LX,α ) ≤ m and, consequently, the second assertion of the theorem is satisﬁed. (b) Let K be an arbitrary set consisting of m natural numbers. This case is easily reduced to the previous one. Namely, there exists unitary transformation U such that S = U ∗ SU, U : ski ki → sii , |ski j | = | sij |, ki ∈ K. (3.9) j =kj

j =i

we complete the proof. Applying the preceding reasoning to the matrix S,

Remark 3.4. (i) Arguing as above, it is easy to show that κ− (LX,α ) = ∞ if m = ∞. (ii) The idea of applying the Gerschgorin theorem is borrowed from the paper of Ogurisu ([18], see also [19]). Let us also mention that our sufﬁcient condition is more explicit than that of Ogurisu, since we apply Gerschgorin’s theorem to the simpler matrix S. Theorem 3.1 enables us to construct an algorithm to determine κ− (LX,α ). Namely, deﬁne the sequence γ = {γk }∞ k=1 by γ1 := α1 + d−1 1 ,

(3.10)

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(i) if γk = 0,

−1 −2 −1 γk+1 := αk+1 + d−1 k+1 + dk − dk γk ,

then

7

k ≥ 1; (3.11)

(ii) if γk = 0,

then

γk+2

γk+1 := ∞ −1 , := αk+2 + d−1 k+1 + dk+2

k ≥ 1.

(3.12)

Theorem 3.5. Let the set X = {xk }∞ k=1 satisfy (1.2). Let the operator LX,α be defined by (1.3) and let the sequence γ = {γk }∞ k=1 be defined by (3.10)–(3.12). Then κ− (LX,α ) = κ− (γ) + N∞ (γ), where κ− (γ) and N∞ (γ) are the number of negative and infinite elements, respectively, in the sequence γ. Proof. Consider two cases. (a)

Let γ1 = α1 + d−1 1 = 0. Setting T11 := γ1 IC and applying Proposition 2.5 to the matrix S deﬁned by (3.1), we get κ− (S) = κ− (γ1 ) + κ− (S2 ), where ⎞ ⎛ −d−1 0 ... γ2 2 ⎜ −d−1 α3 + d−1 + d−1 −d−1 ...⎟ 2 2 3 3 ⎟. S2 := ⎜ −1 −1 ⎝ 0 −d3 α4 + d−1 + d ...⎠ 3 4 ... ... ... ...

Further, if γ2 = 0, then we set T11 = γ2 IC and apply Proposition 2.5 to the matrix S2 . Thus, if γk = 0 for all k ∈ N, i.e., N∞ (γ) = 0, then we obtain κ− (S) = κ− (γ). −1 −1 −1 = 0. Then (b) Assume that γ1 = α1 + d1 −1 γ2 = ∞ and γ3 = α3 + d2 + d3 . 0 −d1 Let T11 := ∈ [C2 ]. Since det T11 = −d−2 −1 1 = 0, −d−1 α + d−1 2 1 1 + d2 by Proposition 2.5, we get κ− (S) = κ− (T11 ) + κ− (S3 ), where ⎛ ⎞ γ3 −d−1 0 ... 3 ⎜ −d−1 α4 + d−1 + d−1 −d−1 ...⎟ 3 3 4 4 ⎟. S3 := ⎜ −1 −1 ⎝ 0 −d4 α5 + d−1 + d ...⎠ 4 5 ... ... ... ... Since κ− (T11 ) = 1, we get κ− (S) = N∞ ({γ1 , γ2 }) + κ− (S3 ). Proceeding as above, we obtain the desired result.

Following [4], consider continued fraction Ak := [αk ; dk−1 , αk−1 , . . . , α1 ]. It is easy to verify by induction that γk = d−1 k + Ak ,

k ≥ 1,

(3.13)

if γk = 0 for all γk ∈ γ. Theorem 3.5 and equality (3.13) yield the following result. Corollary 3.6. The operator LX,α is non-negative if and only if Ak > −d−1 k ,

k ≥ 1.

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3.2. The case of finite number of δ-type interactions Setting αk = 0, k > n, in (1.3), we obtain the operator with δ-interactions on a ﬁnite set. Using Theorems 3.1 and 3.5, we obtain the following description of the negative squares κ− (LX,α ). Namely, deﬁne the sequence

(i) if γ k = 0, then γ k+1

γ 1 := α1 + d−1 1 , −1 −2 −1 αk+1 + dk+1 + d−1 k , k − dk γ := −1 −1 −2 −1 dk+1 + dk − dk γ k ,

k+1 := ∞, (ii) if γ k = 0, then γ

γ k+2 :=

(3.14) k ≤ n − 1, k ≥ n; (3.15)

d−1 k+1

d−1 k+2 ,

αk+2 + + −1 d−1 + d k+1 k+2 ,

k ≤ n − 2, k ≥ n − 1. (3.16)

Corollary 3.7. Let X = {xk }nk=1 ⊂ R be a finite set. Let also the operator = { γk }∞ LX,α be defined by (1.3) and let γ k=1 be the sequence defined by (3.14)–(3.16). Then γ ) + N∞ ( γ ). κ− (LX,α ) = κ− ( Corollary 3.7 has one essential drawback. To obtain κ− (LX,α ), we have to regard inﬁnite number of elements γ n , n ∈ N. But it is possible to overcome this by treating LX,α as an extension of the minimal operator with ﬁnite deﬁciency indices. Namely, deﬁne the matrix S ∈ [Cn ], ⎞ ⎛ α1 + d−1 −d−1 0 ... 0 1 1 −1 ⎟ ⎜ −d−1 α2 + d−1 −d−1 ... 0 1 1 + d2 2 ⎟ ⎜ −1 −1 −1 ⎟. ⎜ S=⎜ 0 −d2 α3 + d2 + d3 ... 0 ⎟ ⎠ ⎝ ... ... ... ... ... −1 0 0 0 . . . αn + dn−1 (3.17) {xk }nk=1

Theorem 3.8. Let X = ⊂ R be a finite set. Let the operator LX,α and the matrix S be defined by (1.3) and (3.17), respectively. Then κ− (LX,α ) = κ− (S). Proof. Consider the operator Lmin deﬁned by (1.5) with X = {xk }nk=1 . Note that Lmin is non-negative and n± (Lmin ) = 2n. The boundary triplet for L∗min might be deﬁned as follows (cf. [13, Section III,§1]) H = ⊕nk=0 Hk , Γ0 = ⊕nk=0 Γk0 , Γ1 = ⊕nk=0 Γk1 , where (3.18) 0 0 H0 = C, Γ0 f = −f (x1 −), Γ1 f = f (x1 −), (3.19) f (xk +) f (xk +) 2 k k Hk = C , Γ0 f = −f (xk+1 −) , Γ1 f = f (xk+1 −) , k ∈ {1, . . . , n − 1}, Hn = C,

Γn0 f = f (xn +),

Γn1 f = f (xn +).

The corresponding Weyl function M (z) is M (z) = ⊕nk=0 Mk (z),

√ M0 (z) = Mn (z) = i z

Mk (z) for k ∈ {1, . . . , n − 1} is given by (3.6).

and

(3.20) (3.21)

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Using (3.18)–(3.21), we obtain the description of (2.1), where ⎛ ⎛ ⎞ −1 1 0 α1 . . . 0 0 ⎜ 0 ⎜ 1 ⎟ 0 1 . . . 0 0 ⎜ ⎜ ⎟ ⎟, D = ⎜ ... ... . . . . . . . . . . . . . . . C=⎜ ⎜ ⎜ ⎟ ⎝ 0 ⎝ 0 0 0 . . . 0 αn ⎠ 0 0 0 0 ... 1 1

9

D(LX,α ) in the form ... 0 ... 0 ... ... . . . −1 ... 0

⎞ 0 0 ⎟ ⎟ ...⎟ ⎟. 1 ⎠ 0

Further, it is easy to verify that the matrix T = CD∗ − DM (0)D∗ has the form ⎞ ⎛ α1 + d−1 0 −d−1 ... 0 0 1 1 ⎟ ⎜ 0 0 0 ... 0 0 ⎟ ⎜ −1 −1 ⎟ ⎜ −d−1 0 α + d + d . . . 0 0 2 1 1 2 ⎟. ⎜ T =⎜ ⎟ . . . . . . . . . . . . . . . . . . ⎟ ⎜ ⎠ ⎝ 0 0 0 ... 0 0 0 0 0 . . . 0 αn + d−1 n

Arguing as in the proof of Theorem 3.1, we conclude the proof. Deﬁne the sequence γ = {γk }nk=1 as follows (i) if γk = 0, then γk+1

(3.22) γ1 := α1 + d−1 1 , −1 −1 −2 −1 αk+1 + dk+1 + dk − dk γk , k ≤ n − 2 := ; −2 −1 αk + d−1 k =n−1 k−1 − dk−1 γk−1 , (3.23)

(ii) if γk = 0, then

γk+1 := ∞, k ∈ {1, . . . , n − 1}, −1 γk+2 := αk+2 + d−1 k ∈ {1, . . . , n − 2}. k+1 + dk+2 , (3.24)

Theorem 3.9. Assume X (1.3) and let the sequence

= {xk }nk=1 . γ = {γk }nk=1

Let the operator LX,α be defined by be defined by (3.22)–(3.24). Then

κ− (LX,α ) = κ− (γ) + N∞ (γ). We omit the proof since it is analogous to that of Theorem 3.5. Proposition 3.10. Corollary 3.7 and Theorem 3.9 are equivalent, i.e., γ ) + N∞ ( γ ) = κ− (γ) + N∞ (γ), κ− ( { γk }∞ k=1

where γ = and γ = {γk }nk=1 are defined by (3.14)–(3.16) and (3.22)– (3.24), respectively. Proof. Since γ k = γk for k < n, it sufﬁces to verify that κ− (γn ) + N∞ (γn ) = κ− ({ γk }∞ γk }∞ k=n ) + N∞ ({ k=n ).

(3.25)

First, assume that γ m < 0 for some m ≥ n. Then, by (3.15), γ m+1 > and hence d−1 m+1 γ k ≥ d−1 k ,

for all k > m.

The latter also yields that in this case κ− ({ γk }∞ k=n ) ∞ γk }k=m ) = 0. N∞ ({

(3.26) ≤

1 and

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m+1 = ∞ and Further, if γ m = 0 for some m > n, then, by (3.16), γ −1 γk }∞ γ m+i = d−1 m+i + dm+i−1 , i ≥ 2. Therefore N∞ ({ k=n ) ≤ 1 and κ− ({ γk }∞ k=m ) = 0. Consider three cases. −1 Let γn ≥ 0. Combining (3.23) with (3.15), we get γ n = γn + d−1 n ≥ dn . Hence γ k satisﬁes (3.26) with m = n, and (3.25) clearly holds. n−1 = 0 and γ n = ∞. Thus, γ k satisﬁes (b) Let γn = ∞. Then γn−1 = γ (3.26) for all k ≥ n, and hence (3.25) holds. (c) Assume now that γn < 0.

(a)

If γ k = 0 for some k ≥ n, then, arguing as above, we arrive at (3.25). Suppose that γ k = 0, k ≥ n. To prove (3.25) it sufﬁces to show that k > 0 for all k ≥ n. γ n+i < 0 for some i > 0. Assume the converse, i.e., γ −1 Denote ξk := d−1 − γ , k ≥ n. Clearly, ξ < d for all k ≥ n. Note that, by k k k k −1 −1 ξ (d − ξ ) . Further, the inequality 0 < ξn < d−1 (3.15), ξk+1 = d−1 k k k n holds k −1 −1 since γn < 0. Moreover, 0 < d−1 − ξ < d < d (see (1.2)) and hence n n n ∗ −1 −1 −1 = ξn + ξn2 (d−1 > ξn + ξn2 d∗ . ξn+1 = d−1 n ξn (dn − ξn ) n − ξn ) −1 Similarly, ξn+1 < d−1 n+1 ≤ d∗ yields 2 ξn+2 > ξn+1 + ξn+1 d∗ > ξn + ξn2 d∗ + ξn2 d∗ = ξn + 2 ξn2 d∗ .

Therefore we get ξn+i > ξn + i ξn2 d∗ , i ∈ N. Hence there exists i0 ∈ N such that −1 ξn + iξn2 d∗ > d−1 ∗ > dn+i ,

i ≥ i0 .

Then ξn+i0 > d−1 n+i0 < 0. This contradiction comn+i0 and consequently γ pletes the proof of (3.25). Combining (a) with (b) and (c), we arrive at the desired result. Remark 3.11. Albeverio and Nizhnik [3] obtained another description of κ− (LX,α ). Namely, assume that the function ϕ is a solution of the problem ϕ (x) = 0, x ∈ / X, ϕ(x) ≡ 1, x < x1 , and (3.27) ϕ(xk +) = ϕ(xk −), ϕ (xk +) − ϕ (xk −) = αk ϕ(xk ) for xk ∈ X. (3.28) According to [3, Theorem 3], κ− (LX,α ) equals the signature of the sequence (ϕ(x1 ), ϕ(x2 ), . . . , ϕ(xn ), (1 + αn dn−1 )ϕ(xn ) − ϕ(xn−1 )).

(3.29)

Note that this result might be deduced from Theorem 3.8 and vise versa. To be precise, let Δk be a k-th order leading principle minor of the matrix S deﬁned by (3.17). Then one can easily verify that Δk =

ϕ(xk+1 ) , dk−1 · . . . · d1

k ∈ {1, . . . , n}.

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4. Operators with δ -type interactions The main result of this Section is the following theorem. Theorem 4.1. Let the set X = {xk }∞ k=−∞ ⊂ R satisfy (1.2). Let the operator LX,β be defined by (1.4) and β = {βk }∞ k=−∞ ⊂ R. Then κ− (LX,β ) = κ− (β). Proof. We divide the proof into several steps. (a)

Consider the minimal operator Lmin (1.5). Since X satisﬁes condition (1.2), we can choose the boundary triplet Π = {H, Γ0 , Γ1 } for L∗min as follows [15, Lemma 1] H = ⊕∞ k=−∞ Hk ,

k Γ0 f = ⊕∞ k=−∞ Γ0 f,

k Γ1 f = ⊕∞ k=−∞ Γ1 f,

(4.1)

where Πk = {Hk , Γk0 , Γk1 }, k ∈ Z, is given by (3.4). The corresponding Weyl function is M (z) = ⊕∞ k=−∞ Mk (z) with Mk (z) deﬁned by (3.6). The domain of the operator LX,β admits the representation D(LX,β ) = D(L∗min ) ker{DΓ1 − CΓ0 } with D and C determined, respectively, by ⎞ ⎛ ⎛ ⎞ ... ... ... ... ... ... ... ... ... ... ... ... ⎜... 0 0 0 0 ...⎟ ⎜ . . . −1 1 0 0 ...⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ . . . 0 β0 0 0 . . . ⎟ ⎜ ⎟ and ⎜ . . . 1 1 0 0 . . . ⎟ . ⎜ . . . 0 0 0 0 . . . ⎟ ⎜ . . . 0 0 −1 1 . . . ⎟ ⎟ ⎜ ⎜ ⎟ ⎝... 0 0 1 1 ...⎠ ⎝... 0 0 0 β1 . . . ⎠ ... ... ... ... ... ... ... ... ... ... ... ... As before, we assume that D is bounded. After straightforward calculations we get the operator T = CD∗ − DM (0)D∗ , ⎛ ⎞ ... ... ... ... ... ... ... ⎜... β0 d−1 −d−1 0 0 0 ...⎟ −1 −1 ⎜ ⎟ 2 −1 ⎜ . . . β0 + β0 d−1 −β0 d−1 0 0 0 ...⎟ −1 ⎜ ⎟ ⎜ . . . −β0 d−1 d−1 + d−1 β1 d−1 −d−1 0 ...⎟ −1 −1 0 0 0 ⎜ ⎟. 2 −1 −1 −1 ⎜... 0 β1 d0 β1 + β1 d0 −β1 d0 0 ...⎟ ⎜ ⎟ −1 −1 −1 −1 ⎝... 0 −d0 −β1 d0 d0 + d1 β2 d−1 ...⎠ 1 ... ... ... ... ... ... ... (4.2) By Theorem 2.4, κ− (LX,β ) = κ− (T ). (b) Note that the matrix T admits the representation T = A + B, where A=

∞ k=−∞

βk (·, e2k−1 )e2k−1 ,

∞

B=

k=−∞

d−1 k−1 (·, bk )bk ,

with bk := e2k−1 + βk e2k − e2k+1 . Since d−1 k > 0, one gets κ− (T ) ≤ κ− (A).

(4.3)

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Let s ∈ Z− ∪ {0} and r ∈ N. Consider the matrix Ts,r ∈ [C2(r−s)+1 ], Ts,r := As,r + Bs,r ,

where

As,r =

r+1

βk (·, e2k )e2k and

k=s−1

Bs,r =

r−1 k=s+1

−1 −1 d−1 k−1 (·, bk )bk + ds−1 (·, ys )ys + dr−1 (·, xr )xr ,

with ys := βs e2s−2 + e2s−1 and xr :=e2r+1 + βr e2r+2 . It is clear that ran(As,r ) ∩ ran Bs,r − d−1 s−1 (·, ys )ys = {0} and hence κ− (Ts,r − d−1 s−1 (·, ys )ys ) = κ− (As,r ). According to the choice of the matrix Ts,r , we get −1 κ− (T ) ≥ κ− (Ts,r − d−1 s−1 (·, ys )ys ) − rank(ds−1 (·, ys )ys )

= κ− (As,r ) − 1.

(4.4)

Combining (4.3) with (4.4), we obtain κ− (A) − 1 ≤ κ− (T ) ≤ κ− (A). If κ− (β) = ∞, then (4.3) yields κ− (T ) = ∞. Therefore κ− (LX,β ) = ∞ and theorem is proven in the case κ− (β) = ∞. (d) Assume now that κ− (β) = m < ∞. Let us show that r r det(Ts,r ) = xr − xs−1 + βk d−1 (4.5) k−1 βk . k=s

k=s

For s = 0, r = 1 equality (4.5) is obvious. Suppose that (4.5) holds with s + 1 < 0 and r > 1. Note that the second row t2 of the matrix Ts,r admits a decomposition

−1 t12 = −βs d−1 s−1 ds−1

t2 = t12 + t22 , where βs+1 d−1 ... 0 . and t22 = 0 d−1 0 ... 0 s s

1 2 1 2 Then det(Ts,r ) = det(Ts,r ) + det(Ts,r ), where Ts,r and Ts,r are matrices 1 2 obtained by substitution of t2 in Ts,r for t2 and t2 , respectively. Adding 1 the second row multiplied by βs , we arrive at the to the ﬁrst row of Ts,r equality 1 ) = βs d−1 det(Ts,r s−1 det(Ts+1,r ).

(4.6)

2 2 ) = (βs + βs2 d−1 It is easily seen that det(Ts,r s−1 ) det(T2(r−s) ), where 2 T2(r−s) ∈ [C2(r−s) ] is an algebraic complement of βs + βs2 d−1 s−1 . Add2 the ﬁrst row multiplied by −βs+1 and ing to the second row of T2(r−s) 2 −1 to the third row the ﬁrst row, we get det(Ts,r ) = (βs + βs2 d−1 s−1 )βs+1 ds 2 det(T2(r−s−1) ). Proceeding analogously, one, ﬁnally, obtains r 2 ) = (βs + βs2 d−1 d−1 (4.7) det(Ts,r s−1 ) k−1 βk . k=s+1

Combining (4.6) with (4.7), we arrive at (4.5).

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r Since κ− (β) < ∞ and the difference (xr − xs−1 ) = k=s dk is r unbounded as either −s or r tends to inﬁnity, (xr − xs−1 + k=s βk ) > 0 for sufﬁciently large −s and r. Besides, we can assume that βs−j > 0 and βr+j > 0 for j > 0. Hence r −1 sgn(det(Ts,r )) = sgn dk−1 βk = (−1)κ− (β) m

= (−1)

k=s m−1

= (−1)

.

Therefore κ− (Ts,r ) = m and, by (4.3), we, ﬁnally, get m = κ− (Ts,r ) ≤ κ− (T ) ≤ κ− (A) = m, i.e., κ− (T ) = m. The proof is completed.

Acknowledgements The authors are grateful to M. M. Malamud for posing the problem and permanent attention to our work. We are especially indebted to A. S. Kostenko for carefully reading of the preliminary version of the manuscript and constructive remarks. We would also like to thank the referee for useful remarks regarding exposition improvement.

References [1] Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence (2005) [2] Albeverio, S., Kurasov, P.: Singular Perturbations of Differential Operators and Schr¨ odinger Type Operators. Cambridge University Press, Cambridge (2000) [3] Albeverio, S., Nizhnik, L.: On the number of negative eigenvalues of one-dimensional Schr¨ odinger operator with point interactions. Lett. Math. Phys. 65, 27–35 (2003) [4] Albeverio, S., Nizhnik, L.: Schr¨ odinger operators with a number of negative eigenvalues equal to the number of point interactions. Methods Funct. Anal. Topol. 9(4), 273–286 (2003) [5] Berezanskii, Yu.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Naukova Dumka, Kiev (1965) [6] Buschmann, D., Stolz, G., Weidmann, J.: One-dimensional Schr¨ odinger operators with local point interactions. J. Reine Angew. Math. 169–186 (1995) [7] Derkach, V.A., Hassi, S., Malamud, M.M., de Snoo, H.S.V.: Generalized resolvents of symmetric operators and admissibility. Methods Funct. Anal. Topol. 6(3), 24–55 (2000) [8] Derkach, V.A., Malamud, M.M.: Generalised resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95(1), 1–95 (1991) [9] Gesztezy, F., Kirsch, W.: One-dimensional Schr¨ odinger operators with interactions singular on a discrete set. J. Reine Angew. Math. 362, 27–50 (1985) [10] Glazman, I.M.: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], Fizmatgiz, Moscow (1963)

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[11] Goloshchapova, N.I., Oridoroga, L.L.: The one-dimensional Sch¨ odinger operator with point δ- and δ -interactions. Math. Notes 84(1), 125–129 (2008) [12] Goloschapova, N., Oridoroga, L.: On the Number of Negative Spectrum of OneDimensional Schr¨ odinger Operators with Point Interactions. arXiv:0903.1180 [13] Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations, Mathematics and its Applications (Soviet Series), vol. 48. Kluwer Academic Publishers Group, Dordrecht (1991) [14] Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966) [15] Kochubei, A.N.: One-dimensional point interactions. Ukrainian Math. J. 41(10), 90–95 (1989) [16] Lancaster, P.: Theory of Matrices [Russian translation]. Nauka, Moscow (1973). English: Academic Press, New York, 1969 [17] Malamud, M.M.: On certain classes of Extensions of Hermitian Operators with gaps. Ukrainian Math. J. 44(2), 215–233 (1992) [18] Ogurisu, O.: On the number of negative eigenvalues of a Schr¨ odinger operator with point interactions. Lett. Math. Phys. 85, 129–133 (2008) [19] Ogurisu, O.: On the number of negative eigenvalues of a Schr¨ odinger Operator with δ-interactions. Methods Funct. Anal. Topol. (To appear) N. Goloschapova Institute of Applied Mathematics and Mechanics NAS of Ukraine R. Luxemburg str. 74 83114 Donetsk, Ukraine e-mail: [email protected] L. Oridoroga Donetsk National University Universitetskaja str. 24 83055 Donetsk, Ukraine e-mail: [email protected] Received: March 5, 2009. Revised: December 14, 2009.

Integr. Equ. Oper. Theory 67 (2010), 15–31 DOI 10.1007/s00020-010-1766-y Published online March 20, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Operator Machines on Directed Graphs Petr H´ajek and Richard J. Smith Abstract. We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X −→ X such that the set A = {x ∈ X : ||Rn x|| → ∞} is non-empty and nowhere norm-dense in X. Moreover, if x ∈ X\A then some subsequence of (Rn x)∞ n=1 converges weakly to x. This answers in the negative a recent conjecture of Prˇ ajiturˇ a. The result can be extended to any Banach space containing an infinite-dimensional, complemented subspace with a symmetric basis; in particular, all ‘classical’ Banach spaces admit such an operator. Mathematics Subject Classification (2000). 47A05. Keyword. Orbits of operators.

1. Introduction Given a Banach space X, a bounded linear operator T on X and x ∈ X, we say that the orbit of x with respect to T is the set orb(x, T ) = {T n x : n ≥ 0}. It is well known that in the ﬁnite-dimensional setting, the orbits of points under operators exhibit ‘regular’ behaviour. Proposition 1.1 (cf. [11, p. 17]). If X is finite-dimensional, then there exist subspaces Z ⊆ Y of X with the property that 1. 2. 3.

if x ∈ X\Y then ||T n x|| → ∞; if x ∈ Y \Z then there is a constant M > 0 such that M −1 ≤ ||T n x|| ≤ M for every n; if x ∈ Z then ||T n x|| → 0.

Both authors are supported by Grant A 100190801 and Institutional Research Plan AV0Z10190503.

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The inﬁnite-dimensional situation is very different. Rolewicz provided simple examples of operators on inﬁnite-dimensional spaces which admit hypercyclic vectors, that is, vectors with orbits which are dense with respect to the norm topology. The study of orbits is connected to the invariant subspace problem. Indeed, an operator T on a Banach space X has no non-trivial, closed, invariant subspaces if and only if span orb(x, T ) is dense for every non-zero x ∈ X. There is a considerable body of literature on operators which admit hypercyclic vectors. In this note, we study operators with more regular orbits, in particular, those which tend to inﬁnity. This type of orbit has received attention from several authors. For example, in a systematic study of orbits of operators on Hilbert space, Beauzamy provides several sufﬁcient conditions for T to admit a dense set of points x satisfying ||T n x|| → ∞ [2, Chapter III]. Broadly speaking, these conditions based on the growth of the sequence ∞ are n −1 . For example, if ||T || < ∞ then T admits such a dense (||T n ||)∞ n=1 n=1 set. Sharp estimates of this nature, applying to general Banach spaces, are given in [9]. We refer the reader to [5,7,8] for additional results on this topic. A given operator can have both regular and highly irregular orbits, and the exact behaviour of orb(x, T ), as x ranges over X, is not so easy to determine. In [10], Prˇ ajiturˇ a makes the following conjecture. Conjecture 1.2 [10, Conjecture 2.9]. Let T be an operator T on a Banach space and let AT = {x ∈ X : ||T n x|| → ∞}. Then AT is dense whenever AT is non-empty. Of course, if ||T n x|| → ∞ for some x then (||T n ||)∞ n=1 is unbounded, so by the uniform boundedness principle, the set of y with the property that supn ||T n y|| = ∞ is a dense Gδ in X. However, clearly this does not say anything about whether ||T n y|| tends to inﬁnity or not. Indeed, the weighted backwards shift operator T on p , 1 ≤ p < ∞, given by 1 (i/(i − 1)) p ei−1 if i > 1 T ei = 0 if i = 1 satisﬁes ||T n || → ∞, but ||T n x|| → ∞ for all x [9, Example 4]. In [2, pp. 66–68], there is an example of an operator T on Hilbert space satisfying ||T n || → ∞, but inf n ||T n x|| = 0 for all x. The object of this note is to show that there is a wide class of Banach spaces which admit operators failing Conjecture 1.2. In fact, by constructing a range of suitable operators, we can impose a reasonable degree of control over the structure of AT . Clearly, AT is always radial, in the sense that if x ∈ AT and λ = 0 then λx ∈ AT . Thus we need only trouble ourselves with what happens to points in the unit sphere SX . We shall consider both real and complex Banach spaces. Recall that if a Schauder basis (ei )∞ i=1 of X is symmetric then there exists an equivalent

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norm || · || on X with the property that ∞ ∞ xi ei = λi xi eπ(i) i=1

i=1

whenever π is a permutation of N and |λi | = 1 for all i. Such a norm is called symmetric; hereafter, whenever we have a Banach space X with a symmetric basis, we shall assume that X is inﬁnite-dimensional and that the associated norm is symmetric. In addition, we shall say that a subset E of a Banach space is symmetric if λx ∈ E whenever x ∈ E and |λ| = 1. Here follows our main result. Theorem 1.3. Let X have a symmetric basis with norm || · || and suppose that Y ⊆ X is a subspace of dimension d, where 2 ≤ d < ∞. Moreover, let E ⊆ SY be closed and symmetric, and let J be a projection of X onto Y . Then there exists an operator R : X −→ X with two properties: 1. 2.

if Jx ∈ E then ||Rn x|| → ∞; n ∞ if Jx ∈ SY \E then there is a subsequence (Rni x)∞ i=1 of (R x)n=1 such ni that R x → x weakly.

Of course, we obtain the claim in the abstract by ensuring that E ⊆ SY in Theorem 1.3 is non-empty and nowhere dense. Roughly speaking, we use the extra dimensions in the complement of Y in X to encode the non-linear information in E. To give an idea of what we mean by this, we can compare, at a distance, this encoding of non-linear information to the standard method of producing an operator on Hilbert space with a prescribed spectrum, namely, by arranging a suitable, countable family of eigenvalues. The proof of Theorem 1.3 is spread across Sects. 2 and 3. First, we will assume that X is either c0 or p , where 1 ≤ p < ∞. Then we indicate how the construction can be adapted to apply to the general case. We expect that it is possible to generalise Theorem 1.3 to incorporate subsets of SY of greater topological complexity, but to do so would go beyond the immediate aims of this paper and would unduly complicate our existing proof. Of course, if X in Theorem 1.3 is complemented in some overspace Z then we obtain a corresponding result about Z. In particular, by considering c0 or p , 1 ≤ p < ∞, we can see that any ‘classical’ Banach space admits an operator T such that AT is non-empty and nowhere dense. Since there is a Banach space with a symmetric basis but containing no isomorphic copy of c0 or p , 1 ≤ p < ∞, [4], it is not possible to obtain Theorem 1.3 by proving it in the cases X = c0 and X = p , and then appealing to complemented subspaces. It is clear from Proposition 1.1 that we cannot hope to obtain results such as Theorem 1.3 if we restrict our attention to operators on ﬁnitedimensional spaces, or operators of ﬁnite rank. We ﬁnish this section by making a slightly more general statement. An operator T , deﬁned on a complex Banach space X, is called a Riesz operator if, for every λ ∈ C\{0}, we can write X as a direct sum of closed subspaces U and V , both invariant for T , and such that U is ﬁnite-dimensional, (T − λI)U is nilpotent and (T − λI)V

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is an isomorphism of V . It follows that every non-zero spectral value of T is an isolated eigenvalue of ﬁnite multiplicity. Such operators have been considered in, for example [3]. By considering those eigenvalues of T of modulus at least 1, of which there are ﬁnitely many, we can write X = U ⊕ V , where U and V are closed invariant subspaces, U is ﬁnite-dimensional and the spectral radius of T V is strictly less than 1. Since U is ﬁnite-dimensional, the sequence (||T n u||)∞ n=1 , where u ∈ U , behaves according to Proposition 1.1. Moreover, if v ∈ V then ||T n v|| → 0. From these observations, we conclude that there exist closed, ﬁnite-codimensional subspaces Z ⊆ Y of X satisfying properties (1)–(3) of Proposition 1.1. In particular, if T is a Riesz operator then AT is simply the complement of a closed, ﬁnite-codimensional subspace, thus AT is an open set which is either empty or dense. Recall that an operator S on X is strictly singular if it is not an into isomorphism when restricted to any inﬁnite-dimensional subspace of X. For example, every compact operator is strictly singular. Suppose that S is strictly singular. If X is complex then S is a Riesz operator [3, p. 66], and we obtain the subspaces Y and Z as above. If X is instead real, we consider a complexiﬁcation XC and the strictly singular operator SC , deﬁned by SC (x + iy) = Sx + iSy, to come to the same conclusion. Either way, it follows that AS is again either empty or dense. We shall consider strictly singular operators once more in Sect. 4.

2. Local Estimates Our map R in Theorem 1.3 is going to be a block diagonal operator on X. In this section, we build the template for the operators acting on the blocks and gather together some basic estimates. Hereafter, we shall assume that all spaces are complex. We simply change the scalar ﬁeld to obtain proofs in the real case. Let m, H ∈ N, ε > 0 and denote by Y the H-dimensional space H p , where 4m ≤ H and 1 ≤ p ≤ ∞. Deﬁne the operators S : Y −→ Y and F : C −→ Y by Sy = (yH , y1 , . . . , yH−1 ) where y = (y1 , . . . , yH ), and F a = (εa, . . . , εa, −εa, . . . , −εa, 0, . . . , 0). m times

m times

In this way, S can be described as a cycle operator and F a ‘feed’ operator. Let R : C ⊕ Y −→ C ⊕ Y be deﬁned by R(a, y) = (a, Sy + F a). We are interested in the behaviour of Rt (a, 0) at time t ∈ N. We can imagine that S drives an airport baggage carousel and F deposits the passengers’ bags onto the moving belt at a ﬁxed set of positions (although the bags have ‘complex mass’). The absolute mass of bags deposited depends on the value of the ﬁrst coordinate. Aided by this analogy, we can see that the result of repeated applications of R to the vector (a, 0) can be viewed as the sum of two bumps: one stationary bump of height εam and base width 2m − 1, and a moving

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bump of height −εam and base width again 2m − 1. The moving bump’s motion is periodic, with period H. Let us denote by P the map (a, y) → y. Lemma 2.1. Let 1 ≤ p ≤ ∞. Firstly, if m ≤ t ≤ H − m then ⎧ ⎨ 2 p−1 −1 εm(p+1)p |a| if p < ∞ P Rt (a, 0) ≥ p+1 ⎩ εm|a| if p = ∞.

(2.1)

Secondly, there exists a constant L, depending only on p, such that at all times t we have (p+1)p−1 |a| if p < ∞ P Rt (a, 0) ≤ Lεm (2.2) Lεm|a| if p = ∞ and if t ≤ m then P Rt (a, 0) ≤

−1

Lεmp t|a| Lεt|a|

if p < ∞ if p = ∞.

(2.3)

Proof. We estimate the norm of the sum of the standing and moving bumps. If p = ∞ we simply measure the absolute height of the sum of the bumps to obtain the values listed above, with L = 1. From now on, we shall assume that p < ∞. Set p+3 p1 1 2 2p+2 + 1 p L= > 2+ . p+1 p+1 For (2.1), we have P Rt (a, 0)p ≥ 2εp |a|p

m sp ds = 0

2εp |a|p p+1 m . p+1

To establish (2.2), we note that the maximum value of the norm is attained when the supports of the standing and moving bumps are disjoint, which occurs if and only if 2m ≤ t ≤ H − 2m. Thus we estimate P Rt (a, 0)p ≤ 4εp |a|p

m+1

sp ds = 0

4εp |a|p 2p+3 εp |a|p p+1 (m + 1)p+1 ≤ m . p+1 p+1

For (2.3), when t ≤ m, we note ⎫ ⎧ t ⎪ ⎪ t+1 2 ⎬ ⎨ P Rt (a, 0)p ≤ 2εp |a|p (m − t)tp + sp ds + (2s)p ds ⎪ ⎪ ⎭ ⎩ 0 0 tp+1 (t + 1)p+1 = 2εp |a|p (m − t)tp + + p+1 2(p + 1) p+2 +1 p p p 2 ≤ 2+ ε mt |a| . p+1

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If X = c0 or X = p , the estimates above sufﬁce to prove Theorem 1.3. The rest of this section is devoted to providing additional estimates for the general case. The reader may wish to assume that X is c0 or p , skip the rest of the section and proceed directly to Section 3, before considering the proof of Theorem 1.3 in full generality. In order to build our operator R on a Banach space X with a symmetric basis, we will need to estimate the norms of certain vectors in X with reasonable precision. In order to do this, we combine the estimates of Lemma 2.1 with a result closely based on a theorem of Tzafriri [12]. We have altered the statement of the next result to suit our purposes. In what follows, d(·, ·) indicates Banach–Mazur distance and, as above, [ · ] denotes norm-closed linear span. Proposition 2.2 [12, Proposition 5]. Let V be a 2n -dimensional normed space with basis (vσ )σ∈G , where G is the set of all functions from {1, . . . , n} to {−1, 1}. Suppose that there are constants K > 0 and r > 2 such that given scalars aσ , σ ∈ G, we have 1s r1 K K −1 |aσ |s ≤ aσ vσ vσ ≤ |aσ |r 1 (2n ) r1 (2n ) s σ∈G σ∈G σ∈G σ∈G where r−1 + s−1 = 1. Then there exists M , dependent on K and r, but independent of V and n, with the property that if we define σ(l)vσ zl = σ∈G

for 1 ≤ l ≤ n, then d([zl ]nl=1 , n2 ) < M . The proof of the next result closely follows that of [12, Theorem 1], although we note that the assumed symmetry of the norm allows us to bypass the Ramsey arguments which feature in [12]. Tzafriri’s notation has also been modiﬁed slightly to suit our requirements. Lemma 2.3. Let X have a normalised symmetric basis (ei )∞ i=1 with conjuand symmetric norm || · ||. Then there exists M > 0 and gate system (e∗i )∞ i=1 p ∈ {1, 2, ∞}, a pairwise disjoint family of finite subsets Fn ⊆ N, n ∈ N, vectors zl,n , 1 ≤ l ≤ n, supported on Fn and permutations πn of Fn with three properties: 1. given n, if a linear operator S on X satisfies Sei = eπn (i) for all i ∈ Fn , then Szl,n = zτ (l),n , where τ is the cycle (1, . . . , n); 2. d([zl,n ]nl=1 , np ) < M for all n; 3. πn has order n. Proof. Deﬁne λ(n) = ||e1 + · · · + en ||

and

μ(n) = ||e∗1 + · · · + e∗n || .

We follow the proof of [12, Theorem 1] in distinguishing three cases. Case I: for every n ∈ N there exists mn ∈ N such that λ(nmn )/λ(mn ) < 2. Put p = ∞. Set k1 = 0 and, given kn , deﬁne kn+1 = kn + nmn . Let Fn = {kn + 1, · · · , kn + nmn }

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and deﬁne zl,n = (ekn +(l−1)mn +1 + · · · ekn +lmn )/λ(mn ) for 1 ≤ l ≤ n, n ∈ N. Finally, deﬁne kn + lmn + r πn (kn + (l − 1)mn + r) = kn + r

if 1 ≤ l < n and 1 ≤ r ≤ mn if l = n and 1 ≤ r ≤ mn .

It is clear that the Fn are pairwise disjoint and properties (1) and (3) hold. Now we prove (2). By the symmetry of the norm, we have ||zl,n || = 1. Since n n n n max |al | ≤ al zl,n ≤ max |al | zl,n l=1 l=1 l=1

l=1

λ(nmn ) n n ≤ max |al | ≤ 2 max |al | l=1 l=1 λ(mn ) for any scalars a1 , . . . , an , we can see that (2) holds for any M > 2. Case II: for every n ∈ N there exists mn ∈ N such that μ(nmn )/μ(mn ) < 2. Now put p = 1 and set kn , Fn and πn exactly as in case I. If we set

∗ = e∗kn +(l−1)mn +1 + · · · e∗kn +lmn /μ(mn ), zl,n then we have

n n ∗ max |al | ≤ al zl,n ≤ 2 max |al | l=1 l=1 n

l=1

∗ (z1,n ) ≥ 12 , and have supjust as above. Let z1,n satisfy ||z1,n || = 1 and z1,n ∗ port contained in {kn + 1, kn + mn }, i.e., the support of z1,n . If we let S be a linear operator satisfying Sei = eπn (i) for i ∈ Fn , and deﬁne zl,n = S l−1 z1,n for 1 < l ≤ n, then it follows by the symmetry of the norm that ||zl,n || = 1 ∗ ∗ (zl,n ) = z1,n (z1,n ) whenever 1 ≤ l ≤ n. By design, we have ensured and zl,n that (1) holds. To check (2), we observe that n n n n n ∗ al zl,n ≤ |al | ≤ 2 λl zl,n ak zk,n ≤ 4 al zl,n l=1

l=1

l=1

k=1

l=1

where λl al = |al | for 1 ≤ l ≤ n. Therefore (2) holds whenever M > 4. Case III: if neither case I nor case II hold then, following the proof of [12, Theorem 1] in case III, we obtain constants K > 0 and r > 2 such that for all n ∈ N and scalars a1 , . . . , an , we have n n 1s n r1 K 1 K −1 |al |s ≤ al en+l ≤ 1 |al |r 1 n r λ(n) ns l=1 l=1 l=1 where r−1 + s−1 = 1. We set p = 2 and Fn = 2n + 1, . . . , 2n+1 . Fix n and let f be a bijection from F = Fn to G, where G is as in Proposition 2.2. Put vσ = ef −1 (σ) for σ ∈ G, and let zl , 1 ≤ l ≤ n, be as in Proposition 2.2. Let τ be the cycle (1, . . . , n), deﬁne a permutation π ˆ on G

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ˆ ◦ f . We have (3). If S is an by π ˆ (σ) = σ ◦ τ −1 , and then set π = f −1 ◦ π operator on X satisfying Sei = eπ(i) then we calculate σ(l)vσ = S σ(l)ef −1 (σ) Szl = S σ∈G

σ∈G

=

σ(l)ef −1 (ˆπ(σ))

σ∈G

=

σ∈G

σ(l)vπˆ (σ) =

(σ ◦ τ )(l)vσ = zτ (l) .

σ∈G

Moreover, by construction, we have ensured that d([zl ]nl=1 , np ) < M .

We remark that we can follow the proof of [12, Theorem 1] a little more to show that the subspaces [zl,n ]nl=1 , n ∈ N, are uniformly complemented in X, that is, they are the images of a sequence of projections which are uniformly bounded in norm. However, we do not require this particular property of the [zl,n ]nl=1 .

3. Proof of Theorem 1.3 Let X, Y, E and J be as in Theorem 1.3. To aid understanding, we shall assume for now that X = c0 or X = p , 1 ≤ p < ∞, with standard unit vector basis (ei )∞ i=1 . It will be convenient to set p = ∞ if X = c0 . At the end of the section, we will deﬁne R on a general Banach space with a symmetric basis. The latter definition uses Lemma 2.3 and is a little more complicated than the former. However, the proof of Proposition 3.1 is largely the same in the general case, and we will make explicit every important difference. As X has a symmetric basis, it is isomorphic to its closed, ﬁnite(d−1) , codimensional subspaces. Moreover, X is isomorphic to the space X2 which denotes the product of d − 1 copies of X, endowed with the norm d−1 2 2 given by ||(x1 , . . . , xd−1 )|| = j=1 ||xj || . We write d2 for the d-dimensional Euclidean space. By considering a suitable isomorphism, Theorem 1.3 follows from the following proposition. Proposition 3.1. Whenever E is a closed, symmetric subset of Sd2 , with 2 ≤ (d−1)

d < ∞, then there exists an operator R : d2 ⊕ X2 two properties: 1. 2.

(d−1)

−→ d2 ⊕ X2

with

if u ∈ E then ||Rn (u, x)|| → ∞; if u ∈ Sd2 \E then there is a subsequence (Rni (u, x)) of (Rn (u, x)) such that Rni (u, x) → (u, x) weakly.

We shall prove Proposition 3.1 with a sequence of lemmas. As in Sect. 2, we shall assume that all spaces are complex. We need only change the ﬁeld to obtain a proof in the real case. First, we consider a map ρ deﬁned on Sd2 by ρ(u, v)2 = 1 − | u , v |2 ,

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where · , · denotes the usual inner product. This function ρ is a pseudometric on Sd2 which, given u ∈ Sd2 , identiﬁes the points λu, |λ| = 1. We shall also deﬁne ρ(u, E) = inf{ρ(u, v) : v ∈ E}. Since E is closed and symmetric, it follows that ρ(u, E) = 0 if and only if u ∈ E. Given v ∈ Sd2 \E, we select an orthonormal basis bv,1 , . . . , bv,d−1 of the perpendicular subspace v ⊥ and deﬁne Δv : d2 −→ d−1 by 2 Δv (u) =

1 ( u , bv,1 , . . . , u , bv,d−1 ). ρ(v, E)

Obviously, ||Δv (u)||2 = ρ(u, v)/ρ(v, E) whenever u ∈ Sd2 , where ||·||2 denotes the usual norm on d−1 . 2 Let Wn = {v ∈ Sd2 : ρ(v, E) ≥ 2−n }. It is a straightforward matter to show that, for each n, we can ﬁnd a n−1 -net of Sd2 , with respect to ρ, which has size of order n2d−1 (or nd−1 if we are considering real Banach spaces). Therefore, there exists an integer K such that, for each n, there are vectors n v1n , . . . vK2 n(2d−1) ∈ Wn ,

with repetitions if necessary, with the property that for any u ∈ Wn , we can ﬁnd vin satisfying ρ(u, vin ) ≤ 2−n . Lemma 3.2. Let u ∈ Sd2 . Firstly ||Δv (u)||2 ≤ 2n

whenever v ∈ Wn .

(3.1)

whenever v ∈ Sd2 \E.

(3.2)

Secondly, if u ∈ E then ||Δv (u)||2 ≥ 1

Finally, if u ∈ E then there exists n0 with the property that whenever n > n0 , there is i in the range 1 ≤ i ≤ K2n(2d−1) , such that ||Δvin (u)||2 ≤ 2n0 +1−n .

(3.3)

Proof. We prove only (3.3). Fix n0 such that u ∈ Wn0 . For n > n0 , we can ﬁnd v = vin such that ρ(u, v) ≤ 2−n . Therefore ρ(v, E) ≥ ρ(u, E) − ρ(u, v) ≥ 2−n0 − 2−n ≥ 2−(n0 +1) and ||Δv (u)||2 ≤ 2n0 +1−n .

It is time to deﬁne our operator R. We take constants mk , Hk ∈ N and εk > 0. The values of these constants will be chosen in due course. For n ∈ N, deﬁne 1 1 n(n − 1) + 1, . . . , n(n + 1) . Fn = 2 2 The Fn form a decomposition of N into blocks of length n. We write ei,n = e 12 n(n−1)+i for 1 ≤ i ≤ n. Let πn be the cyclic permutation of Fn given by πn (i) = i + 1 for i < 12 n(n + 1), and πn ( 12 n(n + 1)) = 12 n(n − 1) + 1. We reproduce the eﬀect of the cyclic and feed operators from Sect. 2 on the spaces

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k [el,Hk ]H l=1 , spanned by the blocks of basis vectors e1,Hk , . . . , eHk ,Hk , for k ≥ 1. Deﬁne an isometric ‘block cyclic’ operator S by eπHk (i) if i ∈ FHk for some Hk Sei = ei otherwise

and extending linearly to X. Observe that if i ∈ FHk then Sel,Hk = eτ (l),Hk , where τ is the cyclic permutation (1, . . . , Hk ). In particular, each space k [el,Hk ]H l=1 is invariant for S. Now deﬁne a sequence of feed operators Fk : k C −→ [el,Hk ]H l=1 by Fk a = aεk

mk

el,Hk − aεk

l=1

2m k

el,Hk .

l=mk +1

To deﬁne our operator R, we require the standard projections Q and Qj of (d−1) d2 ⊕ X2 onto d2 and onto the jth copy of X, respectively. We deﬁne inten gers C1 = 1 and Cn+1 = Cn + K2n(2d−1) for n ≥ 1, and set wk = vk+1−C n (d−1)

whenever Cn ≤ k < Cn+1 . Finally, if u ∈ d2 and x ∈ X2 R(u, x) by QR(u, x) = u and

Qj R(u, x) = SQj (0, x) +

, we can deﬁne

∞ Fk ( u , bwk ,j ) ρ(wk , E)

k=1

for 1 ≤ j ≤ d − 1 and where bwk ,j is the jth element of the given basis of wk⊥ , chosen above. Of course, it is necessary to choose the constants mk , Hk and εk so that R is well-deﬁned. First, let m1 = 1 and H0 = 1. Then set Hk = (5dn +1)Hk−1 , mk = Hk−1 − mk−1 and ⎧ n ⎨ (p+1)p if p < ∞ −1 mk εk = ⎩ n if p = ∞ mk whenever Cn ≤ k < Cn+1 . Our ﬁrst task is to show that, with respect to (d−1) these constants, R is a bounded operator mapping into d2 ⊕ X2 . To tie in k with the notation of Lemma 2.1, let Sk be the restriction of S to [el,Hk ]H l=1 , and set Rk (a, z) = (a, Sk z + Fk a)

and Pk (a, z) = z

k whenever a ∈ C and z ∈ [el,Hk ]H l=1 .

(d−1)

Lemma 3.3. The operator R is bounded and maps into d2 ⊕ X2 . ∞ Proof. It is enough to show that k=1 ρ(wk , E)−1 Fk ( u , bwk ,j ) is absolutely summable for 1 ≤ j ≤ d − 1. We have H k yl el,Hk = ||y||p (3.4) l=1

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k k and || · ||p is the usual norm on H where y = (y1 , . . . , yHk ) ∈ H p p . Take L from Lemma 2.1. Note that Fk a = Sk 0 + Fk a = Pk Rk (a, 0). Therefore, from (2.3) with t = 1, (3.1), (3.4) and the definition of εk , we get p−1 −1 ||Fk ( u , bwk ,j )|| | u , bwk ,j | if p < ∞ k , E) ≤ Lεk mk ρ(w−1 ρ(wk , E) if p = ∞ Lεk ρ(wk , E) | u , bwk ,j |

≤ Ln||Δwk (u)||2 m−1 k

≤ Ln2n m−1 k

whenever Cn ≤ k < Cn+1 . From the definitions of mk and Hk , we obtain mk+1 = Hk − mk = Hk − Hk−1 + mk−1 ≥ 5dn Hk−1 ≥ 5dn mk

(3.5)

whenever Cn ≤ k < Cn+1 . In particular, mk+1 ≥ 5d mk . Therefore ∞ ∞ Cn+1 −1 ||Fk ( u , bwk ,j )|| ≤ Ln2n m−1 k ρ(wk , E) n=1

k=1

k=Cn

≤ Lm−1 1

∞ Cn+1 −1 n2n 5d(k−1) n=1 k=C n

≤ L5d =

∞ Cn+1 −1

n2n 5dn

n=1 k=Cn ∞ n2n L5d K 2n(2d−1) dn 5 n=1

dn ∞ 4 = L5 K n 5 n=1 d

bearing in mind that k − 1 ≥ Cn − 1 ≥ n − 1 whenever Cn ≤ k < Cn+1 . (d−1) Hence R is bounded and R(u, x) ∈ d2 ⊕ X2 . In order to analyse the behaviour of Rm (u, x), it will help to consider separately Rm (u, 0) and Rm (0, x). (d−1)

Lemma 3.4. Given (u, x) ∈ d2 ⊕ X2

, we have

m

Qj R (0, x) = S m Qj (0, x)

(3.6)

and Qj Rm (u, 0) =

∞ Pk Rkm ( u , bwk ,j , 0) ρ(wk , E)

(3.7)

k=1

for 1 ≤ j ≤ d − 1 and for all m. Proof. We proceed by induction. Clearly Qj R(0, x) = SQj (0, x), so (3.6) holds for m = 1. If (3.6) holds for some m ≥ 1 and Rm (0, x) = (0, y) then Qj Rm+1 (0, x) = Qj R(0, y) = SQj (0, y) = SQj Rm (0, x) = S m+1 Qj (0, x). Now Pk Rk (a, 0) = Sk 0 + Fk a = Fk a

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and SQj (u, 0) = 0, so (3.7) holds for m = 1. Assume that (3.7) holds for some k m ≥ 1. Suppose that Rkm (a, 0) = (a, z), where z ∈ [el,Hk ]H l=1 . We observe Pk Rkm+1 (a, 0) = Pk Rk (a, z) = Sk z + Fk a = Sz + Fk a = SPk Rkm (a, 0) + Fk a. remembering that Sk z = Sz. Therefore, if Rm (u, 0) = (u, z ) then Qj Rm+1 (u, 0) = Qj R(u, z ) ∞ Fk ( u , bwk ,j ) = SQj (0, z ) + ρ(wk , E)

k=1

∞ Fk ( u , bwk ,j ) ρ(wk , E) k=1 ∞ ∞ Pk Rm ( u , bw ,j , 0) Fk ( u , bwk ,j ) k k =S + ρ(wk , E) ρ(wk , E)

= SQj Rm (u, 0) +

k=1

= =

∞ k=1 ∞ k=1

SPk Rkm ( u ,

k=1

bwk ,j , 0) + Fk ( u , bwk ,j ) ρ(wk , E)

Pk Rkm+1 ( u , bwk ,j , 0) ρ(wk , E)

as required.

The consequence of Lemma 3.4 is that we can split the analysis of Rm (u, x) into two parts: the ‘block cycle’ and the ‘perturbation’. First, we examine the behaviour of the block cycle. (d−1)

, we have ||Rm (0, x)|| = ||(0, x)|| for all m. Lemma 3.5. Given x ∈ X2 w Hk Moreover, R (0, x) → (0, x). Proof. Given (3.6) and the fact that S is an isometry, the ﬁrst assertion is trivial. Now consider the weak convergence. Let f ∈ X ∗ with ||f || = 1, ε > 0 ∞ and 1 ≤ j ≤ d − 1. If Qj (0, x) = i=1 xi ei , we take k ∈ N such that ∞ < ε. x e i i l=k+1 i∈FH l

Hr Since Hl divides Hr whenever l ≤ r, we can see that πH is the identity for l such l. Therefore, if r ≥ k, we estimate

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|f (Qj RHk (0, x) − Qj (0, x))| = |f (S Hr Qj (0, x) − Qj (0, x))| ⎛ ⎞ ∞ ∞ ⎝ ⎠ = f xi eπHr (i) − xi ei Hl l=r+1 i∈FHl l=r+1 i∈FHl ∞ ≤ 2 xi ei l=r+1 i∈FH l ∞ ≤ 2 xi ei < 2ε l=k+1 i∈FH l

w

by symmetry of the norm. Thus Qj RHk (0, x) → Qj (0, x) whenever 1 ≤ j ≤ d − 1. The weak convergence of RHk (0, x) to (0, x) follows. Now we analyse the behaviour of the perturbation. Ultimately, it is the perturbation that drives the behaviour of the system as a whole. Lemma 3.6. If u ∈ E then ||Rm (u, 0)|| → ∞. On the other hand, if u ∈ Sd2 \E then there exists kn in the range Cn ≤ kn < Cn+1 with the property that H R kn −1 (u, 0) − (u, 0) → 0. Proof. Let u ∈ E and suppose that p < ∞. Assume that mk ≤ m < Hk − mk = mk+1 and Cn ≤ k < Cn+1 . Then by (2.1), (3.2), (3.4), (3.7) and the definition of εk , we have 2

||Rm (u, 0)|| ≥

d−1

2

||Qj Rm (u, 0)||

j=1

≥

d−1 j=1

≥

d−1 j=1

= ≥

||Pk Rkm ( u , bwk ,j , 0)|| ρ(wk , E) 2 p+1

2 p+1 2 p+1

2p−1

2p−1 2p−1

n2

2(p+1)p−1

ε2k mk

| u , bwk ,j | ρ(wk , E)

2 d−1 | u , bwk ,j | ρ(wk , E) j=1

n2

If u ∈ E and p = ∞ then likewise we obtain ||Rm (u, 0)|| ≥ n. Either way, ||Rm (u, 0)|| → ∞.

2 2

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Instead, if u ∈ Sd2 \E then, by (3.3), for every n > n0 there exists kn in the range Cn ≤ kn < Cn+1 , such that ||Δwkn (u)||2 ≤ 2n0 +1−n . By (2.2), (3.4) and the definition of εk , we have (p+1)p−1 Hkn −1 | u , bwkn ,j | ( u , bwkn ,j , 0) ≤ Lεkn mkn Pkn Rkn Lεkn mkn | u , bwkn ,j | ≤ Lρ(wkn , E)n2n0 +1−n . Then we notice that if r ≤ kn − 1, we have H Pr Rr kn −1 ( u , bwr ,j , 0) = 0

if p < ∞ if p = ∞ (3.8)

(3.9)

because RrHr is the identity and Hr divides Hkn −1 whenever r ≤ kn − 1. Now Hkn −1 we have to estimate Pr Rr ( u , bwr ,j , 0) for r ≥ kn + 1. If r ≥ kn + 1 then from (3.5), we have mr ≥ 5d(r−(kn +1)) mkn +1 ≥ 5d(r−(kn +1)) 5dn Hkn −1 . Take l ≥ n such that Cl ≤ r < Cl+1 . We apply (2.3), (3.1) and (3.4) to obtain H Pr Rr kn −1 ( u , bwr ,j , 0) −1 Lεr mpr Hkn −1 | u , bwr ,j | if p < ∞ ≤ if p = ∞ Lεr Hkn −1 | u , bwr ,j | ρ(wr , E)l2l mr l2l ≤ Lρ(wr , E) d(r−(k +1)) dn n 5 5 l l2 ≤ Lρ(wr , E) dn d(l−n) since l − n ≤ r − (kn + 1) 5 5 l2l = Lρ(wr , E) dl (3.10) 5 Combining (3.7) with (3.8), (3.9) and (3.10) gives Hkn −1 ∞ P Rr ( u , bwr ,j , 0) r H k −1 Qj R n (u, 0) ≤ ρ(wr , E) r=1 Hkn −1 ∞ P Rr ( u , b

, 0) r wr ,j = ρ(wr , E) r=kn H ∞ Pr Rr kn −1 ( u , bwr ,j , 0) n0 +1−n + = Ln2 ρ(wr , E) ≤ LHkn −1

r=kn +1 ∞

≤ Ln2n0 +1−n + L

K2l(2d−1)

l=n

l2l 5dl

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∞ dl 4 l 5 l=n

→0 as n → ∞. This concludes the proof.

Proof of Proposition 3.1. Let u ∈ E. By Lemmas 3.5 and 3.6 we have ||Rm (u, x)|| ≥ ||Rm (u, 0)|| − ||Rm (0, x)|| = ||Rm (u, 0)|| − ||(0, x)|| → ∞ as m → ∞. Now suppose u ∈ Sd2 \E. Again by Lemmas 3.5 and 3.6, we can pick suitable kn such that w

RHkn −1 (u, x) = RHkn −1 (u, 0) + RHkn −1 (0, x) → (u, 0) + (0, x) = (u, x). Now we describe the general case. Let M , p ∈ {1, 2, ∞}, Fn , zl,n and πn be as in Lemma 2.3. We can safely assume that M ≥ L, where again L is as in Lemma 2.1. We deﬁne S as above, but with respect to the new Fn and πn . This time, we reproduce the eﬀect of the cyclic and feed operators of Sect. 2, k using the zl,n , rather than the el,n . Deﬁne feeds Fk : C −→ [zl,Hk ]H l=1 by Fk a = aεk

mk

zl,Hk − aεk

2m k

zl,Hk .

l=mk +1

l=1

We deﬁne R as before, but with respect to the new S and Fk . The accompanying constants remain the same. The operators Sk , Rk and Pk are as above, Hk k except that we replace [el,Hk ]H l=1 by [zl,Hk ]l=1 . The proof of Proposition 3.1 in the general case is essentially identical to that given above. We ﬁnish the section outlining the main differences. Lemma 2.3 has been designed to ensure that Sz = Sk z =

Hk

yl zτ (l),Hk

l=1

Hk where z = l=1 yl zl,Hk and τ is the cycle (1, . . . , Hk ). In this way, we retrieve the cyclic behaviour from Sect. 2. The proof of Lemma 3.3 proceeds as before, except that we need to replace (3.4) by H k 1 − 12 yl zl,Hk ≤ M 2 ||y||p M ||y||p ≤ l=1

3 2

and all instances of L by M in the subsequent estimates. Likewise, we have 3 to replace L by M 2 in the proof of Lemma 3.6. It is also necessary to modify the estimates at the beginning of this proof, which apply when u ∈ E. Now, we have p−1 1 2 n ||Rm (u, 0)|| ≥ M − 2 p+1 1

if p = 1 or p = 2, and ||Rm (u, 0)|| ≥ M − 2 n if p = ∞.

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4. Problems Since the operators constructed in this note rely fundamentally on permutations of basis vectors, it makes sense to pose the following question. Problem 4.1. If X is a Banach space with an unconditional basis, does there exist an operator T on X with the property that AT is both non-empty and non-dense? Now, we move away from unconditionality. According to Gowers’s Dichotomy, if an inﬁnite-dimensional Banach space X contains no unconditional basic sequence then it must contain a hereditarily indecomposable (or HI) subspace. It is known that if X is a complex HI space, then every operator on X is of the form λI + S, where S is strictly singular (cf. [6, Theorem 6]). Recently, Argyros and Haydon constructed a real HI space X with the property that every operator on X takes the form λI + K, where K is compact [1]. At the end of the Introduction, we showed that no strictly singular operator can be a counterexample to Conjecture 1.2. Given these facts, the next questions seem natural to us. Problem 4.2. Is it possible to find an operator of the form I + T , where T is strictly singular or even compact, such that AI+T is non-empty and nondense? In particular, can such an operator exist on Argyros–Haydon space? If the questions of Problem 4.2 have negative solutions then this would suggest to us that some kind of unconditional structure is necessary in order to construct such operators. Finally, we make a remark about the title of this note. The operator R constructed in Sect. 3 can be viewed as a machine which acts on a countable family of disjoint cycles. This family of disjoint cycles can be seen as a countable directed graph. We speculate that it may be possible to construct operators with other interesting properties by basing them on more complicated directed graphs. Acknowledgement We thank the referees for several helpful suggestions which enabled us to simplify certain results and improve the presentation of the paper.

References [1] Argyros, S.A., Haydon, R.G.: A hereditarily indecomposable L∞ -space that solves the scalar-plus-compact problem. Preprint. http://arxiv.org/abs/0903. 3921 [2] Beauzamy, B.: Introduction to operator theory and invariant subspaces. NorthHolland Mathematical Library, vol. 42. North-Holland, Amsterdam (1988) [3] Caradus, S.R.: Operators of Riesz type. Pac. J. Math 18, 61–71 (1966) [4] Figiel, T., Johnson, W.B.: A uniformly convex Banach space which contains no p . Composito Math. 29, 179–190 (1977)

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[5] Jung, I.B., Ko, E., Pearcy, C.: Some nonhypertransitive operators. Pac. J. Math 220, 329–340 (2005) [6] Maurey, B.: Banach spaces with few operators. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1249–1297. Elsevier, Amsterdam (2003) [7] M¨ uller, V.: Orbits, weak orbits and local capacity of operators. Integral Equ. Oper. Theory 41, 230–253 (2001) [8] M¨ uller, V.: Orbits of operators. In: Aizpuru-Tom´ as, A., Le´ on-Saavedra, F. (eds.) Advanced Courses of Mathematical Analysis I, pp. 53–79. C´ adiz (2004) [9] M¨ uller, V., Vrˇsovsk´ y, J.: Orbits of linear operators tending to infinity. Rocky Mountain J. Math. 39, 219–230 (2009) [10] Prˇ ajiturˇ a, G.: The geometry of an orbit. Preprint [11] Rolewicz, S.: On orbits of elements. Studia Math. 32, 17–22 (1969) [12] Tzafriri, L.: On Banach spaces with unconditional bases. Israel J. Math. 17, 84–93 (1974) Petr H´ ajek Institute of Mathematics of the AS CR ˇ a 25 Zitn´ 115 67 Praha 1 Czech Republic e-mail: [email protected] Richard J. Smith School of Mathematical Sciences University College Dublin Belfield, Dublin 4 Ireland e-mail: [email protected] Received: May 31, 2009. Revised: January 12, 2010.

Integr. Equ. Oper. Theory 67 (2010), 33–49 DOI 10.1007/s00020-010-1767-x Published online April 13, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Existence Results for Fractional Order Semilinear Integro-Differential Evolution Equations with Infinite Delay Yong Ren, Yan Qin and R. Sakthivel Abstract. This paper is concerned with existence results of mild solutions for fractional order semilinear integro-differential evolution equations (FSIDEEs) and semilinear neutral integro-differential evolution equations (FSNIDEEs in short) with inﬁnite delay in α-norm. Our tools include the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and the Krasnoselskii–Schaefer type ﬁxed point theorem. Mathematics Subject Classification (2000). Primary 34A12; Secondary 34G10. Keywords. Fractional order differential equation, fractional derivative, fractional integral, ﬁxed point theorem, mild solution.

1. Introduction The theory of fractional order differential equations has become an active area of investigation due to their applications in the ﬁelds such as physics, technical sciences and so on. One can see Anguraj [1], Lakshmikantham and his co-authors [14,15], Hu et al. [10], Mophou and N’Gu´er´ekata [18,19], N’Gu´er´ekata [20,21], Podlubny [23], Lin [16], Metzler et al. [17], Zhang [25] and references therein. Recently, Balachandran and Park [2] proved the existence and uniqueness of solutions to the following fractional order semilinear evolution equations by means of the Banach contraction principle and the Krasnoselskii ﬁxed point theorem: q D u(t) − A(t)u(t) = f (t, u(t)) , t ∈ J := [0, T ], u(0) + g(u) = u0 , where T > 0, 0 < q < 1, the derivative Dq is understood here in the Riemann–Liouville sense and A(t) is a bounded linear generator in a Banach space X. And Mophou and N’Gu´er´ekata [18] established the existence and

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uniqueness of mild solutions in α-norm to the following semilinear fractional order differential equations with nonlocal conditions by means of the Banach contraction principle and the Schauder ﬁxed point theorem in a Banach space X: t Dq x(t) + Ax(t) = f t, x(t), 0 K(t, s)x(s) ds , t ∈ J := [0, T ], x(0) = g(x) + x0 , where 0 < q < 1. Very recently, Chang et al. [5] extended the results of [18] to the following semilinear fractional order integro-differential equations with nonlocal conditions in a Banach space X: t Dq x(t) + Ax(t) = f t, x(t), 0 e (t, s, x(s)) ds , t ∈ J := [0, T ], x(0) = g(x) + x0 . However, in many areas of science there has been an increasing interest in the investigation of functional differential equations incorporating memory or aftereffect, i.e., there is the eﬀect of inﬁnite delay on state equations. We refer the reader to Kolmanovskii and Myshkis [12,13], Wu [24] and references therein for a wealth of reference materials on the subject. Therefore, there is a real need to discuss functional differential systems with inﬁnite delay. And the development of the theory of functional differential equations with inﬁnite delays depends on a choice of a phase space. In fact, various phase spaces have been considered and each different phase space has required a separate development of the theory (Hino et al. [9]). The common space is the phase space B proposed by Hale and Kato [7], which is widely applied in functional differential equations with inﬁnite delay, one can see Hern´ andez [8] and references therein. Based on the phase space B, Benchohra et al. [3] proved the existence of mild solutions for the following fractional order functional differential equations with inﬁnite delay: α D y(t) = f (t, yt ), t ∈ J := [0, T ], 0 < α < 1, y(t) = φ(t) ∈ B, t ∈ (−∞, 0]. To the best of our knowledge, there is no work reported on fractional order semilinear integro-differential evolution equations (FSIDEEs) with inﬁnite delay. To close the gap, motivated by the above works, the ﬁrst purpose of this paper is to prove existence results of mild solutions in α-norm for the following FSIDEEs with inﬁnite delay: t Dq x(t) = Ax(t) + f t, xt , 0 a(t, s, xs ) ds , t ∈ J := [0, T ], (1.1) x(t) = φ(t) ∈ B, t ∈ (−∞, 0], where T > 0, 0 < q < 1, A is the inﬁnitesimal generator of a compact semigroup, analytic semigroup S(t), t ≥ 0 on a Banach space X, a and f are functions speciﬁed later. For any function x deﬁned on (−∞, T ] and any t ∈ J, we denote by xt the element of B (will be deﬁned in Sect. 2) deﬁned by xt (θ) = x(t + θ), θ ∈ (−∞, 0].

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Here, xt (·) represents the history of the state from time −∞ up to the present time t. The second purpose of this paper is to prove existence results of mild solutions in α-norm for the following fractional order semilinear neutral integro-differential evolution equations with inﬁnite delay (FSNIDEEs): t Dq [x(t) − g(t, xt )] = Ax(t) + f t, xt , 0 a(t, s, xs ) ds , t ∈ J := [0, T ], x(t) = φ(t) ∈ B, t ∈ (−∞, 0]. (1.2) The main techniques used here include the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and the Krasnoselskii–Schaefer type ﬁxed point theorem. The rest of this paper is organized as follows. In Sect. 2 we give some preliminaries. In Sect. 3, we study existence results of mild solutions for the system (1.1). And existence results of mild solutions for the system (1.2) are given in the last section.

2. Preliminaries In this section, we introduce some preliminaries which are used in the sequel. In this paper, X denotes a Banach space with norm | · |. A : D(A) → X is the inﬁnitesimal generator of a compact analytic semigroup of uniformly bounded linear operators S(t), t ≥ 0. Let 0 ∈ ρ(A). Then, it is possible to deﬁne the fractional power Aα for 0 < α ≤ 1, as a closed linear operator on its domain D(Aα ). Furthermore, the subspace D(Aα ) is dense in X and the expression |x|α := |Aα x|,

x ∈ D(Aα )

deﬁnes a norm on D(Aα ). Hereafter, we denote Xα the Banach space D(Aα ) with the norm |x|α . Then, for each 0 < α ≤ 1, Xα is a Banach space, and Xα → Xβ for 0 < β < α ≤ 1 and the imbedding is compact whenever the resolvent operator of A is compact. For the semigroup S(t), t ≥ 0, we have the following properties [22]: Proposition 2.1. (1) There exists a constant M ≥ 1 such that ||S(t)|| ≤ M, t ∈ J. (2) For any 0 ≤ α ≤ 1, there exists a positive constant Cα such that Cα Aα S(t) ≤ α , 0 < t ≤ T. t (3) S(t) : X → Xα for each t > 0. (4) Aα S(t)x = S(t)Aα x for each x ∈ D(Aα ) and t ≥ 0. (5) A−α is a bounded linear operator in X with D(Aα ) = Im(A−α ). In this paper, we employ an axiomatic definition of the phase space B introduced by Hale and Kato [7]. Let B be a linear space of functions mapping (−∞, 0] to Xα endowed with a seminorm || · ||B and satisfy the following axioms:

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If x : (−∞, T ] → Xα , is such that x0 ∈ B, then for every t ∈ [0, T ], the following conditions hold: (1) xt ∈ B, (2) |x(t)|α ≤ H||xt ||B , (3) ||xt ||B ≤ K(t) sup{|x(s)|α : 0 ≤ s ≤ t} + M (t)||x0 ||B , where H ≥ 0 is a constant, K : [0, T ] → [0, ∞) is continuous, M : [0, ∞) → [0, ∞) is locally bounded and H, K, M are independent of x(·). (B) For the function x(·) in (A), xt is a B-valued function on [0, T ]. (C) The space B is complete.

(A)

Now, we recall the following known definitions. For more details see [11,23]. Definition 2.2. The Riemann–Liouville fractional integral operator of order γ > 0 of a function f ∈ Cα , α ≥ −1 is deﬁned as I γ f (t) =

1 Γ(γ)

t (t − s)γ−1 ds 0

where Γ(·) is the gamma function. m Definition 2.3. If the function f ∈ C−1 , m ∈ N, the fractional derivative of order γ > 0 of a function f (t) in the Caputo sense is given by

dγ f (t) 1 = γ dt Γ(m − γ)

t (t − s)m−γ−1 f m (s) ds,

m − 1 < γ ≤ m.

0

In what follows, we state the following generalization of the Gronwall inequality appeared in [8]. Lemma 2.4. Let u(·), ω(·) : [0, T ] → [0, ∞) be continuous functions. If ω(·) is nondecreasing and there are constants θ > 0 and 0 < q < 1 such that t u(t) ≤ ω(t) + θ 0

u(s) ds, (t − s)1−q

then u(t) ≤ eθ

n

(Γ(q))n tnq /Γ(nq)

n−1 j=0

θT q q

t ∈ J,

j ω(t),

for every t ∈ [0, T ] and every n ∈ N such that nq > 1 and Γ(·) is the Gamma function. Now, we present the Krasnoselskii–Schaefer type ﬁxed point theorem (see [4]). Lemma 2.5. Let Φ1 and Φ2 be two operators such that (i) (ii)

Φ1 is a contraction, and Φ2 is completely continuous.

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Then, either (1) (2)

the operatorequation Φ1 x + Φ

2 x = x has a solution, or the set G = x ∈ Xα : λΦ1 λx + λΦ2 x = x is unbounded for λ ∈ (0, 1).

Let C(J, Xα ) be the Banach space of all continuous functions from J into Xα with the norm ||x||J = sup |x(t)|α . t∈J

Let the space Ω = x : (−∞, T ] → Xα , x|(−∞,0]∈B and x|[0,T ] is continuous .

3. FSIDEEs In this section, we state and prove existence results of mild solutions for the system (1.1). Firstly, we give the definition of mild solutions for the system (1.1). Definition 3.1. A function x ∈ Ω is said to be a mild solution of the system (1.1) if: • •

x(t) = φ(t) ∈ B, t ∈ (−∞, 0], the following integral equation holds for every t ∈ J ⎛ ⎞ t s 1 x(t) = S(t)φ(0) + (t − s)q−1 S(t − s)f ⎝s, xs , a(s, τ, xτ ) dτ ⎠ ds. Γ(q) 0

0

Let us list the following assumptions for some α ∈ (0, 1): (H1)

a : D := {(t, s) ∈ J × J : s ≤ t} × B → Xα is continuous and there exists a constant Ma > 0 such that for all (t, s) ∈ D, x, y ∈ B t (H1.1) 0 [a(t, s, x) − a(t, s, y)] ds ≤ Ma x − yB , α t (H1.2) 0 a(t, s, xs ) ds ≤ Ma (1 + xt B ) ; α

(H2)

f : J ×B×Xα → Xα is continuous. And there exists positive constants Mf , c, d such that for each (t, x, y), (t, x1 , y1 ), (t, x2 , y2 ) ∈ J × B × Xα (H2.1) |f (t, x1 , y1 ) − f (t, x2 , y2 )|α ≤ Mf (||x1 − x2 ||B + |y1 − y2 |α ) , (H2.2) |f (t, x, y)|α ≤ c (||x||B + |y|α ) + d;

Our ﬁrst existence result for the system (1.1) is based on the Banach contraction principle. Theorem 3.2. Assume that (H1.1) and (H2.1) hold. If φ ∈ Xα and MTq Mf (1 + Ma )KT < 1, Γ(q + 1)

(3.1)

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where KT = sup{|K(t)| : t ∈ [0, T ]}, then the system (1.1) has a unique solution on the interval (−∞, T ]. Proof. In order to obtain the existence of mild solutions for the system (1.1). Transform it into a ﬁxed point problem. Consider the operator Φ : Ω → Ω by ⎧ ⎪ ⎨ φ(t), t ∈ (−∞, 0], t 1 (Φx)(t) = S(t)φ(0) + Γ(q) (t − s)q−1 S(t − s) 0 ⎪

⎩ s ×f s, xs , 0 a(s, τ, xτ ) dτ ds, t ∈ J. : (−∞, T ] → Xα be the function deﬁned by Let φ(·) t ∈ (−∞, 0], = φ(t), φ(t) S(t)φ(0), t ∈ J. Let x(t) = y(t) + φ(t), −∞ < t ≤ T. It is obvious that x satisﬁes (3.1) if and only if y0 = 0 and 1 y(t) = Γ(q)

t (t − s)q−1 S(t − s) 0

⎛

×f ⎝s, ys + φs ,

s

⎞ a(s, τ, yτ + φτ ) dτ ⎠ ds,

0 ≤ t ≤ T.

0

Set B = {y ∈ Ω : y0 = 0 ∈ B}, and let || · ||b be the seminorm in B deﬁned by ||y||b = ||y0 ||B + sup{|y(s)|α : 0 ≤ s ≤ T } = sup{|y(s)|α : 0 ≤ s ≤ T }. Thus, (B , || · ||b ) is a Banach space. Deﬁne the operator Φ : B → B by 1 (Φy)(t) = Γ(q)

t (t − s)q−1 S(t − s) 0

⎛

×f ⎝s, ys + φs ,

s

⎞ a(s, τ, yτ + φτ )dτ ⎠ ds,

0 ≤ t ≤ T.

(3.2)

0

Then, the operator Φ has a ﬁxed point is equivalent to Φ has a ﬁxed point, and so we turn to proving that Φ has a ﬁxed point. The aim is to show that Φ is a contraction map of B . In fact, for each t ∈ J, y, y ∈ B , we have |(Φy)(t) − (Φy)(t)|α s ⎤ ⎡ t s M ≤ (t−s)q−1 Mf ⎣||ys −y s ||B + a(s, τ, yτ ) dτ − a(s, τ, y τ ) dτ ⎦ds Γ(q) 0

0

0

α

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≤ ≤

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t (t − s)q−1 Mf (1 + Ma )||ys − y s ||B ds 0

t (t − s)q−1 Mf (1 + Ma )KT sup |y(s) − y(s)|α ds s∈[0,T ]

0 q

MT Mf (1 + Ma )KT ||y − y||b . Γ(q + 1)

Thus, ||Φy − Φy||b ≤

MTq Mf (1 + Ma )KT ||x − y||b . Γ(q + 1)

From (3.1), we see that Φ is a contraction. Therefore, the system (1.1) has a unique solution on the interval (−∞, T ]. Now, we give another existence result for the system (1.1) by means of the nonlinear alternative of the Leray–Schauder ﬁxed point theorem. Theorem 3.3. Assume that (H1.2) and (H2.2) hold. If φ ∈ Xα , then the system (1.1) has at least one solution on the interval (−∞, T ]. Proof. Let Φ : B → B be deﬁned as (3.2). Firstly, we aim to prove that Φ is continuous and completely continuous. We proceed in the following four steps. Step 1. Φ is continuous. The continuities of f and a show that Φ is continuous. Step 2. Φ maps bounded sets into bounded sets in B . Indeed, it is enough to show that there exists a positive constant Λ such that for each y ∈ Bq = {y ∈ B : ||y||b ≤ q}, one has ||Φy||b ≤ Λ. Let y ∈ Bq . From (H1.2) and (H2.2), for each t ∈ [0, T ], we have t ⎛ ⎞ s 1 q−1 ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ⎠ ds |(Φy)(t)|α ≤ (t−s) S(t−s)f Γ(q) 0

0

α

⎛ ⎞ s 1 ≤ (t − s)q−1 S(t − s) Aα f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ⎠ ds Γ(q) 0 0 ⎛ ⎞ t s 1 q−1 ≤ (t − s) S(t − s) f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ ⎠ ds Γ(q) t

0

≤ ≤

Mc Γ(q)

t

0

(t − s)q−1 ||ys + φs ||B + Ma (1 + ||ys + φs ||B ) + d ds

0

M cT q [η + Ma (1 + η) + d] := Λ, Γ(q + 1)

α

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B := η, and where ||ys + φs ||B ≤ ||ys + φs ||B ≤ KT q + MT ||φ|| MT = sup{|M (t)| : t ∈ [0, T ]}. Therefore, we obtain ||Φy||b ≤ Λ. Step 3. Φ maps bounded sets into equicontinuous sets of B . Let t1 , t2 ∈ [0, T ], t1 < t2 , Bq be a bounded set of B as in Step 2 and y ∈ Bq . We have |(Φy)(t2 ) − (Φy)(t1 )|α t 1

1 (t2 − s)q−1 − (t1 − s)q−1 S(t − s) ≤ Γ(q) 0 ⎞ ⎛ s ⎠ ⎝ ×f s, ys + φs , a(s, τ, yτ + φτ ) dτ ds 0 α t ⎛ ⎞ 2 s 1 q−1 ⎝ ⎠ ≤ (t2 − s) S(t − s)f s, ys + φs , a(s, τ, yτ + φτ ) dτ ds Γ(q) 0

t1

≤

M c [η + Ma (1 + η) + d] Γ(q)

t1

α

(t1 − s)q−1 − (t2 − s)q−1 ds

0

M c [η + Ma (1 + η) + d] + Γ(q)

t2 (t2 − s)q−1 ds t1

M c [η + Ma (1 + η) + d] α [(t2 − t1 )α + tα 1 − t2 ] Γ(q + 1) M c [η + Ma (1 + η) + d] (t2 − t1 )α + Γ(q + 1) 2M c [η + Ma (1 + η) + d] (t2 − t1 )α . ≤ Γ(q + 1) ≤

As t2 → t1 the right hand of the above inequality tends to zero. The equicontinuity for the cases t1 < t2 ≤ 0 and t1 ≤ 0 ≤ t2 is obvious. Together with the Arzela–Ascoli theorem, Steps 1–3 show that Φ is continuous and completely continuous. Step 4. In what follows, we show that there exists an open set U ⊆ B with y = λΦy for λ ∈ (0, 1) and y ∈ ∂U. Let y ∈ B and y = λΦy for some λ ∈ (0, 1). Then, for each t ∈ [0, T ], we have ⎛ ⎞ ⎤ ⎡ t s 1 (t − s)q−1 S(t − s)f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ ⎠ ds⎦ . y(t) = λ ⎣ Γ(q) 0

0

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By (H1.2) and (H2.2), for each t ∈ [0, T ], we obtain t ⎛ ⎞ s 1 q−1 ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ⎠ ds |y(t)|α ≤ (t−s) S(t−s)f Γ(q) 0 0 α ⎛ ⎞ t s 1 ≤ (t − s)q−1 S(t − s) Aα f ⎝s, ys + φs , a(s, τ, yτ + φτ )dτ⎠ ds Γ(q) 0 0 ⎛ ⎞ t s 1 q−1 ⎝ ⎠ ≤ (t − s) S(t − s) f s, ys + φs , a(s, τ, yτ + φτ )dτ ds Γ(q) 0

≤

Mc Γ(q)

0

t

α

(t − s)q−1 ||ys + φs ||B + Ma (1 + ||ys + φs ||B ) + d ds.

0

For ||ys + φs ||B ≤ ||ys ||B + ||φs ||B ≤ K(t) sup{|y(s)|α : 0 ≤ s ≤ t} + M (t)||y0 ||B α : 0 ≤ s ≤ t} + M (t)||φ0 ||B + K(t) sup{|φ(s)| ≤ KT sup{|y(s)|α : 0 ≤ s ≤ t} + MT ||φ||B .

(3.3)

If we name u(t) the right-hand of (3.3), we have ||ys + φs ||B ≤ u(t), thus, M c(1 + Ma ) |y(t)|α ≤ Γ(q)

t (t − s)q−1 u(s) ds + 0

M c(Ma + d)T q . Γ(q + 1)

By the above inequality and the definition of u, we get KT M c(Ma + d)T q KT M c(1 + Ma ) u(t) ≤ MT ||φ||B + + Γ(q + 1) Γ(q)

t (t − s)q−1 u(s) ds 0

t (t − s)q−1 u(s) ds,

:= ω(t) + θ 0

where ω(t) = MT ||φ||B + Lemma 2.4, we obtain u(t) ≤ e

KT M c(Ma +d)T q , Γ(q+1)

θ n (Γ(q))n tnq /Γ(nq)

n−1 j=0

θ=

θT q q

j

KT M c(1+Ma ) . Γ(q)

Therefore, from

!, ω(t) := M

t ∈ J.

Thus, we get ||y||J ≤

M c(1 + Ma )T q ! M c(Ma + d)T q M+ := M ∗ . Γ(q + 1) Γ(q + 1)

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Set U = {y ∈ B : ||y||b < M ∗ + 1}. Φ : U → B is continuous and completely continuous. From the choice of U , there is no y ∈ ∂U such that y = λΦ(y), λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray–Schauder type [6], we deduce that Φ has a ﬁxed point in U .

4. FSNIDEEs In this section, we state and prove existence results of mild solutions for the system (1.2). Now, let us give the definition of mild solutions for the system (1.2). Definition 4.1. A function x ∈ Ω is said to be a mild solution of the system (1.2) if: • •

x(t) = φ(t) ∈ B, t ∈ (−∞, 0], the following integral equation holds for every t ∈ J x(t) = S(t) [φ(0) − g(0, φ)] + g(t, xt ) t 1 + (t − s)q−1 AS(t − s)g (s, xs ) ds Γ(q) 0 ⎛ ⎞ t s 1 + (t − s)q−1 S(t − s)f ⎝s, xs , a(s, τ, xτ ) dτ ⎠ ds. Γ(q) 0

0

(4.1) (H3)

There exists a constant β ∈ (0, 1) with α ≤ β ≤ 1, β + q > 1 such that the function g : J × B → Xβ is continuous, and Aβ g : J × B → Xα satisﬁes the Lipschitz condition, that is, there exists a constant Mg such that for any t ∈ J, x, y ∈ B, (H3.1) |Aβ g(t, x) − Aβ g(t, y)|α ≤ Mg ||x − y||B , (H3.2) |Aβ g(t, x)|α ≤ Mg (||x||B + 1), (H3.3) ||A1−β S(t)|| ≤ M1 , t ∈ J.

Our ﬁrst existence result of the mild solution for the system (1.2) is also based on the Banach contraction principle. Theorem 4.2. Assume that (H1.1), (H2.1) and (H3.1) hold. If φ ∈ Xα and # " MTq C1−β Mg KT T β+q−1 −β + Mf (1+Ma )KT < 1, L = ||A ||Mg KT + Γ(q)(β + q − 1) Γ(q + 1) (4.2) then the system (1.2) has a unique solution on the interval (−∞, T ].

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Proof. Consider the operator Φ1 : Ω → Ω by ⎧ φ(t), t ∈ (−∞, 0], ⎪ ⎪ ⎪ ⎨S(t) [φ(0) − g(0, φ)] + g(t, xt ) (Φ1 x)(t) = + 1 t (t − s)q−1 AS(t − s)g (s, x ) ds s ⎪ Γ(q) 0 ⎪ ⎪ ⎩ + 1 t (t−s)q−1 S(t−s)f s, x , s a(s, τ, x ) dτ ds, s 0 τ Γ(q) 0

43

(4.3) t ∈J.

In analogy to Theorem 3.2, consider the operator Φ1 : B → B such as Φ1 y(t) = 0, t ∈ (−∞, 0] and for t ∈ J (Φ1 y)(t) = −S(t)g(0, φ) + g(t, yt + φt ) t 1 + (t − s)q−1 AS(t − s)g s, ys + φs ds Γ(q) 0 ⎛ ⎞ t s 1 + (t−s)q−1 S(t − s)f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ⎠ ds. Γ(q) 0

0

Then, the operator Φ1 has a ﬁxed point is equivalent to Φ1 has a ﬁxed point, and so we turn to proving that Φ1 has a ﬁxed point. The aim is to show that Φ1 is a contraction map of B . In fact, for each t ∈ J, y, y ∈ B , we have (Φ1 y)(t) − (Φ1 y)(t) α ≤ g(t, yt + φt ) − g(t, y t + φt ) α t 1 ds + (t − s)q−1 AS(t − s) g s, ys + φs − g s, y s + φs Γ(q) 0 α t ⎛ ⎛ ⎞ s 1 + (t − s)q−1 S(t − s) ⎝f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ ⎠ Γ(q) 0 0 ⎛ ⎞⎞ s − f ⎝s, y s + φs , a(s, τ, y τ + φτ ) dτ ⎠⎠ ds 0 α ≤ A−β Aβ g(t, yt + φt ) − Aβ g(t, y t + φt ) α t 1 q−1 1−β β ds + (t − s) A S(t − s)A g s, ys + φs − g s, y s + φs Γ(q) 0 s ⎤α ⎡ t s M + (t−s)q−1 Mf ⎣||ys −y s ||B + a(s, τ, yτ ) dτ − a(s, τ, y τ ) dτ ⎦ ds Γ(q) 0

0

≤ ||A−β ||Mg ||yt − y t ||B +

+

M Γ(q)

t 0

1 Γ(q)

t

0

C1−β (t − s)β+q−2 Mg ||ys − y s ||B ds

0

(t − s)q−1 Mf (1 + Ma )||ys − y s ||B ds

α

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≤ ||A−β ||Mg KT sup |y(s) − y(s)|α s∈[0,T ]

C1−β Mg KT + Γ(q) + "

M Γ(q)

t

t

(t − s)β+q−2 sup |y(s) − y(s)|α ds s∈[0,T ]

0

(t − s)q−1 Mf (1 + Ma )KT sup |y(s) − y(s)|α ds s∈[0,T ]

0

≤ ||A−β ||Mg KT +

# C1−β Mg KT T β+q−1 MT q + Mf (1 + Ma )KT ||y − y||b Γ(q)(β + q − 1) Γ(q + 1)

= L||y − y||b .

Thus, ||Φ1 y − Φ1 y||b ≤ L||x − y||b , (4.2) shows that Φ1 is a contraction. Therefore, the system (1.2) has a unique solution on the interval (−∞, T ]. Our second existence result for the system (1.2) is based on the Krasnoselskii–Schaefer type ﬁxed point theorem (Lemma 2.5). Consider the operator Φ : Ω → Ω by ⎧ φ(t), t ∈ (−∞, 0], ⎪ ⎪ ⎪ ⎨S(t) [φ(0) − g(0, φ)] + g(t, xt ) (Φx)(t) = + 1 t (t − s)q−1 AS(t − s)g (s, x ) ds (4.4) s ⎪ Γ(q) 0 ⎪

⎪ t s 1 ⎩ + Γ(q) (t−s)q−1 S(t−s)f s, xs , 0 a(s, τ, xτ ) dτ ds, t ∈ J. 0 In analogy to Theorem 3.2, consider the operator Φ : B → B such as Φy(t) = 0, t ∈ (−∞, 0] and for t ∈ J (Φy)(t) = −S(t)g(0, φ) + g(t, yt + φt ) t 1 + (t − s)q−1 AS(t − s)g s, ys + φs ds Γ(q) 0 ⎛ ⎞ t s 1 + (t−s)q−1 S(t − s)f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ⎠ ds. Γ(q) 0

0

Then, the operator Φ has a ﬁxed point is equivalent to Φ has a ﬁxed point, and so we turn to proving that Φ has a ﬁxed point. Now, we decompose Φ as Φ1 + Φ2 , where (Φ1 y)(t) = −S(t)g(0, φ) + g(t, yt + φt ) t 1 + (t − s)q−1 AS(t − s)g s, ys + φs ds, Γ(q) 0

t∈J

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45

and t 1 (t − s)q−1 S(t − s) (Φ2 y)(t) = Γ(q) 0 ⎞ ⎛ s ×f ⎝s, ys + φs , a(s, τ, yτ + φτ ) dτ ⎠ ds,

t ∈ J.

0

Therefore, the problem is to be transformed to show that the operators Φ1 , Φ2 satisfy the conditions of Lemma 2.5. Lemma 4.3. Assume that (H1.1), (H2.1), (H2.1), (H2.2) and (H3.1) hold. If φ ∈ Xα and " # C1−β Mg KT T β+q−1 −β L1 = ||A ||Mg KT + < 1, (4.5) Γ(q)(β + q − 1) then Φ1 is a contraction and Φ2 is completely continuous. Proof. As shown in Theorem 3.3, Φ2 is completely continuous. Now, we In fact, for each t ∈ J, y, y ∈ B , we have show that Φ1 is a contraction. (Φ1 y)(t) − (Φ1 y)(t) α ≤ g(t, yt + φt ) − g(t, y t + φt ) α t 1 + (t − s)q−1 AS(t − s) g s, ys + φs − g s, y s + φs ds Γ(q) 0 α −β β β A g(t, yt + φt ) − A g(t, y t + φt ) ≤ A α t 1 + (t−s)q−1 A1−β S(t−s)Aβ g s, ys + φs −g s, y s + φs ds Γ(q) 0

α

≤ ||A−β ||Mg ||yt − y t ||B +

1 Γ(q)

t C1−β (t − s)β+q−2 Mg ||ys − y s ||B ds 0

≤ ||A−β ||Mg KT sup |y(s) − y(s)|α s∈[0,T ]

+

C1−β Mg KT Γ(q)

t (t − s)β+q−2 sup |y(s) − y(s)|α ds 0

s∈[0,T ]

" # C1−β Mg KT T β+q−1 ≤ ||A−β ||Mg KT + ||y − y||b Γ(q)(β + q − 1) = L1 ||y − y||b . Thus, ||Φ1 y − Φ1 y||b ≤ L1 ||x − y||b . (4.5) shows that Φ1 is a contraction.

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In order to study the existence results for the system (1.2), we consider the following nonlinear operator equation x(t) = λΦx(t),

0 < λ < 1,

(4.6)

where Φ is deﬁned as (4.4). The following lemma gives a priori estimate for the solution of the above equation. Lemma 4.4. Assume that (H1.2), (H2.2), (H3.2) and (H3.3) hold. If φ ∈ Xα and KT K1 < 1, where KT = sup{|K(t)| : t ∈ [0, T ]}, K1 = M ||A−β ||Mg , then there exists a constant K > 0 such that ||xt ||B ≤ K, t ∈ J. Proof. By (H1.2), (H2.2), (H3.2) and (H3.3), for each t ∈ J, we obtain |x(t)|α ≤ M ||φ||B + ||A−β ||Mg (1 + ||φ||B ) + M ||A−β ||Mg (1 + ||xt ||B ) t M1 M g + (t − s)q−1 (1 + ||xs ||B ) ds Γ(q) 0

+

Mc Γ(q)

t (t − s)q−1 [||xs ||B + Ma (1 + ||xs ||B ) + d] ds 0

t ≤ K0 + K1 ||xt ||B + K2

(t − s)q−1 ||xs ||B ds, 0

where K0 = M ||φ||B + ||A−β ||Mg (1 + ||φ||B ) + M ||A−β ||Mg + M cT q Γ(q+1) (Ma

−β

+ d), K1 = M ||A

||Mg , K2 =

M c(1+Ma ) . Γ(q)

M1 Mg T q Γ(q+1)

+

Therefore, we get

||xt ||B ≤ KT sup{|x(s)|α : 0 ≤ s ≤ t} + MT ||φ||B ≤ MT ||φ||B + KT K0 t + KT K1 sup ||xs ||B + KT K2 0≤s≤t

(t − s)q−1 ||xs ||B ds. 0

Let u(t) = sup{||xs ||B : 0 ≤ s ≤ t}, then the function u(t) is nondecreasing in J, and we have t u(t) ≤ MT ||φ||B + KT K0 + KT K1 u(t) + KT K2

(t − s)q−1 u(s) ds 0

≤

MT ||φ||B + KT K0 KT K2 + 1 − KT K1 1 − KT K1 t (t − s)q−1 u(s) ds,

:= ω(t) + θ 0

t (t − s)q−1 u(s) ds 0

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B +KT K0 where ω(t) = MT ||φ|| , θ = 1−KT K1 t ∈ J, we have

u(t) ≤ e

KT K2 1−KT K1 .

θ n (Γ(q))n tnq /Γ(nq)

n−1 j=0

47

By using Lemma 2.4, for each

θT q q

j ω(t) := K,

which shows the desired result.

Theorem 4.5. Assume that the assumptions of Lemmas 4.3 and 4.4 hold. Then, the system (1.2) has at least one mild solution on J. Proof. Let us take the set y % $ G Φ = y ∈ B : λΦ1 + λΦ2 y = y, for some λ ∈ (0, 1) . λ Then for any y ∈ G Φ , we have by Lemma 4.4 that ||xt ||B ≤ K, t ∈ J, and hence ||y||b = ||y0 ||B + sup{|y(t)|α : 0 ≤ t ≤ T } = sup{|y(t)|α : 0 ≤ t ≤ T } α : 0 ≤ t ≤ T} ≤ sup{|x(t)|α : 0 ≤ t ≤ T } + sup{|φ(t)| ≤ sup{H||xt ||B : 0 ≤ t ≤ T } + sup{|S(t)φ(0)|α : 0 ≤ t ≤ T } ≤ HK + M |φ(0)|α . This implies that G is bounded on J. Consequently, by the Krasnoselskii– Schaefer type ﬁxed point theorem (Lemma 2.5), the operator Φ has a ﬁxed t ∈ (−∞, T ], x is a ﬁxed point of the point y ∗ ∈ B . Since, x(t) = y ∗ + φ(t), operator Φ which is a mild solution of the system (1.2). Acknowledgements The ﬁrst author is a postdoctoral research fellow of University of Tasmania and he would like to thank the Australian Research Council for funding this project through Discovery Project DP0770388 and the University of Tasmania, in particular, Dr. Malgorzata O Reilly for providing a stimulating working environment. Also, he is partially supported by the Great Research Project of Natural Science Foundation of Anhui Provincial Universities (No. KJ2010ZD02).

References [1] Anguraj, A., Karthikeyan, P., N’Gu´er´ekata, G.M.: Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces. Comm. Math. Anal. 6, 31–35 (2009) [2] Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. (2009). doi:10.1016/j.na.2009. 03.005

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[3] Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahaba, A.: Existence results for fractional order functional differential equations with inﬁnite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008) [4] Burton, T.A., Kirk, C.: A ﬁxed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189, 23–31 (1998) [5] Chang, Y.K., Kavitha, V., Mallika Arjunan, M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. (2009). doi:10.1016/j.na.2009.04.058 [6] Granas, A., Dugundji, J.: Fixed Point Theory. Springer-Verlag, New York (2003) [7] Hale, J., Kato, J.: Phase spaces for retarded equations with inﬁnite delay. Funkcial Ekvac. 21, 11–41 (1978) [8] Hern´ andez, E.: Existence results for partial neutral functional integro-differential equations with unbounded delay. J. Math. Anal. Appl. 292, 194–210 (2004) [9] Hino, Y., Murakami, S., Naito, T.: Functional-differential equations with inﬁnite delay. In: Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991) [10] Hu, L., Ren, Y., Sakthivel, R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 79, 507–514 (2009) [11] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) [12] Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht (1992) [13] Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer, Dordrecht (1999) [14] Lakshmikantham, V.: Theory of fractional differential equations. Nonlinear Anal. 60, 3337–3343 (2008) [15] Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008) [16] Lin, W.: Global existence and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007) [17] Metzler, F., Schick, W., Kilian, H.G., Nonnemacher, T.F.: Relaxation in ﬁlled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995) [18] Mophou, G.M., N’Gu´er´ekata, G.M.: Mild solutions for semilinear fractional differential equations. Electron. J. Differ. Equ. 21, 1–9 (2009) [19] Mophou, G.M., N’Gu´er´ekata, G.M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79, 315–322 (2009) [20] N’Gu´er´ekata, G.M.: Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions. In: Differential and Difference Equations and Applications, pp. 843–849. Hindawi Publ. Corp., New York (2006) [21] N’Gu´er´ekata, G.M.: A Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. 70, 1873–1876 (2009) [22] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1963)

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[23] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) [24] Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996) [25] Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 36, 1–12 (2006) Yong Ren School of Mathematics University of Tasmania GPO Box 252C-37 Hobart, TAS 7001 Australia e-mail: [email protected] Yan Qin (B) Department of Mathematics East China University of Science and Technology Shanghai 200237 China e-mail: [email protected] R. Sakthivel Department of Mathematics Sungkyunkwan University Suwon 440-746 South Korea e-mail: [email protected] Received: June 3, 2009. Revised: September 21, 2009.

Integr. Equ. Oper. Theory 67 (2010), 51–56 DOI 10.1007/s00020-010-1769-8 Published online March 16, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Jordan Derivations of Reflexive Algebras Fangyan Lu Abstract. Let L be a subspace lattice on a Banach space X and suppose that ∨{L ∈ L : L− < X} = X or ∧{L− : L ∈ L, L > (0)} = (0). Then each Jordan derivation from AlgL into B(X) is a derivation. This result can apply to completely distributive subspace lattice algebras, J -subspace lattice algebras and reﬂexive algebras with the non-trivial largest or smallest invariant subspace. Mathematics Subject Classification (2000). Primary 47L35; Secondary 17B40, 17B60. Keywords. Jordan derivations, reﬂexive algebras, derivations.

1. Introduction and Preliminaries Throughout, all algebras and vector spaces will be over F, where F is either the real ﬁeld R or the complex ﬁeld C. Given a Banach space X with topological dual X ∗ , by B(X) we mean the algebra of all bounded linear operators on X. The terms operator on X and subspace of X will mean ‘bounded linear map of X into itself’ and ‘norm closed linear manifold of X’, respectively. For A ∈ B(X), denote by A∗ the adjoint of A. For any non-empty subset L ⊆ X, L⊥ denotes its annihilator, that is, L⊥ = {f ∈ X ∗ : f (x) = 0 for all x ∈ L}. A family L of subspaces of X is a subspace lattice if it contains (0) and X, and is complete in the sense that it is closed under the formation of arbitrary closed linear spans (denoted by ∨) and intersections (denoted by ∧). A nest is a totally ordered subspace lattice. Suppose that L is a subspace lattice L on X. For E ∈ L, we deﬁne E− = ∨{F ∈ L : F E}

and E+ = ∧{F ∈ L : F ≤ E}.

Put J (L) = {K ∈ L : K = (0) and K− = X}. Then L is called completely distributive (see [7]) if L = ∨{E ∈ L : E− ≥ L} for all L ∈ L ; L is called a J -subspace lattice (see [9]) if The research was supported by NNSFC (No. 10771154).

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(1) ∨{K : K ∈ J (L)} = X; (2) ∧{K− : K ∈ J (L)} = (0); (3) K ∨ K− = X, ∀K ∈ J (L); (4) K ∧ K− = (0), ∀K ∈ J (L). Given a subspace lattice L on X, the associated subspace lattice algebra AlgL is the set of operators on X leaving every subspace in L invariant, that is, AlgL = {A ∈ B(X) : Ax ∈ E for every x ∈ E and for every E ∈ L}. Dually, if A is a subalgebra of B(X), by LatA we denote the lattice of subspaces of X that are left invariant by each operator in A. An algebra A is reflexive if A = AlgLatA. Clearly, every reﬂexive algebra is of the form AlgL for some subspace lattice L and vice versa. In this paper, we are mainly interested in a certain tractable class of reﬂexive algebras, namely those which are rich in rank one operators. Let x ∈ X and f ∈ X ∗ be non-zero. The rank one operator x ⊗ f is deﬁned by y → f (y)x for y ∈ X. The relevance of J (L) is due to the following lemma. Lemma 1.1. [8]. If L is a subspace lattice, then the rank-one operator x ⊗ f ∈ ⊥ . AlgL if and only if there exists some K ∈ J (L) such that x ∈ K and f ∈ K− ⊥ ⊥ Here K− means (K− ) . Let A be an algebra and M be an A-bimodule. A linear map δ from A into M is called a derivation if δ(AB) = δ(A)B + Aδ(B) for all A, B ∈ A and a Jordan derivation if δ(A2 ) = δ(A)A + Aδ(A) for each A ∈ A. Clearly, a derivation is a Jordan derivation. The converse problem of whether a Jordan derivation is a derivation has received many mathematician’s attention for many years. See [4] and references therein. From the classical result of Bre˘sar [2], we know that each Jordan derivation of semiprime algebras is a derivation. The situation where algebras are semiprime is more involved, but also well understood [3,5,12], and so the problem is now interesting for non-semiprime algebras. Recently, we in [6] proved that Jordan derivations of nest algebras on Banach spaces are derivations, and Benkovic in [1] showed that every Jordan derivation of upper triangular matrix algebras is the sum of a derivation and an antiderivation. In this paper, we shall study Jordan derivations of a class of reﬂexive algebras, namely those reﬂexive algebras AlgL for which ∨{L : L ∈ J (L)} = X or ∧{L− : L ∈ J (L)} = (0). Such a class of reﬂexive algebras includes: • • • •

completely distributive subspace lattice algebras; J -subspace lattice algebras; reﬂexive algebras AlgL for which X− = X; reﬂexive algebras AlgL for which (0)+ = (0).

We shall prove that such Jordan derivations are derivations, which generalizes the result in [6]. The proof is idempotent-free. The idea we shall use is to gather together the kernel and range of a special class of operators. Roughly speaking, we shall prove that for some operators A, δ(A) maps the kernel of A into the range of A. This makes it possible to identify the behavior of the

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Jordan derivation on some special set of operators. We note that this kind of machinery is also eﬀective in describing the structure of the derivation [10] and the Lie derivation [11]. Proposition 1.2. [11, Proposition 1.1] Let E and F be non-zero subspaces of X and X ∗ , respectively. Let Φ : E × F → B(X) be a bilinear map such that Φ(x, f ) ker(f ) ⊆ Fx for all x ∈ E and f ∈ F . Then there exist two linear maps T : E → X and S : F → X ∗ such that Φ(x, f ) = T x ⊗ f + x ⊗ Sf for all x ∈ E and f ∈ F . We close this section by stating some equalities concerning Jordan derivations. Let δ be a derivation of an algebra A into an A-bimodule. Replacing A by A + B in δ(A2 ) = δ(A)A + Aδ(A), we get δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) for all A, B ∈ A. Further, from 2ABA = A(AB + BA) + (AB + BA)A − (A2 B + BA2 ) we see that δ also satisﬁes δ(ABA) = δ(A)BA + Aδ(B)A + ABδ(A) for all A, B ∈ A. This obviously yields δ(ABC + CBA) = δ(A)BC + Aδ(B)C + ABδ(C) + δ(C)BA + Cδ(B)A + CBδ(A) for all A, B, C ∈ A. In what follows, we shall frequently use these identities without implication.

2. Result and Proof The main result in this paper reads as follows. Theorem 2.1. Let L be a subspace lattice on a Banach space X and suppose that ∨{F : F ∈ J (L)} = X or ∧{F− : F ∈ J (L)} = (0). Let δ : AlgL → B(X) be a Jordan derivation. Then δ is a derivation. To prove the theorem, we need some lemmas. Lemma 2.2. Let x⊗f be in AlgL and suppose that f (x) = 1. Then δ(x ⊗ f ) ker(f ) ⊆ Fx. Proof. Indeed, from δ(x ⊗ f ) = δ((x ⊗ f )2 ) = (x ⊗ f )δ(x ⊗ f ) + δ(x ⊗ f )(x ⊗ f ), we easily see that δ(x ⊗ f ) ker(f ) ⊆ Fx.

Lemma 2.3. Let E and L be in J (L) and suppose that E− ≥ L. Let x be in E and f be in L⊥ − . Then δ(x ⊗ f ) ker(f ) ⊆ Fx. Proof. Since E− ≥ L, it follows that E ≤ L. So x⊗f ∈ AlgL. By Lemma 2.2, ⊥ such that we may assume that f (x) = 0. Choose z from L and g from E− g(z) = 1. Then we have that 0 = δ((z ⊗ f )(x ⊗ g)(z ⊗ f )) = δ(z ⊗ f )(x ⊗ f ) + (z ⊗ f )δ(x ⊗ g)(z ⊗ f ).

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So δ(z ⊗ f )x = −f (δ(x ⊗ g)z)z. Hence δ(x ⊗ f ) = δ((x ⊗ g)(z ⊗ f ) + (z ⊗ f )(x ⊗ g)) = δ(x ⊗ g)(z ⊗ f ) + (x ⊗ g)δ(z ⊗ f ) + δ(z ⊗ f )(x ⊗ g) + (z ⊗ f )δ(x ⊗ g) = δ(x ⊗ g)(z ⊗ f ) + (x ⊗ g)δ(z ⊗ f ) −f (δ(x ⊗ g)z)(z ⊗ g) + (z ⊗ f )δ(x ⊗ g). Let y be in ker(f ). Applying the above equation to y, we get δ(x ⊗ f )y = g(δ(z ⊗ f )y)x − g(y)f (δ(x ⊗ g)z)z + f (δ(x ⊗ g)y)z. (2.1) Notice that this equation is valid for all z in L satisfying g(z) = 1. If λ = g(x) = 0, replacing z by λ1 x in (2.1) we see that δ(x ⊗ f )y ∈ Fx. Now suppose g(x) = 0. Then g(z ± x) = 1. Replacing z by z + x and z − x in (2.1), respectively, we have δ(x ⊗ f )y = g(δ((z + x) ⊗ f )y)x − g(y)f (δ(x ⊗ g)(z + x))(z + x) +f (δ(x ⊗ g)y)(z + x)

(2.2)

and δ(x ⊗ f )y = g(δ((z − x) ⊗ f )y)x − g(y)f (δ(x ⊗ g)(z − x))(z − x) +f (δ(x ⊗ g)y)(z − x). Summing those two equations, we get 2δ(x ⊗ f )y = 2(g(δ(z ⊗ f )y)x − g(y)f (δ(x ⊗ g)z)z + f (δ(x ⊗ g)y)z) −2g(y)f (δ(x ⊗ g)x)x = 2δ(x ⊗ f )y − 2g(y)f (δ(x ⊗ g)x)x. So g(y)f (δ(x ⊗ g)x) = 0. Hence by (2.2) and (2.1), δ(x ⊗ f )y = g(δ(z ⊗ f )y)x + g(δ(x ⊗ f )y)x − g(y)f (δ(x ⊗ g)z))(z + x) + f (δ(x ⊗ g)y)(z + x) = g(δ(z ⊗ f )y)x − g(y)f (δ(x ⊗ g)z)x + f (δ(x ⊗ g)y)x − g(y)f (δ(x ⊗ g)z))(z + x) + f (δ(x ⊗ g)y)(z + x) = δ(x ⊗ f )y − 2(g(y)f (δ(x ⊗ g)z) − f (δ(x ⊗ g)y))x. So g(y)f (δ(x ⊗ g)z) − f (δ(x ⊗ g)y) = 0 and hence δ(x ⊗ f )y = g(δ(z ⊗ f )y)x by (2.1), completing the proof. We are now in a position to prove our main result. Proof of Theorem 3.1. We shall give the proof only for the case ∨{F : F ∈ J (L)} = X; the proof for the case ∧{F− : F ∈ J (L)} = (0) is similar. Let F be in J (L). Then there is an element E in J (L) such that F− ≥ E. Hence there is an element L in J (L) such that E− ≥ L. Then by Lemma 2.3 and Proposition 1.2, there are linear maps T : E → X and ∗ S : L⊥ − → X such that δ(x ⊗ f ) = T x ⊗ f + x ⊗ Sf

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for all x ∈ E and f ∈ L⊥ − . Hence for A ∈ AlgL there holds (δ(A) − AT + T A)x ⊗ f = −x ⊗ (δ(A)∗ − SA∗ + A∗ S)f for all x ∈ E and f ∈ L⊥ − . Consequently, there exists a linear map λ : AlgL → F such that δ(A)x = (AT − T A)x + λ(A)x,

A ∈ AlgL, x ∈ E.

Hence δ(AB)x = δ(A)Bx + Aδ(B)x −λ(A)Bx − λ(B)Ax + λ(AB)x,

A, B ∈ AlgL, x ∈ E. (2.3)

In particular, δ(A2 )x = (δ(A)A + Aδ(A))x − 2λ(A)Ax + λ(A2 )x and hence since δ(A2 ) = δ(A)A + Aδ(A) we have λ(A2 )x − 2λ(A)Ax = 0,

A ∈ AlgL, x ∈ E.

(2.4)

Our goal is to show that λ(A) = 0 for all A ∈ AlgL. To do this, ﬁx g ∈ F−⊥ and z ∈ E satisfying g(z) = 1. Let y be in F . If g(y) = 0, putting A = y ⊗ g and x = z in (2.4), we get λ(y ⊗ g)z = 0. If g(y) = 0, putting A = y ⊗ g and x = y in (2.4), we get g(y)λ(y ⊗ g)y = 0. Consequently, λ(y ⊗ g) = 0 for all y ∈ F . Now, for all A ∈ AlgL, x ∈ E, y ∈ F , by (2.3), we have δ(Ay ⊗ g)x = (δ(A)y ⊗ g + Aδ(y ⊗ g))x − g(x)λ(A)y,

(2.5)

δ(y ⊗ gA)x = (δ(y ⊗ g)A + y ⊗ gδ(A))x − g(x)λ(A)y + λ(y ⊗ gA)x,

(2.6)

δ(y ⊗ gAy ⊗ g)x = (δ(y ⊗ g)Ay ⊗ g + y ⊗ yδ(A)y ⊗ g + y ⊗ gAδ(y ⊗ g))x − g(x)λ(y ⊗ gA)y − g(x)g(y)λ(A)y.

(2.7)

Since δ is a Jordan derivation, for all A ∈ AlgL, x ∈ E, y ∈ F , by (2.5) and (2.6) there holds λ(y ⊗ gA)x − 2g(x)λ(A)y = 0

(2.8)

and by (2.7) there holds g(x)λ(y ⊗ gA) − g(x)g(y)λ(A) = 0.

(2.9)

If g(y) = 0 for some non-zero y ∈ F , putting x = z in (2.8) we get λ(y ⊗ gA) z − 2λ(A)y = 0. Hence since y and z are linearly independent, it follows that λ(A) = 0. Now suppose that g(y) = 0 for all non-zero y ∈ F . Putting x = y in (2.8) and (2.9) we have that λ(y ⊗ gA) − 2g(y)λ(A) = 0 and λ(y ⊗ gA) − g(y)λ(A) = 0. It follows that λ(A) = 0. Consequently, λ(A) = 0 for all A ∈ AlgL. Now let A and B be in AlgL. By (2.3), δ(AB)x = (δ(A)B + Aδ(B))x for all x ∈ E. In particular, since F ≤ E, δ(AB)x = (δ(A)B + Aδ(B))x for

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all x ∈ F . Hence, since ∨{F : F ∈ J (L)} = X, it follows that δ(AB) = (δ(A)B + Aδ(B)). The proof is complete.

References [1] Benkovic, D.: Jordan derivations and antiderivations on triangular matrices. Linear Algebra Appl. 397, 235–244 (2005) [2] Bre˘sar, M.: Jordan derivation on semiprime rings. Proc. Am. Math. Soc. 104, 1003–1007 (1988) [3] Bre˘sar, M.: Jordan mappings of semiprime rings. J. Algebra 127, 218–228 (1989) [4] Bre˘sar, M.: Jordan derivations revisited. Math. Proc. Camb. Phil. Soc. 139, 411–425 (2005) [5] Herstein, I.N.: Jordan derivations of prime rings. Proc. Am. Math. Soc. 8, 1104–1110 (1957) [6] Li, J., Lu, F.: Additive Jordan derivations of reﬂexive algebras. J. Math. Anal. Appl. 329, 102–111 (2007) [7] Longstaﬀ, W.E.: Strongly reﬂexive lattices. J. Lond. Math. Soc. 11, 491–498 (1975) [8] Longstaﬀ, W.E.: Operators of rank one in reﬂexive algebras. Can. J. Math. 28, 19–23 (1976) [9] Longstaﬀ, W.E., Panaia, Q.: J-subspace lattice and subspace M-bases. Studia Math. 139, 197–211 (2000) [10] Lu, F.: Derivations of CDC algebras. J. Math. Anal. Appl. 323, 179–189 (2006) [11] Lu, F., Liu, B.: Lie derivations of reﬂexive algebras. Integr. Equ. Oper. Theory 64, 261–271 (2009) [12] Sinclar, A.M.: Jordan homomorphisms and derivations on semisimple Banach algebras. Proc. Am. Math. Soc. 24, 209–214 (1970) Fangyan Lu Department of Mathematics Suzhou University Suzhou 215006 People’s Republic of China e-mail: [email protected] Received: June 23, 2009. Revised: August 18, 2009.

Integr. Equ. Oper. Theory 67 (2010), 57–78 DOI 10.1007/s00020-010-1770-2 Published online March 16, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Kernel of Vector-Valued Toeplitz Operators Nicolas Chevrot Abstract. Let S be the shift operator on the Hardy space H 2 and let S ∗ be its adjoint. A closed subspace F of H 2 is said to be nearly S ∗ -invariant if every element f ∈ F with f (0) = 0 satisfies S ∗ f ∈ F . In particular, the kernels of Toeplitz operators are nearly S ∗ -invariant subspaces. Hitt gave the description of these subspaces. They are of the form F = g(H 2 uH 2 ) with g ∈ H 2 and u inner, u(0) = 0. A very particular fact is that the operator of multiplication by g acts as an isometry on H 2 uH 2 . Sarason obtained a characterization of the functions g which act isometrically on H 2 uH 2 . Hayashi obtained the link between the symbol ϕ of a Toeplitz operator and the functions g and u to ensure that a given subspace F = gKu is the kernel of Tϕ . Chalendar, Chevrot and Partington studied the nearly S ∗ -invariant subspaces for vector-valued functions. In this paper, we investigate the generalization of Sarason’s and Hayashi’s results in the vector-valued context. Mathematics Subject Classification (2000). Primary 47B32, 30D55; Secondary 46C07, 46E40, 47B35. Keywords. Toeplitz operators, de Branges Rovnyak spaces, vector-valued functions.

1. Introduction To begin this section, we present the scalar results of Hitt, Sarason and Hayashi which will be generalized throughout this paper. We denote by H 2 the classical Hardy space of analytic functions on the unit disc D, and by H 2 (Cm ) the Cm -vector-valued Hardy space consisting of m copies of H 2 . The shift S is the operator of multiplication by the variable z and S ∗ is its adjoint. The (closed) S ∗ -invariant subspaces of H 2 are called model subspaces. They are of the form Ku = H 2 uH 2 , where u is an inner function. For ϕ ∈ L∞ , the Toeplitz operator with symbol ϕ is deﬁned by Tϕ f := p+ (ϕf ), where p+ is the orthogonal projection from L2 onto H 2 . Hitt [8] introduced the nearly S ∗ -invariant subspaces: Definition 1.1. A closed subspace F of H 2 is said to be a nearly S ∗ -invariant subspace if every element f ∈ F with f (0) = 0 satisﬁes S ∗ f ∈ F.

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In particular, the kernel of a Toeplitz operator is a nearly S ∗ -invariant subspace. Hitt obtained the complete description of this spaces: Theorem 1.2 (Hitt, 1988). Let F be a non-trivial nearly S ∗ -invariant subspace. Let g be the unique unit-norm function in F, positive at the origin, that is orthogonal to F ∩ zH 2 . Then there exists an inner function u vanishing at zero such that, for all f ∈ F, there exists a unique f0 ∈ Ku and f = gf0 . Furthermore, f 2 = f0 2 . In other words, multiplication by g acts isometrically on Ku . Two questions arise. The ﬁrst one was already posed by Sarason in [12] where he made this remark: “The latter theorem leaves mysterious the relation between the function g and the space Ku . Given a function g of unit norm in H 2 , what are the S ∗ -invariant subspaces Ku that can arise with g in Hitt’s theorem?” 2. Which nearly S ∗ -invariant subspaces are kernels of Toeplitz operators. Sarason obtained the following answer to the ﬁrst question:

1.

Theorem 1.3 (Sarason, 1988). Let g be an outer function of unit norm, and u an inner function with u(0) = 0. We define two analytic functions on the disc: 2π iθ e +z f (z) − 1 1 |g(eiθ )|2 dθ and b(z) := . f (z) := 2π eiθ − z f (z) + 1 0

Then the following statements are equivalent: 1. multiplication by g acts isometrically from Ku to F; 2. bH 2 ⊂ uH 2 (i.e. b = ub0 ); 3. Ku ⊂ (1 − Tb T¯b )1/2 H 2 . The answer to the second question is given by Hayashi in [5–7] and Sarason found an alternative proof in [13]. This answer is expressed in terms of exposed points of the unit ball of H 1 , also called rigid functions. Before stating Hayashi’s result, we need some definitions. With the previous notation, let F = gKu be a nearly S ∗ -invariant space and let b be the function associated to g as in Theorem 1.3. Because log(1 − |b|2 ) is integrable, we can build an outer function a such that |a|2 + |b|2 = 1 a.e. on T. Then (b, a) is called a corona pair (or pair) associated to g. Thanks to Theorem 1.3, b = ub0 . If F is the kernel of a Toeplitz operator, then (b0 , a) is a corona pair associated to the outer function g0 := a/(1 − b0 ). Some pairs, called special pairs, verify an additional property which will be precisely deﬁned in Sect. 5. Admitting this, we can reformulate Hayashi’s result as follows (see also [13]): Theorem 1.4 (Hayashi, 1985). The subspace F = gKu is the kernel of a Toeplitz operator if and only if the pair (b0 , a) is special and g02 is rigid. We would like to generalize the previous theorems to vector-valued functions. The paper is organized as follows. In Sect. 2, we deﬁne the vectoror matrix-valued objects: we recall the inner–outer matricial factorization,

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we comment on the generalization of Theorem 1.2, and we recall the definition of de Branges–Rovnyak spaces, the vector-valued analogue of H(b) := (1 − Tb T¯b )1/2 H 2 appearing in Theorem 1.3. In Sect. 3, we transcribe Sarason’s approach to the vectorial case. We build the analogue of the functions b and u. Thanks to de Branges–Rovnyak spaces, we obtain the matricial version of Theorem 1.3. The matrices do not commute, so we need to modify the original scalar proof given by Sarason. An example illustrates this kind of problem. In Sect. 4, we would like to describe the kernels of Toeplitz operators. We begin with some examples. This allows us to illustrate the difﬁculties due to the dimension, and to establish some notation. We then investigate the descriptions of kernels of Toeplitz operators of ﬁnite dimension. Finally, in Sect. 5, we obtain the full description of the kernels of Toeplitz operators. We establish the desired generalization of Hayashi’s Theorem.

2. Hardy Spaces of Vector-Valued Functions 2.1. Inner–Outer Factorization As usual with Hardy spaces, we identify a function with its radial limits. Let F, G be two subspaces of Cm of dimension r. Nikolskii, in [11, p. 14], calls Θ ∈ H ∞ (F → G) an inner function if its boundary values Θ(ξ) are surjective isometries for a.e. ξ ∈ T. It will be more convenient to say that Θ ∈ H ∞ (Cm → Cm ) is an inner function if its boundary values Θ(ξ) are partial isometries for a.e. ξ ∈ T, with kernel and range independent of ξ a.e. in T. In other words, an inner function is a square-matrix-valued function such that there exist two subspaces F, G of Cm with the same dimension r for which Θ|F ∈ H ∞ (F → G) is an inner function in the sense of Nikolskii. The rank of Θ(ξ) is equal to r for a.e. ξ ∈ T. Here are two examples of inner functions of rank 2. The ﬁrst one will be discussed later (see Theorem 3.7). Let θ be an inner scalar function and a, b ∈ Kzθ verifying |a|2 + |b|2 = 1 a.e. on T. Deﬁne ϕ ∈ H ∞ (C2 → C2 ) and Θ ∈ H ∞ (C3 → C3 ) by the following formulae: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ C C a 0 −b a −b a ⎠. F = ⎝C⎠ , G := ⎝ 0 ⎠ , ϕ := ¯ and Θ := ⎝θ¯b 0 θ¯ θb θ¯ a 0 C 0 0 0 Both ϕ, Θ are inner of rank 2. Note that ϕ ∈ H ∞ (Cm → Cm ) is inner of rank m if and only if det ϕ is inner. Recall the Beurling–Lax Theorem [10]: If a closed subspace M ⊂ H 2 (Cm ) is invariant by the shift, then there exists an inner function Θ such that M = ΘH 2 (Cm ). This description is unique up to multiplication by an unitary matrix. Next, we recall the notion of outer vector-valued function. The outer scalar functions are cyclic vectors for the shift. For g ∈ H 2 (Cm ), we deﬁne G, the smallest S-invariant subspace containing g, by G := span(S k g : k ∈ N). Thanks to Beurling–Lax theorem, there exists Θ, inner with of 1, such that

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G = ΘH 2 (Cm ). We say that g is outer if Θ is a constant matrix. Then G = H 2 (Θ(0)Cm ). It will be useful to write G := Θ(0)Cm . Finally, the function g is a cyclic vector for S in H 2 (G). We extend this construction to deﬁne the outer matrix-valued functions. Let g1 , . . . , gr ∈ H 2 (Cm ), with r ≤ m, be a independent family of vector-valued functions. Let G ∈ H 2 (Cr → Cm ) be the rectangular matrix-valued functions where the columns are (g )≤r . In this case we write G = [g1 , . . . , gr ]. It is said to be outer if G := span(S k g : ≤ r, k ∈ N) = ΘH 2 (Cm ), where Θ is a constant partial isometry of rank r. Then, we will write G = Θ(0)Cm , dim G = r, and G = H 2 (G). Due to the rank theorem, there exists an uni˜ ∈ H 2 (Cr → Cr ) such that tary mapping Θ0 : Cr → G. To G, we associate G ˜ This allows us to translate the properties of square-matrix-valued G := Θ0 G. functions to rectangular ones. For more details about inner–outer factorization of square matrix-valued functions with determinant different from zero, see [9]. In particular the Definition 5.3 in [9] of Beurling left outer function coincides with that of outer given above. The Smirnov–Nevanlinna class N + (Cm → Cm ) of square matrix-valued functions is the set of all matrices with entries in the scalar Smirnov–Nevanlinna class. The Definition 3.1 in [9] of outer function in N + (Cm → Cm ) is that E is outer if det E is outer in N + . The authors shows that all definitions of outer functions are equivalent in H 2 (Cm → Cm ). Theorem 5.4 in [9] says that, given a function F in N + (Cm → Cm ), det F (z) ≡ 0, there exist functions Fi inner and Fo outer (resp. Fi , Fo ) , unique up to a unitary matrix, such that F = Fi Fo (resp. F = Fo Fi ). Furthermore, Theorem 3.1 of [9] will be useful later: Let E ∈ N + (Cm → Cm ) an outer square-matrixvalued function. Then det(z) = 0 for all z ∈ D and E −1 ∈ N + (Cm → Cm ). 2.2. Nearly S ∗ -Invariant Subspaces of H 2 (Cm ) The next result is the description of the nearly S ∗ -invariant subspaces of H 2 (Cm ). For more details, see [3]. Theorem 2.1. Let F ⊂ H 2 (Cm ) be a non-trivial nearly S ∗ -invariant subspace. Let (g1 , . . . , gr ) be a orthonormal basis of

⊥ W := F ∩ F ∩ zH 2 (Cm ) . Then r := dim W ≤ m and there exist an integer r , 1 ≤ r ≤ r, and U ∈ H ∞ (Cr → Cr ) inner, rank U = r , such that F = [g1 , . . . , gr ] H 2 (Cr ) U H 2 (Cr ) = GKU . For all f ∈ F, there exists an unique f0 ∈ KU such that f = Gf0 . Furthermore, f0 H 2 (Cr ) = f H 2 (Cm ) . Because the columns of G form an orthonormal basis of W , the norm of G ∈ H 2 (Cr → Cm ) is 1. For any h ∈ H 2 (Cr ), we deﬁne TG h to be the Fourier projection of the L1 (Cm ) function Gh on H 2 (Cm ). It is an unbounded operator, but, as in the scalar case, it is an isometry on KU .

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2.3. De Branges–Rovnyak Spaces Now, we will recall the definition and the main properties of de Branges– Rovnyak spaces. For more details, see the ﬁrst chapter of [14]. Let H1 and H be two Hilbert spaces and B ∈ L(H1 , H) be a bounded operator. We deﬁne M(B) to be the range space BH1 with the inner product that makes B be a coisometry on H: ∀f, g ∈ H1 ∩ (ker B)⊥ , Bf, Bg M(B) := f, g H1 . For a contraction B, the inclusion is a contraction to H. The from M(B) ∗ 1/2 complementary space H(B) is deﬁned to be M (IdH − BB ) . In the particular case where B is the multiplication by an inner function B, then M(B) = BH 2 (Cm ) and H(B) = KB . In this case, the inner products of M(B) and H(B) coincide with the H 2 inner product and these two spaces are really complementary spaces in the H 2 sense. In this article, H and H1 will be Hardy spaces like H 2 (Cm ) or closed subspaces of H 2 (Cm ) isometrically equivalent to H 2 (Cr ), and B will be the multiplication by a matrix B in the unit ball of H ∞ (Cr → Cm ). The reproducing kernels in H 2 (Cm ) are kλ u := 1−1λz ¯ u for λ ∈ D and m 2 m u ∈ C . Thus, for all f ∈ H (C ), the reproducing kernels verify f, kλ u 2 = f (λ), u Cm . Because the inclusion from H(B) to H 2 is contractive, de Branges–Rovnyak spaces have kernel functions, and a simple calculation shows that Idr − B(z)B(λ)∗ u and f, kλB u H(B) = f (λ), u Cm . kλB u := ¯ 1 − λz Given a symbol B, we write M(B) (resp. H(B)) instead of M(TB ) (resp. H(TB )).

3. Toeplitz Operators Acting as an Isometry on a Model Space In this section, we verify that the tools used by Sarason [12] can be applied to matrix-valued functions. 3.1. A Matricial Intertwining Let (g )≤r be an orthogonal basis of W and let G ∈ H 2 (Cr → Cm ) be the matrix-valued function [g1 , . . . , gr ]. We denote by H 2 (Cm , μG ) the Hardy space of vector-valued functions with the norm 2π 1 2 G(eiθ )q2Cm dθ. qH 2 (Cm ,μG ) := 2π 0

Remember that G = ≤ r, k ≥ 0). Let f = Gq be in H 2 (Cm ). The measure μG will play a role in Sect. 5. This forces q to be in We would like to build from G the functions F and B, the analogues of those appearing in Theorem 1.3. After, we will show that TIdr −B TG∗ is an coisometry from G to H(B), or equivalently, q → (TIdr −B TG∗ G)q is an span(S k g : H 2 (Cm , μG ).

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coisometry from H 2 (Cm , μG ) to H(B). As a consequence, we will obtain the following equality, the key of the proof of the generalization of Theorem 1.3: ∗

Idr − TB TB ∗ = (TIdr −B TG∗ ) (TIdr −B TG∗ ) . We begin by deﬁning F the analytic function on the disc by 1 ∀z ∈ D, F (z) := 2π

2π 0

eiθ + z G(eiθ )∗ G(eiθ )dθ. eiθ − z

Note that, if G = U G , where U is inner of rank m, then the functions F and F are the same. Because the (gk )k≤r form an orthogonal basis of W , the coefﬁcient F (0)i,j is gi , gj H 2 . So, F (0) = Idr . For z0 ∈ D, let u ∈ Cr be an eigenvector of the matrix F (z0 ). Then Re ( F (z0 )u, u ) is a Poisson integral, so the real parts of the eigenvalues of F (z0 ) are G(z0 )u2 ≥ 0. This implies that the moduli of the eigenvalues of F (z0 ) + Idr are greater than 1, so F (z0 ) + Idr is invertible. Next, we deﬁne B, the matrix-valued Herglotz integral of μG , by B(z) := (F (z) + Idr )−1 (F (z) − Idr ). Because F (0) = Idr , the function B vanishes in zero. For all u ∈ Cr , (F (z) ± Idr )u2 = F (z)u2 + u2 ± 2 Re F (z)u, u . Then, because Re F (z)u, u ≥ 0, (F (z) + Idr )u2 ≥ (F (z) − Idr )u2 and B lies in the unit ball of H ∞ (Cr → Cr ). We can therefore consider H(B). Lemma 3.1. For all u, v ∈ Cm we have: B Gkw u, Gkz v H 2 = kw (Idr − B(w)∗ )−1 u, kzB (Idr − B(z)∗ )−1 v H(B) .

Proof. For all u, v ∈ Cm , we express the inner product Gkw u, Gkz v H 2 in terms of F : Gkw u, Gkz v H 2

1 = 2π

2π 0

1 (1 −

we ¯ iθ )(1

− ze−iθ )

G(eiθ )u, G(eiθ )v Cm dθ

1 2π(1 − wz) ¯

−iθ 1 e +w ¯ eiθ + z × + G(eiθ )u, G(eiθ )v Cm dθ 2 e−iθ − w ¯ eiθ − z 1 (F (w)∗ + F (z))u, v Cm . = 2(1 − wz) ¯ =

Because (Idr + B(z)) and (Idr − B(z))−1 commute, F (w)∗ + F (z) = (Idr −B(z))−1 (Idr + B(z)) + (Idr + B(w)∗ )(Idr −B(w)∗ )−1 = 2(Idr − B(z))−1 [Idr − B(z)B(w)∗ ] (Idr − B(w)∗ )−1 .

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B (z): Finally, we interpret Gkw u, Gkz v H 2 in term of inner product of kw

1 (F (w)∗ + F (z))u, v Cm 2(1 − wz) ¯ 1 (Idr − B(z))−1 [Idr − B(z)B(w)∗ ] = 1 − wz ¯ × (Idr − B(w)∗ )−1 u, v Cm

Gkw u, Gkz v H 2 =

B = (Idr − B(z))−1 kw (z)(Idr − B(w)∗ )−1 u, v Cm B = kw (z)(Idr − B(w)∗ )−1 u, (Idr − B(z)∗ )−1 v Cm B = kw (Idr − B(w)∗ )−1 u, kzB (Idr − B(z)∗ )−1 v H(B) .

The following lemma is useful in connection with de Branges–Rovnyak spaces [14, I-5]: Lemma 3.2 (Douglas’s criterion). Let H, H1 and H2 be Hilbert spaces, and let A : H1 → H, B : H2 → H be contractions. We define M(A) := AH1 and M(B) := BH2 . Then M(A) = M(B) is equivalent to AA∗ = BB ∗ . Remember that G = span(S k g : ≤ r, k ∈ N) = span(Gkw u : u ∈ Cr , w ∈ D) and that Θ is an inner function such that G = ΘH 2 (Cm ). (When G is outer, Θ is a constant unitary-matrix) B Lemma 3.3. 1. For all u ∈ Cm , TIdr −B TG∗ maps Gkw u to kw (Idr − ∗ −1 B(w) ) u. 2. If G is outer, then TIdr −B TG∗ is an isometry from G onto H(B). 3. If G = Gi Go , with Gi inner and Go outer, then TIdr −B TG∗ is a coisometry of G to H(B) with null space KGi ∩ G. 4. Define M(TIdr −B TG∗ ) := TIdr −B TG∗ G. Equipped with the inner product

TIdr −B TG∗ h1 , TIdr −B TG∗ h2 := h1 , h2 2 ∀h1 , h2 ∈ G ∪ (ker TIdr −B TG∗ )⊥ , M(TIdr −B TG∗ ) coincides with the de Branges–Rovnyak space H(B). Proof. 1.

We begin by computing the range of Gkw u by TIdr −B TG∗ : (TIdr −B TG∗) Gkw u, kz v H 2 = (Idr −B(z))TG∗ G(z)kw (z)u, v Cm = TG∗ Gkw u, kz (Idr − B ∗ )v H 2 = Gkw u, Gkz (Idr − B(z)∗ )v H 2 B = kw (Idr − B(w)∗ )−1 u, kzB v H(B) B = kw (Idr − B(w)∗ )−1 u, kz v H 2 .

B Therefore, (TIdr −B TG∗ ) sends Gkw u to kw (Idr − B(w)∗ )−1 u.

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The inner product of two functions in G is equal to the inner product of their images in H(B): B (Idr − B(w)∗ )−1 u, TIdr −B TG∗ Gkw u, TIdr −B TG∗ Gkz v H 2 = kw

kzB (Idr − B(z)∗ )−1 v H(B) = Gkw u, Gkz v H 2 . Because G is outer, the functions Gkw u span G, reduced to H 2 (G) and the result follows. 3. The definition of B does not depend on Gi , so TIdr −B TG∗ = TIdr −B TG∗o TG∗i . But TIdr −B TG∗o sends TG∗i G isometrically to H(B), which is dense in Go , so we get the result by continuation. Moreover, ker TIdr −B TG∗ |G = ker TG∗i ∩ G = KGi ∩ G. 4. The last sentence allows us to identify the two de Branges–Rovnyak spaces: M (TIdr −B TG∗ ) := TIdr −B TG∗ G = H(B) = M (Idr − TB TB ∗ )1/2 . Theorem 3.4. As operators on H 2 (Cr ), we have (TIdr −B TG∗ )(TIdr −B TG∗ )∗ = Idr − TB TB ∗ . Proof. The Douglas criterion, Lemma 3.2, implies that M (TIdr −B TG∗ ) = H(B) is equivalent to (TIdr −B TG∗ )(TIdr −B TG∗ )∗ = Idr − TB TB ∗ as opera tors on H 2 (Cr ). 3.2. A Matricial Version of Sarason’s Theorem With the previous notation, Theorem 3.5. Let G = [g1 , . . . , gr ] ∈ H 2 (Cr → Cm ) and let U ∈ H ∞ (Cr → Cr ) be inner of rank r vanishing at zero. Then TG |KU is an isometry ⇐⇒ TB ∗ KU = {0} ⇐⇒ BH 2 (Cr ) ⊂ U H 2 (Cr ). The proof follows Sarason’s ideas, with modiﬁcations to bypass the fact that TB ∗ KU might not be a subspace of KU . Proof. The last equivalence is obvious. Suppose that TB ∗ KU = {0}. Then TG h = TG TIdr −B ∗ h for all h ∈ KU . Thanks to Lemma 3.3, (TIdr −B TG∗ )(TIdr −B TG∗ )∗ = Idr − TB TB ∗ , and we compute the norm of TG h2H 2 : TG h2H 2 = TG TIdr −B ∗ h2H 2 = (TIdr −B TG∗ )(TIdr −B TG∗ )∗ h, h = (Idr − TB TB ∗ )h, h = h, h H 2 = h2H 2 . Thus, TG acts as an isometry on KU . Conversely, suppose that TG |KU is an isometry. Let h ∈ KU . Lemma 3.3 and Theorem 3.4 assert that TIdr −B TG∗ : G → H(B) is a coisometry,

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and (TIdr −B TG∗ )(TIdr −B TG∗ )∗ = Idr − TB TB ∗ on H 2 (Cr ). It follows that TG TIdr −B ∗ h2H 2 = (Idr − TB TB ∗ )h, h H 2 , whose development is: TG h2 − TG TB ∗ h, TG h − TG h, TG TB ∗ h +TG TB ∗ h2 = h2 −TB ∗ h2 . Using the hypothesis TG h = h, we get TG TB ∗ h, TG h + TG h, TG TB ∗ h = TB ∗ h2 + TG TB ∗ h2 .

(3.1)

Now, with Sarason’s trick, we will show that TG TB ∗ h = 0, for h ∈ KU . Remember that U (0) = 0, so Cr ⊂ KU and because B(0) = 0, we get TB ∗ v = 0 for all v ∈ Cm . With c ∈ C and v ∈ Cm , h + cv stays in KU and TB ∗ (h + cv) = TB ∗ h. Replacing h by h + cv in the equality 3.1, we have: TG TB ∗ h, TG (h + cv) + TG (h + cv), TG TB ∗ h = TB ∗ h2 + TG TB ∗ h2 . This is equivalent to 2 Re (c TG TB ∗ h, TG v ) = TB ∗ h2 + TG TB ∗ h2 − 2 Re TG TB ∗ h, TG h . This holds for all c ∈ C, so necessarily Re TG∗ TG TB ∗ h(0), v = 0 2

and

2

TB ∗ h + TG TB ∗ h − 2 Re TG∗ GB ∗ h, h = 0.

(3.2)

The ﬁrst equality holds for all v ∈ Cm , so TG∗ TG TB ∗ h(0) = 0. Replacing h by S ∗k h, which stays in KU , we deduce that TG∗ TG TB ∗ S ∗k h(0) = 0 and so TG∗ TG TB ∗ h = 0. This implies that TB ∗ h ∈ ker TG or TG TB ∗ h ∈ ker TG∗ . We denote f = TG TB ∗ h. Then, f ∈ ker TG∗ ∩ G and there exists q ∈ H 2 (Cm , μG ) such that f = Gq. The norm of f is f 2 = G∗ Gq, q = TG∗ f, q = 0. Finally, TB ∗ KU ⊂ ker TG . The second equality of (3.2) implies the following equivalences: TB ∗ h2 + TG TB ∗ h2 − 2 Re TG∗ TG TB ∗ h, h = 0 ⇐⇒TB ∗ h2 − TG h2 + TG h2 + TG TB ∗ h2 − 2 Re TG∗ TG TB ∗ h, h = 0 ⇐⇒TB ∗ h2 − TG h2 + TG TIdr −B ∗ h2 = 0 ⇐⇒TG TIdr −B ∗ h2 = TG h2 − TB ∗ h2 . But we know that TB ∗ KU ⊂ ker TG , so TG TIdr −B ∗ h2 = TG h2 and TB ∗ h2 = 0. So we get TB ∗ KU = {0} as desired. Corollary 3.6. The operator TIdr −B TG∗ acts on F as division by G. Proof. Let Gh ∈ F. Then, thanks to the last theorem, TB ∗ h = 0. So, TId−B TG∗ Gh = TId−b TG∗ TG TId−B ∗ h and Lemma 3.4 implies that (Id − TB TB ∗ )h = h.

In the original proof, Sarason uses the fact that scalar model spaces Ku (or more generally de Branges spaces [14, II-7]) are stable under the action of T¯b for every symbol b ∈ H ∞ . This does not hold for matrix symbols. The inclusion TB ∗ KU ⊂ KU means that BU H 2 (Cr ) ⊂ U H 2 (Cr ), and so U ∗ BU ∈ H ∞ (Cr → Cr ). This is obvious if B and U commute.

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In this section, we will construct an example in H ∞ (C2 → C2 ) where TB ∗ KU is not contained in KU . The following characterization of (2 × 2)matrix-valued inner functions is due to Garcia, in [4]. Theorem 3.7. Let U ∈ H ∞ (C2 → C2 ). Then U is inner if and only if U is of the form: a −b U= ¯ θb θ¯ a where θ := det U is inner, and a, b ∈ Kzθ verify |a|2 + |b|2 = 1 a.e. on T. Garcia gives an interesting example of an inner function by taking a := (1 + θ)/2, b := −i(1 − θ)/2 and 1 (1 + θ) i(1 − θ) V := . (1 + θ) 2 −i(1 − θ) Taking θ = z, for example, we notice that the entries are outer scalar functions. We can look for U = zV which is still inner and vanishing at zero. b1 b 2 ∈ H ∞ (C2 → C2 ). A calculation shows that U ∗ BU is Let B = b3 b4 equal to:

b1 +b4 + Re (θ)(b1 − b4 )+i Im θ(b3 +b2 )

b2 −b3 + Re (θ)(b3 +b2 )+i Im θ(b1 +b4 )

b3 −b2 + Re (θ)(b3 −b2 )+i Im θ(−b1 +b4 ) −b1 +b4 + Re (θ)(b1 +b4 )+i Im θ(−b3 −b2 )

.

If we suppose that b4 = −b1 and b3 = −b2 , then Re (θ)b1 b2 + Im (θ)b1 ∗ U BU = , −2b2 + Im (θ)b1 − Re (θ)b1 which is not in H ∞ (C2 → C2 ) and so TB ∗ KU ⊂ KU . Remark 3.8. In [12], Sarason establishes an alternative proof of Theorem 2.1 using Corollary 3.4. This approach could be generalized to the vector-valued case.

4. Kernel of Toeplitz Operators 4.1. Some Examples For a nearly S ∗ -invariant subspace F = GKU , we recall that W = F ∩ (F ∩ zH 2 (Cm ))⊥ , and r := dim W ≤ m. If m = 2, then we have two ways to build F with dim F = 2. Example 1. If r = 2, G = [g1 , g2 ] and U = zId2 . Let F be the space a(1 + z)1/2 2 , (a, b) ∈ C . b(1 − z)1/2 Because f (0) = 0 implies f = 0, it is S ∗ -nearly invariant. We see that W = F and F = TG C2 with 1 (1 + z)1/2 0 G(z) = . 0 (1 − z)1/2 2

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The functions 12 (1 + z)1/2 and 12 (1 − z)1/2 are outer in H 2 and G is outer, because its determinant is outer (see Sect. 2.1). Moreover, G = H 2 (C2 ). Is F the kernel of a Toeplitz operator Tϕ ? We will build ϕ ∈ L∞ (C2 → 2 C ) as the following. Remark that G(eit )∗ G−1 (eit ) is diagonal. The diagonal 1 1 terms are e− 2 it and −e 2 it . So, G∗ G−1 lies in L∞ (C2 → C2 ) and then −3 0 z 2 Tz¯G(z)∗ G(z)−1 = p+ a.e. z ∈ T. (4.1) 1 0 −z − 2 Every f in F is of the form f = Ge, with e ∈ C2 , and Tz¯G∗ G−1 f = z G∗ e) = 0. So F is the kernel of Tz¯G∗ G−1 . p+ (¯ Example 2. We modify the previous example to get a nearly S ∗ -invariant subspace which is not the kernel of a Toeplitz operator. Let F and G be deﬁned by 1 1+z 0 a(1 + z) . F= (a, b) ∈ C2 and G(z) = √ 0 1−z b(1 − z) 2 We will show that if it is the kernel of a Toeplitz operator, it is also the kernel of the Toeplitz operator with symbol ϕ(eit ) := G∗ (eit )U ∗ (eit )G−1 (eit ). This symbol is a diagonal matrix. The diagonal terms are e−2it and −1 a.e. eit ∈ T. But a + bz 2 ker Tϕ = : (a, b) ∈ C = F, 0 and so F fails to be the kernel of a Toeplitz operator. Example 3. Let r = 1, G = [g1 ] and dim KU = 2. Let F be deﬁned by a + bz 2 F := : (a, b) ∈ C . 0 1 This space is the nearly S ∗ -invariant F = GKU with G(z) = and 0 U (z) = z 2 . With the notation deﬁned in Sect. 2.1, we have 1 0 1 C G := span(S k G) = H 2 (C2 ), Θ0 = , and G = . 0 0 0 0 ˜ is the Because G = H 2 (G), the function G is outer. As we saw before, G ˜ ˜ (1 × 1)-square matrix such that G = Θ0 G. Here, G = 1. Furthermore, F = ker Tϕ where ∗ ∗ −1 2 ˜ U G ˜ 0 z¯ G 0 ϕ := . = 0 1 0 1 This example illustrates the problem of dimensions : the interesting part lies in G which is a subspace of Cm .

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4.2. The Case r = m This section treats the particular case where r = m seen in the Example 1. First of all, a nearly S ∗ -invariant subspace F which is the kernel of a Toeplitz operator Tϕ has the form F = TG KU with G outer. Remember that the columns of G ∈ H 2 (Cm → Cm ) form an orthogonal basis of

⊥ W := F ∩ F ∩ zH 2 (Cm ) . Because det G ≡ 0, the inner–outer factorization for matrix-valued functions allows us to factorize with an inner function on the right. Let Go be outer, and Gi be inner, such that G = Go Gi . Because U (0) = 0, it follows KU contains Cm and GCm ⊂ ker Tϕ . For all e ∈ Cm , we have Ge ∈ ker Tϕ . So, there exists H ∈ H 2 (Cm → Cm ) such that ϕG = z¯H ∗ . But G = Go Gi , then ϕGo Gi = z¯H ∗ and ϕGo = z¯H ∗ G∗i . Finally, Tϕ (Go e) = 0, which implies that the columns of Go form an orthonormal basis of W . A nearly S ∗ -invariant subspace F which is the kernel of a Toeplitz operator is of the form GKU , with G outer. Following Sarason, the ﬁrst step is to understand what happens if dim ker F = m = r. A new notion, namely the rigid functions, appears to characterize when F is the kernel of a Toeplitz operator. A scalar function f ∈ H 1 is said to be rigid if the only functions in H 1 which have the same argument are of the form cf with c a non-negative constant. When det(F ) = 0, we write (RF , AF ) for the polar decomposition of the matrix F = RF AF . The matrix RF is positive and AF is unitary. The matrix AF is called the argument of F . Definition 4.1. Let F ∈ H 1 (Cm → Cm ) be a square matrix-valued function. Then F is rigid if the only functions which have the same argument AF are of the form RF , where R is a constant hermitian positive matrix. In fact, we can show that rigid functions are exactly the exposed points of the unit ball of H 1 (Cm → Cm ). We recall the definition. Let X be a Banach space and A a closed subset of X. A point x ∈ A is an exposed point of A if there exists L ∈ X ∗ such that L(x) = 1 and Re (L(y)) < 1 for all y ∈ A\{x}. The functional L associated to x is unique. In our case, L is: L : H 1 (Cm → Cm ) −→ C ⎛ 1 H − → tr ⎝ 2π

2π

⎞ H(eit )AF (eit ) dt⎠ ,

0

where tr denote the trace. It is easy to verify that L(F ) = 1 and that F is rigid if and only if L is unique. Moreover, exposed points are extreme points (in the sense of convexity). The extreme points of the unit ball of H 1 (Cm → Cm ) are the outer functions with norm 1. For more results see [1,2]. Before stating the next lemma, we deﬁne precisely the norm ||| · ||| which we shall use. Let (ek )k be the canonical basis of Cm . Then ||| · ||| is the matricial norm deﬁned by m 2 ||| A ||| := k=1 Aek , Aek Cm . Lemma 4.2. Let G ∈ H 2 (Cm → Cm ) and F = TG Cm be a nearly S ∗ -invariant subspace of H 2 (Cm → Cm ). If F is the kernel of a Toeplitz operator, then G2 is rigid and F = ker Tz¯G∗ G−1 .

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Proof. Let ϕ be the symbol of the Toeplitz operator for which F is the kernel. Because G ∈ ker Tϕ , there exists H ∈ H 2 (Cm → Cm ) such that ϕG = z¯H ∗ . We begin by showing that H is outer. Let H = V Ho the inner–outer decomposition of H. Then ϕG = z¯Ho∗ V ∗ , so ϕGV = z¯Ho∗ . Finally, GV ∈ ker Tϕ , but ker T ϕ = GCm , so V is constant. We would like to write Tϕ as the product of two Toeplitz operators, such that the ﬁrst one is injective and the symbol of the second one depends only on G. Because G ∈ H 2 (T, Cm → Cm ) is outer, det G(z) is a outer scalar function in N + and for all z ∈ D, the inverse G(z)−1 has a sense. The function G−1 lives in N + . Now, we consider the polar decomposition of G(z) = AG (z)RG (z) with RG (z) = (G(z)∗ G(z))1/2 positive-definite hermitian and AG (z) unitary. We have G(z)−1 = A∗ (z)R(z)−1

and

G(z)∗ = A∗ (z)R(z).

Because G is outer, G = H 2 (Cm ). We can say that GH ∞ (Cm ) is dense in G. Let f = Gf ∈ GH ∞ (Cm ). We deﬁne ψ ∈ L∞ (T, Cm → Cm ) by ψ = z¯H ∗ (G−1 )∗ G∗ G−1 , and then Tψ f is exactly T(G−1 H)∗ TG∗ z¯G−1 f . We extend by continuity from GH ∞ (Cm ) to H 2 (Cm ). Because G and H are outer, G−1 and G−1 H are outer in N + (Cn → m C ). But ϕG = z¯H ∗ , so we have ϕ(z) = z¯H ∗ (z)G(z)−1 . So, for almost every eit ∈ T, the norm ||| G(eit )−1 H(eit ) ||| is given by: ||| G(eit )−1 H(eit ) ||| = ||| H(eit )H(eit )∗−1 H ∗ (eit )G(eit )−1 ||| = ||| AH (eit )2 eit ϕ(eit ) ||| = ||| ϕ(eit ) ||| . This proves that G−1 H ∈ H ∞ (Cm → Cm ) is outer with the same norm as ϕ. So, the kernel of the Toeplitz operator with symbol G−1 H is trivial. By construction, ker Tz¯G∗ G−1 = F. We shall prove that G2 is rigid. Let J be a function with the same argument as G2 . We can suppose that it is outer, because if not then we can build an outer function with the same argument. Indeed, if J = Ji Je , then −(Id + Ji )2 (Id − Ji∗ )2 = 2Id − Ji2 − Ji∗2 is positive and Ji∗ (Id+Ji )2 J or −Ji∗ (Id−Ji )2 J is outer with the same argument as G2 . Let J1 = J 1/2 = AF (z)RJ (z)1/2 . The hypothesis on J implies that RJ ≡ RG2 , so it is the same with F1 and G. Because J1∗ J1−1 = A2G = G∗ G−1 for a.e. eit ∈ T, then for all u ∈ Cm , Tz¯G∗ G−1 F1 u = 0. But ker Tz¯G∗ G−1 = GCm , so J1 u ∈ GCm . The contradiction follows. By hypothesis, F1 is not a multiple of G by a constant matrix. The following consequence is an interesting characterization of the rigid functions. It is the matricial analogue of a result of Sarason [14, chapter X, p. 70] used to show that 1 + z is rigid. Proposition 4.3. If F ∈ H 2 (Cm → Cm ) is outer, then F 2 is rigid if and only if the Toeplitz operator TF ∗ F −1 has a trivial kernel.

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Proof. Because ker S ∗ = Cm and Tz¯F ∗ F −1 = S ∗ TF ∗ F −1 , then ker TF ∗ F −1 is trivial is equivalent to dim ker Tz¯F ∗ F −1 = m. But, thanks to the previous lemma, this is true only if F 2 is rigid. Here is the matricial analogue of Lemma 1 of [13, p. 161] for an outer square matrix G. Lemma 4.4. Let G ∈ H 2 (Cm → Cm ) be outer and let U ∈ H ∞ (Cm → Cm ) be inner vanishing at zero. Let F = TG KU be the kernel of a Toeplitz operator. Then it is the kernel of TG∗ U ∗ G−1 . Proof. This proof follows Sarason’s. The fact that F is the kernel of a Toeplitz Tϕ allows us to show that G∗ U ∗ G−1 deﬁnes a symbol in L∞ (T, Cm → Cm ). We begin the proof by building an outer function with the same norm as ϕ, then we can suppose that ϕ takes values in the set of norm-1 matrices. Let (ek )1≤k≤m the canonical basis of Cm . Because p+ (ϕgk ) = 0, there exists hk ∈ H 2 (Cm ) such that m ϕGek = z¯hk it it 2 for k ∈ {1, . . . , m}. The norm ||| ϕ(e )G(e ) ||| is equal to k=1 hk , hk , so m log ||| ϕ(eit )G(eit ) ||| = 12 log ( k=1 hk , hk ). m But the function eit → k=1 hk (eit ), hk (eit ) is in H 1 , so it is log-integrable, just as eit → ||| ϕ(eit )G(eit ) ||| . We deduce from ||| ϕ(eit )G(eit )||| ≤ |||ϕ(eit ) ||| ||| G(eit ) ||| that eit → ||| ϕ(eit ) ||| is log-integrable. Via the Poisson kernel (cf [9]), we build ψ ∈ H ∞ (Cm → Cm ) outer with the same norm: ⎛ ⎞ it ∗ it ∗ it 1/2 e + z 1 log ϕ (e ϕ (e ) ψ(z) := exp ⎝ dt⎠ . 2π eit − z T

Because ψ is outer, so is its determinant and the matrix ψ(z) is invertible for z ∈ D and the inverse is in N + . Then the radial limits exist almost everywhere and ψ −1∗ ϕ ∈ L∞ (T, Cm → Cm ). The values ψ −1∗ (eit )ϕ(eit ) are matrices with norm 1 for a.e. eit ∈ T. So, there exists χ ∈ L∞ (T, Cm → Cm ) which takes values in the set of norm-1 matrices such that ϕ = ψ ∗ χ. But, ker Tψ∗ is trivial, so ker Tϕ = ker Tχ and even if we replace ϕ by χ, we can suppose that ϕ take values that are matrix of norm 1. Because p+ (ϕG) = 0, there exists H ∈ H 2 (Cm → Cm ) such that ϕG = ∗ H and H(0) = 0. We note H = Hi Ho , and we will show that Hi = V U with V a unitary constant matrix. Now, we prove that U divides Hi . For all h ∈ KU , Gh ∈ ker Tϕ implies that ϕGh, which is equal to Ho∗ Hi∗ h, lies in zH 2 (Cm ). Because Ho is outer, Hi∗ h ∈ z H 2 (Cm ) and then h is orthogonal to Hi H 2 (Cm ). So, we have KU ⊂ KHi . The converse inclusion holds: Let h ∈ KHi be bounded. Then, there exists H = Hi Ho ∈ H 2 (Cm → Cm ) such that Tϕ Gh = p+ (Ho∗ Hi∗ h). Then

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for all k ∈ H ∞ (Cm ), we have Tϕ Gh, k = Ho∗ Hi∗ h, k = h, Hi Ho k = 0. Because the bounded functions are dense in KHi , we obtain that KHi ⊂ KU and so Hi = V U with V unitary constant. As in the proof of Lemma 4.2, we write ϕ = Ho∗ U ∗ G−1 . Then p+ (ϕf ) = p+ (Ho∗ (G−1 )∗ G∗ U ∗ G−1 f ), and Tϕ = T(G−1 Ho )∗ TG∗ U ∗ G−1 . To conclude, we need to show that G−1 Ho lies in H ∞ (Cm → Cm ). The two functions Ho and G are outer, so Ho−1 G is in the Nevanlinna–Smirnov class. To end the proof, we observe that ||| G−1 Ho ||| = ||| ϕ ||| , which was done at the end of the proof of Lemma 4.2. So ker T(G−1 Ho )∗ is trivial, and ker Tϕ = ker TG∗ U ∗ G−1 . 4.3. The Case r < m Let G ∈ H 2 (Cr → Cm ) be an outer function. With the notation of Sect. 2.1, we consider the space G = H 2 (G), the unitary mapping Θ0 : Cr → G and ˜ with G ˜ ∈ H 2 (Cr → Cr ) outer. G = Θ0 G

Let Θ1 : C m−r → G⊥ be an unitary mapping. Then we decompose H (Cm ) as follows: Θ0 0 H 2 (Cr ) 2 m ⊥ 2 2 ⊥ . H (C ) = G ⊕ G = H (G) ⊕ H (G ) = 0 Θ1 H 2 (Cm−r ) 2

We denote Θ the (m × m)-unitary matrix with diagonal Θ0 , Θ1 . Lemma 4.5. Let F = TG Cr the kernel of a Toeplitz operator. We suppose ˜ 2 ∈ H 1 (Cr → Cr ) is a rigid function and F is the that dim F = r. Then G kernel of the Toeplitz operator with symbol ∗ −1 ˜ G ˜ z¯G 0 (4.2) φ := Θ Θ∗ . 0 Idm−r Proof. Let φ in L∞ (T, Cm → Cm ) be deﬁned as in (4.2). If f ∈ G ⊥ , then Θ∗ f ∈ H 2 ({0Cr } ⊕ H 2 (Cm−r )) and Tφ f = f , so ker Tφ |G ⊥ = {0}. If f ∈ G, then Θ∗ f ∈ H 2 (Cr ⊕ {0Cm−r }). We denote by pr the orthogonal projection from Cm to Cr ⊕{0Cm−r }. Let ˜ r. ϕ = pr Θ∗ φΘpr , then Tϕ is a Toeplitz operator on H 2 (Cr ). Write F˜ = GC ∗ r ∗ ˜ We verify that F = pr (Θ GC ) = pr ΘF is nearly S -invariant of dimension ˜ Then, we apply Lemma 4.2 to F˜ in H 2 (Cr ), which r and that ker Tϕ = F. 2 ˜ is rigid and F˜ = ker T ˜ ∗ ˜ −1 . implies that G z¯G G To conclude, ∗ −1 ˜ ˜ G z¯G 0 Θ∗ φ=Θ 0 Idm−r ˜ 2 is rigid. satisﬁes ker Tφ = F and G

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˜2 ∈ Lemma 4.6. Let F = TG KU be the kernel of a Toeplitz operator. Then G 1 r r H (C → C ) is a rigid function and F is the kernel of the Toeplitz operator with symbol ∗ ∗ −1 ˜ U G ˜ G 0 φ := Θ Θ∗ . 0 Idm−r The proof uses the Lemmas 4.4 and 4.5. The naive idea is to apply Lemma 4.4 to F, considered as a subspace of G = H 2 (G). Then we have dim W = r = dim G. Unfortunately, the range of Tφ |G is a priori not in G. We need to ﬁnd a new symbol φ such that Tφ |G → G and Tφ |G ⊥ → G ⊥ with F = ker Tφ |G . Proof. We begin by building a nearly S ∗ -invariant subspace F which will be the kernel of a certain Toeplitz operator with symbol φ , such that W is of dimension m. It will be more convenient to write F = TG0 KU0 = ker Tφ0 . By hypothesis, G0 ∈ H 2 (Cr ) is outer and its columns form an orthonormal basis of W . Let (er+k )k=1...m−r be an orthonormal basis of G⊥ . Because ˜1 = G0 = H 2 (G), G1 := √12 [(1 + z)1/2 e ]r+1≤≤m is an outer matrix and G 1 ∗ Θ1 G1 is square. This choice of √2 (1 + z), which is a rigid scalar function, ˜ 2 is rigid in H 1 (Cm−r → Cm−r ). Because G, ˜ the diagonal implies that G 1 2 m ˜0, G ˜ 1 is outer, G := [G0 , G1 ] ∈ H (C → Cm ) is outer. matrix with blocks G Let U1 := zIdm−r . It is inner and U1 (0) = 0 and U , the matrix with diagonal blocks U0 and U1 , is of rank m. Finally, we can consider F = TG KU = TG0 KU0 ⊕⊥ TG1 KU1 . But TG0 KU0 ⊂ G and TG1 KU1 ⊂ G ⊥ , so W = W0 ⊕ W1 . We remark that W1 = G1 Cm−r and dim W = m. It remains to show that F is the kernel of a Toeplitz operator with ˜ 2 is rigid, Lemma 4.5 applied to TG KU symbol ϕ. First of all, because G 1 1 1 asserts that this space is the kernel of the Toeplitz with symbol φ1 := Idm−r 0 ∗ Θ ˜ −1 Θ . By hypothesis, F is the kernel of Tϕ0 . Let ϕ := ˜∗G 0 z¯G 1 Idr 0 ϕ0 pG + φ1 pG⊥ , where pG is the matrix Θ Θ∗ of projection from 0 0 H 2 (Cm ) to G = H 2 (G). It is clear that ϕ ∈ L∞ (T, Cm → Cm ) and that F = ker Tϕ . ˜ 2 must Now, we can apply Lemma 4.4 to F , the kernel of Tϕ . So, G 2 ˜ must be rigid. Moreover, be rigid. But it is a block diagonal matrix, so G 0 ˜ −1 Θ∗ . Finally, the ˜ ∗ U ∗ G F is the kernel of Tφ with φ := G∗ U ∗ G−1 = ΘG symbol φ is given by ∗ ∗ −1 ˜ U G ˜ G 0 ∗ 0 0 0 φ = Θ ˜ −1 Θ . ˜∗G 0 z¯G 1

1

The diagonal structure of Θ∗ φ Θ implies that the range of Tφ |G0 is contained in G0 . To conclude, we will show that F = ker Tφ0 where the symbol φ0 is: ∗ ∗ −1 ˜ U G ˜ G 0 0 0 0 φ0 = Θ Θ∗ ∈ L∞ (T, Cm → Cm ). 0 Idm−r

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Applying Lemma 4.4 in G0 = H 2 (G0 ) with dim G0 = r = dim W , we have F is the kernel of TpG0 φ pG0 , so it is the kernel of the Toeplitz operator with ˜ −1 . It remains to complete the symbol on G ⊥ . Then F is the ˜ ∗ U0 G symbol G 0 0 0 kernel of the Toeplitz operator with symbol ∗ ∗ −1 ˜ U G ˜ G 0 0 0 0 φ := Θ (4.3) Θ∗ . 0 Idm−r Let F be a nearly S ∗ -invariant subspace of the form TG KU where G is outer. We can summarize the section by saying that F is the kernel of ˜ 2 is rigid. Furthermore, the formula 4.3 a Toeplitz operator if and only if G express explicitly a symbol depending only on G and U .

5. A Matricial Version of Hayashi’s Theorem The purpose of this section is to obtain a matricial version of Hayashi’s theorem [6]. This section treats mainly the case r = m. We need some results about de Branges–Rovnyak spaces to state the theorem. The ﬁrst chapter of Sarason’s book [14] is general enough for the matricial case. Sarason specializes to the scalar case in the next chapters. The ideas come from the third and fourth chapters. In this section, we adapt some of them to the matricial case. The main problem is the lack of commutativity. In Sect. 3, we built from G an outer function B such that H(B) is unitarily equivalent to H 2 (Cm , μG ). We saw that B is in the unit ball of H ∞ (Cm → Cm ) and a little investigation shows that log(I − B ∗ B) ∈ L1 (T, Cm → Cm ). (Let A be the outer function 2G(F + Id)−1 , then for all z ∈ D, A∗ (z)A(z) + B ∗ (z)B(z) = Id, and the conclusion follows.) In this section, we need to reverse the construction, beginning with a B in the unit ball of H ∞ (Cm → Cm ) verifying log(I − B(eit )∗ B(eit )) ∈ L1 (T, Cm → Cm ). (Such functions are non-extreme points of the unit ball of H ∞ (Cm → Cm ), see Treil’s theorem in [11, p. 85]). We deﬁne A ∈ N + (Cm → Cm ) to be the Poisson integral ⎞ ⎛ 2π it e + z 1 log(I − B(eit )∗ B(eit )) dt⎠ . A(z) := exp ⎝ 2π eit − z 0

By definition, A is outer and A(eit )∗ A(eit ) + B(eit )∗ B(eit ) = Id a.e. on T. Because B is bounded, so is A. The same calculation as the proof of Lemma 3.1 gives: 1 (Id + B ∗ )(Id−B ∗ )−1 + (Id + B)(Id−B)−1 Re (Id + B)(Id−B)−1 = 2 = (Id − B ∗ )−1 [Id − B ∗ B] (Id − B)−1 2 = (Id − B ∗ )−1 A∗ A(Id − B)−1 = R(Id−B) −1 A .

So, for u ∈ Cm , we have Re (Id + B)(Id − B)−1 u, u Cm = A(Id − B)−1 u2Cm

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which is positive, so we can deﬁne the Herglotz integral corresponding to B analogous to F deﬁned Sect. 3. Let μ ∈ M(T) be the positive-definite-valued measure deﬁned by iθ e +z −1 dμ(eiθ ). ∀z ∈ D, F (z) := (Id + B(z))(Id − B(z)) =: iV + eiθ − z T

Then, the Radon–Nikodym derivative of the absolutely continuous compo2 1 m → Cm ). nent of μ is R(Id−B) −1 A ∈ L (T, C −1 We deﬁne G := (Id − B) A. This function is outer, because its determinant is outer. It is the quotient of two outer functions in H ∞ , so G ∈ N + (compute the inverse of (Id−B) with the formula of the comatrix, whose coefﬁcients are polynomials in the coefﬁcients of Id − B, so are outer functions). 2 1 m → Cm ) implies that G ∈ H 2 (Cm → The fact that R(Id−B) −1 A ∈ L (T, C Cm ). If μ is absolutely continuous, then iθ e + z ∗ iθ dθ G (e )G(eiθ ) . F (z) := (Id + B(z))(Id − B(z))−1 = iV + eiθ − z 2π T

Finally, the hypothesis B(0) = 0 implies V = 0. Let (ek )1≤k≤m be the canon dθ ical basis of Cm . The coefﬁcients of F (0) = Id = T G∗ (eiθ )G(eiθ ) are the 2π inner products Gen , Gem H 2 (Cm ) , so G is a matrix which columns form an orthogonal basis and G is of unit norm in H 2 (Cm ). In this context, we say that (B, A) is a corona pair, or a pair. When μ is absolutely continuous, (B, A) is said to be a special pair. Then, the measure μG deﬁned in Sect. 3 is equal to μ and Gf ∈ H 2 (Cm ) is equivalent to f ∈ H 2 (Cm , μ). Recall that G := (Id − B)−1 A is outer, so thanks to Lemma 3.3-2, the operator TId−B TG∗ : H 2 (Cm ) → H(B) is an isometry. To be a special pair means that for all f ∈ H 2 (Cm , μ), TId−B TG∗ Gf = f, the operator TId−B TG∗ represents the division by G. So, we can reformulate Theorem 3.4: Corollary 5.1. Let (B, A) be a pair. Then TId−B TG∗ is an isometry from H 2 (Cm ) to H(B). The pair is special if and only if TId−B TG∗ is surjective. The next Proposition is the analogue of the Proposition 6 in [13]. Proposition 5.2. 1. Let (B, A) be a pair. Then, AH 2 (Cm ) ⊂ H(B). 2. If the pair (B, A) is special and G = (Id − B)−1 A, then AH 2 (Cm ) is dense in H(B) if and only if G2 is rigid. 3. If AH 2 (Cm ) is dense in H(B) (relatively to · H(B) ), then (B, A) is special. Consequently, if AH 2 (Cm ) is dense in H(B), then G2 is rigid. Proof. 1.

Using the property of Toeplitz operators, we obtain easily: ∀f ∈ H 2 (Cm , μ), TId−B TG∗ TG∗−1 G f = TId−B Gf = (Id − B)Gf = Af.

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So, TId−B TG∗ sends the range of TG∗−1 G in AH 2 (Cm ). But, if (B, A) is special pair, then the range of TId−B TG∗ is H(B). So, AH 2 (Cm ) is contained in all H(B). 2. If (B, A) is special, then TId−B TG∗ is a surjective isometry from H 2 (Cm ) to H(B). Because A is outer, AH 2 (Cm ) is dense in H 2 (Cm ), but AH 2 (Cm ) is in H(B) so AH 2 (Cm ) is dense in H(B). Because TId−B TG∗ TG∗−1 G = TA , and TId−B TG∗ is an isometry, AH 2 (Cm ) is dense is equivalent to the range of TG∗−1 G is dense in H 2 (Cm ). This means that the kernel of TG∗ G−1 is reduced to zero. We conclude by Proposition 4.3 that G2 is a rigid function. 3. We need to prove that the range of TId−B TG∗ is H(B). If AH 2 (Cm ) is dense in H(B), then the range of TId−B TG∗ is dense in H(B). But TId−B TG∗ is an isometry, so it is all of H(B). We conclude that (B, A) is a special pair. In the scalar case, for a function b ∈ H ∞ verifying log(1−|b|2 ) ∈ L1 , the polynomials are dense in H(b) [14, II-4]. We verify the matricial analogue, namely that {pu, p ∈ P ol+ , u ∈ Cm } is dense in H(B). Because TA TA∗ ≤ TA∗ TA , and TA∗ TA = Id − TB ∗ TB and TB TB ∗ ≤ TB ∗ TB , Douglas’s Lemma 3.2 implies that M(A) ⊂ M(A∗ ) = H(B ∗ ) ⊂ H(B). Let p be a polynomial and u be a vector in Cm . The range TA∗ pu is of the form qu, where q is a polynomial with the same degree as p. Because {pu, p ∈ P ol+ , u ∈ Cm } is dense in H 2 (Cm ), so it is in M(TA∗ ) = H(B ∗ ). To complete the proof, we need to show that M(A∗ ) is dense in H(B). The link between H(B) and H(B ∗ ) is a corollary of Douglas’s Lemma (see [14] I-8): h is in H(B) is equivalent to TB ∗ h ∈ H(B ∗ ). Because A is outer, ker TA∗ = {0}, so there exists an unique h+ such that TB ∗ h = TA∗ h+ . Moreover, the following formula (see [14] IV-1) holds in the matricial case: + ∀h1 , h2 ∈ H(B), h1 , h2 B = h1 , h2 2 + h+ 1 , h2 2 .

Let h be in H(B) orthogonal to M(A∗ ) (for the inner product of H(B)). Then, for all n ≥ 0, h, TA∗ S ∗n h H(B) = 0, and by unicity, (TA∗ S ∗n h)+ = TA∗ S ∗n h+ . Now, we can show that h is null. For all n ≥ 0, we have 0 = h, TA∗ S ∗n h H(B) = h, TA∗ S ∗n h 2 + h+ , TA∗ S ∗n h+ 2 1 A(eit )h(eit ), h(eit ) Cm + A(eit )h+ (eit ), h+ (eit ) Cm eint dt. = 2π T

The scalar function φ : eit → A(eit )h(eit ), h(eit ) Cm + A(eit )h+ (eit ), h+ (eit ) Cm lies in H01 . The same calculation with TA∗ S ∗n h, h H(B) = 0 implies that φ ∈ H01 , so φ is constant equal to zero, and we obtain the density of {pu, p ∈ P ol+ , u ∈ Cm } in H(B). Corollary 5.3. Let (B, A) be a special pair. If G2 is rigid, and U ∈ H ∞ (Cm → Cm ) is an inner function of rank m, then (U B, A) is special and ((I − U B)−1 A)2 is rigid.

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A consequence of ((I − U B)−1 A)2 being rigid is that (I − U B)−1 A is outer. Proof. Because U is inner and (B, A) is a pair, (BU, A) is a pair too. Once again, Douglas’s Lemma 3.2 allows us to show that H(B) ⊂ H(U B). We have TU B = TU TB ≤ TB , so is Id − TB TB ∗ ≤ I − TU B TB ∗ U ∗ and H(B) ⊂ H(U B). The polynomials are dense in H(B) and in H(U B), so H(B) is dense in H(U B). Thanks to Proposition 5.2, G2 rigid and (B, A) special implies that AH 2 (Cm ) is dense in H(B), so in H(U B). Proposition 5.2 again, tell us that the pair (U B, A) is a special. As a consequence, the corresponding function 2 (I − U B)−1 A is rigid. Lemma 5.4. Let (B, A) be a pair and let U ∈ H ∞ (Cm → Cm ) be an inner function of rank m verifying B = U B0 . Let A := (I − B0 U )G. Then TId−B TG∗ maps the range of the operator TG∗−1 U G onto U A H 2 (Cm ). Proof. By construction, it is clear that A is outer in H ∞ (Cm → Cm ). Because B = U B0 , we obtain the equality (Id − B)U G = (Id − U B0 )U G = U (Id − B0 U )G = U A . So the range of TId−B TG∗ TG∗−1 U G is U TA H 2 (Cm ).

Now, we state the matricial analogue of Hayashi’s theorem. Theorem 5.5. Let F = GKU be a nearly S ∗ -invariant subspace of H 2 (Cm ), where G ∈ H 2 (Cm → Cm ) is outer and U ∈ H ∞ (Cm → Cm ) is inner, verifying U (0) = 0 and rank U = m. We write A := (Id − U B0 )G, A := (Id − B0 U )G and G0 := (Id − B0 )A . Then, F is the kernel of a Toeplitz operator if and only if B = U B0 , the pair (B0 , A ) is special and G2 0 is rigid. Proof. Let F = GKU be the kernel of a Toeplitz operator. Then, Theorem 3.5 implies that B = U B0 , and Lemma 4.4 gives that F = ker TG∗ U ∗ G−1 and so H 2 (Cm ) = F ⊕⊥ TG∗−1 U G H 2 (Cm ). The section I-10 of [14], valid in the matricial case, gives the following orthogonal decomposition: H(B) = KU ⊕⊥B U H(B0 ). The operator TId−B TG∗ is an isometry from H 2 (Cm ) to H(B). Lemma 5.4 tells us that it maps the range of TG∗−1 U G onto U TA H 2 (Cm ). Moreover, it maps F on KU . So, we get the following diagram: TId−B TG∗

⊕⊥ TG∗−1 U G H 2 (Cm ) H 2 (Cm ) = F ↓ ↓ ↓ H(B) = KU ⊕⊥H(B) U H(B0 ).

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It follows that U A H 2 (Cm ) is dense in U H(B0 ). On H(B), TU is an isometry, so A H 2 (Cm ) is dense in H(B0 ). We conclude this implication using Proposition 5.2. The fact that A H 2 (Cm ) is dense in H(B0 ) implies that (B0 , A ) is special and G2 0 is rigid. Conversely, we can reverse the reasoning. The pair (B0 , A ) is special 2 m 2 m and G2 0 is rigid imply that A H (C ) is dense in H(B0 ) and so U A H (C ) is dense in U H(B0 ). The diagram holds to be true, so ⊥

⊥

TId−B TG∗ TG∗−1 U G H 2 (Cm ) = TId−B TG∗ (U H(B0 )) = KU . It follows that F = GKU = TG∗−1 U G H 2 (Cm ) operator TG∗ U ∗ G−1 .

⊥

is the kernel of the Toeplitz

As mentioned in Sarason’s article [13], the proof contains a recipe for constructing a non-trivial proper subspace F ⊂ H 2 (Cm ) which is the kernel of a Toeplitz operator. We repeat the process. We begin with the particular case r = m: Take an outer function G0 ∈ H 2 (Cm → Cm ) such that G2 0 is rigid and with an inner function U ∈ H ∞ (Cm → Cm ) vanishing at zero. The pair associated to G0 is (B0 , A ). Let B = U B0 , G = (Id − B0 U )−1 A and A = (Id − U B0 )G. Thanks to Proposition 5.2, the pair (B, A) is special and G2 is rigid. Then F = GKU is a nearly S ∗ -invariant subspace which is the kernel of the Toeplitz operator with symbol G∗ U ∗ G−1 . With this construction, dim W is equal to m. We can adapt the general case r < m from the particular case r = m. With the previous notation, F = GKU , with G ∈ H 2 (Cr → Cm ) outer, so G = H 2 (G) where G is a subspace of Cm of dimension r. Working in H 2 (G) allow us to apply the previous theorem and the unitary matrix Θ0 to come back in H 2 (Cm ). Acknowledgements The author is grateful to Pr. Thomas Ransford for his advice and to Pr. Andreas Hartmann for the suggestion of the problem. The author also would like to thank Emmanuel Fricain and Isabelle Chalendar.

References [1] Beneker, P., Wiegerinck, J.: The boundary of the unit ball in H 1 -type spaces. In: Function Spaces (Edwardsville, IL, 2002), Contemporary Mathematics, vol. 328, pp. 59–84. Amer. Math. Soc., Providence (2003) [2] Cambern, M.: Invariant subspaces and extremum problems in spaces of vector-valued functions. J. Math. Anal. Appl. 57(2), 290–297 (1977) [3] Chalendar, I., Chevrot, N., Partington, J.R.: Nearly invariant subspaces for backward shifts on vector-valued Hardy spaces. J. Oper. Theory #1767 (to appear) [4] Garcia, S.R.: Conjugation and Clark operators. In: Recent Advances in Operator-Related Function Theory, Contemporary Mathematics, vol. 393, pp. 67–111. Amer. Math. Soc., Providence (2006)

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[5] Hayashi, E.: The solution sets of extremal problems in H 1 . Proc. Am. Math. Soc. 93(4), 690–696 (1985) [6] Hayashi, E.: The kernel of a Toeplitz operator. Integr. Equ. Oper. Theory 9(4), 588–591 (1986) [7] Hayashi, E.: Classification of nearly invariant subspaces of the backward shift. Proc. Am. Math. Soc. 110(2), 441–448 (1990) [8] Hitt, D.: Invariant subspaces of H 2 of an annulus. Pac. J. Math. 134(1), 101–120 (1988) [9] Katsnelson, V.E., Kirstein, B.: On the theory of matrix-valued functions belonging to the Smirnov class. In: Topics in Interpolation Theory (Leipzig, 1994), Operator Theory: Advances and Applications, vol. 95, pp. 299–350. Birkh¨ auser, Basel (1997). http://www.citebase.org/abstract?id=oai:arXiv.org: 0706.1901 [10] Lax, P.D.: Translation invariant spaces. Acta Math. 101, 163–178 (1959) [11] Nikolski, N.K.: Operators, Functions, and Systems: An Easy Reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92. American Mathematical Society, Providence (2002) [12] Sarason, D.: Nearly invariant subspaces of the backward shift. In: Contributions to Operator Theory and its Applications (Mesa, AZ, 1987), Operator Theory: Advances and Applications, vol. 35, pp. 481–493. Birkh¨ auser, Basel (1988) [13] Sarason, D.: Kernels of Toeplitz operators. In: Toeplitz Operators and Related Topics (Santa Cruz, CA, 1992), Operator Theory: Advances and Applications, vol. 71, pp. 153–164. Birkh¨ auser, Basel (1994) [14] Sarason, D.: Sub-Hardy Hilbert spaces in the unit disk. University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10. Wiley, New York (1994) Nicolas Chevrot D´epartement de math´ematiques et de statistique Universit´e Laval Quebec QC G1V 0A6 Canada e-mail: [email protected] Received: June 23, 2009. Revised: December 7, 2009.

Integr. Equ. Oper. Theory 67 (2010), 79–93 DOI 10.1007/s00020-010-1771-1 Published online March 16, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On the Right (Left) Invertible Completions for Operator Matrices Guojun Hai and Alatancang Chen Abstract. Let H1 and H2 be separable Hilbert spaces, and let A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ) be given operators. A necessary A C and sufficient condition is given for (X B ) to be a right (left) invertible operator for some X ∈ B(H1 , H2 ). Furthermore, some related results are obtained. Mathematics Subject Classification (2000). Primary 47A10, 47A53, 47A55. Keywords. Invertible completions, operator matrices, right (left) invertible operator, right (left) Fredholm operator.

1. Introduction The study of operator matrices arises naturally from the following fact: if H is a Hilbert space and we decompose H as a direct sum of two subspaces H1 and H2 , each bounded linear operator T : H −→ H can be expressed as the operator matrix form T11 T12 T = T21 T22 with respect to the space decomposition, where Tij is an operator from Hj into Hi , i, j = 1, 2. One way to study operators is to see them as being composed of simpler operators. The operator matrices have been studied by numerous authors [4–6,8,9,12,14–19,23,25]. This paper is concerned with the right (left) invertibility of 2 × 2 operator matrices. In this paper, H1 and H2 are separable Hilbert spaces. Let B(H1 , H2 ) and K(H1 , H2 ) denote the sets of bounded linear operators and compact operators from H1 into H2 , respectively. When H1 = H2 we write B(H1 , H1 ) = This work was completed with the support of the Specialized Research Foundation for the Doctoral Program of Higher Education (No. 20070126002), the National Natural Science Foundation of China (No. 10962004) and The Scientific Research Foundation for the Returned Overseas Chinese Scholars.

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B(H1 ). If T ∈ B(H1 , H2 ), we use R(T ), N (T ) and T ∗ to denote the range space, the null space and the adjoint of T . For a linear subspace M ⊆ H1 , its closure and orthogonal complement are denoted by M and M⊥ . Write PM for the orthogonal projection onto M along M⊥ and T |M for the restriction of T to M. Let T ∈ B(H1 , H2 ). Recall that a linear operator T + from H2 into H1 is said to be the Moore–Penrose generalized inverse of T if T + satisfies D(T + ) = R(T ) ⊕ R(T )⊥ (D(T + ) denotes the domain of T + ) and the four Moore–Penrose equations: T T + T = T,

T + T = I − PN (T ) ,

T + T T + = T + , T T + = PR(T ) |D(T + ) . The Moore–Penrose generalized inverse T + is uniquely determined and is a closed linear operator. In particular, for any y ∈ R(T ) we have y = T T + y. From the closed graph theorem (see [24]), we know that T + is bounded if and only if R(T ) is closed, and in this case, D(T + ) = H2 (see [3,21]). The following properties of T + are well known (see [3,22]): If R(T ) is closed, then R(T + ) = R(T ∗ ) = N (T )⊥ ,

N (T + ) = N (T ∗ ) = R(T )⊥ .

An operator T ∈ B(H1 , H2 ) is called a right (respectively, left) invertible operator if there exists an operator S ∈ B(H2 , H1 ) such that T S = IH2 (respectively, ST = IH1 ). If T ∈ B(H1 , H2 ) is both left invertible and right invertible, we call it invertible. It is well known [7] that T is right invertible if and only if T is surjective, i.e., R(T ) = H2 . Also, T is left invertible if and only if T x ≥ cx for all x ∈ H1 and some constant c > 0, i.e., R(T ) is closed and N (T ) = {0}. The right (or defect) spectrum σr (T ) of T ∈ B(H1 ) is defined by σr (T ) = {λ ∈ C : T − λI is not right invertible}, whilst the left (or approximate point) spectrum σl (T ) of T ∈ B(H1 ) is defined by σl (T ) = {λ ∈ C : T − λI is not left invertible}. It is evident [1,2,7] that σl (T ) (respectively, σr (T )) is a compact nonempty subset of C, and ∂(σr (T ) ∪ σl (T )) ⊆ σl (T ) ∩ σr (T ), where we write ∂K for the topological boundary of a subset K ⊂ C. We also have from [7] that ¯ ∈ σr (T ∗ ). λ ∈ σl (T ) if and only if λ Let T ∈ B(H1 , H2 ), n(T ) = dim N (T ) and d(T ) = dim R(T )⊥ . If R(T ) is closed and n(T ) < ∞, we call T a left Fredholm operator (or upper semiFredholm operator), and if R(T ) is closed and d(T ) < ∞, then T is called a right Fredholm operator (or lower semi-Fredholm operator) (see [1,17,18,20]). Given an arbitrary operator T ∈ B(H1 ), the right essential spectrum σre (T ) is defined by σre (T ) = {λ ∈ C : T − λI is not right Fredholm}, and the left essential spectrum σle (T ) is defined by σle (T ) = {λ ∈ C : T − λI is not left Fredholm}.

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¯ ∈ σle (T ∗ ). From [1,7] we know that if H1 Clearly, λ ∈ σre (T ) if and only if λ is infinite dimensional then σre (T ) and σle (T ) are compact nonempty subsets of C respectively. See [1,7,13,20] for results about semi-Fredholm operators. For operators A ∈ B(H1 ) and C ∈ B(H2 , H1 ), let N (A|C) = {G ∈ B(H2 , H1 ) : R(AG) ⊆ R(C)}. As is well known [11], an operator G ∈ B(H2 , H1 ) belongs to N (A|C) if and only if there exists H ∈ B(H2 ) such that AG = CH. When A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ) are given operators, we denote by MX (= M (A, B, C; X)) an operator on H1 ⊕ H2 of the form A C X B for X ∈ B(H1 , H2 ). In the case in which C = 0, the invertibility and the left (right) invertibility of MX were characterized in [12] and [17] respectively, and many types of spectra for MX were considered in [4–6,8,9,18,19,25]. In the general case, Takahashi [23] had studied the invertibility of MX . In this paper, we are mainly interested in the right (left) invertibility of MX .

2. Main Results The main results of this paper are contained in the following theorem. Theorem 2.1. Let A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ) be given operators. Then MX is a right invertible operator for some X ∈ B(H1 , H2 ) if and only if R(A) + R(C) = H1 and one of the following conditions holds: (i) N (A|C) contains a non-compact operator; (ii) M0 = M (A, B, C; 0) is a right Fredholm operator and d(M0 ) ≤ n(A) + dim(R(A) ∩ R(C|N (B) )). For the proof of Theorem 2.1, we need some auxiliary results. We begin with: Lemma 2.2 [10]. If H is an infinite dimensional Hilbert space and T is an operator on H, then T is compact if and only if the range of T contains no closed infinite dimensional subspaces. The following lemma will be useful in the sequel. Lemma 2.3 [1,7,20]. Let T ∈ B(H1 , H2 ) be a right (respectively, left) Fredholm operator and K ∈ B(H1 , H2 ) be a compact operator. Then T + K is a right (respectively, left) Fredholm operator with n(T +K)−d(T +K) = n(T )−d(T ). The next result is well known, so its proof will be omitted. Lemma 2.4. Let H1 and H2 be separable Hilbert spaces. If U ⊆ H1 and V ⊆ H2 are closed subspaces with dim U = dim V, then there exists T ∈ B(H1 , H2 ) such that N (T ) = U ⊥ , R(T ) = V, and T |U : U −→ V is unitary. In particular, if U = H1 , then T is left invertible; if V = H2 , then T is right invertible. We now prove the following lemmas.

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Lemma 2.5. Let S ∈ B(H1 ), and let T be a closed linear operator from H1 into H2 . If R(S) ⊆ D(T ), then T S ∈ B(H1 , H2 ). Proof. Note that a linear operator T from H1 into H2 is a closed linear operator if and only if the following holds: If {xn } is a sequence in D(T ) with limn→∞ xn = x0 and limn→∞ T xn = y0 , then x0 ∈ D(T ) and T x0 = y0 (see [24]). Let {xn } be a sequence in D(T S) such that limn→∞ xn = x0 and limn→∞ T Sxn = y0 . It follows from S ∈ B(H1 ) that limn→∞ Sxn = Sx0 . Since T is a closed linear operator and {Sxn } is a sequence in D(T ) with limn→∞ Sxn = Sx0 , we deduce from limn→∞ T Sxn = y0 that Sx0 ∈ D(T ) and T (Sx0 ) = T Sx0 = y0 . Note that Sx0 ∈ D(T ) implies x0 ∈ D(T S), and therefore T S is a closed linear operator. On the other hand, it follows from R(S) ⊆ D(T ) that D(T S) = H1 . Hence the result is a direct consequence of the closed graph theorem. Lemma 2.6. Let A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ) be given operators. Assume that A C M0 = M (A, B, C; 0) = 0 B is a right Fredholm operator on H1 ⊕H2 . Then B is a right Fredholm operator, R(A) + R(C|N (B) ) is a closed subspace, and d(M0 ) = dim(R(A) + R(C|N (B) ))⊥ + d(B), n(M0 ) =n(A) + n(C|N (B) ) + dim(R(A) ∩ R(C|N (B) )). Proof. From [1,20] we can find that if T ∈ B(H1 ), S ∈ B(H1 ) and T S is a right Fredholm operator then T is a right Fredholm operator. Thus the right Fredholmness of A C I 0 I C A 0 = M0 = 0 B 0 B 0 I 0 I implies the right Fredholmness of

I 0

0 B

.

It is easy to see that B is a right Fredholm operator. Therefore M0 has a matrix representation ⎞ ⎛ A C11 C12 ⎝0 0 B11 ⎠ : H1 ⊕ N (B) ⊕ N (B)⊥ −→ H1 ⊕ R(B) ⊕ R(B)⊥ . 0 0 0 Clearly, B11 is an invertible operator. Thus there exists an invertible operator ⎛ ⎞ −1 I −C12 B11 0 U = ⎝0 I 0 ⎠ : H1 ⊕ R(B) ⊕ R(B)⊥ −→ H1 ⊕ R(B) ⊕ R(B)⊥ 0 0 I such that

⎛

A C11 0 U⎝0 0 0

⎞ ⎛ A C12 B11 ⎠ = ⎝ 0 0 0

C11 0 0

⎞ 0 B11 ⎠ . 0

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Since M0 is a right Fredholm operator, U M0 is a right Fredholm operator too, and d(M0 ) = d(U M0 ), n(M0 ) = n(U M0 ). Thus R(A) + R(C11 ) is closed and d(M0 ) = dim(R(A) + R(C11 ))⊥ + d(B), n(M0 ) = dim N (A, C11 ), where N (A, C11 ) denotes the null space of (A, C11 ) : H1 ⊕ N (B) −→ H1 . This, together with N (A) 0 N (A, C11 ) = ⊕ 0 N (C11 ) x ⊕ : x ∈ N (A)⊥ , y ∈ N (C11 )⊥ , Ax = −C11 y y and C11 = C|N (B) : N (B) −→ H1 , shows that d(M0 ) = dim(R(A) + R(C|N (B) ))⊥ + d(B), n(M0 ) =n(A) + n(C|N (B) ) + dim(R(A) ∩ R(C|N (B) )). Obviously, R(A) + R(C|N (B) ) is a closed subspace.

Proof of Theorem 2.1. Necessity. Suppose that MX is a right invertible operator for some X ∈ B(H1 , H2 ). Let H = (N (C) ∩ N (B))⊥ . Then MX as an operator from H1 ⊕ H⊥ ⊕ H into H1 ⊕ H2 has the following operator matrix: A 0 C . X 0 B Clearly, MX =

A X

C B

: H1 ⊕ H −→ H1 ⊕ H2

is a right invertible operator. Thus there exists a bounded linear operator Y F : H1 ⊕ H2 −→ H1 ⊕ H Z E such that

A X

C B

Y Z

F E

=

IH1 0

0 IH2

.

Therefore AY + C Z = IH1 ,

AF + C E = 0,

XF + B E = IH2 .

It follows from AY +C Z = IH1 and R(C) = R(C ) that R(A)+R(C) = H1 . Now we consider two cases. Case I. Suppose that dim H2 < ∞. Then X is a compact operator, and hence by Lemma 2.3 we get that M0 = M (A, B, C; 0) and A C M0 = 0 B

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are right Fredholm operators, and d(M0 ) = d(M0 ) ≤ n(M0 ).

(2.1)

On the other hand, since M0 is a right Fredholm operator and N (C ) ∩ N (B ) = {0}, it follows from Lemma 2.6 that n(M0 ) = n(A) + dim(R(A) ∩ R(C |N (B ) )).

(2.2)

It is easy to see that R(C |N (B ) ) = R(C|N (B ) ) = R(C|N (B) ), which together with (2.1) and (2.2) implies that d(M0 ) ≤ n(A) + dim(R(A) ∩ R(C|N (B) )). Case II. Suppose that dim H2 = ∞. By XF + B E = IH2 and AF + C E = 0, F ) : H −→ H ⊕ H is a left invertible operator and we have that ( E 2 1 F R( ) ⊆ N (A, C ), E where N (A, C ) denotes the null space of (A, C ) : H1 ⊕ H −→ H1 . Hence dim N (A, C ) ≥ dim H2 . From dim H2 = ∞ (this implies dim N (A, C ) = dim H2 ) and Lemma 2.4 it follows that there exists a left invertible operator G ) : H −→ H ⊕ H whose range is equal to N (A, C ). It is clear that (H 2 1 AG = −C H. If G is not compact, it follows from AG = −C H that R(AG) ⊆ R(C ) = R(C). Therefore N (A|C) contains a non-compact operator. If G is compact, let ( R L ) : H1 −→ H1 ⊕ H be the Moore–Penrose gener alized inverse of (A, C ). Then AR+C L = IH1 (because R(A)+R(C ) = H1 ) ⊥ and R(( R L )) = (N (A, C )) . Put R G W = : H1 ⊕ H2 −→ H1 ⊕ H . L H G ) is a left invertible operThen W is an invertible operator. In fact, since ( H ator, there exists an operator (S, T ) : H1 ⊕ H −→ H2 such that SG + T H = IH2 . Thus by AR + C L = IH1 , A C R G IH1 0 A C W = = S T S T L H SR + T L IH2

is an invertible operator, and therefore W is a left invertible operator. Also since G R ⊥ and R( ) = N (A, C ), R( ) = (N (A, C )) H L we have

R G R(W ) = R( ) + R( ) = H1 + H , L H

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which means that W is a right invertible operator. This proves the invertibility of W . From AG = −C H we can see that A C 0 R G IH1 MX W = . = X B XR + B L XG + B H L H Thus XG + B H is right invertible (by the right invertibility of MX ). Note that G is a compact operator, and hence B H is a right Fredholm operator with d(B H) ≤ n(B H) by Lemma 2.3. This, together with IH1 0 0 IH1 A C R G , = 0 B −B L IH2 0 BH L H

shows that M0 =

A C 0 B

: H1 ⊕ H −→ H1 ⊕ H2

is a right Fredholm operator and d(M0 ) = d(M0 ) = d(B H) ≤ n(B H) = n(M0 ). It is obvious that M0 is a right Fredholm operator. Now an argument similar to that in the proof of Case I shows that d(M0 ) ≤ n(A) + dim(R(A) ∩ R(C|N (B) )). Sufficiency. If N (A|C) contains a non-compact operator, then H1 and H2 are infinite dimensional. By Lemma 2.2, there exists a closed subspace M ⊆ H1 with dim M = dim H2 = ∞ such that R(A|M ) ⊆ R(C), and hence R(APM ) = R(A|M ) ⊆ R(C) ⊆ D(C + ). This, together with APM ∈ B(H1 ) and Lemma 2.5, shows that C + APM ∈ B(H1 .H2 ). On the other hand, it follows from Lemma 2.4 that there exists a right invertible operator T ∈ B(H1 , H2 ) such that N (T ) = M⊥ . Define an operator X ∈ B(H1 , H2 ) by X = T + BC + APM . Then MX is a right invertible operator. Indeed, for any u ∈ H1 and v ∈ H2 , since R(A) + R(C) = H1 and R(A|M ) ⊆ R(C), there exist x1 ∈ M⊥ and y1 ∈ H2 such that Ax1 + Cy1 = u. Also, by the right invertibility of T , there exists x2 ∈ M such that T x2 = v − By1 . Let x0 = x1 + x2 and y0 = y1 − C + Ax2 . Then A C x0 u = . X B y0 v This proves the right invertibility of MX . If (ii) holds, put E = R(A)+R(C|N (B) ). From Lemma 2.6 and the right Fredholmness of M0 we can infer that B is a right Fredholm operator, E is closed and dimE ⊥ = d(M0 ) − d(B) < ∞. From R(A) + R(C) = H1 it follows that R(PE ⊥ C) = E ⊥ . Let G = (PE ⊥ C)+ E ⊥ and S = BG ⊕ R(B)⊥ . Then clearly G ⊆ N (B)⊥ and so dim E ⊥ = dim G = dim BG. Therefore dim S = d(M0 ). On the other hand, since d(M0 ) ≤ n(A) + dim(R(A) ∩ R(C|N (B) )), there exists a subspace M ⊆ H1 with dimM = d(M0 ) such that R(A|M ) ⊆ R(C|N (B) ). By dim M = dim S = d(M0 ) < ∞ and Lemma 2.4, there exists

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an operator J : H1 −→ S such that N (J) = M⊥ and J|M : M −→ S is unitary. Define X ∈ B(H1 , H2 ) by J X= : H1 −→ S ⊕ S ⊥ . 0 Then MX as an operator from H1 ⊕ N (B) ⊕ G ⊕ (N (B)⊥ G) into E ⊕ E ⊥ ⊕ S ⊕ S ⊥ has the following operator matrix: ⎞ ⎛ A1 C1 C2 C3 ⎜ 0 0 C4 0 ⎟ ⎟, MX = ⎜ ⎝ J 0 B1 B3 ⎠ 0 0 0 B2 where N (B)⊥ G = {y ∈ N (B)⊥ : y ∈ G ⊥ }. Obviously, C4 is invertible. From the right Fredholmness of B we can infer that B2 is invertible too. Thus there are an invertible operator U ∈ B(H1 ⊕ H1 ) such that ⎛ ⎞ ⎛ ⎞ A1 C1 0 A1 C1 C2 C3 0 ⎜ ⎜ 0 0 C4 0 ⎟ 0 C4 0 ⎟ ⎟=⎜ 0 ⎟. U MX = U ⎜ ⎝ ⎝ J ⎠ 0 B1 B3 J 0 0 0 ⎠ 0 0 0 B2 0 0 0 B2 It follows that MX is a right invertible operator if and only if A1 C1 : H1 ⊕ N (B) −→ E ⊕ S J 0 is a right invertible operator. Now we prove that A1 C1 : H1 ⊕ N (B) −→ E ⊕ S J 0 is a right invertible operator. For any u ∈ E and v ∈ S, it follows from E = R(A) + R(C|N (B) ), R(A|M ) ⊆ R(C|N (B) ) and the definition of J that there exist x1 ∈ M, x2 ∈ M⊥ and y1 ∈ N (B) such that Jx1 = v, Ax2 + Cy1 = u. Since R(A|M ) ⊆ R(C|N (B) ), there exists y2 ∈ N (B) with Ax1 + Cy2 = 0. Note that A1 = A : H1 −→ E, C1 = C|N (B) : N (B) −→ E and N (J) = M⊥ , and hence A1 C1 x1 + x2 u = . J 0 y1 + y2 v From the argument above we get that MX is a right invertible operator. The following is the dual statement of Theorem 2.1. Theorem 2.7. Let A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ) be given operators. Then MX is a left invertible operator for some X ∈ B(H1 , H2 ) if and only if R(B ∗ ) + R(C ∗ ) = H2 and one of the following conditions holds: (i) N (B ∗ |C ∗ ) contains a non-compact operator;

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M0 = M (A, B, C; 0) is a left Fredholm operator and n(M0 ) ≤ d(B) + dim(R(B ∗ ) ∩ R(C ∗ |N (A∗ ) )).

A C ) is a left invertible operator on H ⊕ H if Proof. Note that MX = ( X 1 2 B ∗ ∗ ∗ A X and only if (MX ) = C ∗ B ∗ is a right invertible operator on H1 ⊕ H2 if B∗ C ∗ and only if M (B ∗ , A∗ , C ∗ ; X ∗ ) = X is a right invertible operator on ∗ A∗ H2 ⊕ H1 . Now Theorem 2.7 follows from Theorem 2.1.

As a corollary of Theorem 2.1, we have Corollary 2.8 [17]. Let H1 and H2 be infinite dimensional separable Hilbert spaces. Assume that A ∈ B(H1 ) and B ∈ B(H2 ) are given operators. A 2×2 operator matrix A X LX = 0 B is left invertible for some X ∈ B(H2 , H1 ) if and only if A is left invertible and d(A) = ∞, if R(B) is not closed; n(B) ≤ d(A), if R(B) is closed. Proof. Sufficiency. Note that the left invertibility of A implies the right invertibility of A∗ , and hence R(A∗ ) = H1 . First suppose that R(B) is not closed. Then d(A) = n(A∗ ) = ∞. By Lemma 2.4, there exists a left invertible operator G ∈ B(H2 , H1 ) such that R(G) = N (A∗ ). It is easy to see that G is non-compact (since G is left invertible) and A∗ G = 0. Therefore N (A∗ |0) contains a non-compact operator. Next suppose that R(B) is closed. Then n(B) ≤ d(A). If d(A) = ∞, then by the above argument we get that N (A∗ |0) contains a non-compact operator. If d(A) < ∞, then clearly B ∗ is a right Fredholm operator with d(B ∗ ) ≤ n(A∗ ). This, together with the right invertibility of A∗ , shows that M (A∗ , B ∗ , 0; 0) is a right Fredholm operator with d(M (A∗ , B ∗ , 0; 0)) = d(B ∗ ) ≤ n(A∗ ). Applying Theorem 2.1 to A∗ , B ∗ and C = 0, we conclude that there exists X ∈ B(H2 , H1 ) such that (LX )∗ = M (A∗ , B ∗ , 0; X ∗ ) is right invertible. This implies the sufficiency of Corollary 2.8. Necessity. Assume that LX is left invertible for some X ∈ B(H2 , H1 ). Clearly, A is left invertible and ∗ A 0 M (A∗ , B ∗ , 0; X ∗ ) = X ∗ B∗ is right invertible. It follows from Theorem 2.1 that N (A∗ |0) contains a noncompact operator or M0 = M (A∗ , B ∗ , 0; 0) is a right Fredholm operator with d(B ∗ ) = d(M0 ) ≤ n(A∗ ). Therefore n(A∗ ) = d(A) = ∞ or B ∗ is a right Fredholm operator with n(B) ≤ d(A). Obviously, if R(B) is not closed, then d(A) = ∞; if R(B) is closed, then n(B) ≤ d(A). Recall that an operator T ∈ B(H1 , H2 ) is called Fredholm if it is both left Fredholm and right Fredholm. Using Theorems 2.1 and 2.7, we have

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Corollary 2.9 [23]. Let A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ). Assume that (A, C) : H1 ⊕ H2 −→ H1 and (B ∗ , C ∗ ) : H2 ⊕ H1 −→ H2 are right invertible. Then there exists X ∈ B(H1 , H2 ) such that MX is invertible if and only if both N (A|C) and N (B ∗ |C ∗ ) contain non-compact operators or M0 = M (A, B, C; 0) is Fredholm with n(M0 ) = d(M0 ). Proof. For the proof of sufficiency, see [23]. Now we suppose that there exists X ∈ B(H1 , H2 ) such that MX is an invertible operator. Since (A, C) : H1 ⊕ H2 −→ H1 and (B ∗ , C ∗ ) : H2 ⊕ H1 −→ H2 are right invertible, it follows from Theorems 2.1 and 2.7 that A, B and C satisfy the following conditions: (a) N (A|C) contains a non-compact operator or M0 is a right Fredholm operator with d(M0 ) ≤ n(M0 ), (b) N (B ∗ |C ∗ ) contains a non-compact operator or M0 is a left Fredholm operator with n(M0 ) ≤ d(M0 ). If N (A|C) consists of compact operators only, then clearly M0 is a right Fredholm operator with d(M0 ) ≤ n(M0 ). From the left invertibility of MX (since MX is invertible) and the proof of Theorem 2.1 we can see that M0 is a left Fredholm operator with n(M0 ) ≤ d(M0 ). This implies that M0 is Fredholm with n(M0 ) = d(M0 ). Similarly, if N (B ∗ |C ∗ ) consists of compact operators only, we can show that M0 is Fredholm with n(M0 ) = d(M0 ). Therefore both N (A|C) and N (B ∗ |C ∗ ) contain non-compact operators or M0 = M (A, B, C; 0) is Fredholm with n(M0 ) = d(M0 ). Let A ∈ B(H1 ) and B ∈ B(H2 ). We denote by M (A, B; Y, X) an operator acting on H1 ⊕ H2 of the form A Y M (A, B; Y, X) = , X B where X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ). In [14], a necessary and sufficient condition is obtained for M (A, B; Y, X) to be a positive operator for some X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ), and a necessary and sufficient condition is given for M (A, B; Y, X) to be a projection operator for some X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) in [15]. From Theorems 2.1 and 2.7, we get the following results, concerning the right and left invertibility of M (A, B; Y, X). Corollary 2.10. Let A ∈ B(H1 ) and B ∈ B(H2 ) be given operators. If H1 and H2 are infinite dimensional, then there exist X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) such that M (A, B; Y, X) is a right (respectively, left) invertible operator. Proof. Since H1 and H2 are infinite dimensional, there exists an invertible operator Y ∈ B(H2 , H1 ). It is not difficult to see that R(A) + R(Y ) = H1 (respectively, R(B ∗ ) + R(Y ∗ ) = H2 ) and N (A|Y ) (respectively, N (B ∗ |Y ∗ )) contains a non-compact operator. Applying Theorem 2.1 (respectively, Theorem 2.7) to A, B and Y , we conclude that M (A, B, Y ; X) is a right (respectively, left) invertible operator for some X ∈ B(H1 , H2 ). Corollary 2.11. Let A ∈ B(H1 ) and B ∈ B(H2 ) be given operators.

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(i) If H1 is infinite dimensional and H2 is finite dimensional, then there exist X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) such that M (A, B; Y, X) is a right invertible operator if and only if A is a right Fredholm operator, d(A) ≤ n(A) and d(A) ≤ dim H2 . (ii) If H1 is finite dimensional and H2 is infinite dimensional, then there exist X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) such that M (A, B; Y, X) is a right invertible operator if and only if B is a right Fredholm operator, d(B) ≤ n(B) and d(B) ≤ dim H1 . Proof. (i) Necessity. Suppose that there exist X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) such that M (A, B; Y, X) is a right invertible operator. It follows from Theorem 2.1 that R(A) + R(Y ) = H1 . Therefore (A, Y ) : H1 ⊕ H2 −→ H1 is a right invertible operator. Note that dim H2 < ∞, and hence Y is a compact operator. By Lemma 2.3, we get that A is a right Fredholm operator and d(A) ≤ n(A). On the other hand, since (A, Y ) : H1 ⊕ H2 −→ H1 is a right invertible operator, it follows that d(A) = dim R(PR(A)⊥ Y ). Obviously, dim R(PR(A)⊥ Y ) ≤ dim H2 , and so d(A) ≤ dim H2 . Sufficiency. Since A is a right Fredholm operator and d(A) ≤ dim H2 , there exists an injective operator Y ∈ B(H2 , H1 ) such that R(A) + R(Y ) = H1 . Because dim H2 < ∞, so B and Y are compact operators. This, together with Lemma 2.3, shows that A Y M0 = M (A, B, Y ; 0) = 0 B is a right Fredholm operator with d(M0 ) ≤ n(M0 ) (note that A is a right Fredholm operator with d(A) ≤ n(A)). By the injectivity of Y and Lemma 2.6, we see that n(M0 ) = n(A) + dim(R(A) ∩ R(Y |N (B) )). Hence d(M0 ) ≤ n(A) + dim(R(A) ∩ R(Y |N (B) )).

(ii)

Applying Theorem 2.1 to A, B and Y , we conclude that M (A, B, Y ; X) is a right invertible operator for some X ∈ B(H1 , H2 ). Note that M (A, B; Y, X) is a right invertible operator on H1 ⊕ H2 if and only if M (B, A; X, Y ) is a right invertible operator on H2 ⊕ H1 . In the same way as in Corollary 2.11 (i), we can prove that Corollary 2.11 (ii) holds.

Corollary 2.12. Let A ∈ B(H1 ) and B ∈ B(H2 ) be given operators. (i) If H1 is infinite dimensional and H2 is finite dimensional, then there exist X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) such that M (A, B; Y, X) is a left invertible operator if and only if A is a left Fredholm operator, n(A) ≤ d(A) and n(A) ≤ dim H2 . (ii) If H1 is finite dimensional and H2 is infinite dimensional, then there exist X ∈ B(H1 , H2 ) and Y ∈ B(H2 , H1 ) such that M (A, B; Y, X) is a left invertible operator if and only if B is a left Fredholm operator, n(B) ≤ d(B) and n(B) ≤ dim H1 .

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Proof. Note that M (A, B; Y, X) is a left invertible operator if and only if M (A, B; Y, X)∗ = M (A∗ , B ∗ ; X ∗ , Y ∗ ) is a right invertible operator. This, together with Corollary 2.11, implies (i) and (ii) of Corollary 2.12. The following result is immediate from Theorems 2.1 and 2.7. Corollary 2.13. Let A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ). Then σr (MX ) = {λ ∈ C : R(A − λI) + R(C) = H1 } X∈B(H1 ,H2 )

∪ {λ ∈ C : λ ∈ σre (M0 ), N (A − λI|C) ⊆ K(H2 , H1 )} ∪ {λ ∈ C : N (A − λI|C) ⊆ K(H2 , H1 ),

d(M0 − λI) > n(A − λI) + dim(R(A − λI) ∩ R(C|N (B−λI) ))}, ¯ + R(C ∗ ) = H2 } σl (MX ) = {λ ∈ C : R(B ∗ − λI)

X∈B(H1 ,H2 ) ∗ ¯ ) ⊆ K(H1 , H2 )} ∪ {λ ∈ C : λ ∈ σle (M0 ), N (B ∗ − λI|C ∗ ∗ ¯ ∪ {λ ∈ C : N (B − λI|C ) ⊆ K(H1 , H2 ), ¯ n(M0 − λI) > d(B − λI) + dim(R(B ∗ − λI)

∩ R(C ∗ |N (A∗ −λI) ¯ ))}. From Corollaries 2.10, 2.11 and 2.12, we also get Corollary 2.14. Let A ∈ B(H1 ) and B ∈ B(H2 ) be given operators. (i) If dim H1 = dim H2 = ∞, then σr (M (A, B; Y, X)) = ∅. X,Y

(ii)

If dim H2 < dim H1 = ∞, then σr (M (A, B; Y, X)) = σre (A) ∪ {λ ∈ C : d(A − λI) > dim H2 } X,Y

∪ {λ ∈ C : d(A − λI) > n(A − λI)}. (iii)

If dim H1 < dim H2 = ∞, then σr (M (A, B; Y, X)) = σre (B) ∪ {λ ∈ C : d(B − λI) > dim H1 } X,Y

∪ {λ ∈ C : d(B − λI) > n(B − λI)}. Corollary 2.15. Let A ∈ B(H1 ) and B ∈ B(H2 ) be given operators. (i) If dim H1 = dim H2 = ∞, then σl (M (A, B; Y, X)) = ∅. X,Y

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If dim H2 < dim H1 = ∞, then σl (M (A, B; Y, X)) = σle (A) ∪ {λ ∈ C : n(A − λI) > dim H2 } X,Y

∪ {λ ∈ C : n(A − λI) > d(A − λI)}. (iii)

If dim H1 < dim H2 = ∞, then σl (M (A, B; Y, X)) = σle (B) ∪ {λ ∈ C : n(B − λI) > dim H1 } X,Y

∪ {λ ∈ C : n(B − λI) > d(B − λI)}. Remark 2.16. In Theorem 2.1, d(M0 ) ≤ n(A) + dim(R(A) ∩ R(C|N (B) )) can not be replaced by d(M0 ) ≤ n(M0 ). To see this, let H1 = H2 = 2 and let A ∈ B(H1 ) be an invertible operator. Define C ∈ B(H2 , H1 ) and B ∈ B(H2 ) by C = 0, Bx = (0, x3 , x4 , · · · ), where x = (x1 , x2 , · · · ) ∈ 2 . Clearly, R(A) + R(C) = H1 . It is easy to find that M0 is right Fredholm and 1 = d(M0 ) ≤ n(M0 ) = 2. But, for any X ∈ B(H1 , H2 ), I2 0 A 0 A 0 = −XA−1 I2 X B 0 B is not right invertible, which implies that MX is not right invertible for any X ∈ B(H1 , H2 ). Remark 2.17. In many applications, the entries of block operator matrices are unbounded operators. This paper deals only with the bounded case. We expect that an analogue of Theorem 2.1 holds for the unbounded case. We end with an example illustrating Theorem 2.1. Example. Let H1 = H2 = 2 ⊕ 2 . Define operators A ∈ B(H1 ), B ∈ B(H2 ) and C ∈ B(H2 , H1 ) by 0 0 0 0 T2 0 A= , B= , C= , 0 T1 0 T3 0 I2 where Ti ∈ B(2 ), i = 1, 2, 3, and R(T1 ) = R(T2 ) = 2 . Obviously, R(A) + R(C) = H1 . It follows from n(A) = ∞ that N (A|C) contains a non-compact operator. Using Theorem 2.1 we get that MX is right invertible for some X ∈ B(H1 , H2 ). Indeed, define X ∈ B(H1 , H2 ) by X1 0 X= , 0 0 where X1 ∈ B(2 ) is a right invertible operator. Then clearly MX is a right invertible operator. Acknowledgements The authors are grateful to the referees for valuable comments on this paper.

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References [1] Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers, Dordrecht (2004) [2] Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. Elsevier Science Publishers, Amsterdam (1988) [3] Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003) [4] Cao, X.: Browder essential approximate point spectra and hypercyclicity for operator matrices. Linear Algebra Appl. 426, 317–324 (2007) [5] Cao, X., Guo, M., Meng, B.: Semi-Fredholm spectrum and Weyl’s theorem for operator matrices. Acta Math. Sin. (Engl. Ser.) 22, 169–178 (2006) [6] Cao, X., Meng, B.: Essential approximate point spectra and Weyl’s theorem for operator matrices. J. Math. Anal. Appl. 304, 759–771 (2005) [7] Conway, J.B.: A Course in Functional Analysis. Springer, New York (1985) [8] Djordjevi´c, D.S.: Perturbations of spectra of operator matrices. J. Oper. Theory 48, 467–486 (2002) [9] Djordjevi´c, S.V., Zguitti, H.: Essential point spectra of operator matrices through local spectral theory. J. Math. Anal. Appl. 338, 285–291 (2008) [10] Douglas, R.G.: Banach Algebra Techniques in Operator Theory, 2nd edn. Springer, New York (1998) [11] Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17, 413–415 (1966) [12] Du, H., Pan, J.: Perturbation of spectrums of 2×2 operator matrices. Proc. Amer. Math. Soc. 121, 761–766 (1994) [13] Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, vol. 1. Birkh¨ auser-Verlag, Basel (1990) [14] Hou, J.: On the spectra of the positive completions for operator matrices. J. Oper. Theory 33, 299–315 (1995) [15] Hou, J.: Completion of operator partial matrices to projections. Linear Algebra Appl. 246, 71–82 (1996) [16] Hou, J., Du, H.: Norm inequalities of positive operator matrices. Integral Equ. Oper. Theory 22, 281–294 (1995) [17] Hwang, I.S., Lee, W.Y.: The boundedness below of 2×2 upper triangular operator matrices. Integral Equ. Oper. Theory 39, 267–276 (2001) [18] Li, Y., Du, H.: The intersection of essential approximate point spectra of operator matrices. J. Math. Anal. Appl. 323, 1171–1183 (2006) [19] Li, Y., Sun, X., Du, H.: The intersection of left (right) spectra of 2×2 upper triangular operator matrices. Linear Algebra Appl. 418, 112–121 (2006) [20] M¨ uller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, 2nd edn. Birkh¨ auser-Verlag, Basel (2007) [21] Nashed, M.Z.: Generalized Inverses and Applications. Academic Press, New York (1976) [22] Petryshyn, W.V.: On generalized inverses and on the uniform convergence of (I −βK)n with application to iterative methods. J. Math. Anal. Appl. 18, 417– 439 (1967)

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[23] Takahashi, K.: Invertible completions of operator matrices. Integral Equ. Oper. Theory 21, 355–361 (1995) [24] Weidmann, J.: Linear Operators in Hilbert Spaces. Springer, New York (1980) [25] Zerouali, E.H., Zguitti, H.: Perturbation of spectra of operator matrices and local spectral theory. J. Math. Anal. Appl. 324, 992–1005 (2006) Guojun Hai School of Mathematical Sciences Inner Mongolia University Hohhot 010021 People’s Republic of China e-mail: [email protected] Alatancang Chen School of Mathematical Sciences Inner Mongolia University Hohhot 010021 People’s Republic of China e-mail: [email protected] Received: June 27, 2009. Revised: January 15, 2010.

Integr. Equ. Oper. Theory 67 (2010), 95–107 DOI 10.1007/s00020-010-1772-0 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Group Representations with Empty Residual Spectrum Y. Choi Abstract. Let X be a Banach space on which a discrete group Γ acts by isometries. For certain natural choices of X, every element of the group algebra, when regarded as an operator on X, has empty residual spectrum. We show, for instance, that this occurs if X is 2 (Γ) or the group von Neumann algebra VN(Γ). In our approach, we introduce the notion of a surjunctive pair , and develop some of the basic properties of this construction. The cases X = p (Γ) for 1 ≤ p < 2 or 2 < p < ∞ are more difﬁcult. If Γ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on Γ is necessary. Mathematics Subject Classification (2000). Primary 47A10; Secondary 47C10, 47C15. Keywords. Residual spectrum, surjunctive, group von Neumann algebra, directly ﬁnite, amenable group.

1. Introduction If Γ is a discrete group, and X a Banach space on which Γ acts by translations, elements of the group algebra CΓ deﬁne bounded linear operators on X. The spectral properties of these operators seem little explored when Γ is non-abelian, except if further strong conditions such as self-adjointness, or having positive coefﬁcients, are imposed. In particular, we might consider the residual spectrum of such an operator. In many cases the residual spectrum turns out to be empty, motivating the following question. Question. Given Γ and X as above: does every a ∈ CΓ have empty residual spectrum, when regarded as an operator on X? The answer is known to be yes when Γ is a Moore group, that is, a group all of whose irreducible unitary representations are ﬁnite-dimensional. This

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follows by combining the following theorem of Runde, with the basic observation (see e.g. [11, Proposition 1.3.3]) that every decomposable operator on a Banach space has empty residual spectrum. Theorem 1.1 ([15]). Let G be a locally compact Moore group, X a Banach space, and θ : L1 (G) → B(X) a continuous algebra homomorphism. Then θ(f ) is decomposable for each f ∈ L1 (G). If X is ﬁnite-dimensional, the residual spectrum of any operator is empty (this just follows by counting dimension). We therefore restrict attention to the inﬁnite-dimensional setting. Moreover, it follows from a previous observation of the author ([2, Theorem 1]; but see Remark 1.3 below) that we again obtain a positive answer when X = ∞ (Γ), for arbitrary Γ. In contrast, for X = 1 (Γ) we shall see that the answer depends on our group Γ. The case X = 2 (Γ) is an obvious choice to consider next, and as a consequence of our main result (Theorem 1.2) we shall see that again the answer to our question is “yes” for this choice of X, regardless of our group Γ. We attempt in this article to initiate a systematic approach to such questions, developing some machinery in the more general setting of subalgebras of B(X). To save needless repetition: given a Banach space X and a subalgebra A ⊆ B(X)—which need not be norm-closed nor unital—we say that the pair (A, X) is a surjunctive pair if every T ∈ A has empty residual spectrum. (The terminology will be explained below, see Remark 2.2.) The main result of this article can now be stated as follows. Theorem 1.2. Let Γ be a discrete group, and let X be either the reduced group C∗ -algebra C∗r (Γ), or the non-commutative Lp -space associated to the group von Neumann algebra VN(Γ), for some p ∈ [1, ∞]. Let Γ act by left translations on X in the natural way, and let ı : CΓ → B(X) be the induced algebra homomorphism. Then (ı(CΓ), X) is a surjunctive pair. The proof of this will be given in Sect. 3, and actually yields a stronger result when X is one of the non-commutative Lp -spaces: in these cases, one can replace ı(CΓ) with the group von Neumann algebra. We note that the noncommutative L2 -space associated to VN(Γ) is nothing but 2 (Γ), and the Γ-action referred to above is just the left regular representation of Γ on 2 (Γ). Remark 1.3. We have been lax in describing “the” action of 1 (Γ) on p (Γ) that was mentioned earlier. Unless speciﬁed, we are referring to the homomorphism L• : 1 (Γ) → B(p (Γ)) that is deﬁned by ‘left translation’, viz. La : p (Γ) → p (Γ); (La ξ)(h) := a(g)ξ(g −1 h) (a ∈ 1 (Γ), h ∈ G). g∈Γ

It will be convenient at one point below to consider another algebra homomorphism ρ• : 1 (Γ) → B(p (Γ)), which arises by considering the adjoint of the natural right action of 1 (Γ) on q (Γ) for p−1 + q −1 = 1. Explicitly, ρa : p (Γ) → p (Γ); (ρa ξ)(h) := a(g)ξ(hg) (a ∈ 1 (Γ), h ∈ G). g∈Γ

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(L• is the left regular representation, and ρ• the right regular representation.) Although L• and ρ• do not coincide, in general, they are intertwined by the ‘ﬂip’ operator t : p (Γ) → p (Γ);

t ξ(h) := ξ(h−1 ) (h ∈ Γ);

that is, t ρa = La t for all a ∈ 1 (Γ). It follows that (L• (1 (Γ)), p (Γ)) is a surjunctive pair if and only if (ρ• (1 (Γ)), p (Γ)) is. Let us review the outline of this paper. We collect some general preliminaries in the next section. Section 3 contains the results used to prove Theorem 1.2, as well as some with applications to CCR groups. Finally, we consider the case where A = X = 1 (Γ). Here the question of surjunctivity remains open, although we obtain a partial result in the case where Γ is amenable. We close with some speciﬁc questions which the work here raises. Remark 1.4. In view of Theorem 1.1, it is natural to wonder what can be said for the left-regular representation of L1 (G) on L2 (G) when G is not discrete. We leave this for future work, although—as noted above—in Section 3 we can apply some of our general results to the case where G is CCR.

2. Preliminaries etc. Notation and Other Conventions Notation. If X is a Banach space and A ⊆ B(X) is a subalgebra, we put A := {T + λI : T ∈ A, λ ∈ C}. Note that if A is closed in B(X) with respect to the norm topology, then so is A . Recall that if T : X → X is a bounded linear operator on a Banach space X, then the residual spectrum of T , which we denote by σr (T ), is the set of all λ ∈ σ(T ) such that T − λI is injective with closed range. The points of σ(T )\σr (T ) are ‘permanently singular’, in the following sense: if there exists a Banach space Y , into which X embeds as a closed subspace, together with S ∈ B(Y ) such that S(X) ⊆ X and S|X = T , then σ(T )\σr (T ) ⊆ σ(S). The following lemma is mostly just a translation of the definition (together with an application of the open mapping theorem). We state it as a lemma for later reference, and omit the proof which is straightforward. Lemma 2.1. Let X be a Banach space and A a subalgebra of B(X). Then the following are equivalent: (i) the pair (A, X) is surjunctive; (ii) whenever T ∈ A is such that T : X → X is injective with closed range, T is automatically surjective; (iii) whenever T ∈ A and T : X → X is injective but non-invertible, there exists a sequence (yn ) ⊂ X, such that yn ≥ 1 for all n while T yn → 0.

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Remark 2.2. It is condition (ii) of the preceding lemma which motivates our use of the word ‘surjunctive’. The term is borrowed from a notion from the theory of dynamical systems, introduced by Gottschalk [6] in the context of group actions on certain compact spaces. The corresponding notion for Banach spaces seems not to have been systematically considered. Various Background Remarks and Results In the definition of a surjunctive pair, we did not insist that A is a closed subalgebra (this gives us slightly greater ﬂexibility in the statements of our results). The following lemma shows that this is not an important distinction. Lemma 2.3. Let (A, X) be a surjunctive pair and let A be the closure of A in the norm topology of B(X). Then (A, X) is a surjunctive pair. Proof. Let T ∈ A be injective with closed range. By the open mapping theorem, there exists δ > 0 such that T (x) ≥ δx for all x ∈ X. Since A is norm-dense in A , there exists T1 ∈ A such that T − T1 ≤ δ/3. Then since T1 (x) ≥ (2δ/3)x for all x ∈ X, we see that T1 is injective with closed range; as (A, X) is a surjunctive pair, T1 is therefore invertible in B(X). Note that T1−1 ≤ 3(2δ)−1 , which yields T1−1 T − I ≤ T1−1 · T − T1 ≤

1 3 δ · = . 2δ 3 2

It follows that T1−1 T is invertible, whence T itself is invertible, as required. Remark 2.4. We shall see in Example 3.7 that in general, “norm closure” cannot be replaced by “closure in the strong operator topology”. Lemma 2.5. Let X be a Banach space, A ⊆ B(X) a closed subalgebra. If (A, A ), where we let A act on itself by left multiplication, is surjunctive, then so is (A, X). Proof. Let T ∈ A ⊆ B(X), and suppose that T : X → X is injective but not surjective. Let LT : A → A denote the operator S → T S. Injectivity of T implies that LT is injective. Moreover, LT is not surjective: for if it were, then since I ∈ A there would exist S ∈ A such that T S = I, and so T would be surjective, contrary to our original assumption. Since LT is injective but not surjective, and (A, A ) is a surjunctive pair, LT must have non-closed range; hence there exists a sequence (Sn ) ⊂ A such that Sn = 1 for all n while T Sn → 0. Choose a sequence (xn ) ⊂ X such n . Put yn = n+1 that xn = 1 and Sn xn ≥ n+1 n Sn xn ; then by construction, yn ≥ 1 for all n, while T yn ≤ 2T Sn → 0. Directly Finite Banach Algebras The notion of surjunctive pair is linked closely to the classical notion of a directly ﬁnite ring (sometimes called “Dedekind ﬁnite”, or even just “ﬁnite”). Definition 2.6. A ring R with identity is said to be directly finite if each left-invertible element of R is in fact invertible in R.

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Proposition 2.7. Let (A, X) be a surjunctive pair, where A is a unital subalgebra of B(X). Then A is directly finite. Proof. Let S, T ∈ A be such that ST = I. Since S T x ≥ x for all x ∈ X, we see that T is injective with closed range, and hence by the surjunctivity assumption is invertible. We shall see later that the converse is false: there exist directly ﬁnite, unital Banach algebras A for which the pair (A, A) is not surjunctive. However, if we let A act not on itself but on its dual space A∗ , then we have the following result. Theorem 2.8. Let A be a directly finite, unital Banach algebra. Regard A∗ as a left A-module in the natural way, and let ı : A → B(A∗ ) be the corresponding (isometric, unital) embedding. Then (ı(A), A∗ ) is a surjunctive pair. Proof. By Lemma 2.1, it sufﬁces to show that whenever ı(a) : A∗ → A∗ is injective with closed range, it is surjective. In fact, we do not need the condition of having closed range; injectivity is enough. For, since we may identify ı(a) : A∗ → A∗ with the adjoint of the map Ra : A → A; x → xa, when ı(a) is injective the Hahn-Banach theorem implies that Ra has dense range. Let G denote the group of invertible elements in A; since G is open, there exists u ∈ G and x ∈ A such that u = Ra (x) = xa. Hence 1A = u−1 xa, and so by direct ﬁniteness of A we must have a ∈ G. All this is relevant to our original problem, because of the following old result. Theorem 2.9 (Kaplansky). Let Γ be a discrete group and VN(Γ) its group von Neumann algebra. Then VN(Γ) is directly finite. Since direct ﬁniteness is obviously inherited by unital subalgebras, 1 (Γ) is therefore directly ﬁnite. Combining Theorem 2.8 with Remark 1.3, we conclude that the pair (1 (Γ), ∞ (Γ)) is surjunctive. This had already been observed in previous work of the author [2], which was another source of motivation for the present article. Indeed, the proof of Theorem 2.8 is just an abstract version of the argument of [2]. Remark 2.10. Kaplansky’s original proof of Theorem 2.9 relied on the basic theory of projections in von Neumann algebras. In [12] Montgomery gave a purely C∗ -algebraic proof that CΓ is directly ﬁnite. Inspection of her arguments shows that they extend to the larger algebra VN(Γ), although this seems not to have been written up explicitly in the literature.

3. Results for Directly Finite C∗ -Algebras We start with a small observation, which is purely algebraic. Lemma 3.1. Let A be a unital, directly finite algebra, and let x, y ∈ A. (i) If xy is invertible in A, then so are both x and y. (ii) σ(xy) = σ(yx).

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Proof. To prove (i): if there exists h ∈ A such that hxy = 1A , then y has a left inverse, and so—since A is directly ﬁnite—y is invertible in A. Hence x = xy · y −1 is the product of two invertible elements, so is itself invertible. In particular, if xy is invertible then so is yx. It is a standard, basic result that σ(xy)\{0} = σ(yx)\{0} in any algebra, and thus (ii) follows. The main work needed to prove Theorem 1.2 is contained in the following result, which to the author’s knowledge is new. Theorem 3.2. Let A be a unital, directly finite C∗ -algebra. Then (A, A) is surjunctive. Proof. Let a ∈ A and let La : A → A be given by La (x) = ax for x ∈ A. Suppose that La is injective but not surjective. We shall construct an explicit sequence (yn ) ⊂ A such that yn ≥ 1 for all n and ayn → 0. We may suppose, without loss of generality, that a = 1. Since 0 ∈ σ(a), Lemma 3.1 implies that 0 ∈ σ(a∗ a) ⊆ [0, 1]. Let (fn ) be a sequence in CR [0, 1], to be determined later. Put yn = fn (a∗ a). Then since 0 ∈ σ(a∗ a), we have yn = fn (a∗ a) = sup{|fn (λ)| : λ ∈ σ(a∗ a)} ≥ |fn (0)|; while, using the C∗ -identity and the continuous functional calculus, we obtain ayn = (ayn )∗ ayn 1/2 = fn (a∗ a)a∗ afn (a∗ a)1/2 = a∗ afn (a∗ a)2 1/2 =

sup λ∈σ(a∗ a)

|λ1/2 fn (λ)| ≤ sup λ1/2 |fn (λ)|. 0≤λ≤1

The idea is now clear: take fn to satisfy fn (0) = 1 but to be ‘small outside a small neighbourhood of zero’. For sake of definiteness, put fn (t) = (1 + nt1/2 )−1 : then yn ≥ |fn (0)| = 1 for all n, while λ1/2 1 → 0. = 1/2 1 + n 1 + nλ 0≤λ≤1

ayn ≤ sup This concludes the proof.

Proof of Theorem 1.2. The proof for the case X = C∗r (Γ) is immediate from Theorem 3.2, since C∗r (Γ) is a directly ﬁnite C∗ -algebra. It remains to treat the case of the noncommutative Lp -spaces. Thus, ﬁx 1 ≤ p ≤ ∞, and let X be the noncommutative Lp -space associated to VN(Γ) (see the appendix for the definition). We now make two observations: (i) there is an injective algebra homomorphism ı : VN(Γ) → B(X); (ii) ı has closed range, so that VN(Γ) may be identiﬁed with a closed unital subalgebra of B(X). These observations follow from some very basic noncommutative Lp -theory: as we have been unable to ﬁnd a precise and concise reference for either, we have included sketch proofs and pointers to the literature in the appendix. Note however that in the case X = 2 (Γ)–which, arguably, is the one of greatest interest here—both observations are tautologically true.

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Since VN(Γ) is a unital, directly ﬁnite C∗ -algebra, the pair (VN(Γ), VN(Γ)) is surjunctive by Theorem 3.2. As ı has closed range, by applying Lemma 2.5 we deduce that the pair (ı(VN(Γ)), X), and hence the pair (ı(CΓ), X), is surjunctive. Remark 3.3. It is not clear what can be said for the full group C∗ -algebra C∗ (Γ). In particular, it seems to be unknown if there exists a discrete group Γ such that C∗ (Γ) is not directly ﬁnite. We content ourselves with two remarks: (i) If Γ is amenable, then C∗ (Γ) = C∗r (Γ) is directly ﬁnite. by [1, Theorem 7], (ii) Let F2 denote the free group on n generators. Then ∞ there is an injective *-homomorphism C∗ (F2 ) → n=1 M2n (C); this ∗ larger algebra is directly ﬁnite, and therefore so is C (F2 ). We pause for some notation and definitions. If X is a Banach space, we denote by K(X) the algebra of all compact operators on X. A C∗ -algebra A is said to be CCR or liminal if, for every irreducible ∗-representation π : A → B(Hπ ), we have π(A) ⊆ K(Hπ ). The class of locally compact groups whose reduced C∗ -algebras are CCR has been intensively studied: it clearly includes all compact and all abelian groups, but also includes all connected Lie groups which are either semisimple, nilpotent, or real algebraic. For pointers to the rather large literature, see e.g. [5, Chapter 7]. If A is a C∗ -algebra, not necessarily having an identity element, let A denote its ‘conditional’ unitization with respect to some (hence any) faithful *-representation A → B(H). Theorem 3.4. Let A be a CCR C∗ -algebra. Then (A, A ) is a surjunctive pair. We have included this result here, since the proof goes via Theorem 3.2. The other ingredient in the proof is the following result of Halperin [7].

Theorem 3.5. K(X) is directly finite for any Banach space X. Proof of Theorem 3.4. Let 1 denote the identity of A . By Theorem 3.2, it sufﬁces to prove that A is directly ﬁnite. Suppose otherwise: then there exist a, b ∈ A such that ab = 1 = ba. Let ψ be a pure state on A such that ψ((ba − 1)∗ (ba − 1)) = 0 ,

(1)

and let πψ : A → B(Hψ ) be the corresponding GNS representation. Then πψ is an irreducible *-representation of A (see e.g. [10, Theorem 10.2.3]) and it is easily checked that it is an irreducible *-representation of A; since A is liminal, we have πψ (A ) ⊆ K(Hψ ) . We have πψ (a)πψ (b) = πψ (1) = IHψ , while it follows from (1) and the GNS construction that πψ (b)πψ (a) − IHψ = πψ (ba − 1) = 0. But this contradicts Theorem 3.5, and therefore A must be directly ﬁnite. Going back to Halperin’s result, we can in fact prove something a little stronger, using the same kind of basic spectral theory as is implicit in [7].

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Theorem 3.6. The pair (K(X), X) is surjunctive. Proof. If X is ﬁnite-dimensional, the result is trivially true (all injective endomorphisms of a ﬁnite-dimensional vector space are surjective). Thus we may assume, without loss of generality, that X is inﬁnitedimensional. Let T = λI + K ∈ K(X), where λ ∈ C and K is a compact operator. Suppose that T is injective and non-invertible. Then −λ ∈ σ(K); moreover, since −λ is by assumption not an eigenvalue of K, standard spectral theory for compact operators (see e.g. [14, Theorem 4.25]) forces λ = 0. Since X is inﬁnite-dimensional and K = T is injective, the range of K is inﬁnite-dimensional, and so (by the open mapping theorem) cannot be closed in X. Example 3.7. We can now give the example promised earlier in Remark 2.4. Let X be a Banach space with the approximation property, containing a proper closed subspace isomorphic to itself. For instance, any inﬁnite-dimensional p -space will sufﬁce. We have just seen that (K(X), X) is surjunctive. Now, since X has the approximation property, the closure of K(X) in the strong operator topology is all of B(X). Since X contains a proper isomorphic copy of itself, (B(X), X) is evidently not surjunctive. Remark 3.8. In Theorem 3.4, we cannot relax “CCR” to “GCR”. For instance, the Toeplitz algebra is GCR, but (since it contains nonunitary isometries) is not even directly ﬁnite, and hence by Theorem 3.2 it cannot arise in any surjunctive pair.

4. The Case of p (Γ) for p = 2 Given the results obtained so far, it is tempting to wonder if, whenever A is a unital and directly ﬁnite subalgebra of B(X), the pair (A, X) is surjunctive. The following example, which is a special case of an old construction due to Willis, shows that this is not the case. Proposition 4.1 (see [16, Theorem 2.2]). Let Γ be a discrete group, which contains two elements a and b that generate a free subgroup; let ta and tb be complex scalars of modulus 1, and put x = δe + ta δe + tb δb ∈ CΓ. Let Lx : p (Γ) → p (Γ) denote the left convolution operator. Then: (i) if 1 ≤ p < 2, then Lx has a left inverse in B(p (Γ)), and in particular is injective with closed range; / Lx (1 (Γ)). (ii) if 1 + ta + tb = 0, then δe ∈ Corollary 4.2. If Γ contains a copy of F2 , then (CΓ, 1 (Γ)) is not a surjunctive pair. We are thus faced with the following question: for which groups Γ is the pair (CΓ, 1 (Γ)) surjunctive? Note that by Runde’s result (Theorem 1.1 above), all discrete Moore groups have this property; while Willis’ result shows that any discrete group containing F2 does not have this property. This suggests that amenability may be the distinguishing characteristic; while we

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have not been able to conﬁrm or disprove this guess, we do at least have the following theorem. Theorem 4.3. Let Γ be a discrete amenable group, let 1 ≤ p < ∞, and let a ∈ 1 (Γ). Let CVp (Γ) denote the subalgebra of B(p (Γ)) consisting of all operators that commute with right translations. Suppose that La : p (Γ) → p (Γ) is injective and has complemented range. Then La is invertible in CVp (Γ), and in particular is surjective. Part of the proof uses the following standard property of modules over amenable algebras, whose proof we omit. See [3, Theorem 2.3] for a fairly direct argument. Lemma 4.4. Let A be an amenable Banach algebra, let Y be a right Banach A-module, and V a closed A-submodule of Y . Suppose that (i) there exists a bounded linear projection of Y onto V ; (ii) V is a dual Banach A-module (that is, there exists a left Banach A-module X and a continuous A-module isomorphism V ∼ = X ∗ ). Then there exists a bounded linear projection P : Y → V which is also a right A-module map. It is also convenient to use the following notation: given c ∈ p (Γ), we denote by Lc the bounded operator 1 (Γ) → p (Γ) that is given by left convolution with c. Proof of Theorem 4.3. We start by noting that p (Γ), equipped with the usual right action of Γ, is a dual Banach 1 -module. Let V = La (p (Γ)): by hypothesis this is a closed, right 1 (Γ)-submodule of p (Γ), and it is a dual module, since La deﬁnes a module isomorphism between V and p (Γ). Also, by hypothesis there exists a bounded linear projection of p (Γ) onto V ; hence on applying Lemma 4.4, there exists a bounded linear projection P from p (Γ) onto the closed subspace V , satisfying P (η ∗ b) = P (η) ∗ b

for all η ∈ p (Γ) and all b ∈ 1 (Γ).

In particular, for each η ∈ 1 (Γ), we have P (η) = P (δe ) ∗ η. Since P (δe ) ∈ V = La (p (Γ)), there exists a unique ξ ∈ p (Γ) such that a ∗ ξ = P (δe ). Then a ∗ ξ ∗ a = P (δe ) ∗ a = a, and so by injectivity of La we have δe = ξ ∗ a. We now consider the cases p = 1 and 1 < p < ∞ separately. If p = 1, then ξ ∈ 1 (Γ), so that a is left-invertible in the algebra 1 (Γ); since this algebra is directly ﬁnite, a is invertible in 1 (Γ) with inverse ξ. For the remainder of this proof, we restrict attention to the case 1 < p < ∞. Here, the idea is similar to the case p = 1, but since we only know at the outset that ξ ∈ p (Γ), we need to work harder (and obtain a weaker result). The ﬁrst step is to note that Lξ ∈ CVp (Γ): for, the assumption on a implies there exists δ > 0 such that a ∗ ψp ≥ δψp

for all ψ ∈ p (Γ),

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and in particular, since for every η ∈ 1 (Γ) we have a ∗ ξ ∗ η = P (δe ) ∗ η = P (η), we ﬁnd that ξ ∗ ηp ≤ δ −1 a ∗ ξ ∗ ηp ≤ δ −1 P (η)p ≤ δ −1 P ηp , as claimed. Since Γ is amenable, a result of Herz tells us that there exists Cp > 0 such that T η2 ≤ Cp T p →p η2

for all η ∈ c00 (Γ) and all T ∈ CVp (Γ)

(see [8, Theorem C] and [9, Theorem 5] for details). Each operator in CVp (Γ) can therefore be identiﬁed with a unique operator on 2 (Γ), which, since it commutes with right translations, will be an element of the group von Neumann algebra VN(Γ). This gives us an injective, unital algebra homomorphism from CVp (Γ) into VN(Γ), and since the latter is directly ﬁnite we see that CVp (Γ) is also directly ﬁnite. Since La Lξ = I in CVp (Γ), we conclude that La is invertible in CVp (Γ) with inverse Lξ .

5. Closing Questions and Further Work While the preceding results give complete answers for the regular representation of a discrete group on its 2 -space, we have seen that the problem of determining surjunctivity of (CΓ, p (Γ)) remains unresolved for p = 2 and non-Moore groups. We therefore close with a list of questions which this work has raised, and which we hope may stimulate further investigation. Question 5.1. Let 1 < p < ∞, p = 2. Is (CF2 , p (F2 )) surjunctive? Question 5.2. Let H3 (Z) denote the 3-dimensional, integer Heisenberg group. Is (CH3 (Z), 1 (H3 (Z))) surjunctive? What about p (H3 (Z)) for 1 < p < ∞, p = 2?1 There are several reasons for considering H3 (Z). Firstly, as a two-step nilpotent group it is perhaps the next step into noncommutativity after Moore groups (by results of Thoma, every discrete Moore group has an abelian normal subgroup of ﬁnite index, see [5, Theorem 7.8]). Secondly, in view of the role played by spectral arguments and functional calculus in Section 3, it may be relevant that 1 (H3 (Z)) admits a C k -functional calculus for some k < ∞, by results of Dixmier; however, it is not clear if this can be used to construct approximate eigenvectors, as done for the case of the C∗ -norm. Question 5.3. The result of Runde that was mentioned in the introduction implies, as a very special case, that (1 (Γ), 1 (Γ)) is surjunctive whenever Γ is a discrete Moore group. Is there a more direct proof of this? 1 Since

completing the work in this article, we have learned that both parts of Question 5.2 have a positive answer. This follows from recent, independent work of R. Tessera, see “Left inverses of matrices with polynomial decay”, arXiv 0801.1532v4, Corollary 1.9. The case of discrete solvable groups remains open.

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For instance, if Γ is abelian then surjunctivity can be proved easily by considering the Gelfand transform of 1 (Γ) and noting that all its maximal ideals have bounded approximate identities. It would be interesting to see an analogous argument for Moore groups, which did not require the full machinery deployed in Runde’s article. Acknowledgements The author thanks Matthew Daws for comments on an early version of this article, and for helpful conversations regarding the articles [8,9].

Appendix A. The Action of VN(Γ) on its Noncommutative Lp -Spaces In this short appendix we collect, for convenience, the definition and some basic properties of the noncommutative Lp -spaces associated to a group von Neumann algebra. Rather than refer the reader to sources that treat such constructions in full generality and sophistication, we shall present the background needed to justify the statements in the proof of Theorem 1.2. For more details see [4, §§2–3] or [13, §1]. Throughout this section M is a ﬁxed von Neumann algebra, equipped with a faithful, normal, finite tracial state τ . (In the case M = VN(Γ), the trace τ is the canonical one given by T → T δe , δe .) Definition A.1. Let x ∈ M. For 1 ≤ p < ∞ and x ∈ M, put x(p) := τ ((x∗ x)p/2 )1/p

(2)

and for p = ∞ put x(∞) = x. Then ·(p) deﬁnes a norm on M, and we denote the completion of (M, ·(p) ) by Lp (M, τ ). Remark A.2. It is worth identifying Lp (M, τ ) in the cases p = 1, 2 and ∞. By definition, L∞ (M, τ ) ≡ M. It is also clear from the definition that L2 (M, τ ) is nothing but the Hilbert space associated to M by the GNS construction for the state τ . Finally, it turns out that elements of L1 (M, τ ) correspond to ultraweakly continuous functionals on M, and hence L1 (M, τ ) is naturally identiﬁed with the isometric predual M∗ . (See e.g. [4, Section 2, Th´eor`eme 5].) When M = VN(Γ) and τ is the canonical trace T → T δe , δe , we can therefore identify L2 (VN(Γ), τ ) with 2 (Γ) and L1 (VN(Γ), τ ) with the Fourier algebra A(Γ). Lemma A.3. Let 1 ≤ p ≤ ∞. Then ax(p) ≤ a(∞) x(p) for all a, x ∈ M. Consequently, the left action of M on itself extends to give a continuous left action of M on Lp (M, τ ), and hence a continuous algebra homomorphism ıp : M → B(Lp (M, τ )). We refer to [4] for the proof. (Brieﬂy: for p = ∞ the claim is tautologous, and for p = 1 it is [4, Th´eor`eme 4]. The general case follows from [4, p. 26, Corollaire 3].)

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The following lemma is surely well known to specialists, but the author does not know of any explicit reference for it. Lemma A.4. Let a ∈ M and let 1 ≤ p ≤ ∞. Let C be the closed unital subalgebra of M generated by a∗ a. Then a ≥ sup{ax(p) : x ∈ C, x(p) ≤ 1}. In particular, the embedding ıp : M → B(Lp (M, τ )) has closed range. As the proof needs only basic C∗ -algebraic tools, we give a quick sketch. Without loss of generality, assume that 1 ≤ p < ∞ and that a = a∗ a = 1. Identify C with C(σ(a∗ a)) and let μ be the probability measure on σ(a∗ a) that corresponds to τ |C . Let ε > 0; then by duality and basic measure theory, there exists a continuous positive function f : σ(a∗ a) → [0, ∞) such that f (t) dμ(t) = 1 and ε + tp/2 f (t) dμ(t) ≥ sup |tp/2 | = 1. σ(a∗ a)

σ(a∗ a)

t∈σ(a∗ a)

Put x = f (a∗ a)1/p = f 1/p (a∗ a), and observe that xp(p) = τ ((x∗ x)p/2 ) = τ (f (a∗ a)) = 1 , while, since x = x∗ commutes with a∗ a, axp(p) = τ ((x∗ a∗ ax)p/2 ) = τ ((a∗ ax∗ x)p/2 ) = τ ((a∗ a)p/2 f (a∗ a)) ≥ 1 − ε. Since ε was arbitrary, the result follows.

References [1] Choi, M.D.: The full C ∗ -algebra of the free group on two generators. Pac. J. Math. 87(1), 41–48 (1980) [2] Choi, Y.: Injective convolution operators on ∞ (Γ) are surjective. Can. Math. Bull. (in press) http://arXiv.org/abs/math.FA/0606367 [3] Curtis, P.C. Jr., Loy, R.J.: The structure of amenable Banach algebras. J. Lond. Math. Soc. (2) 40(1), 89–104 (1989) [4] Dixmier, J.: Formes lin´eaires sur un anneau d’op´erateurs. Bull. Soc. Math. France 81, 9–39 (1953) [5] Folland, G.B.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995) [6] Gottschalk, W.: Some general dynamical notions. In: Beck, A. (ed.) Recent Advances in Topological Dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., vol. 318, pp. 120–125. Springer, Berlin (1973) [7] Halperin, I.: On a theorem of Sterling Berberian. C. R. Math. Rep. Acad. Sci. Can. 3(1), 33–35 (1981) [8] Herz, C.: The theory of p-spaces with an application to convolution operators. Trans. Am. Math. Soc. 154, 69–82 (1971) [9] Herz, C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23(3), 91–123 (1973)

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[10] Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. II. Pure and Applied Mathematics, vol. 100. Academic Press, Orlando (1986) [11] Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series, vol. 20. The Clarendon Press, Oxford University Press, New York (2000) [12] Montgomery, M.S.: Left and right inverses in group algebras. Bull. Am. Math. Soc. 75, 539–540 (1969) [13] Pisier, G., Xu, Q.: Non-commutative Lp -spaces. In: Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. North-Holland, Amsterdam (2003) [14] Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill, New York (1991) [15] Runde, V.: Local spectral properties of convolution operators on non-abelian groups. Proc. Edinb. Math. Soc. (2) 39(1), 237–250 (1996) [16] Willis, G.A.: Translation invariant functionals on Lp (G) when G is not amenable. J. Aust. Math. Soc. Ser. A 41(2), 237–250 (1986) Y. Choi D´epartement de math´ematiques et de statistique Pavillon Alexandre-Vachon Universit´e Laval Qu´ebec, QC G1V 0A6 Canada e-mail: [email protected] Received: July 7, 2009. Revised: December 17, 2009.

Integr. Equ. Oper. Theory 67 (2010), 109–121 DOI 10.1007/s00020-010-1773-z Published online March 20, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Positive Decompositions of Selfadjoint Operators Guillermina Fongi and Alejandra Maestripieri Abstract. Given a linear bounded selfadjoint operator a on a complex separable Hilbert space H, we study the decompositions of a as a difference of two positive operators whose ranges satisfy an angle condition. These decompositions are related to the canonical decompositions of the indefinite metric space (H, , a ), associated to a. As an application, we characterize the orbit of congruence of a in terms of its positive decompositions. Mathematics Subject Classification (2000). 47B15, 58B20. Keywords. Selfadjoint operators, congruence of operators, indefinite metric spaces.

1. Introduction Consider L(H) the algebra of linear bounded operators of a complex separable Hilbert space H. Denote by GL(H) the group of invertible operators of L(H). It is well known that a selfadjoint operator a ∈ L(H) can be written as a difference of two positive operators with orthogonal ranges and these operators are uniquely determined by a. The main purpose of this article is to study alternative decompositions of a selfadjoint operator a as a difference of two positive operators whose ranges are not necessarily orthogonal, but satisfy an angle condition. On the other hand, each selfadjoint operator a deﬁnes the following indefinite inner product on H: x, ya = ax, y,

for x, y ∈ H.

If a is also invertible, then (H, , a ) is a Krein space and the canonical decompositions of this space, as a direct orthogonal sum of an a-positive and an a-negative subspaces are described, for example, in the classical books by Bognar [3] and Azizov and Iokhvidov [2]. More generally, for any selfadjoint operator a ∈ L(H), it is possible to associate to every canonical decomposition This work was completed with the support of ANPCyT PICT 2007-00808.

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of (H, , a ) (in this case, as the sum of three subspaces, an a-positive, an a-negative and the nullspace of a), an a-selfadjoint oblique projection with a-nonnegative range and a-nonpositive nullspace, or equivalently, an a-positive reﬂection. We study the relationship between the positive decompositions of a and the canonical decompositions of the indefinite inner product space (H, , a ). We prove that there is a one to one correspondence between the positive decompositions of a and the a-positive reﬂections, when a is injective. As an application, we study the orbit of congruence of a selfadjoint operator a, i.e., the set Oa = {gag ∗ : g ∈ GL(H)}. If a has closed range, it is possible to provide Oa with a structure of differentiable manifold (see [5,6,10]). Moreover, it holds that (GL(H), Oa , πa ) is a ﬁbre bundle, where πa (g) = gag ∗ , for g ∈ GL(H). In this paper we characterize the orbit of a in terms of the positive decompositions of a. When a has closed range, we show that the set of positive decompositions of a is parametrized by the elements of the isotropy group of a, i.e., the set Ia = {g ∈ GL(H) : gag ∗ = a}. The article is organized as follows: Sect. 2 contains some basic results about angles between closed subspaces and a brief survey about equivalence and congruence of operators. In Sect. 3, we collect some definitions and properties of the indefinite metric space (H, , a ), for a selfadjoint operator a. If a is positive and Ha denotes the completion of (H, , a ), then we characterize the operators in L(H) which admit bounded extentions to Ha . Section 4 is devoted to study decompositions of a selfadjoint operator a as a difference of two positive operators such that the minimal angle of their ranges is positive. Any of these decompositions will be called a positive decomposition. We relate the positive decompositions of a to the canonical decompositions of the indefinite metric space (H, , a ). More precisely, we prove that given a positive decomposition of a, there is an associated a-positive reﬂection. Conversely, an a-positive reﬂection determines a positive decomposition of a. If a is injective, we show that there is a bijection between the positive decompositions of a and the set of a-selfadjoint projections with a-positive range and a-negative nullspace. We prove that every positive decomposition of a induces a “pseudo polar decomposition” of a: i.e., a factorization of a as a = αw, where α is positive and w is an a-positive reﬂection. If a = ua |a| is the polar decomposition of a, the a-positive reﬂections w are those of the form w = ua d, where d is |a|-positive (in fact, this turns out to be the polar decomposition of w in the space (H, , |a| )). Finally, if a is injective, given two canonical decompositions of (H, , a ), we prove that the a-positive subspaces and the a-negative subspaces have the same dimension, respectively. In the last section, we characterize the set of congruence of a ﬁxed selfadjoint operator a. It is known that two positive operators are congruent if and only if their ranges are unitarily equivalent. We generalize this fact for selfadjoint operators, by means of their positive decompositions. Also, if a = a1 − a2 is the positive orthogonal decomposition (p.o.d.) of a and g ∈ Ia , it holds that a = ga1 g ∗ − ga2 g ∗ is a positive decomposition of a. When a has closed range, we show that all the positive decompositions of a can be written as a = ga1 g ∗ − ga2 g ∗ , for some g ∈ Ia .

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2. Preliminaries Let L(H) be the algebra of linear bounded operators of a complex separable Hilbert space H, GL(H) the group of invertible operators of L(H) and U(H) the subgroup of GL(H) of unitary operators. Denote by L(H)s the set of selfadjoint operators and L(H)+ the cone of positive operators. An operator v ∈ L(H) is a reflection if v = v −1 and v is symmetry if it is a selfadjoint reﬂection. ˙ Given M and N two closed subspaces of H, then M+N denotes the ˙ direct sum of M and N , and M ⊕ N the orthogonal sum. If M+N = H, we denote by pM//N the oblique projection with range M and nullspace N and pM = pM//M⊥ . Let Q = {q ∈ L(H), q 2 = q} be the set of oblique projections. For every a ∈ L(H), R(a) denotes the range of a, N (a) its nullspace and pa = pR(a) . If a ∈ L(H), we ﬁx the following polar decomposition of a: a = va |a| where |a| = (a∗ a)1/2 is positive and va is a partial isometry from N (a)⊥ onto R(a) with nullspace N (va ) = N (a). If a is selfadjoint, the isometric part of the polar decomposition can be deﬁned to obtain a symmetry: in this case R(a)⊥ = N (a) so that ua = va + pN (a) is a symmetry and a = ua |a| = |a|ua . The following result due to Douglas [8], characterizes the operator ranges inclusion: Theorem 2.1. Consider a, b ∈ L(H). Then R(a) ⊆ R(b) if and only if a = bc, for some c ∈ L(H). Given M and N two closed subspaces of H, the angle or Friedrichs angle between M and N is the angle α(M, N ) ∈ [0, π/2] whose cosine is given by c(M, N ) = sup{|x, y| : x ∈ M ∩ (M ∩ N )⊥ , x ≤ 1, y ∈ N ∩ (M ∩ N )⊥ , y ≤ 1}. The minimal angle or Dixmier angle between M and N is the angle α0 (M, N ) ∈ [0, π/2] whose cosine is given by c0 (M, N ) = sup{|x, y| : x ∈ M, x ≤ 1, x ∈ N , y ≤ 1}. Observe that 0 ≤ c(M, N ) ≤ c0 (M, N ) ≤ 1. The next results about angles can be found in [7]: Theorem 2.2. The following statements are equivalent: 1. c0 (M, N ) < 1, 2. M ∩ N = {0} and M + N is closed. Theorem 2.3. The following statements are equivalent: 1. c(M, N ) < 1, 2. M + N is closed, 3. M⊥ + N ⊥ is closed. Two operator ranges R and S are similar if there exists g ∈ GL(H) such that R = g(S) and unitarily equivalent if g can be taken to be unitary. Operator ranges are similar if and only if they are unitarily equivalent.

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The operators a, b ∈ L(H) are equivalent if there exist g, f ∈ GL(H) such that b = gaf ; the operators a and b are congruent if there exists g ∈ GL(H) such that b = gag ∗ . Proposition 2.4. Normal operators are equivalent if and only if their ranges are unitarily equivalent. Proposition 2.5. Let a, b ∈ L(H)+ , then a and b are equivalent if and only if a and b are congruent. See [9] for the proofs of these facts.

3. The Indefinite Metric Associated to a Selfadjoint Operator Along this paper, we consider a ﬁxed a ∈ L(H)s and the indefinite metric on H induced by a, given by x, ya = ax, y,

x, y ∈ H.

In the following paragraphs, we recall some notions of indefinite inner product spaces. We refer to the classical books of Bognar [3] and Azizov and Iokhvidov [2]. An element x ∈ H is a-positive, a-negative or a-neutral, respect to the indefinite metric , a , if x, xa > 0, x, xa < 0 or x, xa = 0, respectively. The element x is called a-nonnegative (respectively, a-nonpositive), if x is a-positive or a-neutral (respectively, a-negative or a-neutral). A subspace S of H is called a-positive, a-negative or a-neutral if each nonzero element of S is a-positive, a-negative or a-neutral, respectively. Given c ∈ L(H), an operator d ∈ L(H) is an a-adjoint of c if cx, ya = x, dya for all x, y ∈ H; or equivalently ac = d∗ a. Observe that an operator c may admit many, only one or no a-adjoint, depending on whether the equation c∗ a = ah has many, only one or no solution, respectively. By Douglas’ Theorem, this equation has a solution if and only if R(c∗ a) ⊆ R(a). The operator c ∈ L(H) is a-selfadjoint if ac = c∗ a and it is a-positive if cx, xa ≥ 0 for all x ∈ H, or equivalently, ac ∈ L(H)+ . The operator c is an a-expansion (respectively, a-contraction) if cx, cxa ≥ x, xa (respectively, cx, cxa ≤ x, xa ); or equivalently c∗ ac ≥ a (respectively, c∗ ac ≤ a). Given x, y ∈ H, then x and y are a-orthogonal if x, ya = 0. In this case, we write x ⊥a y. Given a subspace S of H, the a-orthogonal subspace of S with respect to the indefinite metric , a is the set S ⊥a = {x ∈ H : x, ya = 0, ∀y ∈ S}. It is easy to see that S ⊥a = a−1 (S ⊥ ) = a(S)⊥ . Observe that S ∩ S ⊥a is not necessarily zero. If S1 , S2 ⊆ S, then S1 ⊕a S2 = S denotes S1 + S2 = S, S1 ∩ S2 = {0} and x, ya = 0 for all x ∈ S1 , y ∈ S2 . Observe that if q ∈ Q, then q is a-selfadjoint if and only if R(q) and N (q) are a-orthogonal.

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A canonical decomposition of (H, , a ) is a decomposition of H as a direct sum H = N (a) ⊕a S + ⊕a S − , +

(1) −

where S is an a-positive closed subspace of H and S is an a-negative closed subspace of H. In particular, if a ∈ L(H)s is injective, a canonical decomposition of H is a decomposition of H as a direct sum H = S ⊕a S ⊥a , where S is a closed subspace of H such that S is a-positive and S ⊥a is a-negative. In this case, each canonical decomposition deﬁnes the projection q = PS//S ⊥a , or equivalently, the reﬂection w = 2q − 1. Observe that q is a-selfadjoint, R(q) = S is a-positive and N (q) = Sa⊥ is a-negative. Conversely, every q ∈ Q, a-selfadjoint, such that R(q) is a-positive and N (q) is a-negative, deﬁnes a canonical decomposition of (H, , a ). Hassi and Norstr¨ om proved that given q ∈ Q, q is an a-expansion (a-contraction) if and only if q is a-selfadjoint and N (q) is a-nonpositive (a-nonnegative) [15, Proposition 5]. The following lemma characterizes those reﬂections associated to canonical decompositions and it is a generalization of [17, Lemma 5.6]: Lemma 3.1. Let q ∈ Q. Then, q is a-selfadjoint, R(q) is a-nonnegative and N (q) is a-nonpositive if and only if the reflection w = 2q − 1 is a-positive. Proof. If q is an a-selfadjoint projection then aq = q ∗ aq, so that if w = 2q − 1 and x ∈ H, wx, x a = qx, qx a − (1 − q)x, (1 − q)x a ≥ 0,

(2)

because R(q) is a-nonnegative and N (q) is a-nonpositive. Thus, w is apositive. Conversely, if w is a-positive then w is a-selfadjoint. Therefore, q = w+1 2 is a-selfadjoint. By (2), if x ∈ R(q), then qx, qx a = wx, x a ≥ 0; so that R(q) is a-nonnegative. In a similar way, N (q) is a-nonpositive. The next proposition gives a characterization of an a-positive reﬂection, in terms of its polar decomposition in (H, , |a| ). Proposition 3.2. Let a = ua |a| be the polar decomposition of a. A reflection w is a-positive if and only if w admits a polar decomposition in (H, , |a| ) given by w = ua d, where d ∈ GL(H) is |a|-positive. Proof. Suppose that w = 2q−1 is an a-positive reﬂection, then aw = |a|ua w ∈ L(H)+ . Therefore, d = ua w ∈ GL(H) is |a|-positive. Then w = ua d, where d is |a|-positive. Since |a|ua = ua |a| = u∗a |a|, then ua is |a|-unitary and w = ua d is the polar decomposition of w in (H, , |a| ). Conversely, suppose that w = ua d, where d is |a|-positive. Then aw = |a|d is positive. When a is positive, the indefinite form , a deﬁnes a semi-inner product on H, and the associated semi-norm, a , is given by xa = x, xa1/2 = a1/2 x,

x ∈ H.

The quotient space (H/N (a), | |a ) is a normed space, where | |a is the associated quotient norm and |x|a = xa , where x = x + N (a), x ∈ H.

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Since (H/N (a), | |a ) is not necessarily complete, denote by Ha the completion of (H/N (a), | |a ). Denote by Π : H → Ha the quotient map. In the context of [4], the pair (Ha , Π) is called a Hilbert space induced by a. In particular, if a is injective, the indefinite form , a deﬁnes an inner product on H. In particular, if a ∈ GL(H)+ , then (H, , a ) is a Hilbert space and the norms and a are equivalent. Remark 3.3. The operator c ∈ L(H) admits an |a|-adjoint if and only if c admits an a-adjoint. In fact, |a|c = f |a| if and only if ac = ua f |a| = ua f ua a, where f ∈ L(H). For c ∈ L(H), such that c admits an a-adjoint, deﬁne c the associated x = cx. Observe that c is well deﬁned: if x, y ∈ x, operator to c on H/N (a) by c¯ then cx = cy, or equivalently, if x − y ∈ N (a) then c(x − y) ∈ N (a). In fact, since ac = d∗ a for some d ∈ L(H), then c(N (a)) ⊆ N (a). Proposition 3.4. If c ∈ L(H) admits an a-adjoint, then c, the associated operator on H/N (a), is well defined and admits a unique bounded extension to H|a| . Proof. Since c admits an a-adjoint, then by Remark 3.3, c admits an |a|adjoint. As we prove above, in this case, c is well deﬁned on H/N (a). From |a|c = ua d∗ ua |a| and [16, Theorem 5.1], there exists h ∈ L(H) such that |a|1/2 c = h|a|1/2 . Since |c¯ x||a| = cx|a| = |a|1/2 cx = h|a|1/2 x ≤ h|x||a| and H/N (a) is dense in H|a| , then c admits a unique bounded extension to H|a| . The above result is similar to [4, Theorem 3.1]. See also [1, Proposition 1.2]. Notice that the a-adjoint of c ∈ L(H) is unique on H|a| , as expected, since H|a| is a Hilbert space. In fact, given d, h ∈ L(H) such that ac = d∗ a = h∗ a, then ad = ah. Therefore, R(d − h) ⊆ N (a), so that h = d.

4. Positive Decompositions In this section we study the decompositions of a selfadjoint operator a as a suitable (in some sense we will establish) difference of two positive operators and the relation of these decompositions with the canonical decompositions of the space (H, , a ), deﬁned in (1). Definition 4.1. Given a ∈ L(H)s and c1 , c2 ∈ L(H)+ , then a = c1 − c2 is a positive decomposition of a if c0 (R(c1 ), R(c2 )) < 1. It is well known that every a ∈ L(H)s admits a unique positive decomposition a = a1 − a2 such that the ranges of a1 and a2 are orthogonal, or equivalently, such that c0 (R(a1 ), R(a2 )) = 0. In fact, consider a1 = |a|+a 2 and a2 = |a|−a . This decomposition will be called the p.o.d. In this case, the 2 operator a1 is the positive part of a, and −a2 its negative part. By Theorems 2.2 and 2.3, the condition c0 (R(c1 ), R(c2 )) < 1 is equivalent to c(N (c1 ), N (c2 )) < 1 and R(c1 )∩R(c2 ) = {0}. If a = c1 −c2 is a positive

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decomposition of a, then N (c1 ) + N (c2 ) is closed and N (c1 ) ∩ N (c2 ) = N (a). Also (N (c1 ) + N (c2 ))⊥ = R(c1 ) ∩ R(c2 ) = {0}, then N (c1 ) + N (c2 ) = H. Lemma 4.2. Consider a = c1 − c2 with c1 , c2 ∈ L(H)+ . Then a = c1 − c2 ˙ is a positive decomposition of a if and only if R(c1 )+R(c 2 ) = R(a). In this ˙ ). In particular, R(a) is closed if and only if R(ci ) case, R(a) = R(c1 )+R(c 2 is closed, for i = 1, 2. Proof. By definition, a = c1 − c2 is a positive decomposition of a if and only if c0 (R(c1 ), R(c2 )) < 1. Then, by Theorem 2.2, R(c1 ) ∩ R(c2 ) = {0} and ⊥ ˙ ˙ R(c1 )+R(c = N (c1 ) ∩ N (c2 ) = 2 ) is closed. Observe that (R(c1 )+R(c 2 )) ˙ N (a). Therefore, R(c1 )+R(c 2 ) = R(a). The converse follows by Theorem 2.2. Suppose that c0 (R(c1 ), R(c2 )) < 1, then the subspace R(c2 ) ⊕ N (a) is ˙ ˙ (a). Consider the oblique projection p1 = closed and H = R(c1 )+R(c 2 )+N pR(c1 )//R(c2 )⊕N (a) ∈ L(H), then p1 a = c1 = ap∗1 so that R(c1 ) ⊆ R(a). In the ˙ same way, R(c2 ) ⊆ R(a). Hence R(c1 )+R(c 2 ) ⊆ R(a). But from a = c1 − c2 , it follows that R(a) ⊆ R(c1 ) + R(c2 ). Remark 4.3. If a = c1 − c2 is a positive decomposition of a, denote by p1 = pR(c1 )//R(c2 )⊕N (a) and p2 = pR(c2 )//R(c1 )⊕N (a) . Since p1 p2 = p2 p1 = 0 then ˙ p1 + p2 ∈ Q and R(p1 + p2 ) = R(c1 )+R(c 2 ) = R(a). Also N (p1 + p2 ) = N (p1 ) ∩ N (p2 ) = N (a). Therefore p1 + p2 = pa . From now on, given a = c1 − c2 a positive decomposition of a, we consider q = p∗1 with p1 = pR(c1 )//R(c2 )⊕N (a)

and

w = 2q − 1.

(3)

Note that q = pN (c2 )∩R(a)//N (c1 ) ∈ Q and w is a reﬂection. The next theorem shows the relation between positive decompositions and a-positive reﬂections. Theorem 4.4. If a = c1 − c2 is a positive decomposition of a, consider w as in (3). Then w is a-positive. Conversely, given an a-positive reflection w, ∈ Q, c1 = aq and c2 = a(q − 1), then a = c1 − c2 is a consider q = w+1 2 positive decomposition of a. Proof. Suppose that a = c1 − c2 is a positive decomposition of a and let q = p∗1 as in (3). Then aq = c1 = q ∗ a, so that q is a-selfadjoint. If x ∈ R(q), then x, xa = aqx, x = c1 x, x ≥ 0, because c1 is positive; so that R(q) is a-nonnegative. In a similar way, N (q) is a-nonpositive, because a(1−q) = −c2 . By Lemma 3.1, w = 2q − 1 is a-positive. Conversely, let w be an a-positive reﬂection and consider q = w+1 2 , c1 = aq and c2 = a(q − 1), then a = c1 − c2 . Note that c1 , c2 ∈ L(H)+ : in fact, if x ∈ H, c1 x, x = aqx, qx + (1 − q)x = aqx, qx ≥ 0, because, by Lemma 3.1, R(q) is a-nonnegative. Similarly for c2 . Observe that R(c1 ) ⊆ R(q ∗ ) and R(c2 ) ⊆ N (q ∗ ), since c1 = q ∗ a and c2 = (q ∗ − 1)a. Therefore, c0 (R(c1 ), R(c2 )) ≤ c0 (R(q ∗ ), N (q ∗ )) < 1; so that a = c1 − c2 is a positive decomposition of a. In particular, if a is injective, we obtain the following correspondence. In case a is not injective, we can consider the operator a ˜ = a|R(a) ∈ L(R(a)).

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Corollary 4.5. Consider a injective. For a = c1 − c2 a positive decomposition of a, define φ(c1 , c2 ) = 2pN (c2 )//N (c1 ) − 1. Then φ is a bijection from the set of positive decompositions of a onto the set of a-positive reflections. Proof. Let a = c1 −c2 be a positive decomposition of a and let q = φ(c1 , c2 ) = 2pN (c2 )//N (c1 ) − 1. By Theorem 4.4, φ(c1 , c2 ) is an a-positive reﬂection. To see that φ is a bijection, consider w an a-positive reﬂection. Deﬁne ϕ(w) = w−1 w+1 w−1 (a( w+1 2 ), a( 2 )). By Theorem 4.4, if c1 = a( 2 ) and c2 = a( 2 ), then w+1 a = c1 − c2 is a positive decomposition of a. Let q = 2 , then φ(ϕ(w)) = φ(aq, a(q − 1)) = w, since N (a(q − 1)) = N (q − 1) = R(q) and N (aq) = N (q) because a is injective. Moreover, if a = c1 −c2 is a positive decomposition of a, then ϕ(φ(c1 , c2 )) = (apN (c2 )//N (c1 ) , a(pN (c2 )//N (c1 ) − 1)) = (c1 , c2 ). Therefore ϕ = φ−1 . Under the hypothesis of the above corollary, let q = pN (c2 )//N (c1 ) . By Theorem 4.4 and Lemma 3.1, R(q) is a-nonnegative and N (q) is a-nonpositive. Moreover, if x ∈ R(q) = N (c2 ) is such that x, xa = 0 1/2 then c1 x = c1 x, x = x, xa = 0. Therefore x ∈ N (c1 ). Since N (c1 ) ∩ N (c2 ) = N (a) = {0}, it follows that x = 0. Hence R(q) is a-positive. Similarly, N (q) is a-negative. In this case, H = N (c2 ) ⊕a N (c1 ) is the canonical decomposition of (H, , a ) determined by the positive decomposition a = c1 − c2 of a. By the previous results, a positive decomposition of a is uniquely determined either by a canonical decomposition of (H, , a ), or an a-positive reﬂection; or equivalently, an a-selfadjoint projection with a-positive range and a-negative nullspace. If a = a1 − a2 is the p.o.d. of a, then |a| = a1 + a2 and a = ua |a|, where ua is a symmetry. In a similar way, each positive decomposition of a induces a decomposition of a as a product of a reﬂection and a positive operator, as shows the following corollary: Corollary 4.6. Suppose that a = c1 − c2 is a positive decomposition of a. If α = c1 + c2 and w = 2q − 1 as in (3), then a = αw where α ∈ L(H)+ and w2 = 1. Conversely, if a = αw, with α ∈ L(H)+ and w2 = 1, consider c1 = a( w+1 2 ) and c2 = c1 − a, then a = c1 − c2 is a positive decomposition of a. Proof. If w = 2q − 1 and α = c1 + c2 ∈ L(H)+ , then w2 = 1 and w∗ α = (2p1 − 1)(c1 + c2 ) = c1 − c2 = a = αw. Conversely, consider a = αw where α ∈ L(H)+ and w2 = 1. Let q = w+1 2 . Since w is a-positive, by Lemma 3.1, q is a-selfadjoint, R(q) is a-nonnegative and N (q) is a-nonpositive. If c1 = aq and c2 = a(q − 1), by Theorem 4.4, it follows that a = c1 − c2 is a positive decomposition of a. By the above corollary, if a = c1 − c2 is a positive decomposition of a, then a = αw where α = c1 + c2 ∈ L(H)+ , w2 = 1 and w is a-positive. In this case, α = (w∗ |a|2 w)1/2 . In fact, α = aw = w∗ a so that α2 = w∗ a2 w = w∗ |a|2 w.

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Remark 4.7. If a ∈ GL(H)s and w is a reﬂection such that a = αw, with α ∈ GL(H)+ , then a = αw is the polar decomposition of a in the space (H, , α ). In fact: w is a symmetry with respect to , α , because w2 = 1 and wx, yα = ax, y = x, ay = αx, wy = x, wyα ,

x, y ∈ H.

It is straightforward to see that α is α-positive. In [11], Gesztesy et al. obtained a generalized formula for the classical polar decomposition of operators in Hilbert spaces. Applying their results we get the following proposition: Proposition 4.8. Consider a ∈ L(H)s and w an a-positive reflexion which is selfadjoint in (H, , ua ). Then, there exist positive operators α and γ such that a = αw = wγ. Moreover, if φ, ϕ are Borel functions on R such that φ(λ)ϕ(λ) = λ, for λ ∈ R, then a = φ(α)wϕ(γ).

(4)

Proof. By Theorem 4.4 and Corollary 4.6, it follows that there exists α L(H)+ such that a = αw. Therefore, a = αw = w2 αw = wγ, with γ wαw. To see that γ is positive observe that γ = wαw = ua w∗ ua αw ua w∗ w∗ αua = ua αua because w = ua w∗ ua and ua αw = |a|. Equation (4) follows from [11, Theorem 2.1].

∈ = =

In particular, the above proposition holds if a is a symmetry. Consider a injective and suppose that H = S ⊕a S ⊥a = S ⊕a S ⊥a are |a| two canonical decompositions of (H, , a ). Denote by S the completion of S respect to , |a| . As a consequence of Proposition 3.2, the next theorem shows that the completion of the a-positive subspaces of the two canonical |a| decompositions have the same Hilbert space dimension; i.e., dim|a| S = |a|

dim|a| S . Moreover, dim S = dim S , when H is separable. The same holds for the a-negative subspaces. Compare this result with [3, Corollary 7.4, Chapter IV]. Theorem 4.9. Consider a injective. Let H = S ⊕a S ⊥a = S ⊕a S ⊥a be canon⊥ ical decompositions of H, then dim S = dim S and dim S ⊥a = dim S a . Proof. Suppose that H = S ⊕a S ⊥a is the decomposition of H given by the p.o.d. a = a1 − a2 . In this case, the associated projection is p1 = pa1 . Consider the oblique projection q = pS//S ⊥a and the reﬂection w = 2q − 1. By Proposition 3.2, w = ua d, where d ∈ GL(H) is |a|-positive. Since w is a reﬂection, ua d = d−1 ua , so that ua dua = d−1 . Observe that ua , d, d−1 are |a|-selfadjoint. Then, by Proposition 3.4 and Remark 3.3, it holds that ua , d, d−1 admit bounded extensions to H|a| , denoted by ua , d, d−1 . Note that d−1 = (d)−1 . Since d−1 is positive in H|a| , then d−1 admits a unique (positive) square root in H|a| , (d−1 )1/2 ∈ GL(H|a| )+ . Note that ua (d)1/2 ua

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is positive in H|a| , because ua is selfadjoint in H|a| and (d)1/2 is positive in H|a| . Also, (ua (d)1/2 ua )2 = ua dua , so that (ua dua )1/2 = ua (d)1/2 ua . Then (d−1 )1/2 = (ua dua )1/2 = ua (d)1/2 ua and (d−1 )1/2 ua = ua (d)1/2 . Hence, 2q − 1 = w = ua d = (d−1 )1/2 ua (d)1/2 = (d−1 )1/2 (2p1 − 1)(d)1/2 = 2(d−1 )1/2 p1 (d)1/2 − 1, since (d−1 )1/2 = [(d)−1 )]1/2 = (d)−1/2 . Therefore, q = (d−1 )1/2 p1 (d)1/2 . Then, R(p1 ) = (d−1 )1/2 R(q) and N (p1 ) = (d−1 )1/2 N (q), so that dim|a| R(p1 ) = dim|a| R(q) and dim|a| N (p1 ) = dim|a| N (q), where dim|a| W is the dimension of a subspace W of H|a| . |a|

|a|

|a|

Then dim|a| R(p1 ) = dim|a| R(q) , because R(q) = R(q) . Since H is separable, it is easy to see that (H, , |a| ) is separable. Hence, H|a| is separable, because (H, , |a| ) is dense in H|a| . In this case, if S is a sub|a|

space of H, then dim|a| S = dim|a| S = dim S, with dim|a| S the cardinal of any maximal orthonormal subset of S in H|a| and dim S is the dimension of S as a subspace of H. Therefore, dim R(p1 ) = dim R(q). Similarly, dim N (p1 ) = dim N (q ∗ ). |a|

In the proof of the previous proposition, we concluded that dim|a| S = dim S for any closed subspace S of H. This holds because the Hilbert space H is separable, so that (H, , |a| ) and (therefore) H|a| are separable. There are examples of inner product spaces E with completions E such that dim E < dim E, where dim E is the cardinal of any maximal orthonormal set (see [12– 14]). Corollary 4.10. Consider a with closed range such that a = a1 − a2 is the p.o.d. of a and a = c1 −c2 is a positive decomposition of a. Then dim R(ai ) = dim R(ci ) and dim N (ai ) = dim N (ci ) for i = 1, 2. Proof. Suppose ﬁrst that a is invertible and consider p1 and q as in the proof of the previous proposition; i.e., p1 = pa1 and q = pN (c2 )//N (c1 ) . By the above proof, there exists g ∈ GL(H)+ such that q = g −1 p1 g so that q ∗ = g ∗ p1 g ∗ −1 . Then, since q ∗ = pR(c1 )//R(c2 ) , we get that dim R(ai ) = dim R(ci ) and dim N (ai ) = dim N (ci ) for i = 1, 2. More generally, if a has closed range and a = c1 −c2 is a positive decomposition of a, notice that ci pa = ci , because N (a) ⊆ N (ci ), for i = 1, 2. Then R(ci |R(a) ) = R(ci ) and a|R(a) = c1 |R(a) − c2 |R(a) is a positive decomposition of a|R(a) . Since a|R(a) ∈ GL(R(a))s , it follows that dim R(ci |R(ai ) ) = dim R(ai |R(ai ) ), where a = a1 − a2 is the p.o.d of a. Then dim R(ci ) = dim R(ai ), for i = 1, 2. Also, dim N (ci |R(a) ) = N (ai |R(a) ). But, N (ci ) = N (ci ) ∩ R(a) ⊕ N (a). Then, dim N (ai ) = dim N (ci ) for i = 1, 2.

5. Congruence of a Selfadjoint Operator Two operators a, b ∈ L(H) are congruent if there exists g ∈ GL(H) such that b = gag ∗ . In this section we study the set of operators in L(H)s which are congruent to a ﬁxed selfadjoint operator a. We characterize this set in terms of the positive decompositions of a.

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The congruence between selfadjoint operators deﬁnes the following natural action of the group GL(H) over the set L(H)s : L : GL(H) × L(H)s → L(H)s ,

Lg a = gag ∗ ,

a ∈ L(H)s ,

g ∈ GL(H).

s

Given a ∈ L(H) , the orbit of a corresponding to the action L is the set Oa of operators which are congruent to a, i.e., Oa = {gag ∗ : g ∈ GL(H)}. Denote by Ia the isotropy group of a, i.e., Ia = {g ∈ GL(H) : gag ∗ = a}. The following result is a consequence of Propositions 2.4 and 2.5, and provides a characterization of Oa , when a is positive. Proposition 5.1. Let a, b ∈ L(H)+ ; then b ∈ Oa if and only if R(a) and R(b) are unitarily equivalent. Remark 5.2. If a, b ∈ L(H)+ , it also holds that b is in the orbit of a if and only if the ranges of their square roots are unitarily equivalent (see [9, Theorem 3.5]). The next result generalizes Proposition 5.1, for selfadjoint operators. Proposition 5.3. Consider a = a1 − a2 the p.o.d of a. Then b ∈ Oa if and only if there exists a positive decomposition b = b1 − b2 such that R(bi ) is (unitarily) equivalent to R(ai ) for i = 1, 2 and dim N (b) = dim N (a). Proof. If b ∈ Oa , then b = gag ∗ for some g ∈ GL(H), so that dim N (b) = dim N (a). Consider bi = gai g ∗ ∈ L(H)+ , for i = 1, 2, then it is easy to see that b = b1 − b2 is a positive decomposition of b. Also, bi ∈ Oai , for i = 1, 2, and by Proposition 5.1, R(bi ) is unitarily equivalent to R(ai ) for i = 1, 2. Conversely, since dim N (b) = dim N (a), there exists a partial isometry v such that v(N (a)) = N (b). By Proposition 5.1 and Remark 5.2, there exist 1/2 1/2 1/2 u1 , u2 ∈ U(H) such that R(bi ) = ui R(ai ), for i = 1, 2. Then bi and 1/2 ui ai u∗i have the same range and nullspace, so that (see [9, Corollary 1]) 1/2 1/2 1/2 1/2 there exists gi ∈ GL(H) such that bi = gi ui ai u∗i or bi ui = gi ui ai for i = 1, 2. Consider p1 = pR(b1 )//R(b2 )⊕N (b) , p2 = pR(b2 )//R(b1 )⊕N (b) and w = p1 g1 u1 pa1 + p2 g2 u2 pa2 + v(1 − pa ). Then w ∈ GL(H). In fact, by Remark 4.3, it is easy to see that w−1 = pa1 u∗1 g1−1 p1 + pa2 u∗2 g2−1 p2 + v ∗ (1 − pb ). On the 1/2 1/2 1/2 1/2 other hand, waw∗ = w(a1 − a2 )w∗ = b1 u1 u∗1 b1 − b2 u2 u∗2 b2 = b. Hence b ∈ Oa . The above result also holds if a = a1 − a2 is any positive decomposition of a; in fact, in the proof of the proposition, it is sufﬁcient to consider the oblique projections pR(a1 )//R(a2 )⊕N (a) and pR(a2 )//R(a1 )⊕N (a) instead of pa1 and pa2 . Corollary 5.4. Let a = a1 − a2 be any positive decomposition of a. Then ˙ Oa = {b1 − b2 : bi ∈ Oai , i = 1, 2; and R(b1 )+R(b 2 ) is unitarily equivalent to R(a)}. Proof. Observe that R(b1 ) + R(b2 ) is an operator range by [9, Theorem 2.2]. If b ∈ Oa then by Proposition 5.3, there exists a positive decomposition

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b = b1 −b2 such that R(bi ) is unitarily equivalent to R(ai ) for i = 1, 2 and N (b) is unitarily equivalent to N (a); or equivalently, bi ∈ Oai , i = 1, 2 and R(b) ˙ is unitarily equivalent to R(a). But, by Lemma 4.2, R(b) = R(b1 )+R(b 2 ). ˙ ) = uR(a), Conversely, let b = b1 − b2 with bi ∈ Oai , i = 1, 2 and R(b1 )+R(b 2 for u ∈ U(H). Then R(b1 ) + R(b2 ) is closed and R(b1 ) ∩ R(b2 ) = {0}, so by Theorem 2.2, c0 (R(b1 ), R(b2 )) < 1. Note that bi ∈ L(H)+ because bi ∈ Oai , for i = 1, 2. Hence b = b1 − b2 is a positive decomposition of b, so that ˙ R(b) = R(b1 )+R(b 2 ) = uR(a) and then N (b) is unitarily equivalent to N (a). Therefore, by Proposition 5.3, b ∈ Oa . If g ∈ Ia , then a = ga1 g ∗ − ga2 g ∗ , where a = a1 − a2 is the p.o.d. of a. It is not difﬁcult to see that a = ga1 g ∗ − ga2 g ∗ is a positive decomposition of a. Therefore, it is natural to ask if all the positive decompositions of a can be written as a = ga1 g ∗ − ga2 g ∗ for some g ∈ Ia . The following proposition shows that this holds if a has closed range. Proposition 5.5. Consider a with closed range and p.o.d. a = a1 − a2 . Then {ga1 g ∗ − ga2 g ∗ : g ∈ Ia } is the set of positive decompositions of a. Proof. Given g ∈ Ia , then a = gag ∗ . It follows easily that a = ga1 g ∗ − ga2 g ∗ is a positive decomposition of a. Conversely, let a = c1 − c2 be a positive decomposition of a. By Corollary 4.10, it holds that dim R(ai ) = dim R(ci ) and dim N (ai ) = dim N (ci ) for i = 1, 2. Therefore, by Remark 5.2, ci ∈ Oai , for i = 1, 2. Then, there exist g1 , g2 ∈ GL(H) such that c1 = g1 a1 g1∗ and c2 = g2 a2 g2∗ . Consider g = g1 pa1 + g2 pa2 + pN (a) . By Remark 4.3, it is not difﬁcult to see that g ∈ GL(H) and g −1 = g1−1 p1 + g2−1 p2 + pN (a) , where p1 = pR(c1 )//R(c2 )⊕N (a) and p2 = pR(c2 )//R(c1 )⊕N (a) . Also, ga1 g ∗ = g1 a1 g1∗ p∗1 = c1 p∗1 = c1 . Similarly, ga2 g ∗ = c2 . Finally, gag ∗ = ga1 g ∗ − ga2 g ∗ = c1 − c2 = a, so that g ∈ Ia . Acknowledgements The authors are very grateful to the referee for his many helpful comments and suggestions.

References [1] Arias, M.L., Corach, G., Gonzalez, M.C.: Lifting properties in operator ranges. Acta Sci. Math. (Szeged) 75, 635–653 (2009) [2] Azizov, T.Ya., Iokhvidov, I.S.: Linear Operators in Spaces with an Indefinite Metric. Wiley, New York (1989) [3] Bognar, J.: Indefinite Inner Product Spaces. Springer, New York (1974) [4] Cojuhari, P., Gheondea, A.: On lifting of operators to Hilbert spaces induced by positive selfadjoint operators. J. Math. Anal. Appl. 304, 584–598 (2005) [5] Corach, G., Maestripieri, A., Stojanoﬀ, D.: Orbits of positive operators from a differentiable viewpoint. Positivity 8, 31–48 (2004) [6] Corach, G., Porta, H., Recht, L.: The geometry of spaces of selfadjoint invertible elements of a C ∗ -algebra. Integral Equ. Oper. Theory 16, 333–359 (1993)

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[7] Deutsch, F.: The angle between subspaces of a Hilbert space. In: Singh, S.P. (ed.) Approximation Theory, Wavelets and Applications, pp. 107–130. Kluwer, Netherlands (1995) [8] Douglas, R.G.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–416 (1966) [9] Fillmore, P.A., Williams, J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971) [10] Fongi, G., Maestripieri, A.: Congruence of selfadjoint operators. Positivity 13(4), 759–770 (2009) [11] Gesztesy, F., Malamud, M., Mitrea, M., Naboko, S.: Generalized polar decompositions for closed operators in Hilbert spaces and some applications. Integral Equ. Oper. Theory 64, 83–113 (2009) [12] Gudder, S.: Inner product spaces. Am. Math. Mon. 81, 29–36 (1974) [13] Gudder, S.: Correction to: ‘Inner product spaces’. Am. Math. Mon. 82, 251– 252 (1975) [14] Gudder, S., Holland, S.: Second correction to: ‘Inner product spaces’. Am. Math. Mon. 82, 818 (1975) [15] Hassi, S., Nordstr¨ om, K.: On projections in a space with an indefinite metric. Linear Algebra Appl. 208/209, 401–417 (1994) [16] Hassi, S., Sebestyen, Z., De Snoo, S.V.: On the nonnegativity of operator products. Acta Math. Hung. 109, 1–14 (2005) [17] Maestripieri, A., Mart´ınez Per´ıa, F.: Decomposition of selfadjoint projections in Krein spaces. Acta Sci. Math. (Szeged) 72, 611–638 (2006) Guillermina Fongi Departamento de Matem´ atica, Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Buenos Aires Argentina e-mail: [email protected] Guillermina Fongi and Alejandra Maestripieri Instituto Argentino de Matem´ atica-CONICET Saavedra 15 3p 1083 Buenos Aires Argentina Alejandra Maestripieri Departamento de Matem´ atica, Facultad de Ingenier´ıa Universisdad de Buenos Aires Buenos Aires Argentina e-mail: [email protected] Received: July 14, 2009. Revised: December 18, 2009.

Integr. Equ. Oper. Theory 67 (2010), 123–149 DOI 10.1007/s00020-010-1774-y Published online April 13, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators Renjin Jiang and Dachun Yang Abstract. Let L be a divergence form elliptic operator with complex bounded measurable coefﬁcients, ω a positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t−1 /ω −1 (t−1 ) for t ∈ (0, ∞). In this paper, the authors introduce the generalized VMO spaces VMOρ,L (Rn ) associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOρ,L (Rn ))∗ = Bω,L∗ (Rn ), where L∗ denotes the adjoint operator of L in L2 (Rn ) and Bω,L∗ (Rn ) the Banach completion of the Orlicz–Hardy space Hω,L∗ (Rn ). Notice that ω(t) = tp for all t ∈ (0, ∞) and p ∈ (0, 1] is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1,L (Rn ))∗ = HL1 ∗ (Rn ), where HL1 ∗ (Rn ) was the Hardy space introduced by Hofmann and Mayboroda. Mathematics Subject Classification (2000). Primary 42B35; Secondary 42B30, 46E30. Keywords. Divergence form elliptic operator, Gaﬀney estimate, Orlicz function, Orlicz–Hardy space, BMO, VMO, CMO, molecule, dual.

1. Introduction John and Nirenberg [22] introduced the space BMO(Rn ), which is deﬁned to be the space of all f ∈ L1loc (Rn ) that satisfy 1 n |f (x) − fB | dx < ∞, f BMO(R ) ≡ sup ball B⊂Rn |B| B 1 where and in what follows, fB ≡ |B| f (x) dx. The space BMO(Rn ) is B proved to be the dual of the Hardy space H 1 (Rn ) by Feﬀerman and Stein in [13]. Sarason [24] introduced the space VMO(Rn ), which is deﬁned to be the space of all f ∈ BMO(Rn ) that satisfy D. Yang (corresponding author) is supported by the National Natural Science Foundation (Grant No. 10871025) of China.

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|f (x) − fB | dx = 0. B

In order to represent H 1 (Rn ) as a dual space, Coifman and Weiss [9] introduced the space CMO(Rn ) as the closure of continuous functions with compact support in the BMO(Rn ) norm and showed that (CMO (Rn ))∗ = H 1 (Rn ). For more general cases, we refer to Janson [18] and Bourdaud [7]. Let L be a linear operator in L2 (Rn ) that generates an analytic semigroup {e−tL }t≥0 with kernels satisfying an upper bound of Poisson type. The Hardy space HL1 (Rn ) and the BMO space BMOL (Rn ) associated with L were deﬁned and studied in [2,11]. Duong and Yan [12] further proved that (HL1 (Rn ))∗ = BMOL∗ (Rn ), where and in what follows, L∗ denotes the adjoint operator of L in L2 (Rn ). Moreover, recently, Deng et al. [10] introduced the space VMOL (Rn ), the space of vanishing mean oscillation associated with L, and proved that (VMOL (Rn ))∗ = HL1 ∗ (Rn ). Also, Auscher and Russ [5] studied the Hardy space HL1 on strongly Lipschitz domains associated with a divergence form elliptic operator L whose heat kernels have the Gaussian upper bounds and regularity, and Auscher et al. [4] treated the Hardy space H 1 associated with the Hodge Laplacian on a Riemannian manifold with doubling measure. Let A be an n × n matrix with entries {aj,k }nj, k=1 ⊂ L∞ (Rn , C) satisfying the uniform ellipticity condition, namely, there exist constants 0 < λA ≤ ΛA < ∞ such that for all ξ, ζ ∈ Cn and almost every x ∈ Rn , λA |ξ|2 ≤ ReA(x)ξ, ξ

and |A(x)ξ, ζ | ≤ ΛA |ξ||ζ|.

(1.1)

Then the second order divergence form operator is given by Lf ≡ div(A∇f ),

(1.2)

interpreted in the weak sense via a sesquilinear form. It is well known that the kernel of the heat semigroup {e−tL }t>0 lacks pointwise estimates in general. From now on, in this paper, we always let L be as in (1.2) and L∗ the adjoint operator of L in L2 (Rn ). Recently, Hofmann and Mayboroda [16,17] studied the Hardy space HL1 (Rn ) and its dual space BMOL∗ (Rn ). Indeed, Hofmann and Mayboroda [16] ﬁrst deﬁned the Hardy space HL1 (Rn ) via its molecular decomposition, then established several maximal function characterizations of HL1 (Rn ), and in particular, showed that (HL1 (Rn ))∗ = BMOL∗ (Rn ). These results were generalized in [20] to the Orlicz–Hardy space Hω,L (Rn ) and its dual space BMOρ,L∗ (Rn ), which contain the Hardy spaces HLp (Rn ) for all p ∈ (0, 1], the space BMOL∗ (Rn ) and the Lipschitz spaces LipL∗ ( p1 − 1, Rn ) for all p ∈ (0, 1) as special cases. Let ω be a positive concave function on (0, ∞) of strictly critical lower type pω ∈ (0, 1] and ρ(t) = t−1 /ω −1 (t−1 ) for t ∈ (0, ∞). Recall that ω(t) = tp for all t ∈ (0, ∞) and p ∈ (0, 1] is a typical example of such positive concave functions. Motivated by [9,18,10,16,20], in this paper, we introduce the generalized VMO spaces VMOρ,L (Rn ) associated with L, and characterize them via tent spaces. Then, we prove that (VMOρ,L (Rn ))∗ = Bω,L∗ (Rn ), where Bω,L∗ (Rn ) denotes the Banach completion of Hω,L∗ (Rn ). When ω(t) = t

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for all t ∈ (0, ∞), we denote VMOρ,L (Rn ) simply by VMOL (Rn ). In this case, our result reads as (VMOL (Rn ))∗ = HL1 ∗ (Rn ). Finally, we show that the space CMO (Rn ) is a subspace of VMOL (Rn ), and if n ≥ 3, then there exists an operator L as in (1.2), constructed in [1,14], such that CMO (Rn ) VMOL (Rn ). Moreover, when A has real entries, or when n = 1, 2 in the case of complex entries, the spaces CMO (Rn ) and VMOL (Rn ) coincide with equivalent norms, which was pointed out to us by the referee. Precisely, this paper is organized as follows. In Sect. 2, we recall some known definitions and notation on the divergence form elliptic operator L and Orlicz functions ω considered in this paper. In Sect. 3, we introduce the generalized VMO spaces VMOρ,L (Rn ) asso∞ ciated with L, and tent spaces Tω,v (Rn+1 + ) and give some basic properties of these spaces. In particular, we characterize the space VMOρ,L (Rn ) via ∞ Tω,v (Rn+1 + ); see Theorem 3.5 below. In Sect. 4, we prove that (VMOρ,L (Rn ))∗ = Bω,L∗ (Rn ), where Bω,L∗ (Rn ) denotes the Banach completion of Hω,L∗ (Rn ); see Theorem 4.10 below. In particular, we have (VMOL (Rn ))∗ = HL1 ∗ (Rn ). Finally, in Proposition 4.12 below, we show that the space CMO (Rn ) is a subspace of VMOL (Rn ), and if n ≥ 3, then there exists an operator L as in (1.2) such that CMO (Rn ) VMOL (Rn ); moreover, as a corollary of Proposition 4.12, we obtain, in Corollary 4.13 below, that when A has real entries, or when n = 1, 2 in the case of complex entries, the spaces CMO (Rn ) and VMOL (Rn ) coincide with equivalent norms. Finally, we make some conventions. Throughout the paper, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol X Y means that there exists a positive constant C such that X ≤ CY ; B ≡ B(zB , rB ) denotes an open ball with center zB and radius rB and CB(zB , rB ) ≡ B(zB , CrB ). Moreover, in what follows, for each ball B ⊂ Rn and j ∈ N, we set U0 (B) ≡ B and Uj (B) ≡ 2j B\2j−1 B. Set N ≡ {1, 2, . . .} and Z+ ≡ N ∪ {0}. For any subset E of Rn , we denote by E the set Rn \E.

2. Preliminaries In this section, we recall some notions and notation on divergence form elliptic operators, describe some basic assumptions on Orlicz functions and also present some basic properties on them. 2.1. Some Notions on the Divergence form Elliptic Operator L A family {St }t>0 of operators is said to satisfy the L2 oﬀ-diagonal estimates, which are also called the Gaﬀney estimates [16], if there exist positive constants c, C and β such that for arbitrary closed sets E, F ⊂ Rn , St f

L2 (F )

−

≤ Ce

dist (E,F )2 ct

β

f L2 (E)

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for every t > 0 and every f ∈ L2 (Rn ) supported in E. Here and in what follows, for any p ∈ (0, ∞] and E ⊂ Rn , f Lp (E) ≡ f χE Lp (Rn ) ; for any sets E, F ⊂ Rn , dist (E, F ) ≡ inf{|x − y| : x ∈ E, y ∈ F }. The following results were obtained in [3,16,15]. Lemma 2.1 [15]. If two families, {St }t>0 and {Tt }t>0 , of operators satisfy the Gaﬀney estimates, then so does {St Tt }t>0 . Moreover, there exist positive constants c, C and β such that for arbitrary closed sets E, F ⊂ Rn , −

Ss Tt f L2 (F ) ≤ Ce

dist (E,F )2 c max{s,t}

β

f L2 (E)

for every s, t > 0 and every f ∈ L2 (Rn ) supported in E. Lemma 2.2 [3,15]. The families {e−tL }t>0 , {tLe−tL }t>0 ,

(2.1)

{(I + tL)−1 }t>0 ,

(2.2)

as well as are bounded on L2 (Rn ) uniformly in t and satisfy the Gaﬀney estimates with positive constants c and C, depending on n, λA , ΛA as in (1.1) only. For the operators in (2.1), β = 1; while in (2.2), β = 1/2. Following [16], set pL ≡ inf{p ≥ 1 : supt>0 e−tL Lp (Rn )→Lp (Rn ) < ∞} and

−tL Lp (Rn )→Lp (Rn ) < ∞ . pL ≡ sup p ≤ ∞ : sup e t>0

It was proved by Auscher [1] that if n = 1, 2, then pL = 1 and pL = ∞, and if n ≥ 3, then pL < 2n/(n + 2) and pL > 2n/(n − 2). Moreover, thanks to a counterexample given by Frehse [14], this range is sharp. Lemma 2.3 [16]. Let k ∈ N and p ∈ (pL , pL ). Then the operator given by setting, for all f ∈ Lp (Rn ) and x ∈ Rn , ⎞1/2 ⎛ 2 k −t2 L dy dt ⎟ ⎜ SLk f (x) ≡ ⎝ f (y) n+1 ⎠ , (t L) e t Γ(x) p

n

is bounded on L (R ). 2.2. Orlicz Functions Let ω be a positive function deﬁned on R+ ≡ (0, ∞). The function ω is said to be of upper type p (resp. lower type p) for certain p ∈ [0, ∞), if there exists a positive constant C such that for all t ≥ 1 (resp. t ∈ (0, 1]) and all s ∈ (0, ∞), ω(st) ≤ Ctp ω(s).

(2.3)

Obviously, if ω is of lower type p for certain p > 0, then limt→0+ ω(t) = 0. So for the sake of convenience, if it is necessary, we may assume that ω(0) = 0.

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If ω is of both upper type p1 and lower type p0 , then ω is said to be of type (p0 , p1 ). Let p+ ω ≡ inf{p > 0 : there exists C > 0 such that (2.3) holds for all t ∈ [1, ∞), s ∈ (0, ∞)}, and p− ω ≡ sup{p > 0 : there exists C > 0 such that (2.3) holds for all t ∈ (0, 1], s ∈ (0, ∞)}. The function ω is said to be of strictly lower type p if for all t ∈ (0, 1) and s ∈ (0, ∞), ω(st) ≤ tp ω(s), and for a such function ω, we deﬁne pω ≡ sup{p > 0 : ω(st) ≤ tp ω(s) holds for all s ∈ (0, ∞) and t ∈ (0, 1)}. + − + It is easy to see that pω ≤ p− ω ≤ pω for all ω. In what follows, pω , pω and pω are called the strictly critical lower type index, the critical lower type index and the critical upper type index of ω, respectively.

Remark 2.4. We claim that if pω is deﬁned as above, then ω is also of strictly lower type pω . In other words, pω is attainable. In fact, if this is not the case, then there exist certain s ∈ (0, ∞) and t ∈ (0, 1) such that ω(st) > tpω ω(s). Hence there exists ∈ (0, pω ) small enough such that ω(st) > tpω − ω(s), which is contrary to the definition of pω . Thus, ω is of strictly lower type pω . We now introduce the following assumption. Assumption (A). Let ω be a positive function deﬁned on R+ , which is of strictly lower type and its strictly lower type index pω ∈ (0, 1]. Also assume that ω is continuous, strictly increasing and concave. Notice that if ω satisﬁes Assumption (A), then ω(0) = 0 and ω is obviously of upper type 1. Since ω is concave, it is subadditive. In fact, let 0 < s < t, then st s+t ω(t) ≤ ω(t) + ω(s) = ω(s) + ω(t). ω(s + t) ≤ t ts t For any concave function ω of strictly lower type p, if we set ω (t) ≡ 0 ω(s)/s ds for t ∈ [0, ∞), then by [25, Proposition 3.1], ω is equivalent to ω, namely, (t) ≤ Cω(t) for all there exists a positive constant C such that C −1 ω(t) ≤ ω t ∈ [0, ∞); moreover, ω is strictly increasing, concave, subadditive and continuous function of strictly lower type p. Since all our results are invariant on equivalent functions, we always assume that ω satisﬁes Assumption (A); otherwise, we may replace ω by ω . For example, if ω(t) = tp with p ∈ (0, 1], then pω = p+ ω = p; if ω(t) = t1/2 ln(e4 + t), then pω = p+ = 1/2. ω Let ω satisfy Assumption (A). A measurable function f on Rn is said to be in the Lebesgue type space L(ω) if ω(|f (x)|) dx < ∞. Rn

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Moreover, for any f ∈ L(ω), deﬁne ⎧ ⎫ ⎨ ⎬ |f (x)| f L(ω) ≡ inf λ > 0 : ω dx ≤ 1 . ⎩ ⎭ λ Rn

Since ω is strictly increasing, we deﬁne the function ρ(t) on R+ by setting, for all t ∈ (0, ∞), ρ(t) ≡

t−1 , ω −1 (t−1 )

(2.4)

where and in what follows, ω −1 denotes the inverse function of ω. Then the types of ω and ρ have the following relation; see [25] for its proof. Proposition 2.5. Let 0 < p0 ≤ p1 ≤ 1 and ω be an increasing function. Then −1 ω is of type (p0 , p1 ) if and only if ρ is of type (p−1 1 − 1, p0 − 1). Throughout the whole paper, we always assume that ω satisﬁes Assumption (A) and ρ is as in (2.4).

3. Spaces VMOρ,L (Rn ) Associated with L In this section, we introduce the generalized vanishing mean oscillation spaces associated with L. We begin with some notions and notation. Let q ∈ (pL , pL ), M ∈ N and ∈ (0, ∞). A function α ∈ Lq (Rn ) is called an (ω, q, M, )L -molecule adapted to B if there exists a ball B such that (1) αLq (Uj (B)) ≤ 2−j |2j B|1/q−1 ρ(|2j B|)−1 , j ∈ Z+ ; (2) For every k = 1, . . . , M and j ∈ Z+ , there holds 1/q−1 j −1 −2 −1 k α ≤ 2−j 2j B ρ 2 B . rB L Lq (Uj (B))

Let and M be as above. We also introduce the space 2 n < ∞ , MM, ω (L) ≡ μ ∈ L (R ) : μMM, (L) ω where μMM, (L) ω

(3.1)

M j 1/2 j −k ≡ sup 2 B(0, 2 ) ρ(|B(0, 2 )|) L μL2 (Uj (B(0,1))) . j

j≥0

k=0

MM, ω (L)

with norm 1, then φ is an (ω, 2, M, )-molNotice that if φ ∈ ecule adapted to B(0, 1). Conversely, if α is an (ω, 2, M, )-molecule adapted to certain ball, then α ∈ MM, ω (L). 2 ∗ Let At denote either (I + t2 L)−1 or e−t L and f ∈ (MM, ω (L)) , the M, ∗ M 2 n dual of Mω (L). We claim that (I − At ) f ∈ L loc (R ) in the sense of distributions. In fact, for any ball B, if ψ ∈ L2 (B), then it follows from the Gaﬀney estimates via Lemmas 2.1 and 2.2 that (I − At )M ψ ∈ MM, ω (L) for all > 0 and any ﬁxed t ∈ (0, ∞). Thus, ! (I − A∗t )M f, ψ ≡ |f, (I − At )M ψ | 2 ≤ C(t, rB , dist (B, 0))f (MM, (L))∗ ψL (B) , ω

which implies that (I − A∗t )M f ∈ L2loc (Rn ) in the sense of distributions.

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Finally, for any M ∈ N, deﬁne n MM ω,L∗ (R ) ≡

"

129

∗ (MM, ω (L)) .

(3.2)

>n(1/pω −1/p+ ω)

Definition 3.1. Let q ∈ (pL , pL ) and M >

n 1 2 ( pω

− 12 ). An element f ∈

q,M n n MM ω,L (R ) is said to be in BMOρ,L (R ) if ⎡ ⎤1/q 2 1 ⎣ 1 |(I − e−rB L )M f (x)|q dx⎦ < ∞, f BMOq,M (Rn ) ≡ sup ρ,L n ρ(|B|) |B| B⊂R B

where the supremum is taken over all balls B of Rn . n Remark 3.2. The spaces BMOq,M ρ,L (R ) were introduced in [20]; moreover, if n n ω(t) = t for all t ∈ (0, ∞), BMOq,M ρ,L (R ) is the space BMOL (R ) introduced q,M by Hofmann and Mayboroda [16]. Since the spaces BMOρ,L (Rn ) coincide for all q ∈ (pL , pL ) and M > n2 ( p1ω − 12 ) [20, Theorem 4.1], in what follows, we n n denote BMOq,M ρ,L (R ) simply by BMOρ,L (R ).

n Let us introduce a new space VMOM ρ,L (R ) as a subspace of n BMOρ,L (R ). Definition 3.3. Let M > n2 p1ω − 12 . An element f ∈ BMOρ,L (Rn ) is said

n to be in VMOM ρ,L (R ) if it satisﬁes the following limiting conditions γ1 (f ) = γ2 (f ) = γ3 (f ) = 0, where ⎛ ⎞1/2 2 1 ⎝ 1 γ1 (f ) ≡ lim sup |(I − e−rB L )M f (x)|2 dx⎠ , c→0 B: rB ≤c ρ(|B|) |B|

⎛

B

1 ⎝ 1 γ2 (f ) ≡ lim sup c→∞ B: rB ≥c ρ(|B|) |B| and

⎛

γ3 (f ) ≡ lim

c→∞

For any f ∈

sup B⊂[B(0,c)]

n VMOM ρ,L (R ),

1 ⎝ 1 ρ(|B|) |B|

⎞1/2 2 −rB L

|(I − e

)M f (x)|2 dx⎠

,

B

⎞1/2 |(I − e−rB L )M f (x)|2 dx⎠ 2

.

B

deﬁne f VMOM n ≡ f BMOρ,L (Rn ) . ρ,L (R )

n We next present some properties of the space VMOM ρ,L (R ). To this end, we ﬁrst recall some notions of tent spaces; see [8]. Let Γ(x) ≡ {(y, t) ∈ Rn+1 : |x − y| < t} denote the standard cone (of + aperture 1) with vertex x ∈ Rn . For any closed set F of Rn , denote by RF the union of all cones with vertices in F , i. e., RF ≡ ∪x∈F Γ(x); for any open ' which is deﬁned by O ' ≡ [R(O )] . set O in Rn , denote the tent over O by O,

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and x ∈ Rn , deﬁne For any measurable function g on Rn+1 + ⎞1/2 ⎛ dy dt ⎟ ⎜ |g(y, t)|2 n+1 ⎠ A(g)(x) ≡ ⎝ t Γ(x)

and

⎛ Cρ (g)(x) ≡

1 ⎜ 1 ⎝ |B| ball B x ρ(|B|)

⎞1/2

|g(y, t)|2

sup

' B

dy dt ⎟ ⎠ t

.

T2p (Rn+1 + )

For p ∈ (0, ∞), the tent space is deﬁned to be the space of all measurable functions g on Rn+1 such that gT p (Rn+1 ) ≡ A(g)Lp (Rn ) < ∞. + 2

+

Let ω satisfy Assumption (A). The tent space Tω (Rn+1 + ) associated to the function ω is deﬁned to be the space of measurable functions g on Rn+1 such + that A(g) ∈ L(ω) with the norm deﬁned by ⎧ ⎫ ⎨ ⎬ A(g)(x) gTω (Rn+1 ) ≡ A(g)L(ω) = inf λ > 0 : ω dx ≤ 1 ; + ⎩ ⎭ λ Rn

the space Tω∞ (Rn+1 + ) is deﬁned to be the space of all measurable functions g on Rn+1 satisfying gT ∞ (Rn+1 ) ≡ Cρ (g)L∞ (Rn ) < ∞. + ω

+

n+1 ∞ ∞ In what follows, let Tω,v (Rn+1 + ) be the space of all f ∈ Tω (R+ ) satisfying η1 (f ) = η2 (f ) = η3 (f ) = 0, where ⎛ ⎞1/2 1 ⎜ 1 dy dt ⎟ |f (y, t)|2 η1 (f ) ≡ lim sup ⎝ ⎠ , c→0 B: rB ≤c ρ(|B|) |B| t

⎛ η2 (f ) ≡ lim

sup

c→∞ B: rB ≥c

and

' B

1 ⎜ 1 ⎝ ρ(|B|) |B| ⎛

η3 (f ) ≡ lim

c→∞

sup B⊂[B(0,c)]

1 ⎜ 1 ⎝ ρ(|B|) |B|

⎞1/2

|f (y, t)|2 ' B

,

⎞1/2

|f (y, t)|2 ' B

dy dt ⎟ ⎠ t

dy dt ⎟ ⎠ t

.

n+1 ∞ ∞ It is easy to see that Tω,v (Rn+1 + ) is a closed linear subspace of Tω (R+ ). n+1 n+1 ∞ ∞ Further, denote by Tω,1 (R+ ) the space of all f ∈ Tω (R+ ) with n+1 2 2 (Rn+1 η1 (f ) = 0, and T2,c + ) the space of all f ∈ T2 (R+ ) with compact supn+1 n+1 2 ∞ ∞ port. Obviously, we have T2,c (R+ ) ⊂ Tω,v (R+ ) ⊂ Tω,1 (Rn+1 + ). Finally, n+1 n+1 ∞ 2 ∞ (R+ ) the closure of T2,c (R+ ) in the space Tω,1 (Rn+1 denote by Tω,0 + ). By [19, Proposition 3.1], we have the following result; see also [10].

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131

n+1 ∞ ∞ (Rn+1 Lemma 3.4. Let Tω,v + ) and Tω,0 (R+ ) be deﬁned as above. Then n+1 n+1 ∞ ∞ Tω,v (R+ ) coincides with Tω,0 (R+ ).

Recall that a measure dμ on Rn+1 is called a ρ-Carleson measure if + ⎫1/2 ⎧ ⎪ ⎪ ⎬ ⎨ 1 1 sup |dμ| < ∞, ⎪ B⊂Rn ρ(|B|) ⎪ ⎭ ⎩ |B| ' B

where the supremum is taken over all balls B of Rn . n We now characterize the space VMOM ρ,L (R ) via tent spaces. Theorem 3.5. Let M, M1 ∈ N and M1 ≥ M > n2 ( p1ω − 12 ). Then the following conditions are equivalent: n (a) f ∈ VMOM ρ,L (R ); n 2 M1 −t2 L ∞ 1 (b) f ∈ MM e f ∈ Tω,v (Rn+1 + ). ω,L (R ) and (t L) Moreover, (t2 L)M1 e−t L f T ∞ (Rn+1 ) is equivalent to f BMOρ,L (Rn ) . 2

ω

+

To prove Theorem 3.5, we need two auxiliary results. Let M ∈ Z+ . In what follows, let CM be the positive constant such that ∞ 2 dt CM t2(M +1) e−2t = 1. (3.3) t 0

The following lemma was established in [20, Proposition 4.6]. Lemma 3.6. Let ∈ (0, ∞) and M > n2 p1ω − 12 . If f ∈ BMOρ,L (Rn ), then for any (ω, 2, M, )L∗ -molecule α, there holds 2 2 ∗ dx dt . f (x)α(x) dx = CM (t2 L)M e−t L f (x)t2 L∗ e−t L α(x) t Rn

Rn+1 +

Definition 3.7. Let M >

n 2

1 pω

−

1 2

M

ρ,L (Rn ) to be . Deﬁne the space VMO

the space of all elements f ∈ BMOρ,L (Rn ) that satisfy the limiting conditions 2 (f ) = γ 3 (f ) = 0, where γ 1 (f ) = γ ⎛ ⎞1/2 1 ⎝ 1 2 |(I − [I + rB L]−1 )M f (x)|2 dx⎠ , γ 1 (f ) ≡ lim sup c→0 B: rB ≤c ρ(|B|) |B| ⎛

B

1 ⎝ 1 γ 2 (f ) ≡ lim sup c→∞ B: rB ≥c ρ(|B|) |B| and γ 3 (f ) ≡ lim

c→∞

sup B⊂[B(0,c)]

⎞1/2 2 |(I − [I + rB L]−1 )M f (x)|2 dx⎠

,

B

⎛ ⎞1/2 1 ⎝ 1 2 |(I − [I + rB L]−1 )M f (x)|2 dx⎠ . ρ(|B|) |B| B

132

R. Jiang and D. Yang

Proposition 3.8. Let M > M

n 1 2 (pω

IEOT

n − 12 ). Then f ∈ VMOM ρ,L (R ) if and only if

ρ,L (Rn ) f ∈ VMO Proof. Recall that it was proved in [20, Lemma 4.1] that f BMOρ,L (Rn ) ∼ sup

B⊂Rn

⎡

1 ⎣ 1 ρ(|B|) |B|

⎤1/2

2 |(I − (I + rB L)−1 )M f (x)|2 dx⎦

. (3.4)

B M

n ρ,L (Rn ). To see f ∈ VMOM Now suppose that f ∈ VMO ρ,L (R ), it sufﬁces to show that ⎛ ⎞1/2 ∞ 2 1 ⎝ |(I − e−rB L )M f (x)|2 dx⎠ 2−k δk (f, B), (3.5) ρ(|B|)|B|1/2 k=0 B

where δk (f, B) ≡

1 1/2 ρ(|B|)|B| rB B ,rB ]} ⎧ ⎫1/2 ⎨ ⎬ 2 × |(I − [I + rB L]−1 )M f (x)|2 dx . ⎩ ⎭ sup

{B ⊂2k+1 B:

∈[2−1 r

(3.6)

B

M

ρ,L (Rn ), by Definition 3.7 and (3.4), we have Indeed, since f ∈ VMO that δk (f, B) f BMOρ,L (Rn ) and for each k ∈ N, lim sup δk (f, B) = lim

sup δk (f, B) = lim

c→∞ B: rB ≥c

c→0 B: rB ≤c

c→∞

sup

δk (f, B) = 0.

B⊂[B(0, c)]

Then by the dominated convergence theorem for series, we have ⎛ ⎞1/2 2 1 ⎝ |(I − e−rB L )M f (x)|2 dx⎠ γ1 (f ) = lim sup c→0 B: rB ≤c ρ(|B|)|B|1/2 B

∞ k=1

2−k lim sup δk (f, B) = 0. c→0 B: rB ≤c

n Similarly, we have that γ2 (f ) = γ3 (f ) = 0, and thus, f ∈ VMOM ρ,L (R ). Let us now prove (3.5). Write ) *−1 M ) *−1 M 2 2 f = I − I + rB L f + I − I − I + rB L f ≡ f1 + f2 .

(3.7)

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Generalized Vanishing Mean Oscillation Spaces

By Lemma 2.2, we have 2 (I − e−rB L )M f1 2 L (B) ∞ M 2 −rB L ≤ (f1 χUk (B) ) I −e

L2 (B)

k=0

ρ(|B|)|B|1/2 ρ(|B|)|B|1/2

∞ k=0 ∞

∞ k=0

133

2k e−c2 f1 χUk (B) L2 (Rn )

e−c2 2kn δk (f, B) 2k

2−k δk (f, B),

(3.8)

k=0

where c is a positive constant and the third inequality follows from the fact that there exists a collection, {Bk,1 , Bk,2 , . . . , Bk,Nk }, of balls such that each nk k ball Bk,i is of radius rB , B(xB , 2k+1 rB ) ⊂ ∪N i=1 Bk,i and Nk 2 . To estimate the remaining term, by the formula that M M 2 I − I − [I + rB L]−1 = j=1

M M! 2 2 (rB L)−j I − [I + rB L]−1 j!(M − j)! (3.9)

(which relies on the fact that (I − (I + r2 L)−1 )(r2 L)−1 = (I + r2 L)−1 for all r ∈ (0, ∞)), and the Minkowski inequality, we obtain 2 (I − e−rB L )M f2 2 L (B)

2 ⎞1/2 ⎡ r ⎤j B M ⎜ s −s2 L ⎦ ⎟ −r 2 L M −j ⎣ − ds f1 (x) dx⎠ ⎝ (I − e B ) 2 e r B j=1 B 0 ⎛

−j M M

rB

rB

···

j=1 k=0 0 M M −j rB

0

rB ···

j=1 k=0 0

0

s1 sj −(krB2 +s21 +···+s2j )L · · · f ds1 · · · dsj e 2 1 2 2 rB rB L (B) ∞ c(2i rB )2 s1 sj − krB 2 +s2 +···+s2 1 j · · · e 2 2 rB rB i=0

×f1 χUi (B) L2 (Rn ) ds1 · · · dsj ∞ c22i ρ(|B|)|B|1/2 e− M 2in δi (f, B) i=0

ρ(|B|)|B|1/2

∞

2−i δi (f, B),

(3.10)

i=0

where c is a positive and in the penultimate inequality, we used constant r r s the fact that 0 B · · · 0 B rs21 · · · r2j ds1 · · · dsj ∼ 1. Combining the estimates B

B

134

R. Jiang and D. Yang

IEOT M

ρ,L (Rn ) ⊂ (3.8) and (3.10), we obtain (3.5), which further implies that VMO M n VMOρ,L (R ). By borrowing some ideas from the proof of [16, Lemma 8.1], then simM n n ilarly to the proof above, we have that VMOM ρ,L (R ) ⊂ VMOρ,L (R ). We omit the details here, which completes the proof of Proposition 3.8. Proof of Theorem 3.5. We ﬁrst show that (a) implies (b). Let f ∈ n 2 M1 −t2 L VMOM e f ∈ ρ,L (R ). By [20, Theorem 6.1], we have that (t L) n+1 2 M1 −t2 L ∞ ). To see that (t L) e f ∈ T (R ), we claim that it sufﬁces Tω∞ (Rn+1 ω,v + + to show that for all balls B, ⎛ ⎞1/2 ∞ 1 ⎜ 2 M1 −t2 L 2 dx dt ⎟ |(t L) e f (x)| 2−k δk (f, B), ⎝ ⎠ t ρ(|B|)|B|1/2 k=0

' B

(3.11) M

n n where δk (f, B) is as in (3.6). In fact, since f ∈ VMOM ρ,L (R ) = VMOρ,L (R ), we have that for each k ∈ N, δk (f, B) f BMOρ,L (Rn ) and

lim sup δk (f, B) = lim

sup δk (f, B) = lim

c→∞ B: rB ≥c

c→0 B: rB ≤c

c→∞

sup

δk (f, B) = 0.

B⊂[B(0, c)]

Then by the dominated convergence theorem for series, we have ⎛ ⎞1/2 1 2 M1 −t2 L 2 dx dt ⎟ ⎜ f (x) η1 (f ) = lim sup (t L) e ⎝ ⎠ c→0 B: rB ≤c ρ(|B|)|B|1/2 t ' B

∞

2−k lim sup δk (f, B) = 0. c→0 B: rB ≤c

k=1

∞ (Rn+1 Similarly, we have η2 (f ) = η3 (f ) = 0, and thus, (t2 L)M1 e−t L f ∈ Tω,v + ). Let us prove (3.11). Write f ≡ f1 + f2 as in (3.7). Then by Lemmas 2.2 and 2.3, similarly to the estimate (3.8), we obtain ⎛ ⎞1/2 2 M1 −t2 L 2 dx dt ⎟ ⎜ f1 (x) (t L) e ⎝ ⎠ t 2

' B

≤

⎛ ∞ k=0

⎞1/2 2 M1 −t2 L 2 dx dt ⎟ ⎜ (f1 χUk (B) )(x) (t L) e ⎝ ⎠ t ' B

f1 L2 (4B) +

∞ k=3

⎞1/2 ⎛r B k 2 dt r ) (2 B ⎠ f1 χUk (B) L2 (Rn ) ⎝ exp − ct2 t 0

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Generalized Vanishing Mean Oscillation Spaces

f1 L2 (4B) +

⎧ ∞ ⎨rB + k=3

ρ(|B|)|B|1/2

∞

⎩

t2 (2k rB )2

,n+1

0

135

⎫1/2 dt ⎬ f1 χUk (B) L2 (Rn ) t ⎭

2−k δk (f, B),

(3.12)

k=0

where c is a positive constant. Applying (3.9), Lemma 2.2 and M1 ≥ M to f2 yields that ⎞1/2 ⎛ 2 2 M1 −t2 L dx dt ⎟ ⎜ f2 (x) (t L) e ⎠ ⎝ t ' B

⎛ ⎞1/2 M 2 2 M1 −t2 L 2 −j dx dt ⎟ ⎜ (rB L) f1 (x) (t L) e ⎝ ⎠ t j=1 ' B

⎛ ⎞1/2 2 2j M ∞ 2 t 2 M1 −j −t2 L dx dt ⎟ ⎜ e (f1 χUk (B) )(x) (t L) ⎝ ⎠ 2 r t B j=1 k=0

' B

⎧ ⎤1/2 ⎡ ⎪ 2 rB 2 2j M ⎨ t dt ⎦ ⎣ 2 ⎪ rB t j=1 ⎩k=0 0

⎤1/2 ⎫ ⎡r ⎪ B 2 2j M ∞ k 2 t dt ⎦ ⎬ (2 rB ) ⎣ + exp − f1 χUk (B) L2 (Rn ) 2 ⎪ rB ct2 t ⎭ j=1 k=3

0

⎡r B ∞ ⎣ f1 L2 (4B) + k=3

ρ(|B|)|B|1/2

∞

t2 (2k rB )2

n+1

⎤1/2 dt ⎦ f1 χUk (B) L2 (Rn ) t

0

2−k δk (f, B).

(3.13)

k=0

The estimates (3.12) and (3.13) imply (3.11), and hence, completes the proof that (a) implies (b). n 2 M1 −t2 L ∞ 1 Conversely, let f ∈ MM e f ∈ Tω,v (Rn+1 + ). By ω,L (R ) and (t L) n [20, Theorem 6.1], we have that f ∈ BMOρ,L (R ). For any ball B, write ⎛ ⎞1/2 2 2 (I −e−rB2 L )M f (x)g(x) dx ⎝ (I −e−rB L )M f (x) dx⎠ = sup

g L2 (B) ≤1 B B 2 −rB L∗ M = sup ) g(x) dx . f (x)(I −e

g L2 (B) ≤1 n R

136

R. Jiang and D. Yang

IEOT

∗

Notice that for any g ∈ L2 (B), (I − e−rB L )M g is a multiple of an (ω, 2, older M, )L∗ -molecule; see [16, p. 43] and [20]. Then by Lemma 3.6 and the H¨ inequality, we obtain 2

⎛ ⎝

⎞1/2

2 −rB L

|(I − e

)M f (x)|2 dx⎠

B

2 2 ∗ 2 ∗ dx dt ∼ sup (t2 L)M1 e−t L f (x)t2 L∗ e−t L (I − e−rB L )M g(x) t

g L2 (B) ≤1 Rn+1 +

⎧ ⎫1/2 ⎪ ∞ ⎪ ⎨ ⎬ 2 2 M1 −t2 L dx dt f (x) (t L) e ⎪ t ⎪ ⎭ k=0 ⎩ Vk (B)

⎧ ⎫1/2 ⎪ ⎪ ⎨ ⎬ 2 2 2 ∗ −t2 L∗ dx dt −rB L∗ M × sup (I − e ) g(x) t L e t ⎪

g L2 (B) ≤1 ⎪ ⎩ ⎭ Vk (B)

≡

∞

σk (f, B)Ik ,

k=0 k B)\(2 k−1 B) for k ∈ N. In what follows, for ' and Vk (B) ≡ (2where V0 (B) ≡ B k−2 k k ≥ 2, let Vk,1 (B) ≡ (2 B)\(2 B × (0, ∞)) and Vk,2 (B) ≡ Vk (B)\Vk,1 (B). For k = 0, 1, 2, by Lemmas 2.2 and 2.3, we obtain

⎧ ⎫1/2 ⎪ ⎪ ⎨ ⎬ 2 2 ∗ −t2 L∗ 2 dx dt −rB L∗ M Ik = sup (I − e ) g(x) t L e t ⎪

g L2 (B) ≤1 ⎪ ⎩ ⎭ Vk (B) 2 ∗ sup (I − e−rB L )M g 2 n 1. L (R )

g L2 (B) ≤1

Now for k ≥ 3, write ⎧ ⎫1/2 ⎪ ⎪ ⎨ ⎬ 2 2 2 ∗ −t2 L∗ dx dt −rB L∗ M Ik sup (I − e ) g(x) t L e t ⎪

g L2 (B) ≤1 ⎪ ⎩ ⎭ Vk,1 (B)

⎧ ⎪ ⎨

+

sup

g L2 (B) ≤1 ⎪ ⎩ Vk,2 (B)

···

⎫1/2 ⎪ ⎬ ⎪ ⎭

≡ Ik,1 + Ik,2 .

Since for any (y, t) ∈ Vk,2 (B), t ≥ 2k−2 rB , by the Minkowski inequality and Lemma 2.2, we obtain

Vol. 67 (2010)

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137

⎫1/2 ⎧ 2 ⎡ ⎤M 2 ⎪ ⎪ r B ⎪ ⎪ ⎨ dx dt ⎬ ⎥ 2 ∗ −t2 L∗ ⎢ ∗ −sL∗ sup ds⎦ g(x) ⎣− L e t L e t ⎪

g L2 (B) ≤1 ⎪ ⎪ ⎪ ⎭ ⎩Vk,2 (B) 0 ⎧ 2 2 rB rB ⎪ ⎨ sup ··· |t2 (L∗ )M +1 ⎪

g L2 (B) ≤1 ⎩ 0

× e−(t

2

0

+s1 +···+sM )L∗

rB 2

Vk,2 (B)

···

sup

g L2 (B) ≤1

0

g(x)|2

⎧ k 2 rB ⎪ ⎨ 2 rB 0

⎪ ⎩ k−2 2

rB

dx dt t

1/2 ds1 · · · dsM t

(t2

4

g2L2 (B)

⎫1/2 ⎪ dt ⎬

+ s1 + · · · + sM )2(M +1) t ⎪ ⎭

× ds1 · · · dsM 2−2kM . Similarly, we have that Ik,1 2−2kM . Combining the above estimates and the fact that ρ is of upper type 1/pω − 1, we ﬁnally obtain ⎛ ⎞1/2 2 1 ⎝ |(I − e−rB L )M f (x)|2 dx⎠ ρ(|B|)|B|1/2 B

∞ k=0 ∞

2−2kM

1 σk (f, B) ρ(|B|)|B|1/2

2−k[2M −n( pω − 2 )] 1

1

k=0

σk (f, B) . k ρ(|2 B|)|2k B|1/2

2 n+1 n 1 1 ∞ ∞ Since (t2 L)M1 e−t L f ∈ Tω,v (Rn+1 ) ⊂ T (R ), by M > − ω + + 2 pω 2 and the dominated convergence theorem for series, we have ⎛ ⎞1/2 2 1 ⎝ |(I − e−rB L )M f (x)|2 dx⎠ γ1 (f ) = lim sup c→0 B: rB ≤c ρ(|B|)|B|1/2 B

∞ k=0

2−k[2M −n(

1 pω

− 12

)] lim sup

c→0 B: rB ≤c

σk (f, B) k ρ(|2 B|)|2k B|1/2

= 0.

n Similarly, γ2 (f ) = γ3 (f ) = 0, which implies that f ∈ VMOM ρ,L (R ), and hence, completes the proof of Theorem 3.5.

Remark 3.9. (1) It follows from Theorem 3.5 that for all M ∈ N and M > M n 1 1 n 2 ( pω − 2 ), the spaces VMOρ,L (R ) coincide with equivalent norms. n n Thus, in what follows, we denote VMOM ρ,L (R ) simply by VMOρ,L (R ); in particular, if ω(t) ≡ t for all t ∈ (0, ∞), then ρ(t) ≡ 1 and we denote n n VMOM ρ,L (R ) simply by VMOL (R ).

138 (2)

R. Jiang and D. Yang

IEOT

When A has real entries, or when the dimension n = 1 or n = 2 in the case of complex entries, the heat kernels always satisfy the Gaussian pointwise estimates [6], in these cases, the space VMOL (Rn ) here coincides with the one introduced by Deng et al. in [10].

4. Dual Spaces of VMOρ,L (Rn ) In this section, we identify the dual spaces of VMOρ,L (Rn ). We begin with some notions and known facts on tent spaces. is called a Tω (Rn+1 Recall that a function a on Rn+1 + + )-atom if there n ' exists a ball B ⊂ R such that supp a ⊂ B and ( B' |a(x, t)|2 dxt dt )1/2 ≤ 1 . ρ(|B|)|B|1/2 Definition 4.1. Let p ∈ (0, 1). The space Tω (Rn+1 + ) is deﬁned to be the space of 0 n+1 ∗ all f = j∈N λj aj in (Tω∞ (Rn+1 )) , where {a j }j∈N are Tω (R+ )-atoms and + 0 {λj }j∈N ∈ 1 . If f ∈ Tω (Rn+1 ), then deﬁne f n+1 ≡ inf{ |λj |}, +

Tω (R+

)

j∈N

where the infimum is taken over all the possible decompositions of f as above. By [16, Lemma 3.1], Tω (Rn+1 + ) is a Banach space. Moreover, from Defin+1 nition 4.1, it is easy to deduce that Tω (Rn+1 + ) is dense in Tω (R+ ); in other n+1 n+1 words, Tω (R+ ) is a Banach completion of Tω (R+ ). The following lemma is just [19, Lemma 4.1]. n+1 Lemma 4.2. Tω (Rn+1 + ) is a dense subspace of Tω (R+ ) and there exists a posn+1 itive constant C such that for all f ∈ Tω (R+ ), f Tω (Rn+1 ) ≤ Cf Tω (Rn+1 ) . +

+

The following theorem was established in [19, Theorem 4.2]; see also [26]. n+1 ∞ ∗ Theorem 4.3. (Tω,v (Rn+1 + )) = Tω (R+ ).

Now, let us recall some notions on the Hardy spaces associated with L. For all f ∈ L2 (Rn ) and x ∈ Rn , deﬁne ⎞1/2 ⎛ 2 dy dt ⎟ ⎜ |t2 Le−t L f (y)|2 n+1 ⎠ . SL f (x) ≡ ⎝ t Γ(x)

The space Hω,L (Rn ) is deﬁned to be the completion of the set {f ∈ L2 (Rn ) : SL f ∈ L(ω)} with respect to the quasi-norm f Hω,L (Rn ) ≡ SL f L(ω) . The Orlicz–Hardy space Hω,L (Rn ) was introduced and studied in [20]. If ω(t) ≡ t for all t ∈ (0, ∞), then the space Hω,L (Rn ) coincides with the Hardy space HL1 (Rn ), which was introduced and studied by Hofmann and Mayboroda [16] (see also [17]). Definition 4.4. Let M ∈ N, M > n2 p1ω − 12 and ∈ n p1ω − p1+ , ∞ . ω

M, An element f ∈ (BMOρ,L∗ (Rn ))∗ is said to be in the space Hω,L (Rn ) if

Vol. 67 (2010)

Generalized Vanishing Mean Oscillation Spaces

139

∞ there exist {λj }∞ j=1 ⊂ C and (ω, 2, M, )L -molecules {αj }j=1 such that f = 0∞ n ∗ ∗ j=1 λj αj in (BMOρ,L (R )) and ⎧ ⎫ ∞ ⎨ ⎬ |λj | Λ({λj αj }j ) ≡ inf λ > 0 : |Bj |ω ≤ 1 < ∞, ⎩ ⎭ λ|Bj |ρ(|Bj |) j=1

where for each j, αj is adapted to the ball Bj . M, If f ∈ Hω,L (Rn ), then its norm is deﬁned by f H M, (Rn ) ≡ inf Λ({λj ω,L αj }j ), where the infimum is taken over all the possible decompositions of f as above. It was proved in [20, Theorem 5.1] that for all M > n2 p1ω − 12 and M, ∈ n p1ω − p1+ , ∞ , the spaces Hω,L (Rn ) and Hω,L (Rn ) coincide with ω equivalent norms. Let us introduce the Banach completion of the space Hω,L (Rn ). Definition 4.5. Let ∈ n p1ω − p1+ , ∞ and M > n2 p1ω − 12 . The space ω 0 M, Bω,L (Rn ) is deﬁned to be the space of all f = j∈N λj αj in (BMOρ,L∗ (Rn ))∗ , M, (Rn ), where {λj }j∈N ∈ 1 and {αj }j∈N are (ω, 2, M, )L -molecules. If f ∈ Bω,L 0 deﬁne f B M, (Rn ) ≡ inf{ j∈N |λj |}, where the infimum is taken over all the ω,L possible decompositions of f as above. M, (Rn ) is a Banach space. By [16, Lemma 3.1] again, we see that Bω,L Moreover, from Definition 4.4, it is easy to deduce that Hω,L (Rn ) is dense in M, Bω,L (Rn ). More precisely, we have the following lemma. Lemma 4.6. Let ∈ n p1ω − p1+ , ∞ and M > n2 p1ω − 12 . Then ω

(1) (2)

M, (Rn ) and the inclusion is continuous. Hω,L (R ) ⊂ Bω,L For any 1 ∈ n p1ω − p1+ , ∞ and M1 > n2 p1ω − 12 , ω M, M1 ,1 Bω,L (Rn ) and Bω,L (Rn ) coincide with equivalent norms. n

the spaces

Proof. From Definition 4.5 and the molecular characterization of Hω,L (Rn ), it is easy to deduce (1). Let us prove (2). By symmetry, it sufﬁces to show M, M1 ,1 (Rn ) ⊂ Bω,L (Rn ). that Bω,L M, (Rn ). By Definition 4.5, there exist (ω, 2, M, )L -molecules Let f ∈ Bω,L 0 {αj }j∈N and {λj }j∈N ⊂ C such that f = j∈N λj αj in (BMOρ,L∗ (Rn ))∗ and 0 j∈N |λj | f B M, (Rn ) . ω,L

M1 ,1 (Rn ) and By (1), for each j ∈ N, we have that αj ∈ Hω,L (Rn ) ⊂ Bω,L M1 ,1 n αj B M1 ,1 (Rn ) αj Hω,L (Rn ) 1. Since Bω,L (R ) is a Banach space, we ω,L 0 M1 ,1 p,1 (Rn ) and f B M1 ,1 (Rn ) ≤ j∈N |λj |αj BL,M obtain that f ∈ Bω,L (Rn ) ω,L

1

M, M1 ,1 (Rn ) ⊂ Bω,L (Rn ), which completes the proof of f B M, (Rn ) . Thus, Bω,L ω,L Lemma 4.6.

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M, Since the spaces Bω,L (Rn ) coincide for all ∈ n p1ω − p1+ , ∞ and ω M, n 1 1 n M > 2 pω − 2 , in what follows, we denote Bω,L (R ) simply by Bω,L (Rn ). Lemma 4.7. (Bω,L )∗ = BMOρ,L∗ (Rn ). Proof. Since (Hω,L (Rn ))∗ = BMOρ,L∗ (Rn ) and Hω,L (Rn ) ⊂ Bω,L (Rn ), by n duality, we have that (Bω,L(Rn ))∗ ⊂ BMO ρ,L ∗ (R ). Conversely, let ∈ n p1ω − p1+ , ∞ , M > n2 p1ω − 12 and f ∈ ω

BMOρ,L∗ (Rn ). For any g ∈ Bω,L (Rn ), by Definition 4.5, there exist 0 {λj }j∈N ⊂ C and (ω, M, 2, )L -molecules {αj }j∈N such that g = j∈N λj αj in 0 (BMOρ,L∗ (Rn ))∗ and j∈N |λj | ≤ 2gBω,L (Rn ) . Thus, |f, g | ≤

|λj ||f, αj |

j∈N

|λj |f BMOρ,L∗ (Rn ) αj Hω,L (Rn )

j∈N

f BMOρ,L∗ (Rn ) gBω,L (Rn ) , which implies that f ∈ (Bω,L (Rn ))∗ , and hence, completes the proof of Lemma 4.7. Let M ∈ N. For all F ∈ L2 (Rn+1 + ) with compact support, deﬁne ∞ 2 dt πL,M F ≡ CM (t2 L)M e−t L F (·, t) , t

(4.1)

0

where CM is the positive constant same as in (3.3). Proposition 4.8. Let M ∈ N. Then the operator πL,M which is initially deﬁned 2 on T2,c (Rn+1 + ), extends to a bounded linear operator (1)

p n L); from T2p (Rn+1 + ) to L (R ), if p ∈ (pL , p

(2)

n from Tω (Rn+1 + ) to Hω,L (R ), if M >

n 2

(3)

n from Tω (Rn+1 + ) to Bω,L (R ), if M >

n 2

(4)

from

∞ Tω,v (Rn+1 + )

n

1 pω

−

1 2

1 pω

−

1 2

; ;

to VMOρ,L (R ).

Proof. (1) and (2) were established in [20]. Let us prove (3). n+1 2 By [19, Lemma 4.7], we have that T2,c (Rn+1 + ) is dense in Tω (R+ ). Let n+1 2 f ∈ T2,c (R+ ). By (2) and Lemma 4.6, we obtain that πL,M f ∈ Hω,L (Rn ) ⊂ Bω,L (Rn ). Moreover, by Definition 4.1, there exist Tω (Rn+1 {a } + )-atoms 0 0 j j∈N ∗ and {λj }j∈N ⊂ C such that f = j∈N λj aj in (Tω∞ (Rn+1 + )) and j |λj | f Tω (Rn+1 ) . In addition, for any g ∈ BMOρ,L∗ (Rn ), by [20, Theorem 6.1], we +

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∗

n+1 ∗ have that (t2 L∗ )M e−t L g ∈ Tω∞ (Rn+1 + ). Thus, by the fact that (Tω (R+ )) = n+1 ∞ Tω (R+ ) [21, Proposition 4.1], we obtain 2

f (x, t)(t2 L∗ )M e−t

2

πL,M (f ), g = CM

=

Rn+1 +

λj CM

j∈N

=

L∗

g(x)

aj (x, t)(t2 L∗ )M e−t

2

dx dt t L∗

g(x)

Rn+1 +

dx dt t

λj πL,M (aj ), g ,

j∈N

which implies that πL,M (f ) = we further obtain πL,M (f )Bω,L (Rn ) ≤

0

j∈N

λj πL,M (aj ) in (BMOρ,L∗ (Rn ))∗ . By (2),

|λj |πL,M (aj )Bω,L (Rn )

j∈N

j

|λj |πL,M (aj )Hω,L (Rn ) f Tω (Rn+1 ) . +

n+1 2 (Rn+1 Since T2,c + ) is dense in Tω (R+ ), we see that πL,M extends to a bounded n+1 linear operator from Tω (R+ ) to Bω,L (Rn ), which completes the proof of (3). 2 Let us now prove (4). By Lemma 3.4, we obtain that T2,c (Rn+1 + ) is n+1 ∞ dense in Tω,v (R+ ). Thus, to prove (4), it sufﬁces to show that πL,M maps 2 n (Rn+1 T2,c + ) continuously into VMOρ,L (R ). n+1 2 Let f ∈ T2,c (R+ ). By (1), we see that πL,M f ∈ L2 (Rn ). Notice that n 1 (3.1) and (3.2) with L and L∗ exchanged implies that L2 (Rn ) ⊂ MM ω,L (R ), n 1 where M1 ∈ N and M1 > n2 p1ω − 12 . Thus, πL,M f ∈ MM ω,L (R ). To n show πL,M f ∈ VMOρ,L (R ), by Theorem 3.5, we only need to verify that 2 ∞ (t2 L)M1 e−t L πL,M f ∈ Tω,v (Rn+1 + ). k B)\(2 k−1 B) ' and Vk (B) ≡ (2For any ball B ≡ B(xB , rB ), let V0 (B) ≡ B

for any k ∈ N. For all k ∈ Z+ , let fk ≡ f χVk (B) . Thus, for k = 0, 1, 2, by Lemma 2.3 and (1), we obtain ⎞1/2 ⎛ 2 dx dt ⎟ ⎜ |(t2 L)M1 e−t L πL,M fk (x)|2 πL,M fk L2 (Rn ) ⎠ ⎝ t ' B

fk T 2 (Rn+1 ) . 2

+

k B)\(2k−2 B×(0, ∞)) and V For k ≥ 3, let Vk,1 (B) ≡ (2k,2 (B) ≡ Vk (B)\Vk,1 (B). We further write fk = fk χVk,1 (B) +fk χVk,2 (B) ≡ fk,1 +fk,2 . By the Minkowski

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inequality, Lemma 2.2 and the H¨ older inequality, we obtain ⎞1/2 ⎛ 2 M1 −t2 L 2 dx dt ⎟ ⎜ πL,M fk,2 (x) (t L) e ⎠ ⎝ t ' B

k 2 ⎞1/2 2 rB 2 2 ds dx dt⎟ (t2 L)M1 e−t L (s2 L)M e−s L (fk,2 (·, s))(x) ⎠ s t k−2 ' 2 rB B ⎛ ⎞1/2 k 2 rB ds 2M1 2M M +M1 −(s2 +t2 )L 2 dx dt ⎟ ⎜ s L e (fk,2 (·, s))(x) t ⎝ ⎠ t s

⎛ ⎜ = CM ⎝

2k−2 rB 2k rB

' B

⎞1/2 ⎛r 2 B t2M1 s2M dt ⎠ ds 2 ⎝ (s2 + t2 )M +M1 fk,2 (·, s)L2 (Rn ) t s

2k−2 rB −2kM1

0 2k rB

2

fk,2 (·, s)L2 (Rn )

ds 2−2kM1 fk,2 T 2 (Rn+1 ) . 2 + s

2k−2 rB

Similarly, we have ⎛ ⎞1/2 2 2 M1 −t2 L dx dt ⎟ ⎜ πL,M fk,1 (x) 2−2kM1 fk,1 T 2 (Rn+1 ) . (t L) e ⎝ ⎠ 2 + t ' B

Combining the above estimates, we ﬁnally obtain that ⎛ ⎞1/2 2 2 1 2 M1 −t L dx dt ⎟ ⎜ πL,M f (x) (t L) e ⎝ ⎠ t ρ(|B|)|B|1/2 ' B

2

1 1/2 ρ(|B|)|B| k=0

+

∞ 2 k=3 i=1

⎛ ⎞1/2 2 2 M1 −t2 L dx dt ⎟ ⎜ πL,M fk (x) (t L) e ⎝ ⎠ t ' B

1 ρ(|B|)|B|1/2

⎛

⎞1/2 2 dx dt ⎟ 2 M1 −t2 L ⎜ πL,M fk,i (x) (t L) e ⎝ ⎠ t ' B

∞ 1 2−2kM1 f fk,i T 2 (Rn+1 ) 2 (Rn+1 ) + k T 2 2 + + ρ(|B|)|B|1/2 ρ(|B|)|B|1/2 k=0 k=3 i=1,2 2

∞ k=0

2−k[2M1 −n( pω −1/2)] 1

1 fk T 2 (Rn+1 ) , 2 + ρ(|2k B|)|2k B|1/2

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n+1 ∞ ∞ (Rn+1 where 2M1 > n(1/pω − 1/2). Since f ∈ Tω,v + ) ⊂ Tω (R+ ), we have

1 ρ(|2k B|)|2k B|1/2

fk T 2 (Rn+1 ) f T ∞ (Rn+1 ) , 2

ω

+

+

and for all ﬁxed k ∈ N, fk T 2 (Rn+1 ) 2

lim sup

c→0 B: rB ≤c

+

ρ(|2k B|)|2k B|1/2

= lim

sup

fk T 2 (Rn+1 ) 2

c→∞ B: rB ≥c

= lim

c→∞

+

ρ(|2k B|)|2k B|1/2 fk T 2 (Rn+1 ) 2

sup B: B⊂[B(0,c)]

+

ρ(|2k B|)|2k B|1/2

= 0.

Thus, by Theorem 3.5, we have πL,M f BMOρ,L (Rn ) ∼ (t2 L)M1 e−t L πL,M f T ∞ (Rn+1 ) f T ∞ (Rn+1 ) , 2

ω

ω

+

+

and by the dominated convergence theorem for series, η1 ((t2 L)M1 e−t L πL,M f ) 2

⎛ ⎞1/2 2 1 ⎝ |(t2 L)M1 e−t L πL,M f (x)|2 dx⎠ = lim sup c→0 B: rB ≤c ρ(|B|)|B|1/2 B

∞

2−k[2M1 −n(1/p−1/2)] lim sup

c→0 B: rB ≤c

k=0

fk T 2 (Rn+1 ) 2

+

ρ(|2k B|)|2k B|1/2

= 0.

Similarly, we have that η2 ((t2 L)M1 e−t L πL,M f ) = η3 ((t2 L)M1 e−t L πL,M f ) = 2 ∞ 0, and thus, (t2 L)M1 e−t L πL,M f ∈ Tω,v (Rn+1 + ), which completes the proof of Proposition 4.8. 2

2

Lemma 4.9. VMOρ,L (Rn ) ∩ L2 (Rn ) is dense in VMOρ,L (Rn ). Proof. Let f ∈ VMOρ,L (Rn ) and M > n2 p1ω − 12 . Then Theorem 3.5 tells ∞ us that h ≡ (t2 L)M e−t L f ∈ Tω,v (Rn+1 + ). Similarly to the proof of Propon+1 2 ∞ (Rn+1 sition 4.8, by Lemma 3.4, there exist {hk }k∈N ⊂ T2,c + ) ⊂ Tω,v (R+ ) such that h − hk T ∞ (Rn+1 ) → 0, as k → ∞. Thus, by Proposition 4.8 (1) 2

ω

+

and (4), we obtain that πL,1 hk ∈ L2 (Rn ) ∩ VMOρ,L (Rn ) and

πL,1 (h − hk )BMOρ,L (Rn ) h − hk T ∞ (Rn+1 ) → 0, ω

+

(4.2)

as k → ∞. Let α be an (ω, 2, M, )L∗ -molecule. Then by the definition of 2 ∗ Hω,L∗ (Rn ), we have that e−t L α ∈ Tω (Rn+1 + ), which, together with n+1 ∗ ∞ Lemma 3.6, the facts that (Tω (R+ )) = Tω (Rn+1 + ) [21, Proposition 4.1]

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and (Hω,L∗ (Rn ))∗ = BMOρ,L (Rn ), further implies that 2 2 ∗ dx dt f (x)α(x) dx = CM (t2 L)M e−t L f (x)t2 L∗ e−t L α(x) t Rn

Rn+1 +

hk (x, t)t2 L∗ e−t

2

= lim CM k→∞

L∗

α(x)

Rn+1 +

=

CM lim C1 k→∞

(πL,1 hk (x))α(x) dx =

Rn

dx dt t

CM πL,1 h, α . C1

Since the set of ﬁnite combinations of molecules is dense in Hω,L∗ (Rn ), we then obtain that f = CCM1 πL,1 h in BMOρ,L (Rn ). Now, for each k ∈ N, let fk ≡ CCM1 πL,1 hk . Then fk ∈ VMOρ,L (Rn ) ∩ 2 n L (R ); moreover, by (4.2), we have f − fk BMOρ,L (Rn ) → 0, as k → ∞, which completes the proof of Lemma 4.9. In what follows, the symbol ·, · in the following theorem means the duality between the space BMOρ, L (Rn ) and the space Bω,L∗ (Rn ) in the sense of Lemma 4.7 with L and L∗ exchanged. Let us now state the main theorem of this paper. Theorem 4.10. The dual space of VMOρ,L (Rn ), (VMOρ,L (Rn ))∗ , coincides with the space Bω,L∗ (Rn ) in the following sense: (1)

For any g ∈ Bω, L∗ (Rn ), deﬁne the linear functional by setting, for all f ∈ VMOρ, L (Rn ), (f ) ≡ f, g .

(4.3)

Then there exists a positive constant C independent of g such that (VMOρ, L (Rn ))∗ ≤ CgBω, L∗ (Rn ) . (2)

Conversely, for any ∈ (VMOρ, L (Rn ))∗ , there exists g ∈ Bω, L∗ (Rn ), such that (4.3) holds and there exists a positive constant C independent of such that gBω, L∗ (Rn ) ≤ C(VMOρ, L (Rn ))∗ .

Proof. By Lemma 4.7, we have (Bω,L∗ (Rn ))∗ = BMOρ,L (Rn ). Definition 3.3 implies that VMOρ,L (Rn ) ⊂ BMOρ,L (Rn ), which further implies that Bω,L∗ (Rn ) ⊂ (VMOρ,L (Rn ))∗ . Conversely, let M > n2 p1ω − 12 and ∈ (VMOρ,L (Rn ))∗ . By Proposi∞ n (Rn+1 tion 4.8, πL,1 is bounded from Tω,v + ) to VMOρ,L (R ), which implies that n+1 ∞ ◦ πL,1 is a bounded linear functional on Tω,v (R+ ). Thus, by Theorem 4.3, n+1 ∞ there exists g ∈ Tω (Rn+1 + ) such that for all f ∈ Tω,v (R+ ), ◦ πL,1 (f ) = f, g . n 2 n Now, suppose that f ∈ VMOρ,L (R )∩L (R ). By Theorem 3.5, we have 2 ∞ that (t2 L)M e−t L f ∈ Tω,v (Rn+1 + ). Moreover, by the proof of Lemma 4.9, we

Vol. 67 (2010) see that f =

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145

πL,1 ((t2 L)M e−t L f ) in BMOρ,L (Rn ). Thus, 2

2 CM ◦ πL,1 ((t2 L)M e−t L f ) C1 2 CM dx dt = . (t2 L)M e−t L f (x)g(x, t) C1 t

(f ) =

(4.4)

Rn+1 +

n+1 2 By [19, Lemma 4.7], T2,c (Rn+1 + ) is dense in Tω (R+ ). Since n+1 2 g ∈ Tω (Rn+1 + ), we may choose {gk }k∈N ⊂ T2,c (R+ ) such that gk → g n+1 in Tω (R+ ). By Proposition 4.8 (3), we have that πL∗ ,M (g), πL∗ ,M (gk ) ∈ Bω,L∗ (Rn ) and

πL∗ ,M (g − gk )Bω,L∗ (Rn ) g − gk Tω (Rn+1 ) → 0, +

as k → ∞. This, together with (4.4), Theorem 4.3, the dominated convergence theorem and Lemma 4.7, implies that 2 CM dx dt (f ) = lim (t2 L)M e−t L f (x)gk (x, t) C1 k→∞ t Rn+1 +

CM = lim C1 k→∞

Rn

∞ 2 ∗ dt f (x) (t2 L∗ )M e−t L (gk (·, t))(x) dx t 0

1 1 = lim f, πL∗ ,M (gk ) = f, πL∗ ,M (g) . C1 k→∞ C1

(4.5)

Since VMOρ,L (Rn ) ∩ L2 (Rn ) is dense in VMOρ,L (Rn ) (Lemma 4.9), we ﬁnally obtain that (4.5) holds for all f ∈ VMOρ,L (Rn ), and (VMOρ,L (Rn ))∗ = 1 n ∗ ∗ n ⊂ C1 πL ,M gBω,L∗ (R ) . In this sense, we have that (VMOρ,L (R )) n Bω,L∗ (R ), which completes the proof of Theorem 4.10. Remark 4.11. If ω(t) ≡ t for all t ∈ (0, ∞), then by Definitions 4.4 and 4.5, we have that Bω,L∗ (Rn ) = Hω,L∗ (Rn ) = HL1 ∗ (Rn ), where HL1 ∗ (Rn ) was the Hardy space introduced by Hofmann and Mayboroda in [16]. By Theorem 4.10, we obtain that (VMOρ,L (Rn ))∗ = (VMOL (Rn ))∗ = HL1 ∗ (Rn ). In what follows, let ω(t) ≡ t for all t ∈ (0, ∞). We denote (ω, 2, M, )L molecule simply by (1, 2, M, )L -molecule. We now compare the space VMOL (Rn ) with the classical space CMO (Rn ) introduced by Coifman and Weiss [9]. Recall that the space BMO(Rn ) is deﬁned to be the set of all f ∈ L1loc (Rn ) that satisfy 1 f BMO(Rn ) ≡ sup |f (x) − fB | dx < ∞, B⊂Rn |B|

B

where fB ≡ f (x) dx. Then the space CMO (Rn ) is deﬁned to be the closure of Cc (R ) (the set of all continuous functions with compact support) in the norm · BMO(Rn ) . 1 |B| B n

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Proposition 4.12. For all n ∈ N, the space CMO (Rn ) is continuously embedded in the space VMOL (Rn ). Moreover, when n ≥ 3, then there exists L as in (1.2) such that CMO (Rn ) is a proper subset of VMOL (Rn ). Proof. Let M > n4 and ∈ (0, ∞). Notice that each (1, 2, M, )L∗ -molecule is a classical H 1 (Rn )-molecule up to a harmless constant; [16, p. 41] or [20, Remark 7.1]. Thus, we have that HL1 ∗ (Rn ) ⊂ H 1 (Rn ) and for all f ∈ HL1 ∗ (Rn ), f H 1 (Rn ) f HL1 ∗ (Rn ) . Since (HL1 ∗ (Rn ))∗ = BMOL (Rn ) and (H 1 (Rn ))∗ = BMO(Rn ), we further obtain that BMO(Rn ) ⊂ BMOL (Rn ), and for all g ∈ BMO(Rn ), gBMOL (Rn ) gBMO(Rn ) . Let us now show that Cc (Rn ) ⊂ VMOL (Rn ). Suppose that f ∈ Cc (Rn ) and B ≡ B(xB , rB ). By an argument as in [1, Section 2.5], we have that for all t > 0, e−tL 1 = 1 in the L2loc (Rn ) sense, that is, for any φ ∈ L2 (Rn ) with compact support, there holds ∗ φ(x) dx = e−tL 1φ(x) dx = e−tL φ(x) dx. Rn

Rn

Rn

This together with Lemma 2.2 and the H¨ older inequality implies that (I − e−rB L )M f L2 (B) 2 −rB L M = sup ) f (x)g(x) dx (I − e

g L2 (B) ≤1 B 2 −rB L M = sup ) (f − fB )(x)g(x) dx (I − e

g L2 (B) ≤1 B ∞ 2k r2 sup exp − 2B f − fB L2 (Uk (B)) gL2 (B) crB

g L2 (B) ≤1 2

k=0

∞ k=0 ∞

f − f2k B L2 (Uk (B)) + |Uk (B)|1/2 |fB − f2k B |

e−2

k

/c

e−2

k

/c kn/2

2

f − f2k B L2 (Uk (B)) ,

k=0

where c is a positive constant. Since f ∈ Cc (Rn ) ⊂ CMO(Rn ), by the dominated convergence theorem for series, [10, Proposition 3.5] and the fact that CMO(Rn ) ⊂ VMO(Rn ), we have 2 1 (I − e−rB L )M f L2 (B) 1/2 c→0 B: rB ≤c |B| ∞ k 1 e−2 /c 2kn lim sup f − f2k B L2 (Uk (B)) = 0. c→0 B: rB ≤c |2k B|1/2 k=0

γ1 (f ) = lim sup

Similarly, we have that γ2 (f ) = γ3 (f ) = 0, which implies that f ∈ VMOL (Rn ), and hence Cc (Rn ) ⊂ VMOL (Rn ). Since Cc (Rn ) is dense in CMO (Rn ), by the

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fact that for all g ∈ CMO(Rn ), gVMOL (Rn ) = gBMOL (Rn ) gBMO(Rn ) ∼ gCMO(Rn ) , we ﬁnally obtain that CMO (Rn ) ⊂ VMOL (Rn ). On the other hand, if n ≥ 3, then there exist L as in (1.2) and p ∈ (1, 2) such that ∇(L∗ )−1/2 is not bounded on Lp (Rn ); [1,14]. By the fact that the Riesz transform ∇(L∗ )−1/2 is bounded on L2 (Rn ) and from HL1 ∗ (Rn ) to L1 (Rn ) [16,20], we obtain that HL1 ∗ (Rn ) is a proper subspace of H 1 (Rn ). Otherwise, HL1 ∗ (Rn ) = H 1 (Rn ), then by the interpolation theorem, ∇(L∗ )−1/2 is bounded on Lp (Rn ) for all p ∈ (1, 2), which contradicts with the fact that ∇(L∗ )−1/2 is not bounded on Lp (Rn ) for some p ∈ (1, 2). Now by Theorem 4.10, we have that (VMOL (Rn ))∗ = HL1 ∗ (Rn ) H 1 (Rn ) = (CMO (Rn ))∗ , which implies that CMO (Rn ) VMOL (Rn ), and hence, completes the proof of Proposition 4.12. From Proposition 4.12, we further deduce the following conclusion. Corollary 4.13. When A has real entries, or when the dimension n = 1 or n = 2 in the case of complex entries, the spaces CMO (Rn ) and VMOL (Rn ) coincide with equivalent norms. To prove Corollary 4.13, we need the following lemma. (We are very grateful to the referee for her/him to tell us Corollary 4.13 and its proof.) Lemma 4.14. Let X and Y be two Banach spaces such that X ⊂ Y with continuous embedding. Assume that X ∗ and Y ∗ coincide with equivalent norms. Then X = Y with equivalent norms. Proof. Let J : X → Y be the inclusion map, which is a continuous linear map by assumption. As is easily seen, its adjoint J ∗ : Y ∗ → X ∗ is the map which, to any φ ∈ Y ∗ , associates its restriction to X. The Hahn-Banach theorem yields that J ∗ is onto. Moreover, it is also one-to-one. Indeed, if this was not true, there would exist φ ∈ Y ∗ identically vanishing on X but not on Y , which would contradict the assumption that X ∗ and Y ∗ coincide with equivalent norms. Thus, J ∗ is a continuous isomorphism, and, by Theorem 4.15 in [23], J is an isomorphism between X and Y , which exactly means that X = Y . From this and Corollaries 2.12 (c) of the open mapping theorem in [23, pp. 49-50], it follows that X and Y coincide with equivalent norms, which completes the proof of Lemma 4.14. Proof of Corollary 4.13. When A has real entries, or when the dimension n = 1 or n = 2 in the case of complex entries, the heat kernel always satisﬁes the Gaussian pointwise estimates in size and regularity [6], and the spaces HL1 ∗ (Rn ) and H 1 (Rn ) coincide with equivalent norms [5,27]. By this and Proposition 4.12, we obtain that CMO (Rn ) ⊂ VMOL (Rn ) and (CMO (Rn ))∗ = H 1 (Rn ) = HL1 ∗ (Rn ) = (VMOL (Rn ))∗ , which together with Lemma 4.14 implies that CMO (Rn ) and VMOL (Rn ) coincide with equivalent norms. This ﬁnishes the proof of Corollary 4.13.

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Acknowledgements The authors sincerely wish to express their deeply thanks to the referee for her/his very carefully reading and also her/his so many valuable and helpful remarks which made this article more readable. In particular, Corollary 4.13 and its proof belong to the referee.

References [1] Auscher, P.: On necessary and sufﬁcient conditions for Lp -estimates of Riesz transforms associated to elliptic operators on Rn and related estimates. Mem. Am. Math. Soc. 186, 1–75 (2007) [2] Auscher, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, (2005) [3] Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., Tchamitchian, Ph.: The solution of the Kato square root problem for second order elliptic operators on Rn . Ann. Math. (2) 156, 633–654 (2002) [4] Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18, 192–248 (2008) [5] Auscher, P., Russ, E.: Hardy spaces and divergence operators on strongly Lipschitz domains of Rn . J. Funct. Anal. 201, 148–184 (2003) [6] Auscher, P., Tchamitchian, P.: Square root problem for divergence operators and related topics. Ast´erisque 249, 1–172 (1998) [7] Bourdaud, G.: Remarques sur certains sous-espaces de BMO(Rn ) et de bmo(Rn ). Ann. Inst. Fourier (Grenoble) 52, 1187–1218 (2002) [8] Coifman, R.R., Meyer, Y., Stein, E.M.: Some new functions and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335 (1985) [9] Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977) [10] Deng, D., Duong, X.T., Song, L., Tan, C., Yan, L.: Functions of vanishing mean oscillation associated with operators and applications. Michigan Math. J. 56, 529–550 (2008) [11] Duong, X.T., Yan, L.: New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Commun. Pure Appl. Math. 58, 1375–1420 (2005) [12] Duong, X.T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005) [13] Feﬀerman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972) [14] Frehse, J.: An irregular complex valued solution to a scalar uniformly elliptic equation. Calc. Var. Partial Differ Equ 33, 263–266 (2008) [15] Hofmann, S., Martell, J.M.: Lp bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ. Mat. 47, 497–515 (2003) [16] Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344, 37–116 (2009) [17] Hofmann, S., Mayboroda, S.: Correction to Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann., to appear (arXiv: 0907.10129)

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[18] Janson, S.: Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J. 47, 959–982 (1980) [19] Jiang, R., Yang, D.: Predual spaces of Banach completions of Orlicz– Hardy spaces associated with operators. J. Fourier Anal. Appl. doi:10.1007/ s00041-010-9123-8 (arXiv: 0906.1880) (2010) [20] Jiang, R., Yang, D.: New Orlicz–Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258, 1167–1224 (2010) [21] Jiang, R., Yang, D., Zhou, Y.: Orlicz–Hardy spaces associated with operators. Sci. China Ser. A 52, 1042–1080 (2009) [22] John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961) [23] Rudin, W.: Functional Analysis, Second Edition. McGraw-Hill, Inc., New York (1991) [24] Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975) [25] Viviani, B.E.: An atomic decomposition of the predual of BMO(ρ). Rev. Mat. Iberoamericana 3, 401–425 (1987) [26] Wang, W.: The predual spaces of tent spaces and some characterizations of λα (Rn ) spaces. Beijing Daxue Xuebao 24, 535–551 (1988) [27] Yan, L.: Classes of Hardy spaces associated with operators, duality theorem and applications. Trans. Am. Math. Soc. 360, 4383–4408 (2008) Renjin Jiang and Dachun Yang (B) School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education 100875 Beijing People’s Republic of China e-mail: [email protected] Present address Renjin Jiang Department of Mathematics and Statistics University of Jyv¨ askyl¨ a P. O. Box 35 (MaD) 40014 Jyv¨ askyl¨ a Finland e-mail: [email protected] Received: July 15, 2009. Revised: November 28, 2009.

Integr. Equ. Oper. Theory 67 (2010), 151–161 DOI 10.1007/s00020-010-1744-4 Published online April 20, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Quasi-wandering Subspaces in the Bergman Space Kei Ji Izuchi, Kou Hei Izuchi and Yuko Izuchi Abstract. Let B be the Bergman shift on the Bergman space L2a over the open unit disk and let I be a nontrivial invariant subspace of L2a . Let PI be the orthogonal projection from L2a onto I. It is proved that PI B(L2a I) is not dense in I if and only if I ∩ D = {0}, where D is the Dirichlet space. It is also discussed some related topics. Mathematics Subject Classiﬁcation (2010). Primary 47A15; Secondary 32A35 47B35. Keywords. Bergman space, invariant subspace, quasi-wandering subspace.

1. Introduction Let T : H → H be a bounded linear operator on a separable Hilbert space H. For an invariant subspace M of H for T , the space M T M is called the wandering subspace for M . In this paper, the space PM T (H M ) is called the quasi-wandering subspace for M , where PM is the orthogonal projection from H onto M . By the deﬁnitions, a quasi-wandering subspace is like as a counterpart of a wandering subspace. Let H 2 be the Hardy space on the open unit disk D. We denote by Tz the operator on H 2 deﬁned by Tz f (z) = zf (z) for f ∈ H 2 . Then Tz is an isometry on H 2 . For a subset E of H 2 , we denote by [E] the smallest invariant subspace of H 2 for Tz containing E. Let M be an invariant subspace of H 2 for Tz with M = {0} and M = H 2 . By the Beurling theorem, M = θH 2 for a nonconstant inner function θ. We have M Tz M = C · θ Tz∗ (H 2

2

and

[M Tz M ] = M.

M ) ⊂ (H M ) and PM Tz (H 2 M ) = C · θ. Hence We also have 2 [PM Tz (H M )] = M and [H 2 M ] = H 2 . In the one variable Hardy space case, a wandering subspace coincides with a quasi-wandering subspace for M .

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Let L2a = L2a (D) be the Bergman space on D. We denote by B the operator on L2a deﬁned by Bf = zf for f ∈ L2a . The operator B is usually called the Bergman shift. For a subset E of L2a , we denote by [E]L2a the smallest invariant subspace for B containing E. Let I be an invariant subspace of L2a for B with I = {0} and I = L2a . Since B is bounded below, BI is an invariant subspace. It is known that the dimension of I BI ranges from 1 to ∞, see [2, 5, 7]. In 1996, Aleman, Richter, and Sundberg [1] proved that [I BI]L2a = I. Diﬀerent proofs of this theorem are given in [12, 13]. The purpose of this paper is to study quasi-wandering subspaces in L2a . By the ﬁrst impression, the quasi-wandering subspace PI B(L2a I) is not so big in I. But this is not correct. In Section 2, we prove that PI B(L2a I) is not dense in I if and only if I ∩ D = {0}, where D is the Dirichlet space on D. If dim(I BI) ≥ 2, then it is known that I ∩ H 2 = {0}, see [9], so in this case PI B(L2a I) is dense in I. If I ∩ H 2 = {0}, then by the Beurling theorem I ∩ H 2 = θH 2 for an inner function θ. In this case, a problem whether I ∩ D = {0} or not concerns deeply with the work of Carleson [3]. If {0} holds. dim(L2a I) < ∞, then trivially I ∩ D = In Section 3, we show that [PI B(L2a I)]L2a = I and this is considered as a counterpart of Aleman, Richter, and Sundberg’s theorem. To prove this fact, in the original manuscript we worked on the Hardy space H 2 (D2 ) over the bidisk D2 , and we used similar techniques given by Sun and Zheng [13]. But the referee pointed out much easier argument of the proof. We shall give this proof. We also discuss some related topics.

2. Quasi-wandering subspaces From now on, I stands for an invariant subspace of L2a with I = {0} and I = L2a . We study a question: when is PI B(L2a I) dense in I? One easily checks that our question is equivalent to the one: does there exist g ∈ I with g ∈ I? g = 0 satisfying B ∗√ Let en (z) = n + 1z n for n ≥ 0. It is known that {en }n≥0 is an orthonormal basis of L2a and √ √ n+1 en+1 , n ≥ 0. (2.1) Ben = n + 1z n+1 = √ n+2 Then √ n ∗ ∗ (2.2) B e0 = 0 and B en = √ en−1 , n ≥ 1, n+1 see [16]. Let D be the Dirichlet space, that is, D is the space of analytic functions f on D satisfying f ∈ L2a . It is well known that f ∈ D if and only if ∞

where f =

∞

(n + 1)|an |2 < ∞,

n=0

n 2 n=0 an z . Note that D ⊂ H , see [14].

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First, we show the following lemma. ∞ Lemma 2.1. Let f ∈ L2a and f = n=0 an en . Then (I − BB∗ )f =

∞ ∞ an a √ n z n ∈ D. en = n + 1 n+1 n=0 n=0

Moreover, if f = 0, then (I − BB∗ )f = 0. Proof. By (2.1) and (2.2), we have BB ∗ f = and

∞

n an en n + 1 n=1

∞ ∞ an a √ n zn. en = (I − BB )f = n + 1 n +1 n=0 n=0 ∗

Since

∞

n=0

|an |2 < ∞, we have

∞ n=0

√

an z n ∈ D. n+1

Theorem 2.2. Let I be an invariant subspace I of L2a for B with I = {0} and I = L2a . Then the following conditions are equivalent. (i) PI B(L2a I) is not dense in I. (ii) There exists f ∈ I with f = 0 satisfying B ∗ f ∈ I. (iii) I ∩ D = {0}. (iv) There exists f ∈ I with f = 0 satisfying f ∈ I. (v) There exists f ∈ I with f = 0 satisfying (I − BB∗ )f ∈ I. Proof. (i) ⇒ (ii) Suppose that PI B(L2a I) is not dense in I. Then there exists f ∈ I with f = 0 satisfying f ⊥ PI B(L2a I). Hence B ∗ f ⊥ L2a I, that is, B ∗ f ∈ I. Thus we get (ii). (ii) ⇒ (iii) Let f ∈ I with f = 0 satisfying B ∗ f ∈ I. Since I is an invariant subspace, (I − BB∗ )f ∈ I. By Lemma 2.1, we have 0 = (I − BB∗ )f ∈ I ∩ D. (iii) ⇒ (iv) Suppose that I ∩ D = {0}. Since D ⊂ H 2 , by the Beurling theorem I ∩D ⊂ I ∩H 2 = θH 2 for an inner function θ. Let h ∈ H 2 with h = 0 satisfying θh ∈ D. In [10], Richter and Shields proved that θh = g1 /g2 for some g1 , g2 ∈ D ∩ H ∞ , where H ∞ is the space of bounded analytic functions on D. We have g1 = θhg2 ∈ I ∩ D ∩ H ∞ . Let f = g12 ∈ I ∩ D ∩ H ∞ . Then we have f = 2g1 g1 ∈ I. (iv) ⇒ (v) Let g ∈ I with g = 0 satisfying g ∈ I. We have g ∈ I ∩ D. Hence zg ∈ I and (zg) = g + zg ∈ I. Let f = (zg) ∈ I. Write g=

∞ n=0

an en =

∞ √ n=0

n + 1an z n .

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Then f=

∞

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∞ √ (n + 1) n + 1an z n = (n + 1)an en .

n=0

n=0

By Lemma 2.1, (I − BB∗ )f =

∞

an en = g ∈ I.

n=0 ∗ (v) ⇒ (ii) Let f ∈ I with f = 0 satisfying (I − BB ∞)f ∈ I. Write ∞ ∗ g = (I − BB )f . Let g = n=0 an en . By Lemma 2.1, f = n=0 (n + 1)an en . Since f + g ∈ I, z(f + g) ∈ I. We have √ ∞ n+1 (n + 2) √ B ∗ z(f + g) = B ∗ an en+1 n+2 n=0

= =

∞

n=0 ∞

(n + 2)

n+1 an en n+2

(n + 1)an en

n=0

= f. Thus we get (ii). (ii) ⇒ (i) Let f ∈ I with f = 0 satisfying B ∗ f ∈ I. Then B ∗ f ⊥ L2a I, so f ⊥ B(L2a I). Hence we get f ⊥ PI B(L2a I). Thus we get (i). Corollary 2.3. Let f be a nonzero function in D and I be the invariant subspace L2a for B generated by f . Then PI B(L2a I) is not dense in I. Since D ⊂ H 2 , we have the following. Corollary 2.4. Let I be an invariant subspace I of L2a for B with I = {0} and I = L2a . If I ∩ H 2 = {0}, then PI B(L2a I) is dense in I. If I ∩ H 2 = {0}, then by [9] dim(I BI) = 1. So if dim(I BI) ≥ 2, then I ∩ H 2 = {0}. Example 2.5. Let {αk }k ⊂ D be an L2a -zero sequence. Suppose that ∞ k=1 (1 − |αk |) = ∞. In [8], Horowitz showed the existence of such a sequence. Let I be the space of functions f ∈ L2a satisfying f (αk ) = 0 for every k ≥ 1. Then I is an invariant subspace of L2a for B and I ∩ H 2 = {0}. By [6, p. 77], dim(I BI) = 1. Suppose that I ∩ D = {0}. Then I ∩ H 2 = θH 2 for an inner function θ. {0}. Carleson It is not known a characterization of θ for which θH 2 ∩ D = [3] and Shapiro-Shields [11] obtained information about the zero sequences of functions D. Let {rn }n be a sequence of positive numbers with rn < 1 in ∞ satisfying n=1 (1 − rn ) < ∞. Carleson proved the following theorem.

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Theorem 2.6. If for every argument {tn }n there exists f ∈ D satisfying f (0) = 1 and f (rn eitn ) = 0 for all n ≥ 1, then ∞ 1 −1−ε log < ∞ for every ε > 0. 1 − rn n=1 Conversely, if ∞ n=1

log

1 −1+ε <∞ 1 − rn

for some ε > 0,

then for every argument {tn }n , there exists f ∈ D satisfying f (0) = 1 and f (rn eitn ) = 0 for all n ≥ 1. It is not so diﬃcult to see that there exists {rn }n satisfying ∞ ∞ 1 −2 (1 − rn ) < ∞, but = ∞. log 1 − rn n=1 n=1 So by Theorem 2.6, there exists a sequence of argument {tn }n such that there are no functions f ∈ D satisfying f (0) = 1

and f (rn eitn ) = 0, n ≥ 1.

This shows the existence of a Blaschke product θ such that θH 2 ∩D = {0}. Let I be the invariant subspace of L2a for B generated by θ. Then by Theorem 2.2, PI B(L2a I) is dense in I. On the other hand, it is known the existence of {rn }n satisfying ∞ 1 − 12 < ∞. log 1 − rn n=1 By Theorem 2.6, for every {tn }n , there exists f ∈ D satisfying f (0) = 1 and f (rn eitn ) = 0, n ≥ 1. Let θ be the Blaschke product with zeros {rn eitn }n . {0}. Let I be the invariant subspace of L2a for B generated Then θH 2 ∩ D = by θ. In this case, by Theorem 2.2, PI B(L2a I) is not dense in I. Suppose that θ is an inner function. If there is h ∈ H 2 with h = 0 satisfying θh ∈ D, by [10] h ∈ D. Hence θ(z) − α h(z) ∈ D 1 − αθ(z) for every α ∈ D. Therefore we have the following corollary. Corollary 2.7. Let θ(z) be an inner function. For α ∈ D, let θα (z) =

θ(z) − α , 1 − αθ(z)

z ∈ D.

Let Iα be the invariant subspace of L2a for B generated by θα . Then PIα0 B(L2a Iα0 ) is dense in Iα0 for some α0 ∈ D if and only if PIα B(L2a Iα ) is dense in Iα for every α ∈ D

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3. Related topics The following is a counterpart of the Aleman, Richter, and Sundberg theorem. Theorem 3.1. Let I be an invariant subspace of L2a for B with I = {0} and 2 2 I = La . Then PI B(La I) L2 = I. a

The referee pointed out that this follows from the following identity. By (2.1) and (2.2), it is not diﬃcult to check the identity. Lemma 3.2. For every nonnegative integers k, n, we have

n z , 0≤n

For an invariant subspace I of L2a for B, I BI is also called an inner space. If I BI is maximal in the family of inner spaces, I BI is said to be a maximal inner space, see [4, 15]. Suppose that I ∩ H 2 = {0}. By [9, p. 598] and [15, Theorem 13], dim(I BI) = 1 and I is generated by a bounded function. Also we have I ∩ H 2 = I, where I ∩ H 2 is the closure of I ∩ H 2 in L2a . Let J be another invariant subspace of L2a with I J. One may ask whether [PI B(J I)]L2a = I or not. Note that [PI B(J I)]L2a = I if and only if [J I]L2a = J. Proposition 3.3. Let I be an invariant subspace of L2a for B with I = {0} and I = L2a . Then we have the following.

(i) If dim(I BI) ≥ 2, then there is an invariant subspace J of L2a satisfying J I and [J I]L2a = J. (ii) If I BI is not a maximal inner space, then there is an invariant subspace J of L2a satisfying J I and [J I]L2a = J.

Proof. (i) We denote by ord(f ) zero’s order of f ∈ L2a at z = 0. Let k = min{ord(f ) : f ∈ I} and I0 = {f ∈ I : ord(f ) ≥ k + 1}. Then I0 is an invariant subspace with I0 I, and there is an invariant subspace I1 with I0 = BI1 . It is easy to see that dim(I I0 ) = 1 and BI ⊂ I0 . Let ϕ ∈ I satisfying I = C · ϕ ⊕ I0 . Since dim(I BI) ≥ 2, we have BI I0 . Hence BI BI1 , so I I1 . Let J = C · ϕ + I1 . Then J is

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an invariant subspace and BJ = I0 . Since ϕ ∈ I and I I1 , I J. Since BJ ⊂ I0 , we have ϕ ⊥ [J I]L2a . This shows that [J I]L2a = J. (ii) Since I BI is not a maximal inner space, there is an invariant subspace J of L2a satisfying I BI J BJ. By Aleman, Richiter, and Sundberg’s theorem, we have I J and (I BI) ⊥ B n (J I) for every n ≥ 1. We also have (I BI) ⊥ (J I). Hence [J I]L2a = J. Here we have the following conjecture. Conjecture 3.4. Suppose that dim(I BI) = 1 and I BI is maximal. Then [J I]L2a = J for every invariant subspace J of L2a with J I. Proposition 3.5. Let I and J be invariant subspace of L2a for B satisfying I ∩ H 2 = θ1 H 2 and J ∩ H 2 = θ2 H 2 for some inner functions θ1 , θ2 . Suppose that θ1 H 2 θ2 H 2 and (θ1 /θ2 )(α) = 0 for some α ∈ D. Then [J I]L2a = J. Proof. Since θ1 H 2 θ2 H 2 , θ1 /θ2 is a nonconstant inner function. Let b(z) =

α−z , 1 − αz

and θ3 = θ1 /(θ2 b). Then θ3 is an inner function and θ1 H 2 θ2 θ3 H 2 ⊂ θ2 H 2 and θ1 = bθ2 θ3 . Let J1 = θ2 θ3 H 2 . Then J1 is an invariant subspace of L2a , I J1 ⊂ J, and bJ1 = I. Since θ 2 θ3 H 2 = θ 1 H 2 + C · θ 2 θ3

1 , 1 − αz

we have 1 , 1 − αz so dim(J1 I) = 1. Let g ∈ J1 I with J1 I = C · g. We have J1 = I + C · θ2 θ3

[g]L2a = [J1 I]L2a ⊂ [J I]L2a ⊂ J.

(4.1)

Let f ∈ J1 . Then g ⊥ I = bJ1 . Since b(b(z)) = z, we have (1 − |α|2 )2 bf g dA = zf (b(z))g(b(z)) dA(z), 0= |1 − αz|4 D D where dA is the normalized area measure on D. Thus we get f ◦b g◦b z (4.2) dA(z) = 0. 2 (1 − αz)2 D (1 − αz) Let (4.3)

J2 =

J1 ◦ b . (1 − αz)2

We denote by Cb the composition operator on L2a deﬁned by (Cb h)(z) = h(b(z)), h ∈ L2a . Since Cb Cb = I, Cb is an invertible operator on L2a . Since (1 − αz)2 is an invertible function in H ∞ , the multiplication operator deﬁned

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by 1/(1−αz)2 is also invertible on L2a . This shows that J2 is a closed subspace of L2a . Since bJ1 ⊂ J1 , we have zJ2 = z

J1 ◦ b (b ◦ b)J1 ◦ b (bJ1 ) ◦ b J1 ◦ b = = ⊂ = J2 . 2 2 2 (1 − αz) (1 − αz) (1 − αz) (1 − αz)2

Hence J2 is an invariant subspace of L2a for B. Since J2 =

θ2 θ3 H 2 ◦ b = ((θ2 θ3 ) ◦ b)H 2 , (1 − αz)2

we have J2 ∩ H 2 = ((θ2 θ3 ) ◦ b)H 2 . By Richter’s result [9], we have dim(J2 BJ2 ) = 1. Since g ∈ J1 , we have g(b(z)) ∈ J2 . (1 − αz)2 By (4.2), J2 BJ2 = C ·

g(b(z)) . (1 − αz)2

Write G(z) =

g(b(z)) . (1 − αz)2

By Aleman, Richter, and Sundberg’s theorem, span {z n G(z) : n ≥ 0} = J2 , where span indicates taking the closed linear span. Then by (4.3), span {bn (z)G(b(z)) : n ≥ 0} = J2 ◦ b = (1 − αz)2 J1 = J1 . Hence

J1 = span bn (z)

g(z) : n ≥ 0 (1 − αb(z))2

= (1 − αz)2 span {bn (z)g(z) : n ≥ 0} ⊂ [g(z)]L2a = [J1 I]L2a

by (4.1)

⊂ J1 .

Thus J1 = [J1 I]L2a . Since J1 I = C · g, we get [PI Bg]L2a = I. Since g ∈ J I, by (4.1) again, J ⊃ [J I]L2a ⊃ (J I) ⊕ [PI Bg]L2a = (J I) ⊕ I = J. Therefore we get J = [J I]L2a .

Corollary 3.6. Let I be an invariant subspace of L2a for B with I = {0} and I = L2a . Suppose that I ∩ H 2 (z) = θ(z)H 2 (z) for a Blaschke product θ. Then for every invariant subspace J of L2a with I J, we have [J I]L2a = J.

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Suppose that dim(I BI) > dim PI B(L2a I). Then there is a nonzero function g in I BI satisfying g ⊥ PI B(L2a I). Then B ∗ g ⊥ (L2a I), that is, B ∗ g ∈ I. Since B ∗ (I BI) ⊥ I, we have a contradiction. Hence we have dim(I BI) ≤ dim PI B(L2a I). We shall give an example satisfying dim(I BI) < dim PI B(L2a I) and

I BI ⊂ PI B(L2a I).

Let

1 (1 − αz)2 be the Bergman kernel for the point α ∈ D. Kα (z) =

Example 3.7. Let α, β ∈ D with α = β, and I = {f ∈ L2a : f (α) = f (β) = 0}. Then I is an invariant subspace of L2a , and L2a I = C · Kα + C · Kβ . Let g = (z − α)(z − β) ∈ I. Since B∗ 1 = 0

and B ∗ z n =

we have g, BKα = and

α β − , 6 2

n n−1 z , n+1 g, BKβ =

n ≥ 1, α β − 6 2

αβ α2 − , 12 6

zg, BKβ =

PI BKα = 0,

PBI BKα = 0

PI BKβ = 0,

PBI BKβ = 0.

zg, BKα =

αβ β2 − . 12 6

Hence and Note that

⎛ det ⎝

α 6

−

β 2

α2 12

− αβ 6 if and only if α = β. This shows that

⎞

β 6

−

α 2

β2 12

−

αβ 6

aPI BKα + bPI BKβ = 0,

⎠=0

a, b ∈ C

if and only if a = b = 0. Therefore we get dim PI B(L2a I) = 2. But by [6, p. 77], dim(I BI) = 1.

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We have z 2 g, BKα = Note that

⎛

α2 β α3 − , 20 12 α2 12

z 2 g, BKβ =

αβ 6

β2 12

2

3

−

αβ 6

β − α12β 20 − if and only if α = β. This fact shows that

αβ 2 12

⎜ det ⎝

α3 20

−

αβ 2 β3 − . 20 12

⎞ ⎟ ⎠=0

aPI BKα + bPI BKβ ⊥ zI if and only if a = b = 0. Thus we get I BI ⊂ PI B(L2a I).

Acknowledgement The authors would like to thank the referee for giving an elementary proof of Theorem 3.1.

References [1] A. Aleman, S. Richter, and C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 117(1996), 275–310. [2] C. Apostol, H. Bercovici, C. Foias, and C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra I, J. Funct. Anal. 63(1985), 369–404. [3] L. Carleson, On the zeros of functions with bounded Dirichlet integrals, Math. Z. 56(1952), 289–295. [4] J. S. Choa and K. Izuchi, A note on maximal inner spaces of the Bergman space, Integral Equations Operator Theory 51(2005), 35–40. [5] H. Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. reine angew. Math. 443(1993), 1–9. [6] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, 199, Springer-Verlag, New York, 2000. [7] H. Hedenmalm, S. Richter, and K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. reine angew. Math. 477(1996), 13–30. [8] C. Horowitz, Zeros of functions in thr Bergman space, Duke Math. J. 41(1974), 693–710. [9] S. Richter, Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc. 304(1986), 585–616. [10] S. Richter and A. Shields, Bounded analytic functions in the Dirichlet space, Math. Z. 198(1988), 151–159. [11] H. S. Shapiro and A. Shields, On the zeros of functions with ﬁnite Dirichlet integral and some related function spaces, Math. Z. 80(1962), 217–229. [12] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. reine angew. Math. 531(2001), 147–189. [13] S. Sun and D. Zheng, Beurling type theorem on the Bergman space via the Hardy space of the bidisk, preprint.

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[14] Z. Wu, Function theory and operator theory on the Dirichlet space, MSRI Publications Vol. 33,“Holomorphic Spaces”, Cambridge Univ. Press, 1998, 179– 199. [15] K. Zhu, Maximal invariant subspaces and Hankel operators on the Bergman space, Integral Equations Operator Theory 31(1998), 371–387. [16] K. Zhu, Operator Theory in Function Spaces, Second Edition, Math. Surveys Monographs Vol. 138, Amer. Math. Soc., 2007. Kei Ji Izuchi Department of Mathematics, Niigata University, Niigata 950-2181, Japan e-mail: [email protected] Kou Hei Izuchi Institute of Basic Science, Korea University, Seoul 136-713, Republic of Korea e-mail: [email protected], [email protected] Yuko Izuchi Aoyama-shinmachi 18-6-301, Niigata 950-2006, Japan e-mail: [email protected] Submitted: December 3, 2008. Revised: October 14, 2009.

Integr. Equ. Oper. Theory 67 (2010), 163–170 DOI 10.1007/s00020-010-1754-2 Published online April 20, 2010 © Birkhäuser/Springer Basel AG 2010

Integral Equations and Operator Theory

Toeplitz Operators on the Dirichlet Space Tao Yu Abstract. In this paper a decomposition of Sobolev space is obtained. Then we prove that a Toeplitz operator on the Dirichlet space is compact only when it is the zero operator. For two Toeplitz operators on the Dirichlet space, we obtain the conditions for that they commute, their product is a Toeplitz operator, and their commutator or semicommutator has ﬁnite rank, respectively. Mathematics Subject Classiﬁcation (2010). Primary 47B35; Secondary 31C25. Keywords. Toeplitz operator, Sobolev space, Dirichlet space, Hardy space.

1. Introduction Let D be the open unit disk in the complex plane C and let dA denote the Lebesgue measure on D, normalized so that the measure of D equals 1. The Sobolev space W 1,2 (D) consists of functions u with weak derivatives in D, for which the norm 2 2 2 12 ∂u + ∂u dA < ∞. u = udA + ∂ z¯ ∂z D D Then W 1,2 (D) is a Hilbert space with the inner product ∂u ∂v ∂u ∂v , , udA v¯dA + + , u, v = ∂z ∂z L2 (D) ∂ z¯ ∂ z¯ L2 (D) D D where the symbol ·, ·L2 (D) represents the inner product in the Lebesgue space L2 (D) respective to the measure dA. The Dirichlet space D is the closed subspace of W 1,2 (D) consisting of all analytic functions vanishing at 0. Then D is a Hilbert space with the inner product h, g = h , g L2 (D) . The work is supported by the National Natural Science Foundation of China (Grant no. 10771064 and 10971195).

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The Sobolev space W 1,∞ (D) is deﬁned by ∂u ∂u W 1,∞ (D) = u ∈ W 1,2 (D) : u, , ∈ L∞ (D) , ∂z ∂ z¯ ∞ where L (D) is the space of essentially bounded measurable functions on D. ∂u The norm in W 1,∞ (D) is deﬁned by u1,∞ = max{u∞ , ∂u ∂z ∞ , ∂ z¯ ∞ }. 1,2 Let P denote the orthogonal projection from W (D) onto D. Given a function φ ∈ W 1,∞ (D), the Toeplitz operator Tφ with symbol φ on D is deﬁned by Tφ (h) = P (φh),

h ∈ D.

For facts on the Dirichlet space and Toeplitz operators one is referred to, for example, [12, 13, 7, 11]. On the classical Hardy space, Brown and Halmos [6] showed that the only compact Toeplitz operator is the zero operator. It is well known that there exist many nontrivial compact Toeplitz operators on the Bergman space. A nice characterization of compact Bergman space Toeplitz operators can be found in [5]. For a special class of Toeplitz operators, Lee [11] proved that a compact Toeplitz operator on the Dirichlet space must be zero. We shall prove that this characterization is true for all Toeplitz operators on the Dirichlet space. We also consider the following problem: What is the relationship between their symbols when two Toeplitz operators (semi-)commute, and the (semi-)commutator of two Toeplitz operators is ﬁnite rank? For the case of the classical Hardy space, Brown and Halmos[6] showed that two Toeplitz operators with bounded symbols commute if and only if either both symbols are analytic, or both symbols are co-analytic, or a nontrivial linear combination of the symbols is constant. On the Bergman space of the unit disk, this problem is more subtle ˘ ckovi˘c[4] charthan its analogues on the Hardy space. In 1991, Axler and Cu˘ acterized commuting Toeplitz operators with harmonic symbols. Ahern and ˘ ckovi˘c [2] gave necessary and suﬃcient conditions for the product of two Cu˘ Toeplitz operators with bounded harmonic symbols to be a Toeplitz operator. Many subsequent works studied these problems for many special classes of symbols, for example, radial symbols or quasi-homogeneous symbols [8]. On the Dirichlet space, Lee in [11] studied the commutativity of two Toeplitz operators with harmonic symbols. In this paper we give suﬃcient and necessary conditions for two Toeplitz operators on the Dirichlet space with W 1,∞ (D) symbols to commute and their product to be a Toeplitz operator respectively. On the classical Hardy space, Axler, Chang and Sarason [3] completely characterize when the semi-commutator of two Toeplitz operators has ﬁnite rank; Ding and Zheng [9] characterize ﬁnite rank commutators. On the Bergman space of the unit disk, Guo, Sun and Zheng [10] gave conditions

Toeplitz Operators on the Dirichlet Space

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for the commutators or semi-commutators of Toeplitz operators with har˘ ckovi˘c and Louhichi [8] characterized monic symbols to be of ﬁnite rank. Cu˘ ﬁnite rank commutators and semi-commutators of Toeplitz operators with quasi-homogeneous symbols. In this paper we give a condition for the commutator or semi-commutator of Toeplitz operators on the Dirichlet space to be of ﬁnite rank. Our main method is to establish a relationship between Toeplitz operators on the Dirichlet space and Toeplitz operators on the Hardy space.

2. A decomposition of Sobolev space Let P0 be the set consisting of all polynomials on the unit disk D in variables z and z¯ which have the following form:

al+j,l z l+j z¯l , j≥−l l≥0

where j and l run over a ﬁnite subset of Z (the set of integers), and a = 0. Then for any n ∈ N (the set of positive integers), we have l+j,l l≥0

al+j,l z l+j z¯l , z n = al+j,l z l+j z¯l , z n l≥0

l≥0

=

al+j,l

l≥0

(l + j)nz l+j−1 z¯l+n−1 dA(z),

D

l+j−1

in above is understood to be 0 when l + j = 0. Now if where (l + j)nz n = j, then obviously the above integral equals to 0. If n = j, then

l+j l n al+j,l z z¯ , z al+j,l (l+j)j|z|2(l+j−1) dA(z) = j al+j,l = 0. = l≥0

For n ∈ N, similarly

l≥0 l≥0

D

l≥0

al+j,l z l+j z¯l , z¯n = 0 also holds.

Let A0 denote the closure of P0 in W 1,2 (D), and let A denote A0 + C. Then both A0 and A are orthogonal to D and D, the space of the conjugates of functions in D. For each polynomial p on D in variables z and z¯, we can write it in the following form:

al+j,l z l+j z¯l + c + an z n + bm z¯m , (2.1) p(z, z¯) = j≥−l l≥0

n>0

m>0

where j, l, n and m run over a ﬁnite subset of Z, and l≥0 al+j,l = 0 for all j. Since the set of all polynomials in z and z¯ is dense in the Sobolev space W 1,2 (D) [14], we have the following decomposition: W 1,2 (D) = A ⊕ D ⊕ D.

(2.2) 2

Let T denote the boundary of D, and let L (T) denote the Lebesgue space relative to the Haar measure on T. Suppose that u ∈ W 1,2 (D). Then

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the trace of u on the boundary of D, denoted by u|T , is in L2 (T), and the trace map u → u|T is bounded. The deﬁnition of trace and the boundedness of the trace map can be found in [1, Chapter 5]. For every polynomial p ∈ P0 , it is clear that p|T = 0. Then by the boundedness of the trace map we have u|T = 0 for every u ∈ A0 . For f ∈ L2 (T), let P[f ] denote the Poisson extension in D of f . For u ∈ W 1,2 (D), there exist analytic functions h1 and h2 with h1 (0) = h2 (0) = 0 and a ¯ 2 + c. Thus we can write u as following constant c, such that P[u|T ] = h1 + h ¯ 2, u = (u0 + c) + h1 + h (2.3) where u0 = u − P[u|T ]. Since a function h in D can be approximated polynomials, from the boundedness of the trace map it follows that the trace of h on the boundary of the unit disk equals to the radial limit of h. By the decomposition (2.2), we have h1 , h2 ∈ D. Since P[f ] = 0 if and only if f = 0 for f ∈ L2 (T), one can see the function u0 in (2.3) to be in A0 . Theorem 2.1. Suppose that u ∈ W 1,2 (D). Then u ∈ A0 if and only if the trace u|T = 0, and u ∈ A if and only if the trace u|T is constant. For every φ ∈ W 1,∞ (D), the multiplication operator Mφ on the Sobolev space W 1,2 (D) is bounded [15]. So we have the following property. Proposition 2.2. Let φ ∈ W 1,∞ (D). Then φA0 ⊂ A0 . Proof. Since z n P0 = z¯n P0 = P0 for all nonnegative integer n, we have that qp ∈ P0 for p ∈ P0 and a polynomial q. Since W 1,∞ (D) ⊂ W 1,2 (D), φ can be approximated by polynomials in W 1,2 (D) [14]. It follows from the boundedness of Mp that φp ∈ A0 . So boundedness of Mφ implies the desired result.

3. Toeplitz operators Let φ ∈ W 1,2 (D). Then the Toeplitz operator Tφ is well deﬁned on W 1,∞ (D)∩ H(D), a dense subset of W 1,2 (D), where H(D) denotes the space of analytic functions in D. If φ ∈ A0 , Tφ is zero on W 1,∞ (D)∪H(D), and so its extension to W 1,2 (D) vanishes. The following result shows that the converse is also true. Theorem 3.1. Suppose that φ ∈ W 1,2 (D). Then the following three statements are equivalent: (i) Tφ = 0 on D. (ii) Tφ is compact on D. (iii) φ is in A0 . Proof. Only (ii) ⇒ (iii) is needed to be proved. The proof is similar to that of the Corollary of [6, Theorem 4]. By the decomposition (2.3), let φ = φ0 + φh , where φ0 ∈ A0 and φh is harmonic in D. Let ∞ ∞

φh (z) = am z¯m + bn z n . m=1

n=0

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√ It is well known that {z j / j}j∈N is an orthogonal basis of the Dirichlet space D. A direct calculation shows that j

j−1 ∞

z am bn √ z j−m + √ z n+j . Tφ √ = j j j m=1 n=0 The compactness of Tφ implies that 2 j−1 ∞

|am |2 (j − m) |bn |2 (n + j) zj Tφ ( √ = + → 0 as j → ∞. ) j j j m=1 n=0 Hence am = bn = 0 for all m and n. So φh = 0, and we get the desired result. Note that Theorem 3.1 generalizes Proposition 9 in [11]. The following lemma gives a relationship between Toeplitz operators on the Dirichlet space and their analogues deﬁned on the Hardy space. Let P˜ denote the orthogonal projection from L2 (T) onto the Hardy space H 2 (T). For f ∈ L∞ (T), the Toeplitz operator T˜f on H 2 (T) with symbol f is deﬁned by T˜f (h) = P˜ (f h) for h ∈ H 2 (T). If φ ∈ W 1,∞ (D), then by the Sobolev embedding theorem [1, Theorem 5.4], the trace φ|T is a continuous function on T. We shall identify functions on the unit circle with their harmonic extensions, deﬁned via Poisson’s formula, into the unit disk. Lemma 3.2. Let φj be in W 1,∞ (D) for j = 1, 2, ..., n, and let (φj )|T denote the trace of φj . Then Tφ1 Tφ2 · · · Tφn (h) = z T˜(φ1 )|T T˜(φ2 )|T · · · T˜(φn )|T (z −1 h) for all h ∈ D. Proof. For φ ∈ W 1,∞ (D), Proposition 2.2 implies that Tφ = TP[φ|T ] . Let −1

P[φ|T ] =

cn z¯−n +

n=−∞

+∞

cn z n .

n=0

Let j be a positive integer. Then TP[φ|T ] (z j ) = It is clear that φ|T (eiθ ) =

+∞

cn−j z n .

n=1

n=−∞ cn e

T˜φ|T (z j−1 ) =

+∞

inθ

. So we have, on H 2 (T), that

+∞

cn−j z n−1 .

n=1 −1

Hence we have Tφ (h) = z T˜φ|T (z h) for every h ∈ D. Assume that the result holds for n = k ≥ 1. Then for φj be in W 1,∞ (D) for j = 1, 2, ..., k + 1 and h ∈ D, Tφ2 Tφ3 · · · Tφk+1 (h) = z T˜(φ2 )|T T˜(φ3 )|T · · · T˜(φk+1 )|T (z −1 h) ∈ D.

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Thus we have

Tφ1 Tφ2 · · · Tφk+1 (h) = Tφ1 z T˜(φ2 )|T T˜(φ3 )|T · · · T˜(φk+1 )|T (z −1 h) = z T˜(φ1 )|T T˜(φ2 )|T · · · T˜(φk+1 )|T (z −1 h).

The lemma then follows by induction.

Lemma 3.3. Let φj,i be in W 1,∞ (D) for (j, i) running in a ﬁnite subset o f N × N. Let T = j i Tφj,i on D, and let T˜ = j i T˜(φj,i )|T on H 2 (T). Then T is of ﬁnite rank if and only if T˜ is of ﬁnite rank. In this case, the rank of T equals the rank of T˜ . Proof. By Lemma 3.2, we have T (h) = z T˜ (z −1 h) for every h ∈ D. Since D is contained in H 2 (T), it is clear that ran(T ) ⊂ z · ran(T˜) (where ran stands for the range). Hence the rank of T less than or equal to the rank of T˜. On the other hand, suppose that T is of ﬁnite rank, and the dimension of ran(T ) is n. Then it follows from Lemma 3.2 that T˜ (z −1 D) = z −1 ran(T ) is an n-dimensional (closed certainly) subspace of H 2 (T). Since z −1 D is dense in H 2 (T), the boundedness of T˜ implies that the dimension of ran(T˜) is n. This completes the proof. On the classical Hardy space, Brown and Halmos gave necessary and suﬃcient conditions for the product of two Toeplitz operators to be a Toeplitz operator and also for their commutator to be zero [6, Theorems 8 and 9]. By BrownHalmos’ Theorems and Lemma 3.3, we obtain the following results. Theorem 3.4. Let φ, ψ ∈ W 1,∞ (D). Then Tφ Tψ = Tψ Tφ if and only if one of the following statements holds: (i) Both φ and ψ are in A ⊕ D. (ii) Both φ and ψ are in A ⊕ D. (iii) A nontrivial linear combination of φ and ψ is in A. Proof. By Lemma 3.3, the equation Tφ Tψ = Tψ Tφ on the Dirichlet space is equivalent to T˜φ|T T˜ψ|T = T˜ψ|T T˜φ|T on the Hardy space. By Brown-Halmos’ Theorem, this is equivalent to that one of the following statements holds: 1) Both φ|T and ψ|T are analytic. 2) Both φ|T and ψ|T are co-analytic. 3) There exist two constants a and b, not both zero, such that aφ|T + bψ|T is zero. By saying a function on the unit circle analytic we mean that its Poisson extension into the unit disc is analytic. For φ ∈ W 1,∞ (D), it follows from the decomposition (2.3) that φ|T is analytic if and only if φ ∈ A⊕ D, that φ|T is co-analytic if and only if φ ∈ A⊕ D, and that φ|T is constant if and only if φ ∈ A. Thus conditions 1), 2) and 3) are respectively equivalent to conditions (i), (ii) and (iii) in the theorem. Theorem 3.5. Let φ, ψ and τ be in W 1,∞ (D). Then Tφ Tψ = Tτ if and only if either ψ ∈ A ⊕ D or φ ∈ A ⊕ D, and φψ|T = τ |T . In this case, Tφ Tψ = Tφψ .

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The proof is similar to that of Theorem 3.4 and we omit it. In [9] Ding and Zheng give a necessary and suﬃcient condition that the commutator of two Toeplitz operators on H 2 (T) has ﬁnite rank. Combining Theorem 1.7 of [9] and Lemma 3.3, it is easy to see the following result. Theorem 3.6. Let φ, ψ ∈ W 1,∞ (D). Then the commutator Tφ Tψ − Tψ Tφ has ﬁnite rank if and only if one of the following statements holds: (i) There is a nonzero analytic polynomial p such that both pφ and in A ⊕ D. (ii) There is a nonzero analytic polynomial q such that both q¯φ and in A ⊕ D. (iii) There are analytic polynomials p1 , p2 , q1 and q2 , at least one of p2 is nontrivial and at least one of q1 and q2 is nontrivial, such

pψ are q¯ψ are p1 and that

p¯1 q1 − p¯2 q2 ∈ A0 , p¯1 φ + p¯2 ψ ∈ A ⊕ D and q1 φ + q2 ψ ∈ A ⊕ D. Proof. By Lemma 3.3, the commutator Tφ Tψ − Tψ Tφ has ﬁnite rank on the Dirichlet space is equivalent to that the commutator T˜φ|T T˜ψ|T − T˜ψ|T T˜φ|T has ﬁnite rank on the Hardy space. By [9, Theorem 1.7], this is equivalent to that one of the following statements holds: 1) There is a nonzero analytic polynomial p such that both pφ|T and pψ|T are analytic. 2) There is a nonzero analytic polynomial q such that both q¯φ|T and q¯ψ|T are co-analytic. 3) There are analytic polynomials p1 , p2 , q1 and q2 , at least one of p1 and p2 is not trivial and at least one of q1 and q2 is not trivial, such that p¯1 (z)q1 (z) = p¯2 (z)q2 (z), p¯1 φ|T + p¯2 ψ|T is co-analytic, and q1 φ|T + q2 ψ|T is analytic. By the same argument as in the proof of Theorem 3.4, we conclude that the conditions 1), 2) and 3) above are respectively equivalent to the conditions (i), (ii) and (iii) in Theorem 3.6. Axler, Chang and Sarason [3] proved that the semi-commutator T˜f T˜g − T˜f g of two Toeplitz operators T˜f and T˜g on the Hardy space H 2 (T) has ﬁnite rank if and only if there exists an analytic polynomial p such that either pf¯ or pg is in H ∞ (D). By Lemma 3.3, we obtain a similar result on the Dirichlet space. Theorem 3.7. Let φ, ψ ∈ W 1,∞ (D). Then the semi-commutator Tφ Tψ − Tψφ of Tφ and Tψ has ﬁnite rank if and only if there exists an analytic polynomial p such that either pφ¯ or pψ is in A ⊕ D. The proof is similar to that of Theorem 3.6 and we omit it. Acknowledgment The author is greatly indebted to the referee for careful reading the manuscript and providing a very helpful, detailed list of suggestions for revisions. He also would like to thank Ying Zhang for his help in revising the manuscript.

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References [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. ˇ Cu˘ ˘ ckovi˘c, A theorem of Brown-Halmos type for Bergman [2] P. Ahern and Z. space Toeplitz operators. J. Funct. Anal., 187 (2001), no. 1, 200–210. [3] S.Axler, S.-Y. A. Chang and D. Sarason, Products of Toeplitz operators, Integral Equations Operator Theory, 1(1978), 285–309. ˇ Cu˘ ˘ ckovi˘c, Commuting Toeplitz operators with harmonic sym[4] S. Axler and Z. bols, Integral Equations Operator Theory, 14 (1991), 1–12. [5] S. Axler and D. Zheng, Compact operators via the Berezin transform, India. Univ. Math. J., 47 (1998), 387–400. [6] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213 (1964), 89–102. [7] G. Cao, Fredholm properties of Toeplitz operators on Dirichlet space, Paciﬁc J. Math., 188 (1999), 209–224. ˇ Cu˘ ˘ ckovi˘c, I. Louhichi, Finite rank commutators and semicommutators of [8] Z. quasihomogeneous Toeplitz operators. Complex Anal. Oper. Theory, 2 (2008), no. 3, 429–439. [9] X. Ding and D. Zheng, Finite rank commutator of Toeplitz operators or Hankel operators, Houston J. Math. 34 (2008), no. 4, 1099–1119. [10] K. Guo, S. Sun and D. Zheng, Finite rank commutators and semicommutators of Toeplitz operators with harmonic symbols. Illinois J. Math. 51 (2007), no. 2, 583–596. [11] Y. J. Lee, Algebraic properties of Toeplitz operators on the Dirichlet space, J. Math. Anal. Appl., 329(2007), 1361–1329. [12] Z. Wu, Hankel and Toeplitz operators on Dirichlet spaces, Integral Equations Operator Theory, 15 (1992), 503–525. [13] Z. Wu, Operator theory and function theory on Dirichlet space, “Holomorphic Spaces” ed. by S. Axler, J. McCarthy and D. Sarason, MSRI (1998), 179–199. [14] T. Yu, Operators on the orthogonal complement of the Dirichlet space, J. Math. Anal. Appl., 357 (2009), 300–306. [15] T. Yu and S. Wu, Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space, Acta Math. Sinica, English Series, 25 (2009), 245–252. Tao Yu Department of Mathematics Zhejiang Normal University Jinhua, Zhejiang, 321004 P. R. China e-mail: [email protected] Submitted: April 29, 2009. Revised: August 29, 2009.

Integr. Equ. Oper. Theory 67 (2010), 171–181 DOI 10.1007/s00020-010-1762-2 Published online April 27, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Distances from Bloch Functions to QK -Type Spaces Zengjian Lou and Wufu Chen Abstract. We establish several formulas for distances from Bloch functions to some QK -type spaces, which generalize similar results of distances from Bloch functions to BM OA by Peter Jones, and to some M¨ obius invariant spaces by Ruhan Zhao. Mathematics Subject Classification (2000). 30D45. Keywords. Bloch functions, QK -type spaces, K-Carleson measure.

1. Introduction Let D = {z : |z| < 1} be the open unit disk of the complex plane C, and ∂D be the boundary of D. Denote by H(D) the set of all analytic functions on D. The Bloch space B is the set of all functions f ∈ H(D) which satisﬁes f B = sup(1 − |z|2 )|f (z)| < ∞. z∈D

It is well known that B is a Banach space if it is equipped with the norm f ∗B = |f (0)| + f B < ∞. The little Bloch space B0 is the space of all functions f ∈ B with the property lim (1 − |z|2 )|f (z)| = 0.

|z|→1

It is well-known that B0 is the closure of polynomials in B. For a ﬁxed point a ∈ D, let 1 , z∈D g(z, a) = log |ϕa (z)| be the Green’s function of D with the pole at a, where ϕa (z) = M¨ obius transformation of D mapping a to zero.

a−z 1−az

is a

This work was supported by NNSF of China (Grant No. 10771130) and Specialized Research Fund for the Doctoral Program of High Education (Grant No. 2007056004).

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For 0 < p < ∞, −2 < q < ∞, 0 < s < ∞ and −1 < q +s < ∞, F (p, q, s) is the space of all functions f ∈ H(D) which satisﬁes f pp,q,s = sup |f (z)|p (1 − |z|2 )q g s (z, a) dA(z) < ∞, a∈D

D

where dA(z) is the normalized area measure on D. The space F0 (p, q, s) consists of all f ∈ F (p, q, s) such that |f (z)|p (1 − |z|2 )q g s (z, a) dA(z) = 0. lim |a|→1

D

It is clear that F (p, q, s) = BM OA and F0 (p, q, s) = V M OA if p = 2, q = 0 and s = 1. We refer the reader to [13] and [6] for more information about F (p, q, s) and F0 (p, q, s). Let K : [0, ∞) → [0, ∞) be a right-continuous and nondecreasing function. For 0 < p < ∞ and −2 < q < ∞, the space QK (p, q) consists of all functions f ∈ H(D) such that p f QK,p,q = sup |f (z)|p (1 − |z|2 )q K(1 − |ϕa (z)|2 ) dA(z) < ∞. a∈D

D

The space QK,0 (p, q) consists of all f ∈ QK (p, q) satisfying lim |f (z)|p (1 − |z|2 )q K(1 − |ϕa (z)|2 ) dA(z) = 0. |a|→1

D

Spaces QK (p, q) and QK,0 (p, q) are ﬁrst introduced in [9], and it was proved there that QK (p, q) ⊂ B and QK,0 (p, q) ⊂ B0 if p = q + 2. There are choices of K, so that QK = B and QK,0 = B0 . We note also that QK (2, 0) = QK and QK,0 (2, 0) = QK,0 which are studied in [3]. If 0 < s < ∞ and K(t) = ts , then QK (p, q) = F (p, q, s), and QK = Qs (see, for example, [1,2,7] and [11] for references). If K(t) = t, in particular, QK coincides with BM OA. For an arc I ⊂ ∂D, denote the normalized arc length of I by |I|, and let S(I) = {rζ ∈ D : 1 − |I| < r < 1, ζ ∈ I}. Let s > 0. A nonnegative measure μ on D is said to be an s-Carleson measure if μ(S(I)) < ∞, sup |I|s I⊂∂D and μ is said to be a vanishing s-Carleson measure if, in addition to the above μ(S(I)) = 0. |I|s |I|→0 lim

We note that 1-Carleson measures are just the classical Carleson measures. Suppose X ⊂ B is an analytic function space. The distance from a Bloch function f to X is deﬁned by distB (f, X) = inf f − g∗B . g∈X

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For f ∈ B and ε > 0, let Ωε (f ) = {z ∈ D : |f (z)|(1 − |z|2 ) ≥ ε}. The following result is due to Peter Jones (see [4], for example). Theorem A. Suppose f ∈ B. The following quantities are equivalent: (1) distB (f, BM OA); χ ε (f ) (z) (2) inf{ε : Ω1−|z| 2 dA(z)is a Carleson measure}, where χ denotes the characteristic function. The following result is obtained by Zhao in [12]. A similar type result in several complex variables can also be found in [10]. Theorem B. Suppose 1 ≤ p < ∞, 0 ≤ q < ∞, 0 < s ≤ 1, and f ∈ B. The following quantities are equivalent: (1) distB (f, F (p, p − 2, s)); χΩε (f ) (z) (2) inf{ε : (1−|z| 2 )2−s dA(z)is a s-Carleson measure}; (3) inf{ε : supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 (1 − |ϕa (z)|2 )s dA(z) < ∞}; (4) inf{ε : supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 g s (z, a) dA(z) < ∞}. Distance from Bloch function to the little Bloch space was studied by Tjani in [5]. The purpose of this paper is to extend above results by considering X = QK (p, p − 2) for p > 1. Our main results are Theorems 1 and 2 in Sect. 3. Throughout this paper, C denotes a constant which may differ at different occurrences. For two functions f and g, f ≈ g means that there are positive constants c and C such that cf ≤ g ≤ Cf .

2. Preliminaries The following result is standard (see, for example, Lemma 4.2.2 in [14]). Lemma 1. Suppose t > −1, s > 0. Then (1 − |w|)t dA(w) 1 ≈ , 2+t+s |1 − zw| (1 − |z|2 )s

for all z ∈ D.

D

We always assume that the nondecreasing function K is differentiable and satisﬁes K(t) = K(1) > 0 if t ≥ 1, and K(2t) ≈ K(t) if t ≥ 0. We assume also that 1 1 (1 − r2 )q K log r dr < ∞ r 0

(otherwise, by a result in [9], QK (p, q) contains constant functions only). The following two constraints on K are needed for our main theorems. 1 ∞ dv dv < ∞; (b) K∗ (v) K∗ (v) 2 < ∞, (a) v v 0

1

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where K(vt) , 0≤t≤1 K(t)

K∗ (v) = sup

0 < v < ∞.

Since K is nondecreasing, we know that K∗ is also nondecreasing. We refer the reader to [3] for more properties about K∗ . A nonnegative measure μ on D is said to be a K-Carleson measure (see [3]) if 1 − |z| K sup dμ(z) < ∞, |I| I⊂∂D S(I)

and μ is said to be a vanishing K-Carleson measure if, in addition to the above 1 − |z| dμ(z) = 0. K lim |I| |I|→0 S(I)

s

For s > 0 and K(t) = t , it is clear that μ is a K-Carleson measure if and only if (1 − |z|2 )s dμ(z) is an s-Carleson measure. The following lemma is Corollary 3.2 in [3]. Lemma 2. Suppose K satisfies the condition (a). Then μ is a K-Carleson measure if and only if sup K(1 − |ϕa (z)|2 ) dμ(z) < ∞. a∈D

D

Both Lemmas 3 and 4 in the following can be found in [8]. Lemma 3. Suppose 1 < p < ∞ and K satisfies the condition (a). Then f ∈ QK (p, p − 2) if and only if the measure |f (z)|p (1 − |z|2 )p−2 dA(z) is a K-Carleson measure. Lemma 4. Suppose 1 < p < ∞, n be a positive integer, and K satisfies the condition (a). Then f ∈ QK (p, p − 2) if and only if sup |f (n) (z)|p (1 − |z|2 )np−2 K(1 − |ϕa (z)|2 ) dA(z) < ∞. a∈D

D

The following lemma plays an important role in the proof of our main theorems, which generalizes Lemma 1 in [12]. Our proof, however, is also very different from the one in [12]. Lemma 5. If K satisfies the condition (b), then K(1 − |ϕa (z)|2 ) K(1 − |ϕa (w)|2 ) dA(z) ≤ C , |1 − zw|4 (1 − |w|2 )2

for all a, w ∈ D.

D

b−z Proof. Recall that for ﬁxed b ∈ D, ϕb (z) = 1−bz is a M¨ obius transformation −1 of D mapping b to zero. It is easy to check that ϕb = ϕb .

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For ﬁxed a, w ∈ D, let λ = ϕw (a). Clearly, both ϕa (z) and ϕλ (ϕw (z)) are M¨ obius transformations of D mapping a to zero. Therefore there is a unimodular constant eiθ (which is − 1−aw 1−aw in our case) such that ϕa (z) = eiθ ϕλ ◦ ϕw (z). Direction computation yields K(1 − |ϕa (z)|2 ) K(1 − |ϕλ ◦ ϕw (z)|2 ) dA(z) = dA(z) |1 − zw|4 |1 − zw|4 D D K(1 − |ϕλ (u)|2 ) |ϕw (u)|2 dA(u) = |1 − wϕw (u)|4 D K (1−|u|2 )(1−|λ|2 ) |1−λu|2 = dA(u) (1 − |w|2 )2 D 1 − |u|2 K(1 − |λ|2 ) dA(u) K∗ ≤ 2 2 (1 − |w| ) |1 − λu|2 D

2

2

2

Note that |λ| = |ϕw (a)| = |ϕa (w)| , and ∞ 1 − |u|2 2 dv dA(u) ≤ K∗ dA(u) ≤ 4 K∗ (v) 2 . K∗ 2 1 − |u| v |1 − λu| D

D

2

Lemma 5 is proved.

Remark. Instead of condition (b), the proof above works if the following condition holds: 1 − |u|2 sup K∗ dA(u) < ∞. |1 − λu|2 λ∈D D

3. Main Theorems We state ﬁrst our main results and corollaries. Recall that for f ∈ B and ε>0 Ωε (f ) = {z ∈ D : |f (z)|(1 − |z|2 ) ≥ ε}. Theorem 1. Suppose 1 < p < ∞, 0 ≤ q < ∞, K satisfies conditions (a) and (b), and f ∈ B. The following four quantities are equivalent: (1) distB (f, QK (p, p − 2)); χΩε (f ) (z) (2) inf{ε : (1−|z| 2 )2 dA(z)is a K-Carleson measure}; (3) inf{ε : supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(g(z, a)) dA(z) < ∞}; (4) inf{ε : supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(1 − |ϕa (z)|2 ) dA(z) < ∞}. Remark. For 0 < s ≤ 1 and K(t) = ts , we know that QK (p, p − 2) = F (p, p − 2, s). Therefore Theorem 1 extends Theorem B.

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Corollary 1. Suppose 0 ≤ q < ∞, K satisfies conditions (a) and (b), and f ∈ B. The following four quantities are equivalent: (1) distB (f, QK ); χΩε (f ) (z) (2) inf{ε : (1−|z| 2 )2 dA(z)is a K-Carleson measure}; (3) inf{ε : supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(g(z, a)) dA(z) < ∞}; (4) inf{ε : supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(1 − |ϕa (z)|2 ) dA(z) < ∞}. Corollary 2. Suppose 1 < p < ∞ and f ∈ B. (1)

If both K1 and K2 satisfy conditions (a), (b) and K1 ≈ K2 , then distB (f, QK1 (p, p − 2)) = distB (f, QK2 (p, p − 2)).

(2)

If p2 > p1 > 1, K satisfies conditions (a) and (b), then distB (f, QK (p2 , p2 − 2)) = distB (f, QK (p1 , p1 − 2)).

Corollary 3. Suppose 1 < p < ∞, 0 ≤ q < ∞, K satisfies conditions (a) and (b), and f ∈ H(D). The following conditions are equivalent: (1) f is in the closure of QK (p, p − 2) in B; χΩε (f ) (z) (2) (1−|z| 2 )2 dA(z) is a K-Carleson measure for any ε > 0; (3) supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(g(z, a)) dA(z) < ∞ for any ε > 0; (4) supa∈D Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(1 − |ϕa (z)|2 ) dA(z) < ∞ for any ε > 0. Corollary 4. Suppose p2 > p1 > 1, K satisfies conditions (a) and (b). Then the closure of QK (p1 , p1 − 2) and QK (p2 , p2 − 2) in B are identical. For the distances from Bloch functions to the space QK,0 (p, p − 2), we have the following theorem which can be compared with a similar result in [5]. Theorem 2. Suppose 1 < p < ∞, 0 ≤ q < ∞, K satisfies conditions (a) and (b), and f ∈ B. The following five quantities are equivalent: (1) distB (f, QK,0 (p, p − 2)); (2) distB (f, B0 ); χΩε (f ) (z) (3) inf{ε : (1−|z| 2 )2 dA(z)is a vanishing K-Carleson measure}; (4) inf{ε : lim|a|→1 Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(g(z, a)) dA(z) = 0}; (5) inf{ε : lim|a|→1 Ωε (f ) |f (z)|q (1 − |z|2 )q−2 K(1 − |ϕa (z)|2 ) dA(z) = 0}. Corollary 5. Let K satisfy conditions (a) and (b). Then f ∈ QK,0 (p, p − 2) χΩε (f ) (z) if and only if the measure (1−|z| 2 )2 dA(z) is a vanishing K-Carleson measure for all ε > 0. The idea of establishing equivalent forms for the distance formula between a Bloch function f and the subspace X ⊂ B is to decompose f properly into two parts so that one part is in the space X, and the Bloch norm of the other part is equivalent to the distance distB (f, X). It is expected that such a decomposition is nonlinear and is not unique.

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For f ∈ B, it is easy to establish the following formula (see, for example, Lemma 4.2.8 in [14]) f (w)(1 − |w|2 ) dA(w), z ∈ D. f (z) = f (0) + (1 − zw)2 w D

Let

Eε (f )(z) = D\Ωε (f )

f (w)(1 − |w|2 ) dA(w) + cε,f (1 − zw)2 w

and

Pε (f )(z) = f (z) − Eε (f )(z), where cε,f is a constant such that Eε (f )(0) = 0. Clearly f (w)(1 − |w|2 ) dA(w), for any z ∈ D. Pε (f )(z) = f (0) − cε,f + (1 − zw)2 w Ωε (f )

Lemma 6. If f ∈ B, then Pε (f ) ∈ B and |Pε (f ) (z)| ≤

Cf B (1 − |z|2 )2

for any z ∈ D.

Proof. By Lemma 1, we have |Pε (f ) (z)| = Ωε (f )

wf ¯ (w)(1 − |w|2 ) dA(w) (1 − zw)4

≤ f B D

1 dA(w) |1 − zw|4

Cf B ≤ . (1 − |z|2 )2 Since for any g ∈ H(D) (see, for example, Theorem 5.1.5 in [14]), we have sup(1 − |z|2 )|g (z)| ≈ |g (0)| + sup(1 − |z|2 )2 |g (z)|, a∈D

it is enough to conclude that Pε (f ) ∈ B.

a∈D

Proof of Theorem 1. Let d1 , d2 , d3 and d4 denote the quantities (1), (2), (3) and (4) in Theorem 1, respectively. We prove the theorem by showing that d1 ≈ d2 , d2 = d4 and d3 = d4 . To prove d1 ≤ Cd2 , we show ﬁrst that Pε (f ) ∈ QK (p, p − 2) if ε > d2 . By Lemma 4, we need only to show that sup |Pε (f ) (z)|p (1 − |z|2 )2p−2 K(1 − |ϕa (z)|2 ) dA(z) < ∞. a∈D

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By Lemmas 6, 5 and Fubini’s theorem, we have sup |Pε (f ) (z)|p (1 − |z|2 )2p−2 K(1 − |ϕa (z)|2 ) dA(z) a∈D

D

≤

Cf p−1 B

sup a∈D

D

sup ≤ Cf p−1 B a∈D

≤

Cf pB

D Ωε (f )

sup

a∈D Ωε (f ) D

≤ Cf pB sup

a∈D Ωε (f )

χ

|Pε (f ) (z)|K(1 − |ϕa (z)|2 ) dA(z) |wf ¯ (w)|(1 − |w|2 ) dA(w)K(1 − |ϕa (z)|2 ) dA(z) |1 − zw|4

K(1 − |ϕa (z)|2 ) dA(z) dA(w) |1 − zw|4

K(1 − |ϕa (w)|2 ) dA(w). (1 − |w|2 )2

(w)

Ωε (f ) Since (1−|w| 2 )2 dA(w) is a K-Carleson measure, by Lemma 2 we know that the last term above is ﬁnite. Hence, by Lemma 4, we know that Pε (f ) ∈ QK (p, p − 2) if ε > d2 . Now since Eε (f )(0) = 0, we have

d1 = distB (f, QK (p, p − 2)) ≤ f − Pε (f )∗B = Eε (f )∗B = Eε (f )B . To conclude d1 ≤ Cd2 , we left to show that Eε (f )B ≤ Cε if ε > d2 , which is the following: f (w)(1 − |w|2 ) |Eε (f ) (z)| = 2 dA(w) (1 − zw)3 D\Ωε (f ) 1 ≤ 2ε dA(w) |1 − zw|3 D

Cε (by Lemma 1) ≤ . 1 − |z|2 We prove d2 ≤ Cd1 by contradiction. Indeed, if d1 < d2 , then there χΩε (f ) (z) exists ε > ε1 > 0 and fε1 ∈ QK (p, p − 2) such that (1−|z| 2 )2 dA(z) is not a K-Carleson measure and f − fε1 ∗B ≤ ε1 . We know for any z ∈ D |f (z)|(1 − |z|2 ) ≤ |fε 1 (z)|(1 − |z|2 ) + f − fε1 ∗B ≤ |fε 1 (z)|(1 − |z|2 ) + ε1 . This implies Ωε (f ) ⊂ Ωε−ε1 (fε1 ) and χΩε (f ) (z) |f (z)|p (1 − |z|2 )p−2 ≤ ε1 . 2 2 (1 − |z| ) (ε − ε1 )p Note that |fε 1 (z)|p (1 − |z|2 )p−2 dA(z) is a K-Carleson measure by Lemma 3, which implies, by above estimate, that measure. This is a contradiction.

χΩε (f ) (z) (1−|z|2 )2 dA(z)

is also a K-Carleson

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(z)

Ωε (f ) To prove d4 ≤ d2 , we start with the assumption that (1−|z| 2 )2 dA(z) is a K-Carleson measure (therefore ε ≥ d2 ). By Lemma 2, we know K(1 − |ϕa (z)|2 ) dA(z) < ∞. sup (1 − |z|2 )2 a∈D

Ωε (f )

Hence

sup

a∈D Ωε (f )

|f (z)|q (1 − |z|2 )q−2 K(1 − |ϕa (z)|2 ) dA(z)

f qB

≤

sup

a∈D Ωε (f )

K(1 − |ϕa (z)|2 ) dA(z) < ∞. (1 − |z|2 )2

Inequality d2 ≤ d4 is obvious from the following estimate and Lemma 2 K(1 − |ϕa (z)|2 ) dA(z) sup (1 − |z|2 )2 a∈D Ωε (f )

≤

1 sup εq a∈D

|f (z)|q (1 − |z|2 )q−2 K(1 − |ϕa (z)|2 ) dA(z).

Ωε (f )

To prove d3 = d4 , we only need to show K(g(z, a)) ≈ K(1 − |ϕa (z)|2 ), or equivalently (since ϕa (z) is a M¨obius transformation of D) 1 K log ≈ K(1 − |z|2 ), z ∈ D. |z| This is obvious because of the assumptions for K, and the following obvious facts 1 1 − e−2 ≤ 1 − |z|2 ≤ 1 ≤ log , if |z| ≤ e−1 ; |z| 1 1 − |z|2 ≤ log ≤ 5(1 − |z|2 ), if e−1 < |z| < 1. |z| The proof of Theorem 1 is completed.

Proof of Theorem 2. It was proved in [3] that if K satisﬁes the condition (a), then there exists a function K1 ≈ K, such that t−c K1 (t) is increasing on (0, 1) for some c > 0. Hence K(1 − |ϕa (z)|2 ) ≈ This implies

K1 (1 − |ϕa (z)|2 ) (1 − |ϕa (z)|2 )c ≤ K1 (1)(1 − |ϕa (z)|2 )c . (1 − |ϕa (z)|2 )c

|f (z)|p (1 − |z|2 )p−2 K(1 − |ϕa (z)|2 ) dA(z)

D

≤C

|f (z)|p (1 − |z|2 )p−2 K1 (1)(1 − |ϕa (z)|2 )c dA(z),

D

which means that F0 (p, p − 2, c) ⊂ QK,0 (p, p − 2).

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It is clear that the space F0 (p, p − 2, c) contains all polynomials (for any c > 0) so does the space QK,0 (p, p − 2). It is well-known that the closure of the set of all polynomials in B is just B0 . It was proved in [9] that QK,0 (p, p − 2) ⊂ B0 . We conclude therefore that the closure of QK,0 (p, p − 2) in B is B0 . This implies that the ﬁrst two quantities in Theorem 2 are equivalent. The proof of the equivalences of the remaining quantities are similar to those in proof of Theorem 1 and we omit the details here. Acknowledgements The authors would like to thank Professor Zhijian Wu (Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487) for a meticulous reading of the manuscript, in particular for the improvement of the English usage and the organization of the manuscript. They also like to thank the referee for valuable comments and suggestions.

References [1] Aulaskari, R., Xiao, J., Zhao, R.H.: On subspaces and subsets of BM OA and U BC. Analysis 15, 101–121 (1995) [2] Aulaskari, R., Stegenga, D., Xiao, J.: Some subclasses of BM OA and their characterization in terms of Carleson measures. Rocky Mountain J. Math. 26, 485–506 (1996) [3] Ess´en, M., Wulan, H., Xiao, J.: Several function-theoretic characterizations of M¨ obuius invariant QK spaces. J. Funct. Anal. 230, 78–115 (2006) [4] Ghatage, P.G., Zheng, D.: Analytic functions of bounded mean oscillation and the Bloch space. Int. Equ. Oper. Theory 17, 501–515 (1993) [5] Tjani, M.: Distance of Bloch functions to the little Bloch space. Bull. Austral. Math. Soc. 74, 101–119 (2006) [6] Wu, Z., Xie, C.: Decomposition theorems for Qp Spaces. Ark. Mat. 40, 383– 401 (2002) [7] Wu, Z., Xie, C.: Q Spaces and Morrey spaces. J. Funct. Anal. 201, 282– 297 (2003) [8] Wulan, H., Zhou, J.: The higher order derivatives of QK type spaces. J. Math. Anal. Appl. 332, 1216–1228 (2007) [9] Wulan, H., Zhou, J.: QK type spaces of analytic functions. J. Funct. Spaces Appl. 4, 73–84 (2006) [10] Xu, W.: Distances from Bloch functions to some M¨ obius invariant function spaces in the unit ball of Cn . J. Funct. Spaces Appl. 7, 91–104 (2009) [11] Xiao, J.: Holomorphic Q Classes. Lecture Notes in Mathematics, vol. 1767. Springer, Heidelberg (2001) [12] Zhao, R.: Distances from Bloch functions to some M¨ obius invariant spaces. Ann. Acad. Sci. Fenn. Math. 33, 303–313 (2008) [13] Zhao, R.: On a general family of function spaces. Ann. Acad. Sci. Fenn. Math. Diss. 105, 1–56 (1996) [14] Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)

Vol. 67 (2010)

Distances from Bloch Functions to QK -Type Spaces

Zengjian Lou (B) Department of Mathematics Shantou University Shantou 515063 Guangdong People’s Republic of China e-mail: [email protected] Wufu Chen Zhujiang College South China Agriculture University Guangzhou People’s Republic of China e-mail: di [email protected] Received: April 9, 2009. Revised: January 28, 2010.

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Integr. Equ. Oper. Theory 67 (2010), 183–202 DOI 10.1007/s00020-010-1775-x Published online March 20, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices Karlheinz Gr¨ochenig, Ziemowit Rzeszotnik and Thomas Strohmer Abstract. The ﬁnite section method is a classical scheme to approximate the solution of an inﬁnite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the ﬁnite section method for unstructured matrices on weighted p -spaces. In particular, the stability of the ﬁnite section method on 2 implies its stability on weighted p -spaces. Our approach uses recent results from the theory of Banach algebras of matrices with oﬀ-diagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a ﬁnite section method that is applicable to large classes of unstructured non-hermitian matrices as well as to least squares problems. Mathematics Subject Classification (2010). 65J10, 47L80. Keywords. Finite section method, stability, inverse-closed Banach algebras, spectral invariance.

1. Introduction Many of the concrete applications of mathematics in science and engineering eventually result in a problem involving linear operator equations. This problem can be usually represented as a linear system of equations (for instance by discretizing an integral equation or because the operator equation is already given on some sequence space) of the form Ax = b,

(1.1)

where A is an inﬁnite matrix A = (akl )k,l∈Z and b belongs to some Banach space of sequences. Solving linear equations with inﬁnitely many variables is a problem of functional analysis, while solving equations with ﬁnitely many variables is one of the main themes of linear algebra. Numerical analysis K.G. and Z.R. were supported by the Marie-Curie Excellence Grant MEXT-CT 2004517154, T.S. was supported by NSF DMS Grant 0511461.

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bridges the gap between these areas [17]. A fundamental problem of numerical analysis is thus to ﬁnd a ﬁnite-dimensional model for (1.1) whose solution approximates the solution of the original inﬁnite-dimensional problem with the desired accuracy. This problem often leads to delicate questions of stability and convergence. A simple and useful approach is the ﬁnite-section method [9,17,20,22]. Let Pn b = (. . . , 0, b−n , b−n+1 , . . . , bn−1 , bn , 0, . . .) be the natural projection onto a 2n + 1-dimensional subspace. We set An = Pn APn

and

bn = Pn b,

(1.2)

and try to solve the ﬁnite system An xn = bn

(1.3)

for properly chosen n. The crucial question is then: what is the relation between the numerical solution xn and the actual solution x? This problem has been analyzed in depth for the case of convolution operators and Toeplitz matrices in the pioneering work of Gohberg, see [9]. Important generalizations and extensions in the Toeplitz setting can be found in [4,5]. Rabinovich et al. [22] derive necessary and sufﬁcient conditions for the convergence of the ﬁnite section method with the limit operator method. This method does not necessarily require any Toeplitz structure, but the arising conditions, while intriguing, are not always easy to verify in practice and are understood mainly for special classes of operators that lead to structured matrices. A general theory for the approximation by ﬁnite sections is based on ottcher, powerful techniques of C ∗ -algebras and has been developed by B¨ Silbermann, and coworkers, see for instance [4,17]. Their framework leads to many attractive and deep results about the applicability of the ﬁnite section method as well as other approximation methods. This line of ideas was continued in [7] in the investigation of quasi-diagonal operators. William Arveson goes a step further and concludes that “numerical problems involving inﬁnite dimensional operators require a reformulation in terms of C ∗ -algebras” [1]. However, C ∗ -algebras have some limitations. It was already pointed out in [17] that C ∗ -algebra techniques do not yield any information about the rate of convergence of the ﬁnite section method. An answer to this question is obviously not only of theoretical interest, but it is important for real applications. For instance, we want to choose n in (1.3) large enough to get a sufﬁciently accurate solution, but n should be small enough to bound the computational complexity, which in general is of order O(n3 ). Theorems about the rate of convergence will give a quantitative indication for how increasing n will impact the accuracy of the solution. Some results about the rate of convergence for the special case of Toeplitz matrices can be found in [10,24,26,27]. The convergence in the p -norm, 1 ≤ p < ∞, is analyzed in [10]. Our starting point is the fact that the matrices arising in wireless communications or in Galerkin methods for PDE are usually almost diagonal, i.e., they possess controlled oﬀ-diagonal decay, but they do not necessarily

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have any additional structure. See, e.g., [29] for the analysis of time-varying channels in wireless communications. In this paper, we present an axiomatic approach to the ﬁnite section method for “almost diagonal” matrices. We analyze the stability and the convergence of the ﬁnite section method for unstructured matrices with some form of oﬀ-diagonal decay, both for positive definite matrices and for non-hermitian matrices. Specifically, we solve the following problems: (a)

We present an axiomatic framework that shows clearly what conditions on the matrix A and the input vector b are sufﬁcient. We identify a large class of matrices, more precisely, Banach algebras of matrices, for which the ﬁnite section method holds. Our main contribution shows that the stability of the ﬁnite sections on 2 implies its stability on weighted p -spaces. (b) We study the ﬁnite section method on weighted p -spaces. If the input vector b belongs to a weighted space pm , then with sufﬁcient oﬀ-diagonal decay of the matrix A, the ﬁnite section method converges in the norm of pm . (c) If the input vector b belongs to a “smaller” weighted q space, then we derive quantitative estimates for the rate of convergence of xn to x in the original weighted p -norm. (d) We study a modiﬁed version of ﬁnite sections [9,11]. This method converges also for non-symmetric matrices and oﬀers an approach for the approximate solution of (inﬁnite-dimensional) least squares problems. The ﬁnite section method for non-symmetric matrices raises a number of rather difﬁcult questions and has motivated a large part of [17]. Since we work with Banach spaces of sequences, the methods will be taken from the theory of Banach ∗-algebras (involutive Banach algebras) instead of C ∗ -algebras which suit only Hilbert spaces. The key property of the matrices that we consider is their oﬀ-diagonal decay; we will rely heavily on recent results from the theory of Banach algebras of matrices. In fact, an important technical part of our analysis is to establish a ﬁnite section property for inﬁnite-dimensional matrix algebras. The paper is organized as follows. In Sect. 2, we derive an axiomatic framework for convergence estimates for the ﬁnite section method. In Sect. 3 we introduce several Banach algebras of inﬁnite matrices and formulate their fundamental properties. In Sect. 4 we combine the axiomatic framework of Sect. 2 with the matrix Banach algebras of Sect. 3 and establish convergence results of the ﬁnite section method on weighted p -spaces. In Sect. 5 we investigate a modiﬁed ﬁnite section method for non-symmetric matrices and for least squares problems.

2. Axiomatization and Quantitative Estimates In this section, we rederive a standard error estimate for the ﬁnite section method on (weighted) p -spaces (e.g., Lemma 7.79 in [5]). This estimate shows which quantities are at the heart of the convergence problem. Based on this

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estimate we then introduce a set of axioms that imply (rates of) convergence of the ﬁnite section method. We will analyze the ﬁnite section method for multidimensional index sets of the form Zd . Clearly our results could also be formulated with respect to other index sets and other projections. We ﬁrst deﬁne the projection Pn in dimension d > 1. We set Cn = [−n, n]d ∩ Zd , the integer vectors in the cube of length 2n centered at the origin. Then the projection Pn is deﬁned by (Pn y)(k) = χ[−n,n]d (k)y(k) = χCn (k)y(k) for k ∈ Zd . Denote the range of Pn by Im Pn and the projection onto its complement by Qn := I − Pn . Since Im Pn is a subspace of 2 (Zd ) of dimension (2n + 1)d , it can be identiﬁed with d C(2n+1) . The ﬁnite section of A is deﬁned to be An = Pn APn restricted to the Im Pn , that is An : Im Pn → Im Pn ⊂ 2 (Zd ). By definition, An is an operator acting on Im Pn , and we can interpret An as d a ﬁnite (2n + 1)d × (2n + 1)d -matrix acting on C(2n+1) . By A−1 n we understand the inverse of this operator on Im Pn , if it exists. Clearly An cannot 2 d be invertible on 2 (Zd ), but the operator A−1 n Pn acts on (Z ). Thus, we p d p can consider its norm on the m (Z ) space, where the m -norm is deﬁned as x pm = xm p (Zd ) , for a positive weight m on Zd . We begin with the standard error estimate for the ﬁnite section method on pm (see Lemma 7.79 in [5]). Lemma 2.1. Assume that A is bounded and invertible on pm and An is invertible on Im Pn for some n ∈ N. If b ∈ pm , Ax = b, and An xn = bn , then p p p x − xn pm ≤ 1 + A−1 n Pn B(m ) A B(m ) Qn x m . Proof. With Ax = b we have the following estimate x − xn pm = A−1 b − An −1 Pn b pm ≤ Pn A−1 b − An −1 Pn b pm + Qn A−1 b pm ≤ An −1 Pn B(pm ) Pn APn A−1 b − Pn b pm + Qn x pm = An −1 Pn B(pm ) Pn AQn A−1 b pm + Qn x pm p p p ≤ 1 + A−1 n Pn B(m ) A B(m ) Qn x m . For a pair of weighted spaces pm and qw we deﬁne the tail function m (2.1) ϕ(n) = Qn , w r where r−1 = max{p−1 − q −1 , 0} with the usual convention 10 = ∞. Remark 2.2. The tail function ϕ is ﬁnite if and only if qw ⊆ pm . Therefore, we shall use this crucial assumption throughout the paper. The following observation can be traced back to [21]. Lemma 2.3. If qw ⊆ pm , then Qn x pm ≤ ϕ(n) Qn x qw .

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Proof. Clearly, Qn x pm = Qn ( m w ) w Qn x p . If p ≥ q, then · p ≤ · q and so m m w Qn x ≤ Qn Qn x pm ≤ Qn wQn x q = ϕ(n) Qn x qw . w w ∞ q If p < q, then r = (p−1 − q −1 )−1 > 1. Thus, we use H¨olders inequality bc p ≤ b q c r and obtain m Qn x pm ≤ Qn wQn x q = ϕ(n) Qn x qw . w r By combining Lemmata 2.1 and 2.3 we obtain immediately a ﬁrst error estimate for the ﬁnite section method. Theorem 2.4. Assume that A is bounded and invertible on pm . If An is invertp ible on Im Pn for n ≥ n0 and supn≥n0 A−1 n Pn B(m ) < ∞, then for every q p solution x ∈ w ⊆ m p p q x − xn pm ≤ ϕ(n) 1 + A−1 n Pn B(m ) A B(m ) Qn x w . In particular, x − xn pm = o (ϕ(n)) ,

for q < ∞

x − xn pm = O (ϕ(n)) ,

for q = ∞.

and Although Theorem 2.4 is elementary, it shows which conditions are necessary for the ﬁnite section method to converge. Clearly the bounded invertibility of A on pm is necessary for a well-posed problem. The quantitative convergence estimates follow from the embeddings of weighted p -spaces. So the crucial assumptions are the invertibility of A on pm and the uniform invertibility of the ﬁnite sections An Pn . Following the standard terminology [20,22], we say that the ﬁnite section method is stable on pm , if there exists n0 ∈ N, such that p sup A−1 n Pn B(m ) < ∞.

n≥n0

Stability is a central concept in the numerical analysis of ﬁnite sections, and a large part of the work of B¨ ottcher, Rabinovich, Roch, Silbermann [5,6,17,22,23] is devoted to understanding the stability of certain classes of matrices and operators. We will study the stability for certain classes of unstructured matrices. Our main contribution is to show that the stability on pm is inherited from the stability on 2 . The key tools are Banach algebras of matrices with oﬀ-diagonal decay. Throughout the paper we denote by A a unital involutive Banach algebra of inﬁnite matrices. The elements of A are matrices A = (akl ), k, l ∈ Zd , addition and multiplication are deﬁned in the natural way, the involution is the usual matrix adjoint, and the unity element of A is the identity matrix. The following definitions formalize some of the properties used in the proof of Theorem 2.4.

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Definition 2.5. (a) We say that A acts boundedly on pm , if there is a constant C > 0 such that A B(pm ) ≤ C A A (b) (c)

for all A ∈ A.

The matrix algebra A is called inverse-closed in B(2 (Zd )), if for every A ∈ A that is invertible on 2 we have that A−1 ∈ A. We say that A is solid, if A ∈ A and the majorization |bkl | ≤ |akl | for all k, l ∈ Zd imply that B ∈ A and B A ≤ A A .

Next we try to build an inﬁnite matrix Ablock from blocks of ﬁnite square matrices {Pn APn }n∈N by stacking them “along the diagonal”. To be more precise, we consider a cube Cn = {−n, . . . , n}d ⊆ Zd and for jn ∈ Z, Jm , m ∈ Zd−1 we set Jn,m = (jn , Jm ) + Cn . The matrix of the ﬁnite section Pn APn shifted “along the diagonal” by (jn , Jm ) can therefore be described as akl for k, l ∈ Cn (AJn,m )(jn ,Jm )+k,(jn ,Jm )+l = (2.2) 0 otherwise. To deﬁne Ablock we ﬁx some n0 ∈ N and choose a sequence{jn }n∈N ⊂ Z and a sequence Jm , m ∈ Zd−1 , such that the union of cubes n≥n0 ,m∈Zd−1 Jn,m forms a disjoint partition of Zd and set Ablock = Ablock = AJn,m . (2.3) n0 n≥n0 m∈Zd−1

Similar “block-diagonal” operators appear under the name stacked operators in [20,23,24], where they are used to prove convergence of a ﬁnite section method for band-dominated operators. Definition 2.6. We say that the ﬁnite section stacking property holds for A if A ∈ A implies Ablock ∈ A. We note that any partition of Zd into translates of the cubes Cn will do the job, as long as at least one translate of each cube Cn occurs. With these definitions in place, we consider the class of Banach algebras A that satisfy the following conditions: (A1) A acts boundedly on pm and qw . (A2) A is inverse-closed in B(2 (Zd )). (A3) A is solid. (A4) A fulﬁls the ﬁnite section stacking property. This allows us to formulate our basic result. Theorem 2.7. Assume that A satisﬁes (A1)–(A4) and that A ∈ A is invertible on 2 . If the ﬁnite section method is stable on 2 , then it is stable on pm . If in addition b ∈ qw ⊆ pm , then x − xn pm = o (ϕ(n)) ,

for q < ∞

x − xn pm = O (ϕ(n)) ,

for q = ∞.

and

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Proof. By (A2) the inverse A−1 is again in A, hence by (A1) A−1 is bounded on both pm and qw . Thus A is invertible on pm and qw . Next, we form Ablock (for n0 satisfying supn≥n0 A−1 n Pn B(2 ) < ∞). By (A4) Ablock ∈ A. By construction the matrix (Ablock )−1 has blocks of the form A−1 n Pn “along the diagonal”, and no rows or columns are zero, because we use a partition of the index set into cubes. Since the ﬁnite section method is assumed to be stable on 2 , Ablock is invertible on 2 , and we obtain that (Ablock )−1 B(2 ) = supn≥n0 A−1 n Pn B(2 ) . Now condition (A2) implies that (Ablock )−1 ∈ A. Finally, (A1) and (A3) imply the stability on pm by −1 block −1 p ) A < ∞. (2.4) sup A−1 n Pn B(m ) ≤ C sup An Pn A ≤ C (A

n≥n0

n∈N

The error estimate follows from Theorem 2.4.

Taking p = q and m = w in the above theorem yields ϕ ≡ 1. Thus, we obtain the following easy consequence of Theorem 2.7. Corollary 2.8. If A satisﬁes (A1)–(A4), A ∈ A and b ∈ pm , then the ﬁnite section method converges in pm , for 1 ≤ p < ∞. Remark 2.9. For p = ∞ only weak∗ convergence can be proved. Theorem 2.7 reduces the understanding of stability to 2 , but even this problem is very difﬁcult, see, e.g., [9,17]. The situation becomes much easier for positive operators, as is shown by the following statement. Corollary 2.10. Assume that A satisﬁes (A1)–(A4). If A ∈ A is positive and invertible on 2 , then the ﬁnite section method is stable on pm . Proof. Let λ− = min σ(A) and λ+ = max σ(A), where σ(A) is the spectrum of A in 2 . Since A is positive and invertible, we have that σ(A) ⊆ [λ− , λ+ ] ⊆ (0, ∞). Therefore, λ− Pn b 22 ≤ APn b, Pn b = An Pn b, Pn b ≤ λ+ Pn b 22 . Consequently, on the invariant subspace Im Pn we obtain that σ(An ) ⊆ [λ− , λ+ ]. In particular, each An is invertible on Im Pn and −1 −1 sup A−1 B(2 ) . n Pn B(2 ) ≤ λ− = A

n∈N

Consequently, the ﬁnite sections are stable on 2 and Theorem 2.7 is applicable. We will discuss the more difﬁcult case of non-positive, non-hermitian matrices in Sect. 5. In the next section we shall exhibit several useful matrix algebras A satisfying conditions (A1)–(A4).

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3. A Class of Banach Algebras of Matrices We consider some typical matrix norms that express various forms of oﬀdiagonal decay. Our approach is partly motivated by the oﬀ-diagonal decay that is observed in various applications, such as signal and image processing, digital communication, and quantum physics. A different description of oﬀdiagonal decay of matrices (and operators) is the notion of band-dominated operators [24]. Oﬀ-diagonal decay is quantiﬁed by means of weight functions. A nonnegative function v on Zd is called an admissible weight if it satisﬁes the following properties: (i) (ii)

v is even and normalized such that v(0) = 1. v is submultiplicative, i.e., v(k + l) ≤ v(k)v(l) for all k, l ∈ Zd .

The assumption that v is even assures that the corresponding Banach algebra is closed under taking the adjoint A∗ . The weight v is said to satisfy the Gelfand–Raikov–Shilov (GRS) condition [8], if lim v(nk)1/n = 1

n→∞

for all k ∈ Zd .

(3.1)

This property is crucial for the inverse-closedness of Banach algebras, see Theorem 4.2 below. The standard weight functions on Zd are of the form b

v(x) = ead(x) (1 + d(x))s , where d(x) is a norm on Rd . Such a weight is submultiplicative, when a, s ≥ 0 and 0 ≤ b ≤ 1; v satisﬁes the GRS-condition, if and only if 0 ≤ b < 1. Consider the following conditions on matrices. 1.

The Jaﬀard class is deﬁned by polynomial decay oﬀ the diagonal. Let As be the class of matrices A = (akl ), k, l ∈ Zd , such that |akl | ≤ C(1 + |k − l|)−s

2.

∀k, l ∈ Zd

(3.2)

with norm A As = supk,l∈Zd |akl |(1 + |k − l|)s . More general oﬀ-diagonal decay. Let v be an admissible weight on Zd that satisﬁes the additional condition v −1 ∗ v −1 ≤ Cv −1 ∈ 1 (Zd ) (v is called subconvolutive). We deﬁne the Banach space Av by the norm A Av = sup |akl |v(k − l) ,

(3.3)

k,l∈Zd

3.

Schur-type conditions. Let v be an admissible weight. The class A1v consists of all matrices A = (akl )k,l∈Zd such that sup |akl | v(k − l) < ∞ and sup |akl |v(k − l) < ∞ (3.4) k∈Zd

l∈Zd

l∈Zd

with norm A A1v = max

⎧ ⎨ sup ⎩k∈Zd

l∈Zd

k∈Zd

|akl |v(k − l), sup

l∈Zd

⎫ ⎬ |akl |v(k − l) . ⎭ d

k∈Z

(3.5)

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The Gohberg–Baskakov–Sj¨ ostrand class. For any admissible weight v we deﬁne the class Cv as the space of all matrices A = (akl )k,l∈Zd such that the norm sup |ak,k−l |v(l) (3.6) A Cv := l∈Zd

k∈Zd

is ﬁnite. An alternative way to deﬁne the norm on Cv is A Cv = inf{ α 1v : |akl | ≤ α(k − l)}.

(3.7)

This type of “convolution-dominated” decay of matrices plays an important role in the discretization of pseudodifferential operators with respect to Gabor frames [12,15] and in mobile communications [29]. 5. Further generalizations are due to Sun [30]. Roughly speaking, Sun’s class amounts to an interpolation between Cv and Av or between A1v and Av .

4. Matrix Algebras and Convergence Estimates In this section, we show that the matrix Banach algebras introduced in the previous section satisfy the postulates (A1)–(A4) put forth in Sect. 2. This in turn will immediately yield various convergence results of the ﬁnite section method. We start with a summary of the elementary properties of these Banach algebras. Lemma 4.1. Let v be an admissible weight and A be one of the algebras As for s > d, Av , A1v , Cv . Then A has the following properties: (a) Both A1v and Cv are involutive Banach algebras with the norms deﬁned in (3.5) and (3.6). Av and As , s > d, can be equipped with an equivalent norm so that they become involutive Banach algebras. (b) If A ∈ A, then A is bounded on 2 (Zd ). (c) If A ∈ A and |bkl | ≤ |akl | for all k, l ∈ Zd , then B ∈ A and B A ≤ A A . Proof. Properties (a) and (c) are easy and follow directly from the definition of the matrix norms. The statements about As and Av are proven in [14]. (b) is a consequence of Schur’s test. In particular, by Lemma 4.1(c) the indicated Banach algebras are solid, and so they satisfy automatically condition (A3). Our next theorem states that the matrix algebras introduced above are inverse-closed and thus fulﬁll (A2), as long as v satisﬁes the GRS-condition. The precise formulation is slightly more complicated, because we need to be a bit pedantic about the weights. Theorem 4.2 (Inverse-closedness). Let v be an admissible weight that satisﬁes the GRS-condition, i.e., limn→∞ v(nk)1/n = 1 for all k ∈ Zd . (a) Assume that v −1 ∗ v −1 ≤ Cv −1 ∈ 1 (Zd ), then Av is inverse-closed in B(2 (Zd )). In particular As for s > d possesses this property.

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(b) If v(k) ≥ C(1 + |k|)δ for some δ > 0, then A1v is inverse-closed in B(2 (Zd )). (c) Cv is inverse-closed in B(2 (Zd )) for arbitrary admissible weights with the GRS-property. Remark 4.3. The inverse-closedness is the key property and lies rather deep. While for C ∗ -(sub)algebras this property is inherent, for Banach algebras it is always hard to prove. Inverse-closedness for As is due to Jaﬀard [18] and Baskakov [2,3], a simple proof is given in [30]. For Av it was proved by Baskakov [3] and reproved in a different way in [14]. The result for Cv with v ≡ 1 is due to Gohberg–Kaeshoek–Woerdeman [11] and was rediscovered by Sj¨ ostrand [25]. This matrix algebra also plays an important role in the limit operator method in [22]. The case of arbitrary weights is due to Baskakov [3], see also [19, Chapter 5] and the references therein. Baskakov also proved that Cv and Av are inverse-closed in B(p ) for 1 ≤ p < ∞. The algebra A1v was treated by one of us with Leinert [13]. More general conditions were obtained in [15,30]. We note that A1v with constant weight v ≡ 1 is not inverse-closed in B(2 ) [31]. To assure condition (A1) we present some sufﬁcient conditions for the bounded action of A on pm . Clearly the weight m depends on the submultiplicative weight v used to parameterize the oﬀ-diagonal decay. Thus we need to discuss moderate weights. Let v be an admissible weight. The class of v-moderate weights is

Mv =

m(k + l) ≤ Cv(l), m > 0 : sup m(k) d k∈Z

∀ l ∈ Zd .

(4.1)

For example, if a, s ∈ R are arbitrary and 0 ≤ b ≤ 1 and d(x) is a norm on b b Rd , then m(x) = ead(x) (1 + d(x))s is e|a|d(x) (1 + d(x))|s| -moderate. The Banach algebras considered above act boundedly on the entire range of pm for 1 ≤ p ≤ ∞ and a family of moderate weights associated to v. The following lemma provides some explicit sufﬁcient conditions on m for Av , A1v or Cv to act boundedly on pm . Lemma 4.4. Let v be an admissible weight. (a) A1v acts boundedly on pm (Zd ) for 1 ≤ p ≤ ∞ and m ∈ Mv . (b) If v0 (k) := v(k)/(1 + |k|)s is submultiplicative for some s > d, then Av acts boundedly on pm (Zd ) for 1 ≤ p ≤ ∞ and m ∈ Mv0 . d (c) Av acts boundedly on ∞ v (Z ). p d (d) Cv acts boundedly on m (Z ) for 1 ≤ p ≤ ∞ and m ∈ Mv . Proof. For completeness we sketch the easy proof.

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(a) First, let p = 1, c ∈ 1m (Zd ) and A ∈ A1v . Then, since m(k) ≤ Cv(k − l)m(l), we obtain 1 Ac m = akl cl m(k) ≤ C |akl | |cl |v(k − l)m(l) k∈Zd l∈Zd k∈Zd l∈Zd ⎛ ⎞ ⎝ sup ≤C |akl |v(k − l)⎠ |cl |m(l) ≤ C A A1v c 1m . l∈Zd

l∈Zd

k∈Zd

Next, let p = ∞ and c ∈ ∞ m . Then, as before Ac ∞ = sup akl cl m(k) ≤ C sup |akl | |cl |v(k − l)m(l) m k∈Zd k∈Zd l∈Zd l∈Zd ≤ C sup |cl |m(l) sup |akl |v(k − l) ≤ C A A1v c ∞ . m l∈Zd

k∈Zd

l∈Zd

The boundedness on pm (Zd ) for 1 < p < ∞ now follows by interpolation. (b) and (d) follow from the easy embeddings Av → A1v0 , Cv ⊆ A1v and from (a). d (c) uses the subconvolutivity of v. Let A ∈ Av and c ∈ ∞ v (Z ). Then, −1 −1 v(l) . Consequently, |akl | ≤ A Av v(k − l) and |cl | ≤ c ∞ v v(k) = sup a c Ac ∞ kl l v k∈Zd l∈Zd 1 1 ≤ A Av c ∞ v(k) ≤ C A Av c ∞ sup , v v v(k − l) v(l) d k∈Z d l∈Z

because (v −1 ∗ v −1 )(k) ≤ Cv(k)−1 .

The ﬁnal condition to be checked is the ﬁnite section stacking property (A4). We prove that (A4) holds for the indicated Banach algebras. Lemma 4.5. If v is an admissible weight, then the ﬁnite section stacking property holds for Av , A1v , and Cv . Proof. We denote the entries of Ablock by bkl , k, l ∈ Zd , and the entries of A by akl . For Av we simply observe that, by the definition of Ablock , for every k and l we can ﬁnd j ∈ Zd such that bkl is either equal to ak+j,l+j or to zero. Thus, |bkl |v(k − l) ≤ |ak+j,l+j |v(k + j − l − j), and therefore, Ablock Av = sup |bkl |v(k − l) ≤ sup |akl |v(k − l) = A Av . k,l∈Zd

k,l∈Zd

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To deal with A1v , we notice that for every k there is a j ∈ Zd such that |bkl | ≤ |ak+j,l+j | independent of l (that is, each row of Ablock is dominated by a row of A). Thus, |bkl |v(k − l) ≤ |ak+j,l+j |v(k − l) = |ak+j,l |v(k + j − l), l∈Zd

l∈Zd

and therefore, sup k∈Zd

l∈Zd

|bkl |v(k − l) ≤ sup

k∈Zd

l∈Zd

|akl |v(k − l).

l∈Zd

An analogous argument applied to the columns yields that Ablock A1v ≤ A A1v . In the case of Cv , we use that for every l and k there is a j ∈ Zd such that |bk,k−l | ≤ |ak+j,k−l+j | (that is, each entry on the lth diagonal of Ablock is dominated by some entry on the lth diagonal of A). Thus, sup |bk,k−l | ≤ sup |ak+j,k+j−l | ≤ sup |ak,k−l |,

k∈Zd

and therefore, Ablock Cv =

k∈Zd

l∈Zd

sup |bk,k−l |v(l) ≤

k∈Zd

k∈Zd

l∈Zd

sup |ak,k−l |v(l) = A Cv .

k∈Zd

We now combine the ﬁndings of this section with Corollary 2.10 into a coherent result. Theorem 4.6. Let A be one of the Banach algebras As , Av , A1v or Cv , where v is an admissible weight that satisﬁes the GRS-condition, i.e., limn→∞ v(nk)1/n = 1, for all k ∈ Zd . Moreover, assume that (a) if A = As , then m, w are v0 -moderate, where v0 (k) := (1 + |k|)δ and 0 < δ < s − d, (b) if A = Av , then v −1 ∗ v −1 ≤ Cv −1 ∈ 1 (Zd ) and m, w are v0 -moderate, where v0 (k) := v(k)/(1 + |k|)s is submultiplicative for some s > d, (c) if A = A1v , then v(k) ≥ C(1 + |k|)δ for some δ > 0 and m, w are v-moderate, (d) if A = Cv , then m, w are v-moderate. Then, for every positive operator A ∈ A the ﬁnite section method is stable on pm . For an input vector b ∈ qw (Zd ) ⊆ pm (Zd ) the error estimate of the ﬁnite section method is x − xn pm = o (ϕ(n)) ,

for q < ∞

x − xn pm = O (ϕ(n)) ,

for q = ∞.

and

As in Corollary 2.8, by taking p = q and m = w in the above theorem, we obtain the following Corollary 4.7. If A and b satisfy the assumptions of Theorem 4.6, then the ﬁnite section method converges in pm , for 1 ≤ p < ∞.

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Since the Theorem 4.6 does not incorporate Lemma 4.4(c), let us state an explicit result using this part of the lemma. For rapidly growing weight functions v we then obtain superalgebraic convergence of the ﬁnite section method. Theorem 4.8. Let v be an admissible subconvolutive weight that satisﬁes the GRS-condition and set 1/2 v(k)−2 . ϕ(n) = |k|>n

Then, for every positive operator A ∈ Av , the error estimate of the ﬁnite 2 section method with input b ∈ ∞ v ⊆ is x − xn 2 = O(ϕ(n)). Proof. Lemma 4.1(b) and Lemma 4.4(c) assure that Av acts boundedly on 2 and on ∞ v , thus (A1) holds for Av with p = 2, q = ∞, m ≡ 1 and w = v. By Lemma 4.1(c), Theorem 4.2(a) and Lemma 4.5, conditions (A2)–(A4) hold as well. Thus Corollary 2.10 holds, and the tail function ϕ, as deﬁned in (2.1), is given by m 1/2 1 = Q v(k)−2 . ϕ(n) = Qn = n w r v 2

Remark 4.9. If v(k) = (1 + |k|)s , then ϕ(n) = −s+d/2

n

|k|>n

|k|>n (1

+ |k|)−2s

1/2

∼

, and we recover the result of [27].

5. The Modified Finite Section Method and Least Squares In the previous section we derived quantitative estimates for the convergence of the ﬁnite section method under the assumption that the matrices are positive definite. This assumption was crucial within the axiomatic framework of Sect. 2, since it implied the stability of the ﬁnite section method without much hassle. For non-hermitian matrices it is a fundamental and difﬁcult problem to guarantee the convergence of the ﬁnite section method, even in a purely qualitative sense [6,9,10,20,22]. In principle, the solution of Ax = b with a non-hermitian A could be computed by solving the normal equations A∗ Ax = A∗ b. One then applies the ﬁnite section method to the positive definite system A∗ Ax = A∗ b and invokes the results from the previous sections. This idea works for banded matrices because the computation of Pn A∗ APn requires only a ﬁnite number of steps. For general inﬁnite matrices, however, Pn A∗ APn cannot be computed exactly, and this approach is not directly feasible for numerical purposes. Furthermore, it is not difﬁcult to see that if A is not positive, (Pn A∗ Pn APn )−1 (even if it exists) may not converge strongly to (A∗ A)−1 . ∗ A natural remedy would be to approximate the entries (A A)kl by “properly” truncating the sums j a∗kj ajl . The idea is to replace Pn A∗ APn

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by Pn A∗ Pr APn for suitable r. This modiﬁed matrix can be computed with a ﬁnite number of operations. The difﬁculty lies in choosing the right truncation level. Truncating the sums too early can result in singular normal equations, while truncating very late may unreasonably increase the computational costs. We will address these issues below and derive a version of the ﬁnite section method that converges in pm for non-hermitian, unstructured matrices. Consider the system Ax = b where A is an invertible, but not necessarily hermitian matrix. We set Ar,n = Pr APn ,

and

br,n = A∗r,n b,

(5.1)

and try to solve the system A∗r,n Ar,n xr,n = br,n

(5.2)

for appropriately chosen r and n. In general, we will need r > n and approximate A by a rectangular matrix Ar,n . We will refer to (5.1)–(5.2) as the modiﬁed ﬁnite section method. Let us denote Bn := Pn A∗ APn

and Dr,n := Pn A∗ Pr APn = A∗r,n Ar,n ,

(5.3)

where both matrices are restricted to Im Pn . Clearly, Bn , n ∈ N, is the sequence of ﬁnite sections of A∗ A and Dr,n is an approximation of Bn . Lemma 5.1. If A and A∗ are bounded on pm , where 1 ≤ p < ∞, then there exists a sequence R(n) ∈ N, such that for every sequence r(n) ≥ R(n) lim Bn − Dr(n),n B(pm ) = 0,

n→∞

Proof. Let Qr = I − Pr . Then we may rewrite Bn − Dr,n as Bn − Dr,n = Pn A∗ Qr APn .

(5.4)

Therefore, Bn − Dr,n B(pm ) ≤ A∗ B(pm ) Qr APn B(pm ) . Since for each n ∈ N the operator APn is of ﬁnite rank, and Qr → 0 strongly in pm as r → ∞, we can ﬁnd a sequence R(n) such that lim QR(n) APn B(pm ) = 0.

n→∞

To ﬁnish the argument, we use that Qr(n) = Qr(n) QR(n) , whenever r(n) ≥ R(n). For the matrix algebras of Sects. 2 and 3 we may choose r(n) ≈ n and Ar,n to be almost a square matrix. Lemma 5.2. Assume that the matrix algebra A satisﬁes (A1)–(A4) and that the space of banded matrices is dense in A, e.g., A = Cv . If limn→∞ r(n)−n = ∞, then Bn − Dr(n),n B(pm ) → 0 for 1 ≤ p ≤ ∞ and m ∈ Mv .

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The same statement holds for A ∈ Av and m ∈ Mv0 [v0 is deﬁned in Lemma 4.4(b)]. Proof. Let AL be the banded truncation of A deﬁned by (AL )kl = akl for |k − l| ≤ L and (AL )kl = 0 for |k − l| > L. By (A1), (A3), and the density of banded matrices, A − AL B(pm ) ≤ A − AL A → 0 for L → ∞. Likewise for A = Av ⊆ Cv0 we have A − AL B(pm ) ≤ A − AL Cv0 → 0. If r(n) − n ≥ L, then Qr(n) Ar(n)−n Pn = 0 by computing the individual entries. Consequently Qr(n) APn B(pm ) = Qr(n) (A − Ar(n)−n )Pn B(pm ) ≤ A − Ar(n)−n B(pm ) → 0 since r(n) − n → ∞. As in the proof of Lemma 5.1, this implies that Bn − Dr(n),n B(pm ) → 0. By rewriting Lemma 2.1 and Theorem 2.4 for non-hermitian matrices, we ﬁnd that the modiﬁed ﬁnite section method still converges, provided that the sequence Dr(n),n is stable on pm (for a suitable sequence r(n)). As in the analysis of the usual ﬁnite section method, our axiomatic approach makes the veriﬁcation of these conditions much easier. Lemma 5.3. Let A be a Banach algebra of inﬁnite matrices satisfying (A1)– (A4) and 1 ≤ p < ∞. If A ∈ A is invertible on 2 , then there exists a sequence R(n) ∈ N, such that, for every sequence r(n) ≥ R(n), Dr(n),n is invertible on Im Pn and −1 Pn B(pm ) < ∞. sup Dr(n),n

n≥n0

Proof. Let In be the identity matrix on Im Pn . Since A is invertible on 2 , A∗ A is positive and invertible on 2 . Since A∗ A ∈ A, Corollary 2.10 implies that the sequence Bn Pn is stable on pm , i.e., supn∈N Bn−1 Pn B(pm ) < ∞. Thus, by Lemma 5.1, there is a sequence R(n) ∈ N such that Bn−1 Pn (Bn − Dr(n),n ) B(pm ) ≤

1 2

for every r(n) ≥ R(n).

(5.5)

Therefore In − Bn−1 Pn (Bn − Dr(n),n ) is invertible on Im Pn . We now use the −1 −1 −1 identity Dr,n = In − Bn−1 Pn (Bn − Dr,n ) Bn on Im Pn to deduce that Dr(n),n is invertible on Im Pn . Furthermore, we have −1 Pn B(pm ) sup Dr(n),n

n∈N

−1 ≤ sup 1 − Bn−1 Pn (Bn − Dr(n),n ) B(pm ) sup Bn−1 Pn B(pm ) < ∞, n∈N

for every r(n) ≥ R(n).

n∈N

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The main result of this section provides a quantitative estimate for the rate of convergence of the modiﬁed ﬁnite section method. Recall that is the tail function. ϕ(n) = Qn m w r Theorem 5.4. Let A be a Banach algebra satisfying (A1)–(A4) with 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. If A ∈ A is invertible on 2 and b ∈ qw ⊆ pm , then there exists a sequence R(n) ∈ N such that for every r(n) ≥ R(n) the sequence xr(n),n converges to x in the norm of pm , with the error estimate x − xr(n),n pm = o (ϕ(n)) ,

for q < ∞

x − xr(n),n pm = O (ϕ(n)) ,

for q = ∞.

and

In particular, for b ∈ on pm .

pm

the modiﬁed ﬁnite section method converges

Proof. Using Lemma 5.3 we may assume that Dr,n is invertible (thus, r ≥ n). We split the error x − xr,n into three terms as follows: −1 ∗ Ar,n b pm x − xr,n pm = (A∗ A)−1 A∗ b − Dr,n

≤ (A∗ A)−1 A∗ b−Bn−1 Pn A∗ b pm + Bn−1 Pn A∗ b−Bn−1 A∗r,n b pm −1 ∗ + Bn−1 A∗r,n b − Dr,n Ar,n b pm = I pm + II pm + III pm .

To deal with the ﬁrst term, we observe that the vector Bn−1 Pn A∗ b is exactly the result of the ﬁnite section method applied to the normal equation A∗ Ax = A∗ b. Since A∗ A ∈ A is positive and A∗ b ∈ qw ⊆ pm , Corollary 2.10 is applicable and implies that I pm = o(ϕ(n)), for 1 ≤ q < ∞, and that I pm = O(ϕ(n)), for q = ∞. Since Ar,n = Pr APn , the second term can be estimated by II pm = v Bn−1 Pn (A∗ b−A∗ Pr b) pm ≤ sup Bn−1 Pn B(pm ) A∗ B(pm ) Qr b pm . n∈N

As in the proof of Lemma 5.3, estimate (2.4) yields supn∈N Bn−1 Pn B(pm ) < ∞. Thus, Lemma 2.3 implies that II pm = o(ϕ(r)) , for 1 ≤ q < ∞, and that II pm = O(ϕ(r)), for q = ∞. However, since r ≥ n, we also obtain that II pm = o(ϕ(n)), for 1 ≤ q < ∞, and that II pm = O(ϕ(n)), for q = ∞. For the third term, we have the following estimate: using (5.4) −1 −1 III pm = (Bn−1 −Dr,n )Pn A∗r,n b pm = Dr,n Pn (Dr,n −Bn )Bn−1 Pn A∗r,n b pm −1 ≤ Dr,n Pn B(pm ) Pn A∗ Qr APn Bn−1 Pn A∗r,n b pm −1 ≤ Dr,n Pn B(pm ) Pn A∗ B(pm ) Qr APn Bn−1 Pn A∗r,n b pm .

Thus, by Lemma 2.3 we get that −1 Pn B(pm ) Qr APn Bn−1 Pn A∗r,n b qw III pm ≤ Cϕ(r) Dr,n −1 ≤ Cϕ(r) Dr,n Pn B(pm ) Qr APn B(qw ) Bn−1 Pn A∗r,n b qw .

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To deal with the term Bn−1 Pn A∗r,n b qw , we observe that estimate (2.4), applied in the case of qw , yields that supn∈N Bn−1 Pn B(qw ) < ∞. Therefore, Bn−1 Pn A∗r,n b qw ≤ C uniformly in n and r. This leads to −1 III pm ≤ Cϕ(r) Dr,n Pn B(pm ) Qr APn B(qw ) .

Lemma 5.3 now guarantees that Dr(n),n is invertible and furthermore −1 Pn B(pm ) < ∞ whenever r(n) ≥ R(n). In addition, the proof supn∈N Dr(n),n of Lemma 5.1 assures that the sequence R(n) can be chosen such that limn→∞ Qr(n) APn B(qw ) = 0 for r(n) ≥ R(n) and 1 ≤ q < ∞. Therefore, for such a sequence, we get that III pm = o(ϕ(r(n))). Thus, since r(n) ≥ n, we have proved that III pm = o(ϕ(n)), for 1 ≤ q < ∞. In the case when q = ∞, we simply use that Qr APn B(qw ) is bounded and we obtain that III pm = O(ϕ(n)). For the convergence in pm , we take p = q and m = w as in Remark 2.9. In particular, Theorem 5.4 holds for the matrix algebras discussed in Sect. 3. If A = Cv or if the class of banded matrices is dense in A, then Theorem 5.4 also holds for p = ∞. Finally we use the modiﬁed ﬁnite section method to approximate solutions of inﬁnite-dimensional least squares problems. Assume that A has a left-inverse on 2 (Zd ) and let b ∈ 2 (Zd ). We denote the solution of min Ax − b 2

(5.6)

x

by xLS , and remark that xLS is given by xLS = (A∗ A)−1 A∗ b. We approximate xLS by computing xn = (A∗r,n Ar,n )−1 A∗r,n b. An inspection of the proofs in this section shows that all steps of the proofs still go through if we replace invertibility of A by the assumption that A has a leftinverse (the key observation is that A∗ A is still positive and invertible on 2 ). We therefore obtain the following consequence. Corollary 5.5. Let A be a Banach algebra satisfying (A1)–(A4) with 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. Consider the least squares problem min Ax − b 2 . If A ∈ A is left-invertible on 2 and b ∈ qw ⊆ pm , then there exists a sequence R(n) ∈ N such that for every r(n) ≥ R(n) the sequence xr(n),n converges to xLS in the norm of pm , with the error estimate xLS − xr(n),n pm = o (ϕ(n)) ,

for q < ∞

xLS − xr(n),n pm = O (ϕ(n)) ,

for q = ∞.

and

Remark 5.6. The ﬁnite section methods described in this paper have been successfully used in various applied problems, such as in frame theory for the numerical computation of dual frames, or in wireless communications for the calculation of transmit and receive pulses [26,28]. Remark 5.7. It is well-known that, from a numerical viewpoint, the solution of the normal equations should be avoided whenever the condition number of the matrix is large. As an alternative to the normal equations one often

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uses matrix factorization methods. Since Dr,n is invertible, the matrix Ar,n has full rank (2n + 1)d , and one could apply a QR-factorization of Ar,n or some other factorization and compute an approximate solution to Ax = b in that way. This idea raises a number of interesting questions in inﬁnite matrix algebras: For instance, assume we can factorize a matrix A ∈ A into A = QR, where Q is unitary and R is upper triangular, do the individual components Q and R also belong to A? How about other matrix factorizations such as the LU -decomposition? We will address these questions in a forthcoming paper. Acknowledgements We would like to thank the referees for helpful suggestions and an improved formulation of Lemma 5.2. A more leisurely version of this paper was already posted in [16].

References [1] Arveson, W.: C ∗ -algebras and numerical linear algebra. J. Funct. Anal. 122(2), 333–360 (1994) [2] Baskakov, A.G.: Wiener’s theorem and asymptotic estimates for elements of inverse matrices. Funktsional. Anal. i Prilozhen. 24(3), 64–65 (1990) [3] Baskakov, A.G.: Estimates for the elements of inverse matrices, and the spectral analysis of linear operators. Izv. Ross. Akad. Nauk Ser. Mat. 61(6), 3–26 (1997) [4] B¨ ottcher, A., Silbermann, B.: The ﬁnite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols. Math. Nachr. 110, 279–291 (1983) [5] B¨ ottcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Springer, Berlin (1990) [6] B¨ ottcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer, New York (1999) [7] Brown, N.P.: Quasi-diagonality and the ﬁnite section method. Math. Comp. 76(257), 339–360 (electronic) (2007) [8] Gelfand, I., Raikov, D., Shilov, G.: Commutative Normed Rings (Translated from the Russia). Chelsea Publishing Co., New York (1964) [9] Gohberg, I., Fel’dman, I.: Convolution Equations and Projection Methods for Their Solution, Vol. 41. American Mathematical Society, Providence, R.I., Translated from the Russian by F. M. Goldware, Translations of Mathematical Monographs (1974) [10] Gohberg, I., Goldberg, S., Kaashoek, M.: Basic Classes of Linear Operators. Birkh¨ auser, Basel (2003) [11] Gohberg, I., Kaashoek, M.A., Woerdeman, H.J.: The band method for positive and strictly contractive extension problems: an alternative version and new applications. Integr. Equ. Oper. Theory 12(3), 343–382 (1989) [12] Gr¨ ochenig, K.: Time-frequency analysis of Sj¨ ostrand’s class. Revista Mat. Iberoam. 22(2), 703–724 (2006) [13] Gr¨ ochenig, K., Leinert, M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17, 1–18 (2004)

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[14] Gr¨ ochenig, K., Leinert, M.: Symmetry of matrix algebras and symbolic calculus for inﬁnite matrices. Trans. Am. Math. Soc. 358, 2695–2711 (2006) [15] Gr¨ ochenig, K., Rzeszotnik, Z.: Banach algebras of pseudodifferential operators and their almost diagonalization. Ann. Inst. Fourier (Grenoble) 58(7), 2279– 2314 (2008) [16] Gr¨ ochenig, K., Rzeszotnik, Z., Strohmer, T.: Quantitative estimates for the ﬁnite section method. http://arxiv.org/pdf/math.FA/0610588 [17] Hagen, R., Roch, S., Silbermann, B.: C ∗ -Algebras and Numerical Analysis, Volume 236 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (2001) [18] Jaﬀard, S.: Propri´et´es des matrices “bien localis´ees” pr`es de leur diagonale et quelques applications. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 7(5), 461–476 (1990) [19] Kurbatov, V.: Functional Differential Operators and Equations, volume 473 of Mathematics and Its Applications. Kluwer, Dotrecht (1999) [20] Lindner, M.: Inﬁnite Matrices and Their Finite Sections: An Introduction to the Limit Operator Method (Frontiers in Mathematics). Birkh¨ auser, Basel, (2006) [21] Pr¨ ossdorf, S.: Some Classes of Singular Equations. North Holland, Amsterdam, German original: Birkh¨ auser, Basel and Stuttgart 1974 (1977) [22] Rabinovich, V., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory, Volume 150 of Operator Theory: Advances and Applications. Birkh¨ auser, Basel (2004) [23] Rabinovich, V.S., Roch, S., Silbermann, B.: Fredholm theory and ﬁnite section method for band-dominated operators. Integr. Equ. Oper. Theory 30(4), 452– 495 (1998) [24] Rabinovich, V.S., Roch, S., Silbermann, B.: Algebras of approximation sequences: ﬁnite sections of band-dominated operators. Acta Appl. Math. 65(1–3), 315–332 (2001) (Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday) [25] Sj¨ ostrand, J.: Wiener type algebras of pseudodifferential operators. In: S´emi´ naire sur les Equations aux D´eriv´ees Partielles, 1994–1995, pages Exp. No. IV, ´ 21. Ecole Polytech., Palaiseau (1995) [26] Strohmer, T.: Rates of convergence for the approximation of dual shift-invariant systems in 2 (Z). J. Four. Anal. Appl. 5(6), 599–615 (2000) [27] Strohmer, T.: Four short stories about Toeplitz matrix calculations. Linear Algebra Appl. 343/344, 321–344 (2002) (Special issue on structured and inﬁnite systems of linear equations) [28] Strohmer, T.: Approximation of dual Gabor frames, window decay, and wireless communications. Appl. Comput. Harmonic Anal. 11(2), 243–262 (2001) [29] Strohmer, T.: Pseudodifferential operators and Banach algebras in mobile communications. Appl. Comput. Harmonic Anal. 20(2), 237–249 (2006) [30] Sun, Q.: Wiener’s lemma for inﬁnite matrices. Trans. Am. Math. Soc. 359(7), 3099–3123 (electronic) (2007) [31] Tessera, R.: The Schur algebra is not spectral. Monatsh. Math. (to appear)

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Karlheinz Gr¨ ochenig Faculty of Mathematics University of Vienna Nordbergstrasse 15 1090 Vienna, Austria e-mail: [email protected] Ziemowit Rzeszotnik Mathematical Institute University of Wroclaw 2/4, Pl. Grunwaldzki 50-384 Wroclaw Poland e-mail: [email protected] Thomas Strohmer Department of Mathematics University of California Davis, CA 95616-8633 USA e-mail: [email protected] Received: July 17, 2009. Revised: February 4, 2010.

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Integr. Equ. Oper. Theory 67 (2010), 203–213 DOI 10.1007/s00020-010-1776-9 Published online March 16, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

A General Christoffel-Darboux Type Formula Hugo J. Woerdeman Abstract. In this note we provide a general expression for multivariable stable polynomials that reduces to Christoffel-Darboux type formulas. As a result we obtain new proofs for existing Christoffel-Darboux formulas based on Schur complements. In some cases we are able to refine existing results. Mathematics Subject Classification (2000). Primary 47B35; Secondary 65F30. Keywords. Multivariable stable polynomial, Multivariable Toeplitz matrices, Christoffel-Darboux type expressions.

1. Introduction The classical Christoffel-Darboux formula for orthogonal polynomials of the unit circle is the following: n−1 − − p (w) p (z)← p(z)p(w) − ← = pj (z)pj (w). (1.1) 1 − zw j=0 − p (z) = z n p(z), and pj (z) are of degree j. Here p(z) = pn (z) is of degree n, ← The starting point is a positive measure on the unit circle, and the polynomip− als ← p−0 , . . . , ← n are the ﬁrst n + 1 orthogonal polynomials corresponding to the measure. One may use the Christoffel-Darboux formula to show that all the roots of p lie outside the unit disk. Another reason this formula is interesting is that it gives an integral kernel for the projection operator to the subspace p− spanned by the polynomials ← p−0 , . . . , ← n . In this paper we derive a very gen− − eral formula for the expression p(z)p(w) − ← p (w) (in fact, we do it for p (z)← operator valued polynomials). The main tool we use is Schur complements. Subsequently, we recover some of the Christoffel-Darboux type formulas that exist in the literature, and provide new proofs for these results. In some cases it leads to reﬁnement of existing results in that we are able to be more The author gratefully acknowledges support via NSF grant DMS-0901628.

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speciﬁc about the functions that appear in the Hermitian sums of squares representation (which in the classical case corresponds to the right hand side of (1.1)).

2. Christoffel-Darboux Type Formulas We use the following notation. As usual we let N0 , Z, C, T, D denote the sets of nonnegative integers, integers, complex numbers, complex numbers of modulus one, and complex numbers of modulus less than one, respectively. In addition, we denote D = D ∪ T. Let d ≥ 1. For z = (z1 , . . . , zd ) ∈ Cd and k = (k1 , . . . , kd ) ∈ Zd , we let z k = z1k1 · · · zdkd , where for negative ki we have zi = 0. When n = (n1 , . . . , nd ) ∈ Nd0 we let n denote the set n = n1 × · · · × nd , where = {0, . . . , } ( ∈ N0 ). When P (z) = k∈n Pk z k , with Pk : H → H, we say that P is an operator valued polynomial of degree at most n. We say d that P is stable when P (z) is invertiblefor z ∈ D . With P we associate its adjoint P ∗ , which is given by P ∗ (z) = k∈n Pk∗ z −k , and its reverse which is ← − given by P (z) := z n P ( z1 )∗ . Recall that if H1 and H2 are Hilbert spaces and A B M= (2.1) : H1 ⊕ H2 → H1 ⊕ H2 B∗ C is a positive semidefinite operator, then there exists a unique contraction G : ran (C) → ran (A) such that B = A1/2 GC 1/2 . Here ran (L) denotes the closure of the range of L, and L1/2 denotes the positive semidefinite square root of the positive semidefinite operator L. The Schur complement S of M supported on H1 is deﬁned to be the positive semidefinite operator A1/2 (1 − GG∗ )A1/2 . An alternative way to deﬁne the Schur complement of M supported on H1 is via f f A B Sf, f = inf , : g ∈ H 2 ; B∗ C g g that is, it is the largest positive semidefinite operator which may be subtracted from A in (2.1) such that the resulting operator matrix remains positive semidefinite. A few words are required on our notation. We will typically index rows and columns of (operator) matrices and vectors with subsets of Nd0 . When performing algebraic operations with the matrices and vectors, the appearing subset(s) of Nd0 should be ordered according to a single chosen total ordering of Nd0 (e.g., the lexicographical order). The speciﬁc choice for the ordering of Nd0 , however, is not important for the results. For an operator matrix M with rows and columns indexed by K(⊆ Nd0 ) and for Λ ⊆ K, we write S(M ; Λ) for the Schur complement supported on rows and columns labelled by elements of Λ. We will view S(M ; Λ) as a matrix whose rows and columns are indexed by K (thus of the same size as M ); its potentially nonzero entries lie in the rows and columns indexed by Λ. Thus, for instance, when Λ1 , Λ2 ⊆ K the

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expression S(M ; Λ1 ) − S(M ; Λ2 ) is well deﬁned. Finally, we denote Λ (z) = row(z k )k∈Λ , Λ ⊂ Nd0 . Theorem 2.1. Let n ∈ Nd0 and P (z) = k∈n Pk z k and R(z) = k∈n Rk z k be operator valued stable polynomials of degree at most n so that P (z)P (z)∗ = R(z)∗ R(z),

z ∈ Td .

(2.2)

Put F (z) = P (z)∗−1 P (z)−1 = R(z)−1 R(z)∗−1 , (2.3) j d and let F (z) = j∈Zd Fj z , z ∈ T , be its Fourier expansion. We let T = (Fk−l )k,l∈n . Then ← − ← − P (z)P (w)∗ − R (z) R (w)∗ = n (z)(S(T −1 ; n \ {n})−S(T −1 ; n\{0}))n (w)∗ . (2.4) Note that if P (z) is a stable scalar valued polynomial, one may choose R(z) = P (z) to satisfy (2.2). More generally, one can think of P (z) and R(z) as the left and right stable factor of a trigonometric polynomial Q(z)(= F (z)−1 ). The main result in [15] yields a characterization for when a positive definite valued Q(z) has a left (right) stable factorization. This result in [15] concerns factorization along commutative subspace lattices, and is based on the general factorization result in [14]. We need a few lemmas. Lemma 2.2. Assume that the operator matrix (Aij )2i,j=1 : H1 ⊕H2 → H1 ⊕H2 and the operator A22 are invertible. Then S = A11 − A12 A−1 22 A21 is invertible and −1 −1 A11 A12 S ∗ = . (2.5) A21 A22 ∗ ∗ Proof. Follows directly from the factorization 0 A11 − A12 A−1 22 A21 0 A22 −1 I A11 A12 I −A12 A22 = A21 A22 −A−1 0 I 22 A21

0 . I

(2.6)

Lemma 2.3. Let the operator matrix M = (Aij )2i,j=1 : H1 ⊕ H2 → H1 ⊕ H2 be positive definite. If ∗−1 L11 A11 A12 L11 = , A21 A22 L21 0 then

A11 A21

A12 A22

−1 =

L11 L21

0 L22

L∗11 0

L∗21 , L∗22

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for some invertible L22 . In fact, L22 L∗22 = A−1 22 . Similarly, if

∗

A11 A12 −1 ∗ U12 U22 , = 0 U22 A21 A22 then

A11 A21

A12 A22

−1

U11 = 0

U12 U22

0 , ∗ U22

∗ U11 ∗ U12

∗ = A−1 for some invertible U11 . In fact, U11 U11 11 .

Proof. Easy to check.

Proof of Theorem 2.1. As F (z)P (z) = P (z)∗−1 and P (z) is stable, we have that F−l Pl = P0∗−1 , Fk−l Pl = 0, k ∈ n \ {0}. (2.7) l∈n

l∈n

Letting L11 = P0 and L21 = col(Pk )k∈n\{0} we get from (2.7) that ∗−1 L11 L11 T = . L21 0 Applying Lemma 2.3 we get that ∗ L11 0 L11 −1 T = 0 L21 L22

L∗21 , L∗22

(2.8)

where L22 L∗22 = [(Fk−l )k,l∈n\{0} ]−1 . Next, since R(z)F (z) = R(z)∗−1 , we also have that Rl F−l = R0∗−1 , Rl Fk−l = 0, k ∈ n \ {0}. (2.9) l∈n

Letting

∗ U22

= R0 and

l∈n ∗ U12

∗ U12

= row(Rn−k )k∈n\{n} we get that

∗−1 ∗ U22 T = 0 U22 .

Applying Lemma 2.3 we get that

∗ U11 U12 U11 −1 T = ∗ 0 U22 U12

0 ∗ U22

,

(2.10)

∗ where U11 U11 = [(Fk−l )k,l∈n\{n} ]−1 . Next observe that ← − L11 U12 n (z) = P (z), n (z) = R (z). L21 U22

Now equating the right hand sides of (2.8) and (2.10), and hitting the equation with n (z) on the left and with n (w)∗ on the right, we get that 0 0 P (z)P (w)∗ +n (z) (w)∗ 0 L22 L∗22 n ∗ ← − ← − U11 U11 0 = n (z) (2.11) n (w)∗ + R (z) R (w)∗ . 0 0

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Observing, using Lemma 2.2, that ∗ 0 0 U11 U11 = S(T −1 ; n \ {n}), 0 0 0

0 L22 L∗22

207

= S(T −1 ; n \ {0}),

equation (2.4) follows.

Let us derive the classical one-variable Christoffel-Darboux formula from the above result. n n Proposition 2.4. Let P (z) = k=0 Pk z k : H → H and R(z) = k=0 Rk z k : H → H be single variable stable operator valued polynomials of degree at most n so that P (z)P (z)∗ = R(z)∗ R(z), z ∈ T. Then there exist operator-valued j (j) polynomials Gj (z) = k=0 Gk z k , j = 0, . . . , n − 1, of degree j so that ← − ← − P (z)P (w)∗ − R (z) R (w)∗ = (1 − zw) Gj (z)Gj (w)∗ , z, w ∈ C. n−1 j=0

(2.12) In fact, the polynomials Gj (z), j = 0, . . . , n − 1, can be found as follows. Put (2.13) F (z) = P (z)∗−1 P (z)−1 = R(z)−1 R(z)∗−1 , and let F (z) = j∈Z Fj z j , z ∈ T, be its Fourier expansion. We let Tm = (Fk−l )m k,l=0 . The coefficients of the polynomials Gj may be found via a Chole−1 sky factorization of Tn−1 as follows: ⎞⎛ ⎞ ⎛ (0) (n−1) (0)∗ ··· G0 0 G0 G0 ⎜ ⎟ ⎜ .. ⎟ .. −1 .. .. ⎟⎜ ⎟. (2.14) Tn−1 =⎜ . . . ⎠⎝ . ⎠ ⎝ (n−1)

0

Gn−1

(n−1)∗

G0

···

(n−1)∗

Gn−1

Proof. In the notation of the proof of Theorem 2.1 we have that −1 ∗ U11 U11 = [(Fk−l )k,l∈n\{n} ]−1 = Tn−1 = [(Fk−l )k,l∈n\{0} ]−1 = L22 L∗22 .

Next, notice that 0 n (z) 0 Similarly,

n (z)

0 −1 n−1 (w)∗ . (w)∗ = zwn−1 (z)Tn−1 L22 L∗22 n ∗ U11 U11 0

0 −1 n−1 (w)∗ . (w)∗ = n−1 (z)Tn−1 0 n

Combining these observations with (2.11) we get ← − ← − −1 P (z)P (w)∗ − R (z) R (w)∗ = (1 − zw)n−1 (z)Tn−1 n−1 (w)∗ . Using the factorization (2.14), equation (2.12) now follows.

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The above one-variable Christoffel-Darboux formula is closely related to the block matrix Gohberg-Semencul formula [10], as is explained in, for instance, [12]. Next, let us derive the two-variable Christoffel-Darboux type formula of Geronimo and Woerdeman [9, Proposition 2.2.3]. Proposition 2.5. [9] Let P (z) =

n1 n2

Pk1 ,k2 z1k1 z2k2 , R(z) =

k1 =0 k2 =0

n1 n2

Rk1 ,k2 z1k1 z2k2

k1 =0 k2 =0

be stable operator-valued polynomials (with underlying Hilbert space H), i.e., 2 taking invertible operator values for z = (z1 , z2 ) ∈ D , and let P (z)P (z)∗ = R(z)∗ R(z), z ∈ T2 . Then there exist operator-valued polynomials Gi and Fi (with underlying Hilbert space H) such that n 1 −1 ← − ← − P (z)P (w)∗ − R (z) R (w)∗ = (1 − z1 w1 ) Gj (z)Gj (w)∗ j=0

+(1 − z2 w2 )

n 2 −1

Fj (z)Fj (w)∗ .

(2.15)

j=0

The polynomials Fi and Gi can be constructed directly from Schur complements of the Toeplitz operator with symbol F := P ∗−1 P −1 = R−1 R∗−1 . The Gi can be chosen to have degree at most (i, n2 ), i = 0, . . . , n1 − 1, and the Fj can be chosen to have degree at most (n1 , j), j = 0, . . . , n2 − 1. Proof. We will use (2.4). Note that in the two variable case we have that n\{n} = (n1 −1×n2 ) ∪ (n1 ×n2 −1), n\{0} = (n1 \{0}×n2 )∪(n1 ×n2 \{0}). Next we observe that S(T −1 ; n \ {n}) is equal to the operator matrix [(Fk−l )k,l∈n\{n} ]−1 padded with zeros. It follows from [17, Theorem 1.5] (see [9, Theorem 2.4.1] for the scalar case) that [(Fk−l )k,l∈n\{n} ]−1 has zeros in locations (k, l) = ((k1 , k2 ), (l1 , l2 )) where (k1 , l2 ) = (n1 , n2 ) and where (k2 , l1 ) = (n2 , n1 ). But then we may apply [8, Corollary 2.7] and obtain that S(T −1 ; n \ {n}) = S(T −1 ; n1 − 1 × n2 ) + S(T −1 ; n1 × n2 − 1) −S(T −1 ; n1 − 1 × n2 − 1).

(2.16)

Similarly, S(T −1 ; n \ {0}) = S(T −1 ; n1 \ {0} × n2 ) + S(T −1 ; n1 × n2 \ {0}) −S(T −1 ; n1 \ {0} × n2 \ {0}). Next, let Z1 and Z2 be the shift matrices so that 1 (n (z1 , z2 ) − n (0, z2 )), n (z1 , z2 )Z2 z1 1 = (n (z1 , z2 )−n (z1 , 0)) z2

n (z1 , z2 )Z1 =

(2.17)

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for all z1 , z2 . Using that S(T −1 ; Λ) is equal to the operator matrix [(Fk−l )k,l∈Λ ]−1 sitting in the rows and columns corresponding to Λ, one sees that S(T −1 ; ν + Λ) consists of the same matrix [(Fk−l )k,l∈ν+Λ ]−1 = [(Fk−l )k,l∈Λ ]−1 but now sitting in the rows and columns corresponding to ν + Λ. Using this it is not hard to see that S(T −1 ; n1 \ {0} × n2 ) = Z1 S(T −1 ; n1 − 1 × n2 )Z1∗ , S(T −1 ; n1 × n2 \ {0}) = Z2 S(T −1 ; n1 × n2 − 1)Z2∗ , S(T −1 ; n1 \ {0} × n2 \ {0}) = Z1 Z2 S(T −1 ; n1 × n2 − 1)Z2∗ Z1∗ . Putting these observations together with (2.4) we get that ← − ← − P (z)P (w)∗ − R (z) R (w)∗ = (1 − z1 w1 )n (z)S(T −1 ; n1 − 1 × n2 )n (w)∗ + 1 − z2 w2 )n (z)S(T −1 ; n1 × n2 − 1)n (w)∗ −(1−z1 z2 w1 w2 )n (z)S(T −1 ; n1 −1×n2 − 1)n (w)∗ . One may write 1 − z1 z2 w1 w2 = 1 − z1 w1 + (1 − z2 w2 )z1 w1 . Rewriting, now yields that ← − ← − P (z)P (w)∗ − R (z) R (w)∗ = (1 − z1 w1 )n (z)[S(T −1 ; n1 − 1 × n2 ) −S(T −1 ; n1 − 1 × n2 − 1)]n (w)∗ + (1 − z2 w2 )n (z)[S(T −1 ; n1 × n2 − 1) −S(T −1 ; n1 \ {0} × n2 − 1)]n (w)∗ . ˜ ≥ 0 whenever K ˜ ⊂ K, we have that As S(M ; K) − S(M ; K) S(T −1 ; n1 − 1 × n2 ) − S(T −1 ; n1 − 1 × n2 − 1) = CC ∗ , S(T −1 ; n1 × n2 − 1) − S(T −1 ; n1 \ {0} × n2 − 1) = DD∗ for some triangular operator matrices C and D. But then writing n (z)CC ∗ n (w)∗ =

n 1 −1 j=0

Gj (z)Gj (w)∗ , n (z)DD∗ n (w)∗

n 2 −1

Fj (z)Fj (w)∗ ,

j=0

where the coefﬁcients of the polynomials Gj and Fj are in the jth row of C and D, respectively, we get the result. It should be noted that one also could have used the equality 1 − z1 z2 w1 w2 = 1 − z2 w2 + (1 − z1 w1 )z2 w2 , which would have resulted in different polynomials Fj and Gj . These two ways of writing correspond exactly to the two choices that exist in the context of two-evolution scattering systems; see [5, Theorem 5.5] (in particular, equations (5.8) and (5.9)). This phenomenon was also observed in [13, Theorem 5.1]. It seems worthwhile to investigate if this also implies that in the two variable Nevanlinna–Pick interpolation problem there are two extreme solutions to Agler’s matrix equation (see equation (11.50) in [3, Theorem 11.49]; the original result is from [1] (scalar case) and [2]) that gives the necessary

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and sufﬁcient condition for solvability. If so, it may be so that in the two variable case of [6, Theorem 6.1] one only needs to consider these two extreme solutions to sweep through the set of all solutions in that theorem (see the Remark after the statement of [6, Theorem 6.1]). The scalar case of (2.15) has been proved by Cole and Wermer [7] without explicitly constructing the polynomials Fi and Gi . They used it to deduce Andˆ o’s theorem [4] (a two-variable generalization of von Neumann’s inequality) and Agler’s decomposition for rational inner functions. As noticed by Knese [13], one obtains Agler’s decomposition (via Rudin’s approximation) for any scalar-valued function from the bidisk algebra. His proof [13] of the scalar case of (2.15) uses reproducing kernel Hilbert spaces and is also nonconstructive. We now obtain a Christoffel-Darboux type formula for 3 or more variables. Theorem 2.6. Let n ∈ Nd0 and P (z) = k∈n Pk z k and R(z) = k∈n Rk z k be operator valued stable polynomials of degree at most n so that P (z)P (z)∗ = R(z)∗ R(z), z ∈ Td . Then there exist analytic functions Gi and Fi such that d

⎡ n 1 −1 ← − ← − (1−zi wi ) ⎣(1−z1 w1 ) Gj (z)Gj (w)∗ P (z)P (w)∗ − R (z) R (w)∗ = i=3

j=0

+ (1−z2 w2 )

n 2−1

⎤

Fj (z)Fj (w)∗ ⎦ .

(2.18)

j=0

The functions Fi and Gi can be constructed directly from Schur complements of the Toeplitz operator associated with F := P ∗−1 P −1 = R−1 R∗−1 . The Gi can be chosen to be polynomials in (z1 , z2 ) of degree at most (i, n2 ), i = 0, . . . , n1 − 1, and the Fj can be chosen to be polynomials in (z1 , z2 ) of degree at most (n1 , j), j = 0, . . . , n2 − 1. Proof. Deﬁne pz1 ,z2 (z3 , . . . , zd ) = P (z1 , . . . , zd ), rz1 ,z2 (z3 , . . . , zd ) = R(z1 , . . . , zd ). Let M1 = M1 (z1 , z2 ), M2 = M2 (z1 , z2 ) : H2 (Td−2 ; H) → H2 (Td−2 ; H) be the Toeplitz operators with symbols pz1 ,z2 and rz1 ,z2 , respectively. With respect }, where z˜ = (z3 , . . . , zd ), we have the to the standard basis {˜ z k : k ∈ Nd−2 0 matrix representations M1 (z1 , z2 ) = M2 (z1 , z2 ) =

n2 n1 k1 =0 k2 =0 n1 n2 k1 =0 k2 =0

z1k1 z2k2 (P(k1 ,k2 ,k−l) )k,l∈Nd−2 , 0

z1k1 z2k2 (R(k1 ,k2 ,k−l) )k,l∈Nd−2 . 0

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The operator-valued polynomials M1 = M1 (z1 , z2 ), M2 = M2 (z1 , z2 ) satisfy the conditions of Proposition 2.5, and thus we obtain the existence of polynomials fj , j = 0, . . . , n2 − 1, of degree at most (n1 , j), and gi , i = 0, . . . , n1 − 1, of degree at most (i, n2 ), so that ←− ←− M1 (z1 , z2 )M1 (w1 , w2 )∗ − M2 (z1 , z2 )M2 (w1 , w2 )∗ = (1 − z1 w1 )

n 1 −1

gj (z1 , z2 )gj (w1 , w2 )∗

j=0

+(1 − z2 w2 )

n 2 −1

fj (z1 , z2 )fj (w1 , w2 )∗ .

(2.19)

j=0

Let Si , i = 3, . . . , d, be the multiplication operator on H2 (Td−2 ) with symbol z ) = (˜ z k )k∈Nd−2 . Then, denoting z = (z1 , z2 , z˜) zi , i = 3, . . . , d, and denote (˜ 0 and w = (w1 , w2 , w), ˜ we have k ˜ ∗= z˜k w ˜ P (z)P (w)∗ . (˜ z )M1 (z1 , z2 )M1 (w1 , w2 )∗ (w) k∈Nd−2 0

Also, (˜ z )(S3 · · · Sd )M1 (z1 , z2 )M1 (w1 , w2 )∗ (S3 · · · Sd )∗ (w) ˜ ∗ k = z˜w˜ ˜zk w ˜ P (z)P (w)∗ . k∈Nd−2 0

Taking the difference, we obtain k k z˜k w ˜ P (z)P (w)∗ − z˜w˜ ˜zk w ˜ P (z)P (w)∗ = P (z)P (w)∗ . k∈Nd−2 0

k∈Nd−2 0

←− ← − Similar equations hold above if we replace M1 and P by M2 and R , respectively. z )gi (z1 , z2 ), Fj (z) = (˜ z )fj (z1 , z2 ), which are anaNext, put Gi (z) = (˜ lytic for z˜ = (z3 , . . . , zd ) ∈ Dd−2 . Now if multiply (2.19) on the left with S3 · · · Sd and on the right with S3∗ · · · Sd∗ , and subtract the result from (2.19) we obtain (2.18). In (2.18) one can multiply with P (z)−1 and P (z)−1∗ , and use Rudin’s approximation (see [16, Chapter 5]) to retrieve the scalar case of Theorem 1.2 in [11]. Acknowledgements It was a pleasure to discuss the topic in the paper with Professors Anatolii Grinshpan, Dmitry Kaliuzhnyi-Verbovetskyi, and Victor Vinnikov. The author thanks the anonymous referee for pointing out the references [14] and [15] to him.

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References [1] Agler, J.: Some Interpolation Theorems of Nevanlinna–Pick Type (1988) [2] Agler, J.: On the Representation of certain holomorphic functions defined on a polydisc. In: de Branges, L., Gohberg, I., Rovnyak, J. (eds.) Topics in Operator Theory: Ernst D. Hellinger Memorial Volume. Operator Theory: Advances and Applications, vol. 48, pp. 47–66. Birkh¨ auser, Basel (1990) [3] Agler, J., McCarthy, J.E.: Nevanlinna–Pick interpolation on the bidisk. J. Reine Angew. Math. 506, 191–204 (1999) [4] Andˆ o, T.: On a pair of commutative contractions. Acta Sci. Math. (Szeged) 24, 88–90 (1963) [5] Ball, J.A., Sadosky, C., Vinnikov, V.: Scattering systems with several evolutions and multidimensional input/state/output systems. Int. Eqs. Oper. Theory 52(3), 323–393 (2005) [6] Ball, J.A, Trent, T.T.: Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna–Pick interpolation in several variables. J. Funct. Anal. 157(1), 1–61 (1998) [7] Cole, J.B., Wermer, J.: Ando’s theorem and sums of squares. Indiana Univ. Math. J. 48(3), 767–791 (1999) [8] Dritschel, M.A., Woerdeman, H.J.: Outer factorizations in one and several variables. Trans. Am. Math. Soc. 357(11), 4661–4679 (electronic) (2005) [9] Geronimo, J.S., Woerdeman, H.J.: Positive extensions, Fej´er-Riesz factorization and autoregressive filters in two variables. Ann. Math. (2) 160(3), 839–906 (2004) [10] Gohberg, I.C., Ha˘ınig, G.: Inversion of finite Toeplitz matrices consisting of elements of a noncommutative algebra. Rev. Roumaine Math. Pures Appl. 19, 623–663 (1974) [11] Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D., Vinnikov, V., Woerdeman, H.J.: Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality. J. Funct. Anal. 256, 3035–3054 (2009) [12] Kailath, T., Vieira, A., Morf, M.: Inverses of Toeplitz operators, innovations, and orthogonal polynomials. SIAM Rev. 20(1), 106–119 (1978) [13] Knese, G.: Bernstein-Szeg˝ o measures on the two dimensional torus. Indiana Univ. Math. J. 57(3), 1353–1376 (2008) [14] Larson, D.R.: Nest algebras and similarity transformations. Ann. Math. (2) 121(3), 409–427 (1985) [15] Moore, R.L., Trent, T.T.: Factorization along commutative subspace lattices. Int. Eqs. Oper. Theory 25(2), 224–234 (1996) [16] Rudin, W.: The extension problem for positive-definite functions. Illinois J. Math. 7, 532–539 (1963) [17] Woerdeman, H.J.: Estimates of inverses of multivariable Toeplitz matrices. Oper. Matrices 2(4), 507–515 (2008)

Vol. 67 (2010)

General Christoffel-Darboux Formula

Hugo J. Woerdeman Department of Mathematics Drexel University 3141 Chestnut Street Philadelphia, PA 19104, USA e-mail: [email protected] Received: July 27, 2009. Revised: January 26, 2010.

213

Integr. Equ. Oper. Theory 67 (2010), 215–246 DOI 10.1007/s00020-010-1777-8 Published online March 20, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Unicity of the Integrated Density of States for Relativistic Schr¨ odinger Operators with Regular Magnetic Fields and Singular Electric Potentials Viorel Iftimie, Marius M˘antoiu and Radu Purice Abstract. We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schr¨ odinger operators with magnetic ﬁelds and scalar potentials introduced in Iftimie et al. (Publ Res Inst Math Sci 43(3):585–623, 2007; Topics in applied mathematics and mathematical physics, Editura Academiei Romˆ ane, 2008), the ﬁrst one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator deﬁned on the entire space. In this way one generalizes the results of Doi et al. (Math Z 237:335–371, 2001) and Iftimie (Publ Res Inst Math Sci 41(2):307–327, 2005) for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus developed in Iftimie et al. (Publ Res Inst Math Sci 43(3):585–623, 2007), as well as a Feynman-Kac-Itˆ o formula for L´evy processes (Ichinose and Tamura, Commun Math Phys 105(2):239–257, 1986; Iftimie et al. Topics in applied mathematics and mathematical physics, Editura Academiei Romˆ ane, 2008). In addition, in case when both the magnetic ﬁeld and the scalar potential are periodic, one also proves the existence of the IDS. Mathematics Subject Classification (2000). Primary 81Q10, 47G30; Secondary 47A60, 47D08. Keywords. Integrated density of states, relativistic Schr¨ odinger operators, pseudodifferential operators, magnetic ﬁelds, Feynman-Kac-Ito formula.

VI and RP acknowledges partial support from the Contract no. 2-CEx06-11-18/2006. MM was supported by the Fondecyt Grant No. 1085162 and by the N´ ucleo Cientifico ICM P07-027-F “Mathematical Theory of Quantum and Classical Systems”.

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1. Introduction We specify ﬁrst the class of operators we consider. For d ≥ 2 we set BC ∞ (Rd ) := {f ∈ C ∞ (Rd ) | ∂ α f ∈ L∞ (Rd ), ∀ α ∈ Nd }, and ∞ Cpol (Rd ) := {f ∈ C ∞ (Rd ) | ∂ α f is polynomially bounded, ∀ α ∈ Nd }. d The magnetic ﬁeld B = 12 j,k=1 Bjk dxj ∧ dxk satisﬁes:

Hypothesis (i). dB = 0, Bjk = −Bkj ∈ BC ∞ (Rd ). Using the transversal gauge one constructs a vector potential A = d ∞ d j=1 Aj dxj , with Aj ∈ Cpol (R ), such that dA = B. The circulation of A through the segment [x, y], x, y ∈ Rd , can be written as 1

A = −x − y, Γ (x, y), A

dsA((1 − s)x + sy). (1.1)

A

Γ (x, y) := 0

[x,y]

In some papers [25,27] one proposes the following quantization of a classical observable a : T ∗ Rd → R: A x+y , ξ u(y), (1.2) dy d¯ξ eix−y,ξ+Γ (x,y) a OpA (a)u (x) := 2 Rd Rd

where u ∈ S(R ), d¯ξ := (2π)−d dξ and the oscillatory integral makes sense if, for example, a ∈ S m (Rd ). A symbolic calculus for the operators deﬁned by (1.2), essential for the present work, has been developed in [21]. The quantization (1.2) has the ∞ (Rd ), then important physical property of being gauge covariant: if ϕ ∈ Cpol A and A = A + dϕ deﬁne the same magnetic ﬁeld B and d

OpA (a) = eiϕ OpA (a)e−iϕ . There exists another approach for quantization in the presence of a magnetic ﬁeld [13,16–19,28,29]. One deﬁnes OpA (a) by the Weyl quantization of the symbol T ∗ Rd (x, ξ) → a (x, ξ − A(x)) ∈ R, but in this way gauge covariance is lost, as shown in [21] for a(ξ) = ξ := (1 + |ξ|2 )1/2 . One notices, however, that both quantizations lead to the same magnetic non-relativistic Schr¨ odinger operator. We are concerned in the present paper with the case a(ξ) = ξ − 1

(1.3)

for which the two quantizations do not coincide. As shown in [21,22], the operator OpA (a) in L2 (Rd ) is essentially selfadjoint on S(Rd ). One denotes by HA its closure; then HA ≥ 0 and its domain is the magnetic Sobolev space of order 1: 1 HA := {u ∈ L2 (Rd ) | (Dj − Aj )u ∈ L2 (Rd ), 1 ≤ j ≤ d}.

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odinger operator with magnetic field. One We call HA the relativistic Schr¨ should remark that another candidate exists for this concepts, the operator [(D − A)2 + 1]1/2 − 1 (cf. [12] for instance), but this one cannot be deduced from a quantization which systematically applies to a whole class of symbols. For the scalar potential V , let us ﬁrst consider the following condition. Hypothesis (ii). V : Rd → R, V = V+ − V− , V± ≥ 0, V± ∈ L1loc (Rd ), and the operator of multiplication by V− is form-bounded with respect to H0 , with relative bound strictly less than 1. In other words, there exist α ∈ [0, 1) and β ≥ 0 such that 1/2 1/2 V− |u|2 dx ≤ α H0 u 2 + β u 2 , u ∈ D(H0 ) = H1/2 (Rd ),

(1.4)

Rd

where · is the norm of L2 (Rd ) and Hs (Rd ) is the usual Sobolev space of order s ∈ R. We are going to show in Sect. 4 that under the assumptions (i) and (ii), the form sum ·

H ≡ H(A; V ) := HA + V is well-deﬁned. The operator H will be self-adjoint and lower semi-bounded in L2 (Rd ). In particular, HA = H(A; 0). To use the Feynman–Kac–Itˆ o formula from Sect. 4 we will need a stronger hypothesis, involving Kato’s class Kd associated to the operator H0 , deﬁned as follows: The semigroup generated by H0 is given by convolution with a function pt (deﬁned in Sect. 3); a function W ∈ L1loc (Rd ), W ≥ 0, belongs to Kd if ⎡ ⎤ t ⎣ ps (x − y)W (y)dy ⎦ ds = 0. lim sup (1.5) t0 x∈Rd

0

Rd

In particular, if W ∈ L∞ (Rd ), W ≥ 0, then W ∈ Kd . In [36,6,9] one shows that W ∈ Kd veriﬁes (1.4) for any α > 0. For our main results we shall need a stronger assumption on V . Hypothesis (ii ). V : Rd → R, V = V+ − V− , and V− ∈ Kd .

V± ≥ 0,

V± ∈ L2loc (Rd )

To deﬁne the integrated density of states (IDS) we need a family F of bounded open subsets of Rd , satisfying: Hypothesis (iii). For any m ∈ N∗ , there exists Ω ∈ F such that the ball B(0; m) centered in the origin, of radius m, is contained in Ω. Hypothesis (iv). For any > 0, there exists m0 ∈ N∗ such that if Ω ∈ F and B(0, m0 ) ⊂ Ω, we have |{x ∈ Rd | dist(x, ∂Ω) < 1}| < |Ω|, where we set |Ω| for the Lebesgue measure of Ω.

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Let us mention some basic references concerning IDS [5,8,11,14,32] and [10] that is closer related to our work. There are two definitions of IDS. The ﬁrst one [5,8] uses the operator HΩ induced by H on Ω ∈ F, with Dirichlet boundary conditions (it is deﬁned in Sect. 6, where we prove that HΩ has compact resolvent on L2 (Ω)). IDS is the function ρ : R → R+ ,

ρ(λ) := lim

Ω→Rd Ω∈F

NΩ (λ) , |Ω|

(1.6)

where NΩ (λ) is the number of eigenvalues of HΩ smaller than λ. The second definition [8,14] uses the fact (proved in Sect. 5) that the operator 1Ω Eλ (H)1Ω belongs to I1 , i.e. is trace-class. Here 1Ω is the operator of multiplication by the characteristic function of Ω, and Eλ (H)is the spectral projection of H corresponding to the interval (−∞, λ], λ ∈ R. Then IDS is also deﬁned by ρ(λ) := lim

Ω→Rd Ω∈F

tr[1Ω Eλ (H)1Ω ] . |Ω|

(1.7)

The existence of the limits (1.6) and (1.7) and their equality are both nontrivial problems. In order to solve them one uses the notion of density of states for H, for which we also have two different definitions. We are going to see in Sects. 5 and 6 that for any f ∈ C0 (R) (continuous function with compact support on R) the operators f (HΩ ) and 1Ω f (H)1Ω belong to I1 . By the Riesz–Markov Theorem for any Ω ∈ F there exist Borel measures μD Ω and μΩ on R, such that −1 , |Ω| tr [1 f (H)1 ] = f dμΩ . |Ω|−1 trf (HΩ ) = f dμD Ω Ω Ω R

R

One notices that the two expressions in (1.6) and (1.7) are exactly the distribution functions of these two measures: −1 NΩ (λ), μD Ω ((−∞, λ]) = |Ω|

μΩ ((−∞, λ]) = |Ω|−1 tr [1Ω Eλ (H)1Ω ] .

If Borel measures μD , μ on R exist such that lim

F Ω→Rd

D μD Ω =μ ,

meaning that for any f ∈ C0 (R) and that if B(0; m0 ) ⊂ Ω, then D f dμD f dμ < , Ω − R

R

lim

F Ω→Rd

μΩ = μ,

any > 0 there exists m0 ∈ N∗ such f dμΩ − f dμ < , R

R

each of them is called the density of states of H. The main result of this article is the equivalence of these definitions: Theorem 1.1. Under assumptions (i), (ii ), (iii) and (iv), the density of states μD exists if and only if the density of states μ exists. In addition, if one of them exists, then μD = μ.

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For the non-relativistic Schr¨ odinger operator, such a result has been obtained in [10] for V− = 0 and in [20] for V− = 0. There are several results concerning the existence and unicity of IDS for non-relativistic Schr¨ odinger operators without magnetic ﬁeld (see [10] for references). The case of a constant magnetic ﬁeld has been treated in [14] (V ∈ C ∞ (Rd ) periodic) or in [15] (V random potential, eventually unbounded from below). We also mention the results in [2–4,7,14] and references therein. The existence of IDS has been proved in [20] for periodic magnetic ﬁelds and scalar potentials. The second goal of this paper is to extend this last result to the relativistic case. We consider a lattice Γ in Rd , generated by a base e1 , . . . , ed : ⎧ ⎫ d ⎨ ⎬ Γ= αj ej | αj ∈ Z, 1 ≤ j ≤ d . ⎩ ⎭ j=1

Let us also denote by F a fundamental domain of Rd with respect to Γ; for instance ⎫ ⎧ d ⎬ ⎨ tj ej | 0 ≤ tj < 1, 1 ≤ j ≤ d . F = ⎭ ⎩ j=1

We make the following hypothesis: Hypothesis (v). V and Bjk , 1 ≤ j, k ≤ d are Γ-periodic functions. Theorem 1.2. Under the hypothesis (i), (ii ), (iii), (iv) and (v), the integrated density of states of H exists and for each f ∈ C0 (R) we have lim

F Ω→Rd

|Ω|−1 tr [1Ω f (H)1Ω ] = |F |−1 trΓ f (H),

(1.8)

where trΓ is the Γ-trace in the sense of Atiyah [1]. The plan of this paper is as follows: In Sect. 2 we review ﬁrst some properties of the magnetic pseudodifferential calculus, established in [21,22]. Some reﬁned results about commutators are obtained and one overlines approximation by regularisations (using the magnetic convolution) and cut-oﬀs. In Sect. 3 we present the Feller semi-group deﬁned by the free Hamiltonian H0 and the associated L´evy process. The diamagnetic inequality (3.10) will be a consequence of a Feynman–Kac–Itˆo formula for the relativistic Hamiltonian HA . Section 4 is devoted to the construction of the relativistic Schr¨ odinger operator H = H(A; V ). Using the Feynman–Kac–Itˆ o formula, representing the semi-group generated by H, we prove the important fact that C0∞ (Rd ) is an essential domain for the form associated to H and we present some consequences regarding commutators and covariance under gauge transformations. In Sect. 5 we estimate the trace norm of some operators of the form 1Ω f (H)1Ω , Ω being a bounded open subset of Rd and f : R → C a suitable

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function. The hypothesis V ∈ Kd is essential, allowing us to use some estimations for the integral kernel of the semi-group generated by H(0; −V− ) (cf. [36,9]). In Sect. 6 one deﬁnes HΩ as a pseudo-self-adjoint operator on L2 (Rd ), using a result in [33] on monotone sequences of quadratic forms. One also estimates the I1 -norm of operators of the form f (HΩ ). Section 7 is dedicated to the proof of Theorem 1.1. The main difﬁculty is the I1 -norm estimate of operators of the form 1Ω (H +λ)−m 1Ω −(HΩ +λ)−m . Then, using ideas from [10] ﬁnishes the proof. In Sect. 8 we prove Theorem 1.2, on the lines of the proof of Theorem 1.6 from [20].

2. The Magnetic Pseudodifferential Calculus Let us recall ﬁrst some properties of operators deﬁned by (1.2), proved in [21]. We are going to assume everywhere that B = dA fulﬁlls hypothesis (i). Definition 2.1. Let m ∈ R. (a)

A function f ∈ C ∞ (R2d ) belongs to the symbol space S m (Rd ) if for any α, β ∈ Nd there is a constant Cα,β > 0 such that | ∂xα ∂ξβ f (X)| ≤ Cα,β ξm−β , X = (x, ξ) ∈ R2d .

The space S m (Rd ) is endowed with the usual Fr´echet topology. (b) S −∞ (Rd ) := ∩m∈R S m (Rd ) is endowed with the projective limit topology. (c) A symbol f ∈ S m (Rd ) is called elliptic if for some positive constants C, R one has |f (X)| ≥ Cξm , (d)

∀X = (x, ξ) ∈ R2d , |ξ| ≥ R.

We call principal symbol of an operator of the form OpA (f ), where f ∈ S m (Rd ), any element f0 ∈ S m (Rd ) such that f − f0 ∈ S m−1 (Rd ).

Proposition 2.2. Let f ∈ S m (Rd ) and g ∈ S m (Rd ). (a) OpA (f ) is a continuous linear operator on S(Rd ) and on S (Rd ). (b) OpA (f ) is the formal adjoint of OpA (f ), i.e. OpA (f )u, v = u, OpA (f )v , ∀u, v ∈ S(Rd ). L2 (Rd )

(c)

L2 (Rd )

There exists a unique element f ◦B g ∈ S m+m (Rd ) such that OpA (f )OpA (g) = OpA (f ◦B g). Moreover, a principal symbol of OpA (f )OpA (g) is f g.

Proposition 2.3. Let f ∈ S 0 (Rd ). Then OpA (f ) ∈ B(L2 (Rd )), and its norm in B(L2 (Rd )) is dominated by a semi-norm of f in S 0 (Rd ).

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Definition 2.4. Let s ∈ R+ , ps (ξ) := ξs ∈ S s (Rd ), Ps := OpA (ps ). s (Rd ) (a) An element u ∈ L2 (Rd ) belongs to the magnetic Sobolev space HA 2 d s d if Ps u ∈ L (R ). HA (R ) is a Hilbert space for the norm 1/2 u s,A := Ps u 2L2 (Rd ) + u 2L2 (Rd ) , (2.1) s (Rd ). and S(Rd ) is dense in HA −s d s (b) HA (R ) will be the dual of HA (Rd ) endowed with the natural norm.

Remark 2.5. If s ∈ N, we have s HA (Rd ) = {u ∈ L2 (Rd ) | (D − A)α u ∈ L2 (Rd ), ∀ α ∈ Nd , |α| ≤ s},

and a norm equivalent to (2.1) is given by ⎞1/2 ⎛ u s,A = ⎝ (D − A)α u 2L2 (Rd ) ⎠ . |α|≤s

We used the notation (D − A)α = (D1 − A1 )α1 . . . (Dn − An )αn . Proposition 2.6. For each s, m ∈ R and each f ∈ S m (Rd ), s d s−m (R ), HA (Rd ) . OpA (f ) ∈ B HA Proposition 2.7. Let p ∈ S m (Rd ) be real and elliptic, m ≥ 0. We assume p(X) ≥ 0 for |ξ| ≥ R (R > 0 large enough). Then the operator OpA (p), defined on S(Rd ), is essentially self-adjoint in L2 (Rd ). Its closure P will be m (Rd ). a lower semi-bounded self-adjoint operator on the domain HA Remark 2.8. This proposition applies to the case p(X) = ξ − 1. The corresponding operator, denoted by HA , will be a lower semi-bounded self-adjoint operator on L2 (Rd ) with domain 1 (Rd ) = {u ∈ L2 (Rd ) | (Dj − Aj )u ∈ L2 (Rd ), 1 ≤ j ≤ d}. HA

(2.2)

In fact, adapting arguments from [17] (where the quantiﬁcation OpA is used), one can show that HA ≥ 0. The next result has been proved in [21] for B admitting a vector potential with bounded derivatives and for the general case of hypothesis (i) in [23]. Proposition 2.9. Let us consider verified the hypothesis of Proposition 2.7. (a) If λ ∈ R, λ < inf σ(P ), then (P − λ)−1 is the closure in L2 (Rd ) of an operator OpA (p(λ) ), with p(λ) ∈ S −m (Rd ). If in addition λ ≤ infp − 1, then a principal symbol of OpA (p(λ) ) is (p − λ)−1 . (b) If m > 0, p ≥ 1 and P ≥ 1, then for every s ∈ R, P s is the closure in L2 (Rd ) of a operator OpA (q(s) ), q(s) ∈ S sm (Rd ), which admits ps like principal symbol. In the remaining part of this section we are going to prove two properties of commutators of magnetic pseudo-differential operators, as well as applications to approximation by regularization or cut-oﬀ.

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Proposition 2.10. Let m ∈ R or m = −∞ and g ∈ S m (Rd ), m ∈ R. Then

(a) f ◦B g − g ◦B f ∈ S m+m −1 (Rd ), ∀ f ∈ S m (Rd ). (b) Let m ∈ R and M be a subset of S m (Rd ) formed by symbols independents of x ∈ Rd , such that the set {∂ξ1 f, . . . , ∂ξd f | f ∈ M } is bounded in S m (Rd ). Then the set {f ◦B g − g ◦B f | f ∈ M } is bounded in S m+m (Rd ). Proof. (a) It follows from the Proposition 2.2(c). To verify (b), one uses the composition formula from [21] for f ∈ M , in which the integral is oscillatory: B (f ◦ g)(X) = dY ¯ dZ ¯ e−2i[Y,Z] ω B (x, y, z) f (ξ − η) g(X − Z), R2d R2d

X ∈ R2d ,

(2.3)

where X = (x, ξ), Y = (y, η), Z = (z, ζ) are points in R2d , dY ¯ = π −d dY , d [Y, Z] = η, z − ζ, y (·, · is the scalar product in R ) and ω B (x, y, z) = e−4iFB (x,y,z) , where FB ∈ C ∞ (R3d ), depending only on the magnetic ﬁeld B and its ﬁrst order derivatives, are of the form: [Dj (x, y, z)yj + Ej (x, y, z)zj ] , Dj , Ej ∈ BC ∞ (R3d ). (2.4) 1≤j≤d

Using the Leibnitz–Newton formula f (ξ − η) = f (ξ) −

1 (∂j f )(ξ − tη)dt

ηj

1≤j≤d

0

and the fact that 1 ◦B g = g, one writes (2.3) as (f ◦B g)(X) = f (ξ)g(X) + ρf (X),

(2.5)

where ρf (X) = −

1

1≤j≤d 0

dt

dY ¯ dZ ¯ ηj e−2i[Y,Z] ω B (x, y, z) (∂j f ) (ξ − tη)

R2d R2d

× g(X − Z).

(2.6)

We use the identity ηj e−2i[Y,Z] = −

1 ∂zj e−2i[Y,Z] 2i

to integrate by parts with respect to zj . We also use (2.4) as well as yk e−2i[Y,Z] =

1 ∂ζk e−2i[Y,Z] , 2i

zk e−2i[Y,Z] = −

1 ∂ηk e−2i[Y,Z] 2i

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to integrate by parts with respect to ζk and ηk . This gives ρf (X) =

1

1≤j≤d 0

⎡

×⎣

dY ¯ dZ ¯ e−2i[Y,Z] ω B (x, y, z)

dt R2d R2d

Djk (x, y, z)(∂ξk g)(X − Z)(∂j f )(ξ − η)

1≤k≤d

+t

Ejk (x, y, z)(∂j ∂k f )(ξ − tη)g(X − Z)

1≤k≤d

⎤

− (∂j f )(ξ − tη)(∂xj g)(X − Z)⎦ ,

Djk , Ejk ∈ BC ∞ (R3d ).

By hypothesis, the sets {∂j f | f ∈ M }, 1 ≤ j ≤ d are bounded in S m (Rd ). Using the standard integration by parts procedure with respect to y, z, η, ζ, starting from the equality N 1 2N −2i[Y,Z] = 1 − Δz e−2i[Y,Z] , N ∈ N η e 4 and its analogs, by eliminating the monomials in y and z, as above, one obtains the estimation 1 −2Nη dY ¯ dZz ¯ ζ−2Ny η−2Nz |ρf (X)| ≤ p(f ) q(g) dt 0 −2Nζ

× y

R2d R2d

· ξ − tηm ξ − ζm ,

where Nη , Ny , Nz , Nζ are natural integers which must be chosen in a suitable way in order to have absolute convergencemof dthe integrals, p(f ) = pj (∂j f ), pj is a continuous semi-norm on S (R ) and q a continuous

1≤j≤d

semi-norm on S m (Rd ). Since ξ − tηm ≤ Cξm η|m| ,

ξ − ζm ≤ Cξm ζ|m | ,

C ∈ R+ ,

one can choose Nη = Nζ = d, Ny = d + |m |, Nz = d + |m| and get

|ρf (X)| ≤ C0 p(f ) q(g)ξm+m ,

C0 > 0 constant.

Analogously one estimates the derivatives of ρ and obtains that the set {ρf | f ∈ M } is bounded in S m+m (Rd ). In the same way one can show that g ◦B f = f g + ρf and {ρf ; f ∈ M } is bounded in S m+m (Rd ). Definition 2.11. Let u ∈ S (Rd ), f ∈ S(Rd ). One calls magnetic convolution of u with f , the function u A f ∈ C ∞ (Rd ) deﬁned by A

(u A f )(x) := u(y), eix−y,Γ

(x,y)

f (x − y),

x ∈ Rd ,

where ·, · is the duality bracket between S(R ) and S (R ). d

d

(2.7)

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Remark 2.12. Using the equality f (x − y) = out that u A f = OpA (f)u.

Rd

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eix−y,ξ f(ξ)dξ, one ﬁnds

To regularize a distribution by means of the magnetic convolution, one uses a standard δ-sequence. Let us consider a function θ ∈ C0∞ (Rd ), θ ≥ 0, supp θ ⊂ B(0; 1), Rd θ(x)dx = 1. For j ≥ 1 one deﬁnes θj (x) := j d θ(jx), x ∈ −1 ξ), Rd . Then θj ∈ C0∞ (Rd ), supp θj ⊂ B(0; 1/j), θj ∈ S(Rd ), θj (ξ) = θ(j d d A ξ ∈ R . For u ∈ S (R ) we set Rj u := u θj . Proposition 2.13. (a) If u ∈ E (Rd ), Rj u ∈ C0∞ (Rd ) and suppRj u ⊂ {x ∈ Rd | dist(x, supp u) ≤ 1/j}. (b) If u ∈ L∞ (Rd ), Rj u ∈ L∞ (Rd ) and Rj u L∞ (Rd ) ≤ u L∞ (Rd ) . (c) If u ∈ L2 (Rd ), Rj u ∈ L2 (Rd ) and limj→∞ Rj u = u on L2 (Rd ). (d) Let P = OpA (p), p ∈ S 1/2 (Rd ). If u ∈ L2 (Rd ) and P u ∈ L2 (Rd ), then limj→∞ P Rj u = P u in L2 (Rd ). Proof. Properties (a) and (b) are evident. (c) Since θ ∈ S(Rd ) ⊂ S −∞ (Rd ) ⊂ S 0 (Rd ) and Rj = OpA (θj ), Proposition 2.3 shows that Rj u ∈ L2 (Rd ). But A (Rj ) u(x) = θ(z) u (x − z/j) e−(i/j)z,Γ (x,x−z/j) dz

Rd

and u(x) = Rd θ(z) u(x) dz. Using twice the Dominated Convergence Theorem and the continuity of u in mean, one gets θ(z) u(· − z/j) − u(·) L2 (Rd ) + Rj u − u L2 (Rd ) ≤ Rd

A + u(·) e−(i/j)z,Γ (·,·−z/j) − 1 L2 (Rd ) dz → 0. j→∞

(d) The sequence {j 1/2 ∂k θj }j≥1 is bounded in S −1/2 (Rd ) for 1 ≤ k ≤ d. One applies Proposition 2.10 (b) with M = {j 1/2 θj ; j ≥ 1}, m = −∞, m = − 21 , m = 12 , f (ξ) = j 1/2 θj (ξ), g = p and deduces that the set {j 1/2 (θj ◦B p − p ◦B θj ) | j ≥ 1} is bounded in S 0 (Rd ). By Proposition 2.3, there is a constant C > 0 such that Rj P − P Rj B(L2 (Rd )) ≤ Cj −1/2 , j ≥ 1. Thus limj→∞ (P Rj u − Rj P u) = 0 in L2 (Rd ) and limj→∞ Rj P u = P u in L2 (Rd ) implies the conclusion. Proposition 2.14. Let P = OpA (p), p ∈ S m (Rd ), m ≤ 1 and ϕ ∈ BC ∞ (Rd ). Let us suppose that there exists N ∈ N, N > m + d − 1 such that |MN | < ∞, where MN :=

∪

supp ∂ α ϕ.

|α|=N +1

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Then there is a constant C > 0 independent on ϕ, operators Sα , Sα ∈ B(L2 (Rd )), 1 ≤ |α| ≤ N , independent on ϕ and operators TN , TN ∈ I2 (the Hilbert-Schmidt space on L2 (Rd )) such that TN I2 + TN I2 ≤ C max ∂ α ϕ L∞ (Rd ) |MN |1/2

(2.8)

|α|=N +1

and

[ϕ, P ] := ϕP − P ϕ =

(∂ α ϕ)Sα + TN =

1≤|α|≤N

Sα (∂ α ϕ) + TN ,

(2.9)

1≤|α|≤N

with the convention that the sums in (2.9) do not exist if N = 0. Proof. Using (1.2) and Taylor’s formula

ϕ(x) − ϕ(y) = −

1≤|α|≤N

(y − x)α α (∂ ϕ)(x) + rN (x, y), α!

where

rN (x, y) = −

|α|=N +1

(y − x)α N!

1 (1 − t)N (∂ α ϕ)(x + t(y − x))dt, 0

1 one obtains the ﬁrst equality from (2.9) with Sα = − α! OpA (Dξα p) and

1 TN u(x) := − N!

1

dt(1 − t)N

|α|=N +1 0

× (Dξα p)

A

dy dξ ¯ eix−y,ξ+Γ

(x,y)

(2.10)

Rd

x+y , ξ (∂ α ϕ)(x + t(y − x))u(y), 2

u ∈ S(Rd ).

Since for |α| ≥ 1 one has Dξα p ∈ S 0 (Rd ), by Proposition 2.3 one has Sα ∈ B(L2 (Rd )). But, using in (2.10) the identity x − y2s eix−y,ξ = (1 − Δξ )s (eix−y,ξ ), for s ∈ N, to integrate by parts, one sees that for any s ∈ N, TN can be written as an integral operator with kernel 1 KN (x, y) := − N!

1

dt(1 − t)N

|α|=N +1 0

A

dξ ¯ eix−y,ξ+Γ Rd

× x − y−2s (1−Δξ )s Dξα p

(x,y)

x+y , ξ (∂ α ϕ)(x+t(y−x)). 2

Denoting by 1M the characteristic function of a set M ⊂ Rd , one obtains for each s ∈ N −2s

|KN (x, y)| ≤ Cs x − y

1 max ∂ ϕ L∞ (Rd )

1MN (x + t(y − x))dt,

α

|α|=N +1

0

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with Cs a constant depending on s but not on ϕ. Choosing s > d/4 one gets TN I2 = KN L2 (Rd ×Rd ) ≤ Cs max ∂ α ϕ L∞ (Rd ) |α|=N +1

⎡

× sup ⎣ t∈[0,1]

⎤1/2 x − y−4s 1MN (x + t(y − x)) dx dy ⎦

.

R2d

The integral in the previous formula equals z−4s 1MN (x + tz)dxdz = 1MN (y)dy z−4s dz R2d

Rd

= |MN |

Rd

z−4s dz.

Rd

It follows that the norm TN I2 is bounded by the right member of the inequality (2.8). In the same way one gets the second equality from (2.9) and the corre sponding bound for TN I2 . A ﬁrst application of the Proposition 2.14 concerns cut-oﬀ approximations. Let ψ ∈ C0∞ (Rd ), 0 ≤ ψ ≤ 1, supp ψ ⊂ B(0; 2), ψ = 1 on B(0; 1). For j ≥ 1 one sets ψj (x) := ψ(x/j), x ∈ Rd . Proposition 2.15. Let u ∈ L2 (Rd ) and P = OpA (p), p ∈ S m (Rd ), m ≤ 1. (a) ψj u ∈ L2 (Rd ) and limj→∞ ψj u = u in L2 (Rd ). (b) If P u ∈ L2 (Rd ), then P (ψj u) ∈ L2 (Rd ) and limj→∞ P (ψj u) = P u in L2 (Rd ). Proof. (a) is trivial. (b) follows if for some constant C > 0 one obtains the inequality [ψj , P ] B(L2 (Rd )) ≤ Cj −1 ,

j ≤ 1.

(2.11)

For this one applies Proposition 2.14 with ϕ = ψj and N = d + 1. Since supp ∂ α ψj ⊂ B(0; 2j), ∀ α ∈ Nd , there is a constant C1 > 0 such that (j) (j) |MN |1/2 ≤ C1 j d/2 , ∀ j ≥ 1, where MN := ∪|α|=N +1 supp ∂ α ψj . On the other hand (∂ α ψj )(x) = j −|α| (∂ α ψ)(x/j), hence there is a constant C2 > 0 such that ∂ α ψj L∞ (Rd ) ≤ C2 j −|α| , for any j ≥ 1 and for all α ∈ Nd with 1 ≤ |α| ≤ N + 1. Then (2.11) follows from (2.8) and (2.9).

3. The Feller Semigroup In this section!we are going to recall some well-known properties of the semigroup e−tH0 t≥0 , where H0 is the free relativistic Hamiltonian for A = 0. H0 is self-adjoint on L2 (Rd ), H0 ≥ 0 and its domain is the standard Sobolev space H1 (Rd ) ≡ H01 (Rd ). Its restriction to S(Rd ) is the operator Op0 (p), with

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p(X) = ξ − 1. By the L´evy–Khincin formula (see for example [31]), there exists a measure n on Rd such that eiy,ξ −1−iy, ξ1B(0;1) (y) dn(y), ∀ ξ ∈ Rd . (3.1) ξ−1 = − Rd

Cf. [17] one has the explicit formula dn(y) = 2(2π)−

d+1 2

|y|−

d+1 2

K d+1 (|y|)dy, 2

(3.2)

where Kν is the modiﬁed Bessel function of the third kind and order ν, for which the next inequalities are veriﬁed for some positive constant C: 0 < Kν (r) ≤ C max r−ν , r−1/2 e−r , ∀ r > 0, ∀ ν > 0. (3.3) By [6,19], for t > 0, the operator e−tH0 is given by the convolution with the function √ 2 2 t et−ξ |x |+t dξ (3.4) pt (x) := (2π)−d " |x|2 + t2 Rd " d+1 d+1 − d−1 − t = 2 2 π 2 t e (|x|2 + t2 )− 4 K d+1 |x|2 + t2 , x ∈ Rd . 2

One veriﬁes the properties

pt (x) > 0,

pt (x)dx = 1

(3.5)

Rd −tH0

and the fact that e on the Banach space

can be extended as a well-deﬁned bounded operator

C∞ (Rd ) :=

# $ f ∈ C(Rd ) | lim f (x) = 0 , |x|→∞

equipped with the norm · ∞ . One also checks easily the Feller semi-group properties for the family of these extensions. By [6,9], this Feller semi-group is generated by a L´evy process. More precisely, on the space Ω of the “c`adlag” functions on [0, ∞) (Rd -valued, continuous to the right, having left limits), endowed with the smallest σ-algebra F for which all the coordinate functions Ω ω → Xt (ω) := ω(t) ∈ Rd are measurable, one can deﬁne for each x ∈ Rd a probabilistic measure Px such that Px {X0 = x} = 1 and the random variables Xt1 − Xt0 , . . . , Xtn − Xtn−1 are independent with distributions pt1 −t0 , . . . , ptn −tn−1 for each 0 = t0 < t1 < · · · < tn < ∞. If we denote by Ex the expectation with respect to the probability Px , then for any f ∈ C∞ (Rd ) and t ≥ 0 we have −tH0 f (x) = Ex (f ◦ Xt ) , x ∈ Rd . (3.6) e By the L´evy–Itˆo Theorem [19,24] one has t+ Xt = x + 0 Rd

%X (ds dy), yN

(3.7)

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where %X (ds dy) := NX (ds dy) − N X (ds dy), N X (ds dy) := Ex (NX (ds dy)) = dsdn(y) N and NX is a “counting measure” on [0, ∞) × Rd , deﬁned by NX ((t, t ] × B) := {s ∈ (t, t ] | Xs = Xs− , Xs − Xs− ∈ B}, where 0 < t < t and B is a Borel subset of Rd . Using the procedure from [19] o (where one works with the quantiﬁcation OpA ), one gets a Feynman–Kac–Itˆ formula for the Hamiltonian HA . For u ∈ L2 (Rd ), x ∈ Rd and t ≥ 0 one has −tHA (3.8) u (x) = Ex (u ◦ Xt ) e−S(t,X) , e where t+ S(t, X) := i

%X (dsdy) N

0 Rd

&1

' A(Xs− + ry)dy, y

0

t

X (dsdy) N

+i

&1

(3.9) '

[A(Xs + ry) − A(Xs )]dr, y

0 Rd

.

0

Let us remark that from (3.8) and (3.6) one obtains the diamagnetic inequality for the relativistic hamiltonian HA : −tH A e u ≤ e−tH0 |u|, ∀ u ∈ L2 (Rd ), ∀ t ≥ 0. (3.10) (3.10) implies another proof of the fact that HA ≥ 0: e−tH0 is a contraction, thus e−tHA is a contraction too, which implies HA ≥ 0. Once again from (3.10), it follows that for any λ > 0, r > 0 and u ∈ L2 (Rd ) one has (HA + λ)−r u ≤ (H0 + λ)−r |u|. (3.11) This inequality is deduced using the fact that for any lower semi-bounded self-adjoint operator H in a complex Hilbert space H, for any r > 0 and any λ ∈ R such that λ + inf σ(H) > 0, one has −r

(H + λ)

1 = Γ(r)

∞

tr−1 e−λt e−tH dt,

(3.12)

0

where Γ is the Euler function of the second kind.

4. The Hamiltonian H(A; V ) We denote by hA the quadratic form associated to HA : 1/2 1/2 1/2 , u, v ∈ D(hA ) := D(HA ). hA (u, v) := HA u, HA v L2 (Rd )

(4.1)

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To a function W ∈ L1loc (Rd ), W ≥ 0, one assigns a quadratic form qW : qW (u, v) := W (x)u(x)v(x)dx, Rd

u, v ∈ D(qW ) := {f ∈ L2 (Rd ) | W 1/2 f ∈ L2 (Rd )}.

(4.2)

These forms are symmetric, closed and positive. We set hA (u) := hA (u, u),

qW (u) := qW (u, u).

The next result is known [22], but for convenience we are going to include a proof. Proposition 4.1. We assume hypothesis (i) and (ii). Then the sesquilinear form h = h(A; V ) := hA + qV+ − qV− is well-defined on D(hA ) ∩ D(qV+ ), being symmetric, closed and lower semi-bounded. Thus it defines a lower semi·

bounded self-adjoint operator on L2 (Rd ), denoted by H = H(A; V ) := HA +V (in the sense of forms). Proof. The form hA + qV+ , deﬁned on D(hA ) ∩ D(qV+ ), is densely deﬁned, symmetric, closed and positive. The conclusion of the Proposition would follow if we show that the form qV− is (hA + qV+ )-bounded, with relative bound < 1. · We denote by H+ := HA + V+ the unique self-adjoint operator ≥ 0 associated to the form hA + qV+ . Since C0∞ (Rd ) ⊂ D(hA ) ∩ D(qV+ ) we can use the version from [26] of Kato-Trotter formula t n t e−tH+ = s − lim e− n HA e− n V+ , ∀ t ≥ 0. (4.3) n→∞

Combining with (3.10) and (3.12) we infer that for every r > 0, λ > 0 and f ∈ L2 (Rd ) one has (H+ + λ)−r f ≤ (H0 + λ)−r |f |. (4.4) Let g ∈ L2 (Rd ), λ > 0 large enough, u := (H0 + λ)−1/2 g. By using the hypothesis (ii), there exists α ∈ (0, 1), β ≥ 0 and α ∈ (0, 1) such that 1/2

qV− (u) ≤ α H0 u 2L2 (Rd ) + β u 2L2 (Rd ) 1/2

= α H0 (H0 + λ)−1/2 g 2L2 (Rd ) + β (H0 + λ)−1/2 g 2L2 (Rd ) ≤

α+

β λ

(4.5) g 2L2 (Rd ) ≤ α g 2L2 (Rd ) .

For v ∈ D(hA ) ∩ D(qV+ ) we set f := (H+ + λ)1/2 v and g := |f |. Using (4.4) with r = 1/2, (4.5) and the explicit form of qV− , we get qV− (v) = qV− [(H+ + λ)−1/2 f ] ≤ qV− [(H+ + λ)−1/2 g] ≤ α g 2L2 (Rd ) = α (H+ + λ)1/2 v 2L2 (Rd ) = α [hA (v) + qV+ (v) + λ v 2L2 (Rd ) ].

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The Feynman–Kac–Itˆo formula (3.8) can be extended to the Hamiltonian H (cf. [19]). Proposition 4.2. Under hypothesis (i) and (ii), for any u ∈ L2 (Rd ) and all t ≥ 0, we have t −tH u (x) = Ex (u ◦ Xt ) e−S(t,X)− 0 (V ◦Xs )ds , x ∈ Rd . (4.6) e By using ideas from [34] and Propositions 2.13, 2.15 and 4.2, we are going to prove Proposition 4.3. Under hypothesis (i), (ii), C0∞ (Rd ) is an essential domain for the form h. Proof. Due to Hypothesis (ii) the form h and the operator H are well-deﬁned. d 1. Let us ﬁrst suppose that V− = 0. We show that D(h) ∩ L∞ comp (R ) is −H an essential domain for h. It is known that the range R(e ) is an essential domain for h. By Proposition 4.2, for any u ∈ L2 (Rd ) |e−H u| ≤ e−H(0,0) |u|,

a.e. on Rd

(4.7)

∞

d

the function on the right hand side being of class L (R ). Let u ∈ D(h) ∩ L∞ (Rd ), ψ and ψj as in Proposition 2.15 and uj := ψj u, j ≥ 1. Then d uj ∈ L∞ comp (R ) ∩ D(qV+ ),

lim uj = u in L2 (Rd ) and lim qV+ (uj − u) = 0.

j→∞

j→∞

Let us notice that we have the equality hA (v, w) = (HA + 1)1/2 v, (HA + 1)1/2 w

L2 (Rd )

− (v, w)L2 (Rd )

(4.8)

1/2

for any v, w ∈ D(hA ) = D(HA ) = D[(HA + 1)1/2 ]. The operator HA + 1 is deﬁned, by Proposition 2.8, by the magnetic pseudo-differential operator OpA (ξ), so by the point (b) of Proposition 2.9, (HA + 1)1/2 is deﬁned by an operator OpA (q), where q ∈ S 1/2 (Rd ) and q − ξ1/2 ∈ S −1/2 (Rd ). Since u ∈ D[(HA + 1)1/2 ], we shall have OpA (q)u ∈ L2 (Rd ); using Proposition 2.15 (b) we infer that OpA (q)uj belongs to L2 (Rd ) and limj→∞ OpA (q)uj = OpA (q)u in L2 (Rd ). Since OpA (q)−OpA (ξ1/2 ) ∈ B[L2 (Rd )], by Proposition 2.3, it fol1/2 1/2 lows that OpA (ξ1/2 )uj ∈ L2 (Rd ), so uj ∈ HA (Rd ) = D(HA ) = D(hA ). Also using (4.8), we get lim hA (uj − u) = 0, so lim uj = u in D(h).

j→∞

j→∞

C0∞ (Rd )

2. We prove that is an essential domain for h(A; V+ ). Obviously d C0∞ (Rd ) ⊂ D(h). Let u ∈ D(h) ∩ L∞ comp (R ) and Rj u, j ≥ 1, deﬁned as in Proposition 2.13. Then Rj u ∈ C0∞ (Rd ), lim Rj u = u in L2 (Rd ), j→∞

lim OpA (q)Rj u = OpA (q)u

j→∞

in L2 (Rd ),

where q has been deﬁned above. It follows that limj→∞ hA (Rj u − u) = 0.

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On the other hand, suppRj u ⊂ {x ∈ Rd | dist(x, suppu) ≤ 1}, |(Rj u)(x)− u(x)| ≤ 2 u L∞ (Rd ) ,

x ∈ Rd

and there is a subsequence (Rjk u)k≥1 such that (Rjk u)(x) −→ u(x) a.e. x ∈ k→∞

Rd . Using the Dominated Convergence Theorem we see that limk→∞ qV+ (Rjk u − u) = 0, thus limk→∞ Rjk u = u in D(h). 3. In order to end the proof we have to notice that qV− is relatively bounded with respect to h(A; V+ ) and consequently any convergent sequence from D(h(A; V+ )) is also convergent in D(qV− ). Corollary 4.4. Under hypothesis (i) and (ii), a vector u ∈ D(h) belongs to D(H) if and only if OpA (p)u + V u ∈ L2 (Rd ), where p(ξ) := ξ − 1. Moreover Hu = OpA (p)u + V u for any u ∈ D(H). Proof. Let u ∈ D(h). Since C0∞ (Rd ) is an essential domain for h, u ∈ D(H) if and only if there exists f ∈ L2 (Rd ) such that h(u, v) = (f, v)L2 (Rd ) , ∀ v ∈ C0∞ (Rd ); if this is the case, then Hu = f . By Proposition 4.1, V± u ∈ L1loc (Rd ) and ∀ v ∈ C0∞ (Rd ),

qV± (u, v) = V± u, v,

(4.9)

where we denoted by ·, · the duality bracket between D(Rd ) and D (Rd ). Let 1/2 {uj }j≥1 ⊂ C0∞ (Rd ) such that limj→∞ uj = u in D(h). Then limj→∞ HA uj = 1/2 HA u in L2 (Rd ). Using Proposition 2.2 (a), we get 1/2

1/2

hA (u, v) = lim (HA uj , HA v)L2 (Rd ) = lim (HA uj , v)L2 (Rd ) j→∞

j→∞

= lim Op (p)uj , v = Op (p)u, v, A

A

j→∞

∀ v ∈ C0∞ (Rd ).

(4.10)

The Proposition follows immediately from the equality h(u, v) = OpA (p)u, v + V u, v,

v ∈ C0∞ (Rd ),

which is a consequence of (4.9) and (4.10).

(4.11)

Proposition 4.5. Suppose hypothesis (i) and (ii) are true. Let ϕ ∈ BC ∞ (Rd ) such that |M | < ∞, where M := ∪|α|=d+2 supp ∂ α ϕ. If u ∈ D(H), then ϕu ∈ D(H). Moreover the commutator [ϕ, H], which is well-defined on D(H), can be extended to an element of B[L2 (Rd )]. (b) There exists a constant C > 0, independent of ϕ, operators (a)

Sα , Sα ∈ B[L2 (Rd )], 1 ≤ |α| ≤ d + 1, independent of ϕ and operators T, T ∈ I2 , such that T I2 + T I2 ≤ C max ∂ α ϕ L2 (Rd ) |M |1/2 |α|=d+2

(4.12)

232

V. Iftimie, M. M˘ antoiu and R. Purice and [ϕ, H] =

(∂ α ϕ)Sα + T =

1≤|α|≤d+1

(c)

Sα (∂ α ϕ) + T .

IEOT

(4.13)

1≤|α|≤d+1

One has [(H + λ)−1 , ϕ] = (H + λ)−1 [ϕ, H](H + λ)−1 , ∀ λ ∈ R, λ > −infσ(H).

(4.14)

Proof. (a) Let u ∈ D(H). Then u ∈ D(qV+ ) ∩ D(hA ). It follows that ϕu ∈ D(qV+ ) and 1/2 1/2 u ∈ D(HA ) = D (HA + 1)1/2 = HA (Rd ). Since ϕ ∈ S 0 (Rd ) and OpA (ϕ) is the operator of multiplication by ϕ, by Proposition 2.6, 1/2 1/2 ; ϕu ∈ HA (Rd ) = D HA thus ϕu ∈ D(h). By Proposition 2.10 (a) it follows that [OpA (p), ϕ] ∈ B[L2 (Rd )], p being given by Corollary 4.4. Therefore, computing in D (Rd ), we get OpA (p)(ϕu)+V (ϕu) = ϕ[OpA (p)u+V u]+[OpA (p), ϕ]u ∈ L2 (Rd ). (4.15) From Corollary 4.4 we deduce that ϕu ∈ D(H). In addition, the equality (4.15) shows that [ϕ, H] = [ϕ, OpA (p)] on D(H),

(4.16)

which implies the last statement of point (a). (b) follows from (4.16) and proposition 2.14 with m = 1 and N = d + 1. (c) is trivial. We close this section with a result on gauge covariance of the operator H. Proposition 4.6. Assume hypothesis (i) and (ii). Let A be a vector potential ∞ % = A−dϕ for some real function (Rd ) and let A for B with components in Cpol ∞ d ϕ ∈ Cpol (R ). We denote by U the unitary operator of multiplication by e−iϕ on L2 (Rd ). Then % V ). U H(A; V ) U −1 = H(A;

(4.17)

Proof. We notice ﬁrst that from the equality & ' 1 ϕ(x) − ϕ(y) = x − y, (∇ϕ)((1 − s)x + sy)ds 0

and from Definition 1.2, one gets the relation % [e−iϕ OpA (a)(eiϕ w)](x) = OpA (a)w (x), for any a ∈ S m (Rd ) and any w ∈ S(Rd ).

∀ x ∈ Rd

(4.18)

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% V )]; cf. Corollary 4.4, u ∈ D[h(A; % V )] and Let u ∈ D[H(A; %

OpA (p)u + V u ∈ L2 (Rd ), where p(ξ) := ξ − 1. From (4.18) we deduce that %

OpA (p)(eiϕ u) + V (eiϕ u) = eiϕ [OpA (p)u + V u] ∈ L2 (Rd ).

(4.19)

Let us show that eiϕ u ∈ D[h(A; V )]. Obviously eiϕ u ∈ D(qV+ ). We notice now that (2.2) implies that w ∈ D(HA% ) if and only if eiϕ w ∈ D(HA ). From (4.18) 1/2 1/2 we get U HA U −1 = HA% , so U HA U −1 = HA% and then U −1 [D(hA% )] = D(hA ). It follows that eiϕ u ∈ D(hA ), so eiϕ u ∈ D[h(A; V )]. Using Corollary 4.4 and equality (4.19), we deduce U −1 u ∈ D[H(A; V )] as well as (4.17).

5. Trace Estimations Proposition 5.1. Suppose hypothesis (i) and (ii) are verified. There exists μ ≥ 1, only depending on V− , such that for all λ ≥ λ0 := max{−infσ(H) + 1, μ},

r ≥ r0 := d + 1,

there exists C > 0 such that for every bounded open subset Ω of Rd we have 1Ω (H + λ)−r ∈ I2 and 1Ω (H + λ)−r I2 ≤ C|Ω|1/2 .

(5.1)

We denoted by 1Ω both the characteristic function of Ω and the associated multiplication operator on L2 (Rd ). Proof. We use (3.12) and Proposition 4.2 to obtain that for any f ∈ L2 (Rd ), λ ≥ λ0 and r ≥ r0 one has −r

|(H + λ)

1 f| ≤ Γ(r)

∞

tr−1 e−λt e−tH(0;−V− ) |f |dt.

(5.2)

0

Since V− ∈ Kd , by Theorem 1.5 from [36] (or Theorem 2.9 from [9]), for any t > 0 the operator e−tH(0,−V− ) has an integral kernel satisfying: For any ρ, ρ > 1, ρ1 + ρ1 = 1, one can choose positive constants M, b such that

0 ≤ e−tH(0,V− ) (x, y) ≤ M ebt sup [pt/2 (z)]1/ρ [pt (x − y)]1/ρ , ∀ t > 0, x, y ∈ Rd , z∈Rd

(5.3) where pt is deﬁned by (3.4). Using (3.4) and (3.3) it follows that there exists an absolute constant C > 0 such that 2 2 1/2 d+1 d+2 pt (x) ≤ Ctet (|x|2 + t2 )− 2 +(|x|2 + t2 )− 4 e−(|x| +t ) , ∀t > 0, x ∈ Rd . (5.4)

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We choose ρ = 4 and ρ = C1 > 0:

4 3

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in (5.3). From (5.4) it follows that for some

3d 3d sup [pt/2 (z)]3/4 ≤ C1 t− 4 + t− 8 ,

∀t > 0.

(5.5)

z∈Rd

Using (5.2), (5.5), (5.3) and (5.4), we get the inequality −r |[(H + λ) f ](x)| ≤ L(x − y)|f (y)|dy =: (T |f |)(x), f ∈ L2 (Rd ), x ∈ Rd , Rd

(5.6) where ∞ L(x) := C2

3d 3 3d e−(λ−b−1/4)t tr− 4 t− 4 + t− 8

(5.7)

0

1 2 2 1/2 d+1 d+2 × (|x|2 + t2 )− 8 + (|x|2 + t2 )− 16 e− 4 (|x| +t ) dt

− d+1 4

≤ C2 |x|

− d+2 8

+ |x|

− |x| 4

∞

e

3d 1 3 3d e−(λ−b− 4 )t tr− 4 t− 4 + t− 8 dt,

0

where C2 is a positive constant. Choosing μ = b + 12 , the assumptions insure the convergence of the last integral. It follows that L ∈ L2 (Rd ), ∀d ≥ 2. The integral operator 1Ω T is Hilbert-Schmidt, since ⎡ ⎤1/2 |1Ω (x)L(x − y)|2 dxdy ⎦ = L L2 (Rd ) |Ω|1/2 . (5.8) 1Ω T I2 = ⎣ Rd Rd

The conclusion of the Proposition follows from (5.6), (5.8) and Theorem 2.13 from [53]. Corollary 5.2. Under the assumptions of Proposition 5.1, for any m ≥ 2r0 there exists C > 0 such that for any Ω ⊂ Rd open and bounded, we have 1Ω (H + λ)−m 1Ω ∈ I1 and 1Ω (H + λ)−m 1Ω I1 ≤ C|Ω|.

(5.9)

Proof. We choose r ≥ r0 , s ≥ r0 , r + s = m. Then 1Ω (H + λ)−m 1Ω I1 ≤ 1Ω (H + λ)−r I2 (H + λ)−s 1Ω I2 , and we use Proposition 5.1 to conclude.

Corollary 5.3. Let f ∈ L∞ (R), suppf ⊂ (−∞, a], a ∈ R. Under the assumptions of Proposition 5.1, ∃C > 0 such that for any Ω ⊂ Rd open and bounded we have 1Ω f (H)1Ω ∈ I1 and 1Ω f (H)1Ω I1 ≤ C|Ω|.

(5.10)

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Proof. We use the equality 1Ω f (H)1Ω = 1Ω (H + λ)−r (H + λ)2r f (H)(H + λ)−r 1Ω , r ≥ r0 and Proposition 5.1, taking into account the fact that H, being lower semibounded, satisﬁes (H + λ)2r f (H) ∈ B[L2 (Rd )].

6. The Operator HΩ Assume hypothesis (i) and (ii) for a while; let H = H(A; V ) be the operator constructed in Sect. 4. Let Ω be an open subset of Rn and Ωc its complement. For n ∈ N, n ≥ 1, we set Hn := H + n1Ωc , which is a self-adjoint operator on L2 (Rd ) with domain D(Hn ) = D(H). The associated quadratic form hn (u, v) := h(u, v) + n(1Ωc u, 1Ωc v),

u, v ∈ D(hn ) = D(h)

(6.1)

is symmetric, lower semi-bounded and closed. We also have h ≤ hn ≤ hn+1 , ∀n ≥ 1. We are going to identify L2 (Ω) with the closed subspace of L2 (Rd ) whose elements are null on Ωc . The operator 1Ω will be the orthogonal projection of L2 (Rd ) on L2 (Ω). To the monotone sequence of forms {hn }n≥1 deﬁned by (6.1) one assigns the form hΩ deﬁned on # $ D(hΩ ) := u ∈ ∩n≥1 D(hn ) | sup hn (u, u) < ∞ = D(h) ∩ L2 (Ω) (6.2) n≥1

by the equality hΩ (u, v) = lim hn (u, v) = h(u, v), n→∞

u, v ∈ D(hΩ ).

(6.3)

The form hΩ is not densely deﬁned but, by Theorem 4.1 from [33], it is lower bounded and closed, deﬁning a unique pseudo-self-adjoint operator HΩ on L2 (Rd ); we have D(HΩ ) ⊂ L2 (Ω), HD(HΩ ) ⊂ L2 (Rd ) and HΩ , considered as an operator in L2 (Ω), is self-adjoint. In addition, limn→∞ Hn = HΩ in strong resolvent sense. We denote by CH (R) the set of functions f : [mf , ∞) → R, where mf < inf σ(H) (maybe depending on f ), f continuous and limt→∞ f (t) = 0. Since inf σ(Hn ) and inf σ(HΩ ) are smaller or equal than inf σ(H), one can deﬁne for any f ∈ CH (R) the operators f (Hn ), f (HΩ ) ∈ B L2 (Rd ) . 2 Thesecond one is deﬁned as follows: f (HΩ )|L (Ω) is the operator from B L2 (Ω) associated to HΩ (seen as a self-adjoint operator in L2 (Ω)) by ⊥ the usual functional calculus, while f (HΩ ) = 0 on L2 (Ω) 2. Then we have limn→∞ f (Hn ) = f (HΩ ) for the strong convergence in B L (Rd ) . We have

f (HΩ ) = 1Ω f (HΩ ) = f (HΩ )1Ω . In particular, the properties above are checked for the function f (t) = (t + λ)−1 ,

λ > − inf σ(H),

(6.4)

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deﬁned on a neighborhood of σ(H). Then f (H) = (H + λ)−1 ,

f (HΩ ) = (HΩ + λ)−1 .

Lemma 6.1. If we assume (i) and (ii), for any Ω ⊂ Rd open bounded set, the operator HΩ has compact resolvent. Proof. It will be enough to show that any M ⊂ D(HΩ ), bounded for the graph norm deﬁned by HΩ , is relatively compact in L2 (Ω). The set M will be bounded in D(hΩ ), thus also bounded in L2 (Ω) and D(h). Hence the set MA := (HA + 1)1/2 M is bounded in L2 (Rd ) and M = (HA + 1)−1/2 MA . Let χ ∈ C0∞ (Rd ),

0 ≤ χ ≤ 1,

χ = 1 in a neighborhood of Ω.

By (3.11) one has |χ(HA + 1)−1/2 f | ≤ χ(H0 + 1)−1/2 |f |,

∀f ∈ L2 (Rd ).

(6.5)

Since R(H0 + 1)−1/2 = H1/2 (Rd ), the operator χ(H0 + 1)−1/2 is compact on L2 (Rd ). By Pitt’s Theorem [30] and by (6.5), the operator χ(HA + 1)−1/2 is also compact on L2 (Rd ). Since χM = M , the set M is relatively compact in L2 (Ω). Proposition 6.2. Assume hypothesis (i) and (ii ) are verified. For any λ ≥ λ0 , r ≥ r0 (λ0 and r0 as in Proposition 5.1), there is a constant C > 0 such that for any open subsets U, Ω of Rd such that U ∩ Ω is bounded we have 1U (HΩ + λ)−r ∈ I2 and the next inequality holds: 1U (HΩ + λ)−r I2 ≤ C|U ∩ Ω|1/2 .

(6.6)

Proof. By using the inequality (5.2) for Hn and the fact that s − lim (Hn + λ)−r = (HΩ + λ)−r , n→∞

one obtains that −r

|(HΩ + λ)

1 f| ≤ Γ(r)

∞

tr−1 e−λt e−tH(0;−V− ) |f |dt,

∀f ∈ L2 (Rd ).

0

The proof is completed in the same way as for Proposition 5.1, since 1U (HΩ + λ)−r = 1U 1Ω (HΩ + λ)−r = 1U ∩Ω (HΩ + λ)−r . Corollary 6.3. Under the assumptions of Proposition 6.2, for any m ≥ 2r0 , ∃C > 0 such that for any Ω ⊂ Rd bounded and open one has (HΩ +λ)−m ∈ I1 and (HΩ + λ)−m I1 ≤ C|Ω|.

(6.7)

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Proof. We use the identity (HΩ + λ)−m = 1Ω (HΩ + λ)−r (HΩ + λ)−s 1Ω , where r ≥ r0 , s ≥ r0 , r + s = m,

as well as Proposition 6.2 with U = Ω.

Corollary 6.4. Let f ∈ C0 (R). Under the assumptions of Proposition 6.2, there exists a constant C > 0 such that for any Ω ⊂ Rd open and bounded, we have f (HΩ ) ∈ I1 and f (HΩ ) I1 ≤ C|Ω|.

(6.8)

Proof. For any g ∈ CH (R), since s − limn→∞ g(Hn ) = g(HΩ ), one obtains for each Ω ⊂ Rd open set, the inequality g(HΩ ) B[L2 (Rd )] ≤ sup |g|. R

(6.9)

We choose / suppf. g(t) := (t + λ)m f (t), where m ≥ 2r0 , λ ≥ λ0 , −λ ∈ Then / suppg and f (t) = (t + λ)−m g(t), ∀t ∈ R. g ∈ CH (R), −λ ∈ It follows that f (HΩ ) = (HΩ + λ)−m g(HΩ ), so (6.8) is a consequence of (6.7) and (6.9).

7. Proof of Theorem 1.1 Lemma 7.1. Assume hypothesis (i) and (ii ). Let λ > − inf σ(H), Ω ⊂ Rd an open bounded set and ϕ ∈ BC ∞ (Rd ), ϕ = 1 on Ωc . Then one has (H + λ)−1 − (HΩ + λ)−1 = (H + λ)−1 − (HΩ + λ)−1 ϕ + [H, ϕ](HΩ + λ)−1 = ϕ − (H + λ)−1 [H, ϕ] (H + λ)−1 − (HΩ + λ)−1 .

(7.1)

Proof. The function ϕ veriﬁes the assumptions of Proposition 4.5, so the operator of multiplication by ϕ leaves D(H) invariant and [H, ϕ] ∈ B[L2 (Rd )]. Using (4.14) for Hn and the equality [Hn , ϕ] = [H, ϕ],

∀ n ≥ 1,

where Hn := H + n1Ωc , we deduce that (H + λ)−1 − (Hn + λ)−1 = (H + λ)−1 n1Ωc · ϕ(Hn + λ)−1 = (H + λ)−1 n1Ωc (Hn + λ)−1 ϕ +(H + λ)−1 n1Ωc (Hn + λ)−1 [H, ϕ](Hn + λ)−1 = (H + λ)−1 − (Hn + λ)−1 [ϕ + [H, ϕ] (Hn + λ)−1 ].

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The ﬁrst equality in (7.1) follows from the formula above in the limit n → ∞, taking into account the relation s − lim (Hn + λ)−1 = (HΩ + λ)−1 n→∞

and the fact that the sequence {(Hn + λ)−1 }n≥1 is bounded in B[L2 (Rd )]. The second equality in (7.1) follows in the same way. The next Proposition is basic for proving Theorem 1.1 Proposition 7.2. Assume hypothesis (i) and (ii ). For any λ ≥ λ0 and m ∈ N, m ≥ 4r0 (λ0 and r0 as in Proposition 5.1), there exists C > 0 such that for any bounded open subset Ω of Rd % 1/2 , 1Ω (H + λ)−m 1Ω − (HΩ + λ)−m I1 ≤ C|Ω|1/2 |Ω|

(7.2)

% := {x ∈ Rd | dist(x, ∂Ω) < 1}. where Ω Proof. We use (6.4) and infer that 1Ω (H + λ)−m 1Ω − (HΩ + λ)−m = 1Ω (H + λ)j−m+1 (H + λ)−1 − (HΩ + λ)−1 (HΩ + λ)−j 1Ω . 0≤j≤m−1

(7.3) We denote by Ej the general term of the sum. Let Ω ⊂ R bounded and open. By taking the convolution of the characteristic function of a neighborhood of Ωc by a function from C0∞ (Rd ) with the support included in a small neighborhood of the origin, one constructs a real function ϕ ∈ BC ∞ (Rd ) such % and such that ∂ α ϕ L2 (Rd ) ≤ that 0 ≤ ϕ ≤ 1, ϕ = 1 on Ωc , ϕ = 0 on Ω \ Ω d Cα , ∀α ∈ N , with Cα independent of Ω. We estimate ﬁrst the I1 -norm of Ej for 2r0 ≤ j ≤ m−1. We use the ﬁrst equality form (7.1) and write Ej = Ej + Ej , where Ej and Ej correspond to the two terms of the sum ϕ + [H, ϕ]. We have Ej I1 = 1Ω (H + λ)j−m+1 (H + λ)−1 − (HΩ + λ)−1 ϕ(HΩ + λ)−j 1Ω I1 ≤ 1Ω (H + λ)j−m+1 (H + λ)−1 − (HΩ + λ)−1 d

× B[L2 (Ω)] 1U (H + λ)−j/2 I2 (HΩ + λ)−j/2 1Ω I2 , % Using (6.9) and Proposition 6.2 we get where U := Ωc ∪ Ω. % 1/2 |Ω|1/2 Ej I1 ≤ C1 |Ω|

(7.4)

for some positive constant C1 , independent of Ω. To estimate the I1 norm of Ej , we write it as Ej = Ej,α + Ej,0 , 1≤|α|≤d+1 and Ej,0 correspond to the decomposition of [H, ϕ] in where the terms Ej,α the second of the inequalities (4.13). Using Propositions 4.5 and 6.2 we obtain

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inequalities, in which the constants are independent of Ω: Ej,α I1 = 1Ω (H + λ)j−m+1 (H + λ)−1 − (HΩ + λ)−1 × Sα (∂ α ϕ)(HΩ + λ)−j−1 1Ω I1 ≤ 1Ω (H +λ)j−m+1 (H +λ)−1 − (HΩ + λ)−1 Sα (∂ α ϕ) B[L2 (Rd )] × 1Ω% (HΩ + λ)−j/2 I2 (HΩ + λ)−j/2−1 1Ω I2 % 1/2 |Ω|1/2 , 1 ≤ |α| ≤ d + 1 ≤ C |Ω| and

Ej,0 I1 = 1Ω (H +λ)j−m+1 (H +λ)−1 −(HΩ +λ)−1 T (HΩ + λ)−j−1 1Ω I1 ≤ 1Ω (H + λ)j−m+1 (H + λ)−1 − (HΩ + λ)−1 × B[L2 (Rd )] T I2 (HΩ + λ)−j−1 1Ω I2 % 1/2 |Ω|1/2 . ≤ C |Ω|

Thus we have % 1/2 |Ω|1/2 . Ej I1 ≤ C2 |Ω|

(7.5)

Taking (7.4) into account we get % 1/2 , Ej I1 ≤ C|Ω|1/2 |Ω|

2r0 ≤ j ≤ m − 1,

(7.6)

for some constant C > 0 independent of Ω. Let us assume now that 0 ≤ j ≤ 2r0 − 1; then m − j − 1 ≥ 2r0 . We use % + E % where, as before, E % now the second equality in (7.1) to write Ej = E j j j % correspond to the two terms in the sum ϕ + [H, ϕ]. The I1 -norm of and E j % is estimated as above, writing E j %j,α %j = %j,0 E E +E , 1≤|α|≤d+1

% E j,α

% E j,0

where the terms and correspond to the decomposition of [H, ϕ] in the ﬁrst equality in (4.13). By (6.9) and Propositions 4.5 and 5.1, we get for Ω- independent constants % I E j,α 1

= 1Ω (H + λ)j−m (∂ α ϕ)Sα (H + λ)−1 − (HΩ + λ)−1 (HΩ + λ)−j 1Ω I1 j−m

j−m

≤ B[L2 (Rd )] ≤ 1Ω (H + λ) 2 I2 (H + λ) 2 1Ω% I2 × (∂ α ϕ)Sα (H + λ)−1 − (HΩ + λ)−1 (HΩ + λ)−j 1Ω B[L2 (Rd )] % 1/2 , 1 ≤ |α| ≤ d + 1 ≤ C |Ω|1/2 |Ω| and %j,0 E I1

= 1Ω (H + λ)j−m T (H + λ)−1 − (HΩ + λ)−1 (HΩ + λ)−j 1Ω I1 ≤ 1Ω (H + λ)j−m I2 T I2 (H + λ)−1 − (HΩ + λ)−1 % 1/2 . × (HΩ + λ)−j 1Ω B[L2 (Rd )] ≤ C |Ω|1/2 |Ω|

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So we have %j I ≤ C|Ω|1/2 |Ω| % 1/2 . E 1

(7.7)

% , we introduce another auxiliary real function To estimate the I1 -norm of E j % ψ = 1 in a neighborhood of Ω, ψ ∈ C0∞ (Rd ), 0 ≤ ψ ≤ 1, supp ψ ⊂ Ω ∪ Ω, such that for any α ∈ Nd one has ∂ α ψ L∞ (Rd ) ≤ Cα , with Cα independent of Ω. We have ˜j I = 1Ω (H +λ)j−m+1 ϕ[(H +λ)−1 −(HΩ +λ)−1 ](HΩ +λ)−j 1Ω I E 1

1

≤ C 1Ω (H +λ)

j−m+1

−1

ϕ(H +λ)

j−m+1

ψ I1 +C 1Ω (H +λ)

ϕψ I1 ,

where we used the fact that ψ1Ω = 1Ω and we denoted by C various constants independent of Ω. % using Proposition 5.1 as above, we get Since supp (ϕψ) ⊂ Ω, % 1/2 . 1Ω (H + λ)j−m+1 ϕψ I1 ≤ C|Ω|1/2 |Ω|

(7.8)

For the last term that has to be estimated we use Proposition 4.5 and write ϕ(H + λ)−1 = (H + λ)−1 ϕ − (H + λ)−1 (∂ α ϕ)Sα (H + λ)−1 1≤|α|≤d+1 −1

−(H + λ)

T (H + λ)−1 .

One gets immediately the inequalities % 1/2 , 1Ω (H + λ)j−m ϕψ I1 ≤ C|Ω|1/2 |Ω| % 1/2 1Ω (H + λ)j−m (∂ α ϕ)Sα (H + λ)−1 ψ I1 ≤ C|Ω|1/2 |Ω| and % 1/2 , 1Ω (H + λ)j−m T (H + λ)−1 ψ I1 ≤ C|Ω|1/2 |Ω| which gives % 1/2 . 1Ω (H + λ)j−m+1 ϕ(H + λ)−1 ψ I1 ≤ C|Ω|1/2 |Ω|

(7.9)

From (7.8) and (7.9) we obtain % I ≤ C|Ω|1/2 |Ω| % 1/2 E j 1

(7.10)

which, together with (7.7), implies the inequality % 1/2 , Ej I1 ≤ C|Ω|1/2 |Ω|

0 ≤ j ≤ 2r0 + 1.

The relation (7.2) follows from (7.3), (7.6) and (7.11).

(7.11)

Theorem 1.1 is a direct consequence of the next Proposition: Proposition 7.3. Now we assume that the hypothesis (i), (ii ), (iii) and (iv) are fulfilled. Then for any f ∈ C0 (R) and > 0, there exists m0 ∈ N∗ such that |tr[1Ω f (H)1Ω ] − trf (HΩ )| ≤ |Ω| for any Ω ∈ F with B(0, m0 ) ⊂ Ω.

(7.12)

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Proof. One uses ideas of [10] (see also [20]). Let λ0 and r0 be the constants from Proposition 5.1. We set a := λ0 + 1, m0 := 4r0 . It will be enough to prove (7.12) for the real functions f ∈ C0 (R) such that supp f ⊂ [−a + 12 , ∞). The functions ( 1 −a + , ∞ t → (a + t)m0 f (t) ∈ R 2 and [0, 2] τ → τ −m0 f (τ −1 − a) ∈ R are continuous. For any > 0 there is a polynomial P with real coefﬁcients such that −m τ 0 f (τ −1 − a) − P (τ ) ≤ , ∀ τ ∈ [0, 2]. Therefore

1 (a + t)m0 f (t) − P ≤ , a+t

Let −m0

Q (t) := (a + t)

P

1 ∀ t ≥ −a + . 2

1 . a+t

Then in form-sense −(a + H)−m0 ≤ f (H) − Q (H) ≤ (a + H)−m0 , so −1Ω (a+H)−m0 1Ω ≤ 1Ω f (H)1Ω −1Ω Q (H)1Ω ≤ 1Ω (a+H)−m0 1Ω , Ω ∈ F. Using Corollaries 5.2 and 5.3 we obtain |tr[1Ω f (H)1Ω ] − tr[1Ω Q (H)1Ω ]| ≤ tr[1Ω (a + H)−m0 1Ω ] ≤ C1 |Ω|, (7.13) where C1 is a constant independent on and Ω ∈ F. In the same way, using Corollaries 6.3 and 6.4, one shows that for some constant C2 , independent on and Ω ∈ F, one has |trf (Ω) − trQ (HΩ )| ≤ tr(a + HΩ )−m0 ≤ C2 |Ω|. Inequality (7.12) follows from (7.13), (7.14), (7.2) and hypothesis (iv).

(7.14)

8. Proof of Theorem 1.2 Let us suppose that hypothesis (i), (ii ), (iii), (iv) and (v) are veriﬁed. Since the proof of Theorem 1.2 is very close to that of Theorem 1.6 from [20], we shall not indicate all the details. Let us notice ﬁrst that the hypothesis (i) and (v), the proofs of Proposition 5.1 and Corollary 5.2 from [20] show that there exists a constant magnetic ﬁeld 1 0 0 0 B0 = Bjk dxj ∧ dxk , Bjk = −Bkj ∈R 2 1≤j,k≤d

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and a vector potential A(p) =

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(p)

Aj dxj ,

1≤j≤d (p) Aj 0

∞ belong to Cpol (Rd ) and are Γ-periodic, such that where the components 0 (p) 0 B − B = dA . Since B = dA with 1 0 A0 = A0j dxj , A0j (x) = Bkj xk , 2 1≤j≤d

(p)

1≤k≤d

0

we have B = d(A +A ), so, by Proposition 4.6, we will assume in the sequel that the vector potential deﬁning the magnetic ﬁeld B is A := A(p) + A0 . For γ ∈ Γ we deﬁne the function ϕγ : Rd → R, 1 0 ϕγ (x) := A0j (γ)xj = Bkj γk xj 2 1≤j≤d

1≤j,k≤d

(so dϕγ = A0 (γ)) and the unitary operators of multiplication with eiϕγ on L2 (Rd ) denoted by Uγ . Let us put (Lγ u) (x) := u(x − γ) and Tγ := Uγ Lγ . The operators Tγ are the magnetic translations [37]. Lemma 8.1. Let us suppose that (i), (ii’) and (v) are verified. Then the operator H = H(A; V ) constructed in Sect. 4 commutes with Tγ , i.e. HTγ = Tγ H,

∀ γ ∈ Γ.

(8.1)

Proof. By Proposition 4.6, we have H(A; V )Uγ = Uγ H(A − dϕγ ; V ). So we only need to show that Lγ H(A; V ) = H(A − dϕγ ; V )Lγ ,

∀ γ ∈ Γ.

(8.2)

Since ΓA (x − γ, y − γ) = ΓA (x, y) − A0 (γ) = ΓA−dϕγ (x, y), x, y ∈ Rd , it follows that for any a ∈ S m (Rd ), Γ-periodic in x, and for any w ∈ S(Rd ), we have Lγ OpA (a)w = OpA−dϕγ (a)(Lγ w).

(8.3) 2

Let u ∈ D(H(A; V )), so u ∈ D(h(A; V )) and Op (p)u + V u ∈ L (R ) with p(ξ) := ξ − 1. From (8.3) we have OpA−dϕγ (p) (Lγ u) + V (Lγ u) = Lγ OpA (p)u + V u ∈ L2 (Rd ). (8.4) A

d

Let us show that Lγ u ∈ D(h(A − dϕγ ; V )). Obviously Lγ u ∈ D(qV+ ). From (2.2) it follows that w ∈ D(HA ) if and only if Lγ w ∈ D(HA ) = D(HA−dϕγ ). 1/2

1/2

−1 From (8.3) we deduce that Lγ HA L−1 γ = HA−dϕγ , so Lγ HA Lγ = HA−dϕγ . It follows that Lγ [D(hA )] = D(hA−dϕγ ), so Lγ u ∈ D(hA−dϕγ ) and then

Lγ u ∈ D [h(A − dϕγ ; V )] . From Corollary 4.4 and equality (8.4) we get Lγ u ∈ D [H(A − dϕγ ; V )] as well as (8.2).

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The family {Tγ }γ∈Γ satisﬁes Tα Tβ = e−iϕβ (α) Tα+β , α, β ∈ Γ, so it does not form a group. However, using [1] as a model (cf. also [20]), one can deﬁne a Γ-trace for a class of operators on B(L2 (Rd )) commuting with the magnetic translations Tγ . Definition 8.2. An operator S ∈ B(L2 (Rd )) belongs to I1Γ if Tγ S = STγ , d ∀ γ ∈ Γ and if for every ϕ, ψ ∈ L∞ comp (R ) one has ϕSψ ∈ I1 . d One can show that for all ϕ, ϕ , ψ, ψ ∈ L∞ comp (R ) such that Lγ (ϕψ) = Lγ (ϕ ψ ) = 1, ∀S ∈ I1 , γ∈Γ

γ∈Γ

we have the equality tr(ϕSψ) = tr(ϕ Sψ ). This justiﬁes Definition 8.3. Let S ∈ I1Γ . We call Γ-trace of S the expression trΓ S := tr(ϕSψ), d where ϕ, ψ ∈ L∞ Lγ (ϕψ) = 1. comp (R ) and γ∈Γ

One can prove (see [20]) Lemma 8.4. Let S = S ∗ ∈ I1Γ . Then KS , the integral kernel of S, is a locally integrable function on Rd × Rd , its restriction to the diagonal of Rd × Rd is well-defined and locally integrable and one has trΓ S = KS (x, x)dx, (8.5) F

where F is a fundamental domain of Rd with respect to Γ. From Corollary 5.3 and Lemma 8.1 it follows that for any f ∈ C0 (R) one has f (H) ∈ I1Γ . From Lemma 8.4 we know that the restriction to the diagonal of the integral kernel Kf (H) exists as a locally integrable function. Then for any Ω ∈ F one has tr(1Ω f (H)1Ω ) = Kf (H) (x, x)dx. (8.6) Ω

By the proof of Theorem 1.6 in [20] we get 1 1 lim Kf (H) (x, x)dx = Kf (H) (x, x)dx. |F | Ω→Rd ,Ω∈F |Ω| Ω

Then (1.9) follows from (8.6), (8.7) and (8.5).

F

(8.7)

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References [1] Atiyah, M.: Elliptic operators, discrete groups and von Neumann algebras. Ast´erisque 32–33, 43–72 (1976) [2] Bellissard, J.: Non commutative methods in semiclassical analysis course given at the CIME (1991). In: Transition to Chaos in Classical and Quantum Mechanics. Lecture Notes in Mathematics, vol. 1589. Springer, Berlin (1994) [3] Bellissard, J.: Gap labelling theorems for Schr¨ odinger’s operators in from number theory to physics. In: J.M., Moussa, P., Waldschmidt, M. (eds.) Les Houches March, vol. 89, pp. 538–630. Springer, Luck (1993) [4] Briet, Ph., Raikov, G.D.: The integrated density of states in strong magnetic ﬁelds. J. Funct. Anal. 237, 540–564 (2006) [5] Carmona, R., Lacroix, J.: Spectral theory of random Schr¨ odinger operators. Birkh¨ auser, Basel (1990) [6] Carmona, R., Masters, W.C., Simon, B.: Relativistic Schr¨ odinger operators: asymptotic behavior of eigenfunctions. J. Funct. Anal. 91, 117–143 (1990) [7] Combes, J.M., Hislop, P.D., Klopp, F., Raikov, G.D.: Global continuity of the integrated density of states for random Landau Hamiltonians. Commun. P.D.E. 29, 1187–1213 (2004) [8] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer, Berlin (1987) [9] Demuth, M., van Casteren, J.A.: Stochastic Spectral Theory for Self-Adjoint Feller Operators. Birkh¨ auser, Basel (2000) [10] Doi, S., Iwatsuka, A., Mine, T.: The uniqueness of the integrated density of states for the Schr¨ odinger operator with magnetic ﬁelds. Math. Z. 237, 335–371 (2001) [11] Figotin, A., Pastur, L.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992) [12] Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schr¨ odinger operators. J. Am. Math. Soc. 21(4), 925–950 (2008) [13] G´erard, C., Martinez, A., Sj¨ ostrand, J.: A mathematical approach to the eﬀective Hamiltonian in perturbed periodic problems. Commun. Math. Phys. 142, 217–244 (1991) [14] Helffer, B., Sj¨ ostrand, J.: Equation de Schr¨ odinger avec champ magn´etique et ´equation de Harper in LNP, vol. 345, pp. 118–197. Springer, Berlin (1989) [15] Hupfer, T., Leschke, H., M¨ uller, P., Warzel, S.: Existence and uniqueness of the integrated density of states for Schr¨ odinger operators with magnetic ﬁelds and unbounded random potentials. Rev. Math. Phys. 13, 1581–1587 (2001) [16] Ichinose, T.: The nonrelativistic limit problem for a relativistic spinless particle in an electromagnetic ﬁeld. J. Funct. Anal. 73(2), 233–257 (1987) [17] Ichinose, T.: Essential selfadjointness of the Weyl quantized relativistic Hamiltonian. Ann. Inst. H. Poincar´e, Phys. Th´eor. 51(3), 265–297 (1989) [18] Ichinose, T., Tamura, H.: Path integral for the Weyl quantized relativistic Hamiltonian. Proc. Japan Acad. Ser. A Math. Sci. 62(3), 91–93 (1986) [19] Ichinose, T., Tamura, H.: Imaginary-time path integral for a relativistic spinless particle in an electromagnetic ﬁeld. Commun. Math. Phys. 105(2), 239–257 (1986)

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[20] Iftimie, V.: Uniqueness and existence of the integrated density of states for Schr¨ odinger operators with magnetic ﬁeld and electric potential with singular negative part. Publ. Res. Inst. Math. Sci. 41(2), 307–327 (2005) [21] Iftimie, V., M˘ antoiu, M., Purice, R.: Magnetic pseudodifferential operators. Publ. Res. Inst. Math. Sci. 43(3), 585–623 (2007) [22] Iftimie, V., M˘ antoiu, M., Purice, R.: Estimating the number of negative eigenvalues of a relativistic Hamiltonian with regular magnetic ﬁeld. In: Topics in Applied Mathematics and Mathematical Physics. Editura Academiei Romˆ ane, Stolova (2008) [23] Iftimie, V., M˘ antoiu, M., Purice, R.: A Beals-type criterion for magnetic pseudodifferential operators (Preprint ArXiv) [24] Ikeda, V., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland, Amsterdam (1981) [25] Karasev, M.V., Osborn, T.A.: Symplectic areas, quantization and dynamics in electromagnetic ﬁelds. J. Math. Phys. 43(2), 756–788 (2002) [26] Kato, T., Masuda, K.: Trotter’s product formula for nonlinear semigroups generated by the subdifferentials of convex functionals. J. Math. Soc. Japan 30, 169–178 (1978) [27] M˘ antoiu, M., Purice, R.: The magnetic Weyl calculus. J. Math. Phys. 45(4), 1394–1417 (2004) [28] Nagase, M., Umeda, T.: Weyl quantized Hamiltonians of relativistic spinless particles in magnetic ﬁelds. J. Funct. Anal. 92, 136–164 (1990) [29] Pascu, M.: On the essential spectrum of the relativistic magnetic Schr¨ odinger operator. Osaka J. Math. 39(4), 963–978 (2002) [30] Pitt, L.D.: A compactness criterion for linear operators on function spaces. J. Operat. Theory 1, 49–54 (1979) [31] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 4. Academic Press, New York (1979) [32] Shubin, M.A.: The density of states of selfadjoint elliptic operators with almost periodic coefﬁcients. Am. Math. Soc. Transl. (2) 118, 307–339 (1982) [33] Simon, B.: A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28, 377–385 (1978) [34] Simon, B.: Maximal and minimal Schr¨ odinger forms. J. Operat. Theory 1, 37–47 (1979) [35] Simon, B.: Trace ideals and applications, London Math. Soc. LNS, vol. 35. Cambridge University Press, Cambridge (1979) [36] van Casteren, J.A.: A pointwise inequality for generalized Schr¨ odinger semigroups. In: Symposium “Partial differential equations”, Holzhau, TeubnerTexte zur Mathematik, pp. 298–312 (1988) [37] Zak, J.: Magnetic translation group. Phys. Rev. 134(6A) (1964)

Viorel Iftimie, Marius M˘ antoiu and Radu Purice Institute of Mathematics Simion Stoilow of the Romanian Academy P. O. Box 1-764, Bucharest Romania e-mail: [email protected]

246

V. Iftimie, M. M˘ antoiu and R. Purice

Marius M˘ antoiu Universidad de Chile Las Palmeras 3425 Casilla 653, Santiago Chile e-mail: [email protected] Radu Purice Laboratoire Europ´een Associ´e CNRS Math-Mode, Franco-Roumain Bucharest, Romania e-mail: [email protected] Received: July 30, 2009. Revised: December 17, 2009.

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Integr. Equ. Oper. Theory 67 (2010), 247–256 DOI 10.1007/s00020-010-1778-7 Published online March 17, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Almost Invariant Half-spaces of Algebras of Operators Alexey I. Popov Abstract. Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if T Y ⊆ Y + F for some finite-dimensional “error” F . In this paper, we study subspaces that are almost invariant under every operator in an algebra A of operators acting on X. We show that if A is norm closed then the dimensions of “errors” corresponding to operators in A must be uniformly bounded. Also, if A is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then A has an invariant half-space. Mathematics Subject Classification (2000). 47A15, 47L10. Keywords. Operator algebras, Almost invariant subspace, Half-space.

1. Introduction The notion of an almost invariant subspace was recently introduced in [2]. If T is an operator on a Banach space X then a subspace Y of X is called almost invariant under T if there exists a finite-dimensional subspace F of X such that T Y ⊆ Y + F. (1) Clearly, if Y has finite dimension or finite codimension then Y is almost invariant under every operator on X. Definition 1.1. [2] A subspace Y ⊆ X is called a half-space if Y is both of infinite dimension and of infinite codimension. The question whether every operator on a Banach space has an almost invariant half-space was posed in [2]; it was solved there for certain classes of operators. Just as the studies of transitive algebras generalize the Invariant Subspace Problem for a single operator, the purpose of this paper is to introduce and study the notion of a subspace that is simultaneously almost invariant under every operator in a given algebra of operators.

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Throughout the paper, X is a Banach space. The term “subspace” refers to a norm-closed linear subspace of X, while the term “linear subspace” refers to a subspace that is not necessarily closed. Whenever we say that A is an algebra of operators, we mean that A is an algebra of operators on X. Also, given a sequence (xi ), we write [xi ] for the closed linear span of (xi ). Definition 1.2. Let C ⊆ L(X) be an arbitrary collection of operators and Y ⊆ X a subspace of X. We call Y almost invariant under C, or C-almost invariant if Y is almost invariant under every operator in C. Like in the case of a single operator, every subspace that is not a halfspace is automatically almost invariant under every collection C of operators on X. In Sect. 2, we study the finite-dimensional “errors” F appearing in formula (1) corresponding to operators in an algebra A. We prove that if A is an algebra without invariant half-spaces then for an A-almost invariant half-space Y these finite-dimensional subspaces cannot be the same (Proposition 2.2). On the other hand, we prove (Theorem 2.7) that if A is norm closed then these finite-dimensional subspaces cannot be “too far apart”; the dimensions of these subspaces must be uniformly bounded. In Sect. 3, the invariant subspaces of algebras having almost invariant half-spaces are investigated. It is proved that if A is a norm-closed algebra generated by a single operator then existence of an A-almost invariant half-space implies existence of an A-invariant half-space (Theorem 3.6). This theorem then is generalized to the case of a commutative algebra generated by a finite number of operators (Theorem 3.7). Also, the question of whether W OT are the same is investigated the almost invariant half-spaces of A and A (Corollary 3.2 and an example after Theorem 3.6). In this paper we will occasionally refer to some standard facts about invariant subspaces of operators and algebras of operators. For a general account on invariant subspaces and transitive algebras, see [3]. A good review of this topic can be found in [1].

2. Finite-dimensional “Errors” of Almost Invariant Half-spaces Observe that the finite-dimensional subspace F appearing in Eq. (1) is by no means unique. However the minimal dimension of a subspace satisfying this condition is unique. Some simple properties of a subspace of minimal dimension satisfying (1) are collected in the following lemma. Lemma 2.1. Let Y ⊆ X be a subspace, C be a collection of bounded operators on X and G ⊆ X be a finite-dimensional space of the smallest dimension such that T Y ⊆ Y + G for all T ∈ C. Then (i) Y + G = Y ⊕ G; (ii) if P : Y ⊕ G → G is the projection along Y then span P T (Y ) = G; T ∈C

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(iii) if C consists of a single operator, that is C = {T }, and if P : Y ⊕ G → G is the projection along Y then P T (Y ) = G. Moreover, in this case G can be chosen so that G ⊆ T Y . Proof. (i) Suppose that there exists a non-zero g ∈ G ∩ Y . Build g2 , . . . gn ∈ G such that {g, g2 , . . . , gn } is a basis of G. Denote G1 = span {g2 , . . . , gn }. It is clear that T Y ⊆ Y + G1 for all T ∈ C. However dim G1 < dim G. (ii) Define F = span {g ∈ G : v + g ∈ T Y for some v ∈ Y, T ∈ C}. Clearly F = span T ∈C P T (Y ). We claim that T Y ⊆ Y + F for all T ∈ C. Indeed, if y ∈ Y and T ∈ C then T y = v + g for some v ∈ Y and g ∈ G. By definition of F we get: g ∈ F , hence T y ∈ Y + F . Since F ⊆ G and G has the smallest dimension among the spaces with the property T Y ⊆ Y + G for all T ∈ C, we get G = F . (iii) The first part of this statement follows immediately from (ii). Let’s prove the “moreover” part. Let g1 , . . . , gn be a basis of G. By (ii), there exist u1 , . . . , un and y1 , . . . , yn in Y such that T ui = yi + gi (i = 1, . . . , n). Put fi = T ui and F = [fi ]ni=1 . Then clearly F ⊆ T Y . Also Y + F = Y + G, so that T Y ⊆ Y + F . From the minimality of G we obtain that dim F = dim G. The following example shows that T ∈C P T (Y ) may not be a linear space even in the case when C is an algebra of operators. Example. Let X = 2 (Z). Define T, S ∈ L(X) by T e0 = e1 ,

T e−1 = e2 ,

T ei = 0 if i = 0, −1,

and Se0 = e3 , 2

Sei = 0 if i = 0.

2

Since T = S = T S = ST = 0, the algebra A generated by T and S consists exactly of the operators of form aT + bS where a and b are arbitrary scalars. Let Y = [ei ]i0 . Then clearly AY ⊆ Y + F where F = span {e1 , e2 , e3 }, and F is the space of the smallest dimension satisfying this condition. If P : Y ⊕ F → F is the projection along Y then R∈A P R(Y ) is not a linear space. If it were, it would have been equal to F , since it contains the basis of F . However the vector e2 + e3 is not in this union. Suppose Y is a half-space that is almost invariant under a collection C of operators on X, that is, formula (1) holds for every operator T in C with some F . One may ask if it is possible that F does not depend on T . The following simple reasoning shows that in case of algebras of operators, this can only happen if the algebra already has a common invariant half-space. Proposition 2.2. Let Y ⊆ X be a half-space and A an algebra of operators. Suppose that there exists a finite-dimensional space F such that for each T ∈ A we have T Y ⊆ Y + F . Then there exists a half-space that is invariant under A.

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Proof. Let G be a space of the smallest dimension such that T Y ⊆ Y + G for all T ∈ A. We claim that Y + G is invariant under every operator in A. Denote A(Y ) = T ∈A T Y . Clearly A(Y ) is invariant under A. Hence, so is span A(Y ). Denote Z = Y + span A(Y ). Since T Y ⊆ span A(Y ) for every T ∈ A, we obtain that Z is invariant under A. By Lemma 2.1(ii), if P : Y ⊕ G → G is a projection along Y then P (span A(Y )) = G. Hence Y ⊕ G = Y ⊕ P (span A(Y )) = Y + span A(Y ) = Z, so that Y ⊕ G is invariant under A. Definition 2.3. Let T ∈ L(X) be an arbitrary operator and Y ⊆ X be a linear subspace. We will write dY,T for the smallest n such that there exists F with T Y ⊆ Y + F and dim F = n. The following observation is obvious. Lemma 2.4. Let T ∈ L(X) be an operator and Y ⊆ X be a subspace. Let q : X → X/Y be a quotient map. Then Y is T -almost invariant if and only if (qT )|Y is of finite rank. Moreover, dim(qT )(Y ) = dY,T . To proceed, we need the following two auxiliary lemmas. Lemma 2.5. Let Y ⊆ X be a linear subspace and {ui }N i=1 be a collection of lin∩ Y = {0}. Let {vi }N early independent vectors in X such that [ui ]N i=1 i=1 ⊆ X is linearly be arbitrary. Then for all but finitely many α we have {vi +αui }N i=1 ∩ Y = {0}. independent and [vi + αui ]N i=1 Proof. Let F = span {ui , vi : i = 1, . . . , N }. Let G = (Y + F )/Y . Denote xi = ui + Y ∈ G, zi = vi + Y ∈ G. Then the set {xi }N i=1 is linearly independent. Clearly, to establish the lemma it is enough to prove that the set {zi + αxi }N i=1 is linearly independent for all but finitely many α. Denote M = dim G. Let {bi }M i=1 be a basis of G such that bi = xi for all 1 i N . Denote the coordinates of vectors zi in this basis by zij . Let A be the M × M -matrix with first N rows consisting of the coordinates of zi (i = 1, . . . , N ), the last M –N rows being zero rows: ⎡ ⎤ z11 z12 · · · z1,M −1 z1M ⎢ . .. ⎥ ⎢ .. ··· . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ zN 1 zN 2 · · · zN,M −1 zN M ⎥ ⎢ ⎥ A=⎢ 0 ··· 0 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ .. ⎥ ⎢ .. ⎣ . ··· . ⎦ 0

0

···

0

0

Since the spectrum of A is finite, det(A + αI) = 0 for all but finitely many α. For these α, the rows of A + αI must be linearly independent. In particular, the first N rows are linearly independent. However the first N rows are exactly the representations of the vectors zi + αxi in the basis {bi }M i=1 . Lemma 2.6. Let Y ⊆ X be a linear subspace and T ∈ B(X). Let f1 , . . . , fn ∈ T Y be such that no non-trivial linear combination of {f1 , . . . , fn } belongs to Y . Then n dY,T .

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Proof. Let q : X → X/Y be the quotient map. Then qf1 , . . . , qfn are linearly independent. Since qf1 , . . . , qfn ∈ (qT )(Y ), we get n dim(qT )(Y ) = dY,T by Lemma 2.4. The following theorem is the main statement in this section. Recall that, according to our convention, the term “subspace” stands for a norm-closed subspace. Theorem 2.7. Let S be a subspace of L(X). Suppose that Y is a linear subspace of X that is almost invariant under S. Then sup dY,S < ∞.

S∈S

Proof. For every S ∈ S, fix a subspace FS ⊆ X such that SY ⊆ Y + FS and dim FS = dY,S . By Lemma 2.1, Y +FS is a direct sum. Fix PS : Y ⊕FS → FS dY,S dY,S the projection along Y . Also fix a basis (fiS )i=1 of FS and a tuple (giS )i=1 in Y such that (PS S)giS = fiS (this can be done by Lemma 2.1(iii)). Suppose that the statement of the theorem is not true. Then there exists a sequence of operators (Sk ) ⊆ S such that the sequence (dY,Sk )∞ k=1 is strictly increasing. Without loss of generality, Sk = 1. We will inductively construct a sequence (ak ) of scalars such that the following two conditions are satisfied for every m.

m (i) If Tm = k=1 ak Sk then Nm := dY,Tm dY,Sm . (ii) Let Cm =

sup

Nm bi giTm ·

b1 ,...,bNm ∈[−1,1] i=1

max (fiTm )∗ ,

i=1,...,Nm

∗ m where (fiTm )∗ is the ith biorthogonal functional for (fiTm )N i=1 in FTm , and

1 1 ,..., . Dm = min 1, C1 · PT1 Cm · PTm

Then 0 < a1

1 2

and 0 < am+1 <

1 2m+1 Dm

1 2.

for all m 1.

Indeed, on the first step put a1 = Suppose that a1 , . . . , am have been constructed. Define Dm as in (ii). Denote for convenience N = dY,Sm+1 . Let S S ui = fi m+1 and vi = Tm gi m+1 , i = 1, . . . , N . By Lemma 2.5 we can find 1 0 < α < 2m+1 Dm such that no non-trivial linear combination of vectors from the set {vi +αui }N i=1 is contained in Y . Put am+1 = α. This makes both conditions (i) and (ii) satisfied for m + 1. Indeed, condition (ii) is satisfied immediS ately. Let’s check condition (i). Denote for convenience yi = gi m+1 . Observe: for each i = 1, . . . , N , we have Tm+1 yi = Tm yi + αSm+1 yi = vi + αui + wi where wi is some vector in Y . Since no linear combination of {vi + αui }N i=1 is contained in Y , the same is true for {Tm+1 yi }N i=1 . Condition (i) now follows from Lemma 2.6.

∞ Denote S = k=1 ak Sk . By condition (ii), ak 21k for all k ∈ N, so

∞ that S is well-defined. For every m ∈ N, denote Rm = k=m+1 ak Sk , so that 1 for all m ∈ N. S = Tm + Rm . By condition (ii), we get: Rm < Cm ·P m

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Clearly, S ∈ S. By assumptions of the theorem, SY = Y ⊕ FS . Denote n = dim FS < ∞. Pick m ∈ N such that Nm > n and put zi = SgiTm , m i = 1, . . . , Nm . Since Nm > n, there exists a sequence (bi )N i=1 of scalars such that maxi |bi | = 1 and z := Consider y =

Nm

bi zi ∈ Y.

i=1

Nm

Tm i=1 bi gi .

We have

Tm y = Sy − Rm y = z − Rm y, hence (PTm Tm )y = −(PTm Rm )y. Clearly, for each i = 1, . . . , Nm , we have bi = (fiTm )∗ (PTm Tm y). Let k be such that |bk | = 1. Then 1 = |bk | = (f Tm )∗ (PT Tm y) (f Tm )∗ · PT Tm y m

k

k

m

Nm T ∗ T ∗ m m bi giTm = (fk ) · PTm Rm y (fk ) · PTm · Rm · i=1

PTm · Rm · Cm

1 Cm = 1 < PTm Cm · PTm

which is a contradiction.

3. Invariant Subspaces of Algebras Having Almost Invariant Half-spaces In this section we study some connections between the invariant subspaces of an algebra of operators and the almost invariant half-spaces of this algebra. In particular, we establish that if a norm-closed algebra generated by a single operator has an almost invariant half-space then it has an invariant half-space. Then we generalize this to commutative algebras generated by a finite number of operators. Also, we study the question when the almost invariant half-spaces of an algebra and of its WOT-closure are the same. It is well-known that the invariant subspaces of an algebra of operators coincide with those of the WOT-closure of this algebra. Remarkably, the same statement holds for almost invariant half-spaces, provided that the algebra is norm closed. Proposition 3.1. Let Y be a subspace of X and A an algebra of operators acting on X. Let N ∈ N be such that dY,T N for all T ∈ A. Then dY,T N for all T ∈ A

W OT

. SOT

Proof. It is enough to prove that if T ∈ A SOT

then dY,T N . Suppose this is

not true. Let T ∈ A be an operator with dY,T N +1. Let F ⊆ X be such that dim F = dY,T and T Y ⊆ Y ⊕ F . Fix N + 1 linearly independent vectors +1 N +1 (fi )N i=1 in F . By Lemma 2.1(iii), there exist (ui )i=1 ⊆ Y such that for each

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i = 1, . . . , N + 1 we have T ui = yi + fi for some yi ∈ Y . Since Y ∩ F = {0}, +1 [T ui ]N i=1 ∩ Y = {0} and T u1 , . . . , T uN +1 are linearly independent. SOT

Fix a net (Tα ) ⊆ A such that Tα → T . Let q : X → X/Y be the +1 quotient map. Since [T ui ]N i=1 ∩ Y = {0} and T u1 , . . . , T uN +1 are linearly +1 independent, the collection {(qT )ui }N i=1 is linearly independent. Observe +1 that if ε > 0 is sufficiently small then each collection {vi }N i=1 satisfying vi − (qT )ui < ε as i = 1, . . . , N + 1 is again linearly independent. Fix α0 such that for all α α0 we have (Tα − T )(ui ) < ε for all i = +1 1, . . . , N + 1. Then (qTα − qT )(ui ) < ε, so that the collection {(qTα )ui }N i=1 is linearly independent for all α α0 . By Lemma 2.4, this, however, implies that dY,Tα N + 1 for all α α0 which contradicts the assumptions. Corollary 3.2. Let A be a norm-closed algebra of operators on X and Y be W OT a half-space in X. Then Y is A-almost invariant if and only if Y is A almost invariant. Proof. If Y is A-almost invariant then by Theorem 2.7 there exists N ∈ N such that dY,S < N for all S ∈ A. By Proposition 3.1 the same is true for W OT

all S ∈ A . This implies that Y is A statement is evident.

W OT

-almost invariant. The converse

We will show later in this section that the condition of A being norm closed is essential here. The following lemma is standard: Lemma 3.3. Let X and Y be Banach spaces and T ∈ L(X, Y ) be of finite rank. Then dim(Range T ) = codim (ker T ). We will now introduce some notations. If Y and Z are two subspaces of X and Y ⊆ Z then the symbol codim Z Y will stand for the codimension of Y in Z. Let T ∈ L(X) be an operator and Y ⊆ X be a half-space. Consider two procedures of constructing new linear spaces: DT (Y ) = {y ∈ Y : T y ∈ Y } UT (Y ) = Y + T Y

“going downwards”, “going upwards”.

Clearly DT (Y ) ⊆ Y ⊆ UT (Y ). Lemma 3.4. Let Y ⊆ X be a half-space and T ∈ L(X). If Y is T almost invariant then both DT (Y ) and UT (Y ) are half-spaces. Moreover, codim Y DT (Y ) = codim UT (Y ) Y = dY,T . Proof. The statement about UT (Y ) follows immediately from the definition of an almost invariant subspace. Let’s verify the statement about DT (Y ). Clearly we only need to verify the “moreover” part. Let T Y ⊆ Y + F where F is such that dim F = dY,T . By Lemma 2.1(i), we have Y + F = Y ⊕ F . Let P : Y ⊕ F → F be the projection along Y . Then DT (Y ) = ker(P T |Y ). By Lemma 3.3 we get codim Y DT (Y ) = dim(Range P T |Y ) = dim F = dY,T .

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The following lemma is the key statement of this chapter. Lemma 3.5. Suppose that Y is a half-space in X that is almost invariant under an operator T ∈ L(X). If dY,T > 0 and dDTk (Y ),T dY,T and dUTk (Y ),T dY,T for all k ∈ N then dY,T m m for all m ∈ N. 0 Proof. Denote for convenience N = dY,T . Fix f10 , . . . , fN ∈ X such that 0 N T Y ⊆ Y ⊕ F where F = [fi ]i=1 . In particular, dim F = dY,T . Suppose that dDTk (Y ),T N and dUTk (Y ),T N for all k ∈ N. Denote Y0 = Y and Yk = DTk (Y ), k 1. Since Yk = DT (Yk−1 ) for all k 1, it follows that T Yk ⊆ Yk−1 as k 1. k We claim that for each k 1 there exists an N -tuple (f1k , . . . , fN ) in Yk−1 such that

(i) Yk ⊕ Fk = Yk−1 where Fk = [fik ]N i=1 , and (ii) if Pk : Yk ⊕ Fk → Fk is the projection along Yk then (Pk−1 T )fik = fik−1 for all i = 1, . . . , N (if k = 1 then we assume P0 : Y ⊕ F → F is the projection along Y ). Let k = 1. By Lemma 2.1(iii), for each i = 1, . . . , N , we can find fi1 ∈ Y such that (P0 T )fi1 = fi0 . Then (ii) is satisfied. Write F1 = [fi1 ]N i=1 . Since Y1 ∩F1 = {0} by definition of Y1 and dim F1 = N = codim Y Y1 by Lemma 3.4, Y1 ⊕ F1 = Y . Suppose the claim is true for k 0. Then Yk ⊕ Fk = Yk−1 . Since T Yk ⊆ Yk−1 and dYk ,T N = dim Fk , we get dYk ,T = N . Then from Lemma 2.1(iii) for each i = 1, . . . , N there exists fik+1 ∈ Yk such that (Pk T )fik+1 = fik , so that (ii) is satisfied for k+1. To show (i), write Fk+1 = [fik+1 ]N i=1 and observe: Yk+1 ∩ Fk+1 = {0} by definition of Yk+1 and dim Fk+1 = N = codim Yk Yk+1 by Lemma 3.4. Observe that from condition (ii) of this claim we have: for each k 1 there exists y ∈ Y such that T k fik = y + fi0 . That is, fik is a kth “preimage” of fi0 . It follows that any f ∈ F has a kth “preimage” in Yk−1 . Denote Z0 = Y , Zk = UTk (Y ), k 1. That is, Zk = UT (Zk−1 ). In particular, T Zk−1 ⊆ Zk for all k 0. We claim that Zk = Y ⊕ F ⊕ T F · · · ⊕ T k−1 F . Indeed, for k = 0 this is obvious. Suppose the claim is true for k 1. Let’s prove that Zk+1 = Y ⊕ F ⊕ T F · · · ⊕ T k F . We have Zk+1 = UT (Zk ) = Zk + T Zk = (Y ⊕ F ⊕ T F · · · ⊕ T k−1 F ) + (T Y + T F + T 2 F + · · · + T k F ) = (Y ⊕F ⊕T F · · ·⊕T k−1 F )+T k F since T Y ⊆ Y ⊕F . That is, Zk+1 = Zk +T k F . We only have to prove that this sum is direct. We have dim T k F N since dim F = N . On the other hand, T Zk ⊆ Zk+1 = Zk + T k F . Since dZk ,T N for all k 0, we get dim T k F = N = dZk ,T . By Lemma 2.1(i), the sum must be direct. Observe that in particular, this means that if f ∈ F is non-zero then T k f ∈ Zk+1 \Zk (k 0). Let u ∈ F be a non-zero vector, m ∈ N be arbitrary, and k ∈ {1, . . . , m}. Put uk to be the kth “preimage” of u, that is, such a vector in Yk−1 that T k uk = vk + u for some vk ∈ Y . Then T m uk = T m−k T k uk = T m−k vk + T m−k u. Since vk ∈ Y , it follows that T m−k vk ∈ Zm−k . Also since u = 0,

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we get T m−k u ∈ Zm−k+1 \Zm−k . Since Zm−k ⊆ Zm−k+1 we have T m uk ∈ Zm−k+1 \Zm−k . This means that T m Y contains m vectors {T m (uk )}m k=1 such that no non-zero linear combination of these vectors belongs to Y . By Lemma 2.6 we get: dY,T m m. As an immediate corollary we get: Theorem 3.6. Let T ∈ L(X) be an operator and A the norm-closed algebra generated by T . If A has an almost invariant half-space then A has an invariant half-space. Proof. If dY,T = 0 then there is nothing to prove. Let dY,T > 0. Since A is norm closed, supS∈A dY,S < ∞ by Theorem 2.7. In particular, supm∈N dY,T m < ∞. By Lemma 3.5 we obtain that either dDTk (Y ),T < dY,T or dUTk (Y ),T < dY,T for some k ∈ N. Applying this finitely many times we get a half-space Z such that dZ,T = 0. Since Z is T -invariant, it is A-invariant. This theorem allows us to get an earlier promised example of a (not necessarily closed) algebra A whose almost invariant half-spaces are different W OT (and even from those of the norm closure of A). This from those of A example also shows that, unlike in case of invariant subspaces, there exists an operator whose almost invariant half-spaces are different from those of the norm-closed algebra generated by this operator. Example. Let D be a Donoghue operator on 2 . That is, D is a backward ∞ shift with non-zero weights (wn )∞ n=1 which satisfy conditions (|wn |)n=1 is monotone decreasing and is in 2 (see, e.g., [3, Section 4.4] for the properties of Donoghue operators). Put A = {p(D) : p is a polynomial such that p(0) = 0}. It was proved in [2] that D has an almost invariant half-space. Then A has an almost invariant half-space. However all the invariant subspaces of D are finite dimensional (see [3, Theorem 4.12]) and therefore D has no invariant half-spaces. By Theorem 3.6, A

·

has no almost invariant half-spaces.

The following result is a generalization of Theorem 3.6. Theorem 3.7. Let A be a norm-closed algebra generated by a finite number of pairwise commuting operators. If A has an almost invariant half-space then A has an invariant half-space. Proof. Let A be generated by pairwise commuting operators T1 , . . . , Tn and let Y be an A-almost invariant half-space. We will prove that there exists a half-space that is invariant under Tk for each k = 1, . . . , n. First, observe that if T ∈ A then both DT (Y ) and UT (Y ) are A-almost invariant because codim Y DT (Y ) < ∞ and codim UT (Y ) Y < ∞ by Lemma 3.4. Next, we claim that if S ∈ A is such that Y is S-invariant then DT (Y ) and UT (Y ) are again S-invariant. Indeed, let y ∈ DT (Y ), then T (Sy) = ST y ∈ Y since T y ∈ Y and Y is S-invariant. Hence Sy ∈ DT (Y ), so that DT (Y ) is S-invariant. Let u + v ∈ UT (Y ), with u ∈ Y and v ∈ T y

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for some y ∈ Y . Then S(u + v) = Su + ST y = Su + T Sy ∈ UT (Y ) since Su, Sy ∈ Y , so that UT (Y ) is S-invariant. For k = 1, . . . , n, denote by Ak the norm-closed algebra generated by Tk . Clearly Ak ⊆ A and hence every A-almost invariant half-space is Ak -almost invariant for all k = 1, . . . , n. Apply a finite sequence of procedures DT1 and UT1 to Y to obtain a T1 -invariant half-space Y1 , as in the proof of Theorem 3.6. By the discussion above, Y1 is A-almost invariant. Apply a finite sequence of procedures DT2 and UT2 to Y1 to obtain a T2 -invariant half-space. Then Y2 is T1 - and T2 -invariant and still A-almost invariant. Repeat this procedure n − 2 more times to get an A-invariant half-space. Acknowledgements The author wants to express his deep gratitude to V. Troitsky for many useful discussions and suggestions and wishes to thank A. Tcaciuc for useful comments.

References [1] Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory, Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence, RI (2002) [2] Androulakis, G., Popov, A.I., Tcaciuc, A., Troitsky, V.G.: Almost invariant half-spaces of operators on Banach spaces. Integral Equ. Oper. Theory 65(4), 473–484 (2009) [3] Radjavi, H., Rosenthal, P.: Invariant Subspaces, 2nd edn. Dover Publications, Inc., Mineola, NY (2003) Alexey I. Popov Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB T6G 2G1, Canada e-mail: [email protected] Received: August 19, 2009. Revised: September 14, 2009.

Integr. Equ. Oper. Theory 67 (2010), 257–277 DOI 10.1007/s00020-010-1780-0 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Spectral Estimates and Basis Properties for Self-Adjoint Block Operator Matrices Michael Strauss Abstract. In the ﬁrst part of this manuscript a relationship between the spectrum of self-adjoint operator matrices and the spectra of their diagonal entries is found. This leads to enclosures for spectral points and in particular, enclosures for eigenvalues. We also consider graph invariant subspaces, and their corresponding angular operators. The existence of a bounded angular operator leads to basis properties of the ﬁrst component of eigenvectors of operator matrices for which the corresponding eigenvalues lie in a half line. The results are applied to an example from magnetohydrodynamics. Mathematics Subject Classification (2000). Primary 47B25; Secondary 34L15, 47A10, 47A11, 47A15. Keywords. Schur complement, eigenvalue estimates, graph invariant subspace, angular operator, Bari basis, magnetohydrodynamics.

1. Introduction Throughout, A is a self-adjoint operator acting on a Hilbert space H1 and is bounded from below, C is a self-adjoint operator acting on a Hilbert space H2 and is bounded from above, and B is a densely deﬁned closed operator from H2 to H1 . We shall be concerned with the spectral properties of block operator matrices of the form A B (1.1) : H1 × H2 → H1 × H2 M0 := B∗ C with domain Dom(M0 ) = (Dom(A) ∩ Dom(B ∗ )) × (Dom(B) ∩ Dom(C)). Operator matrices of this form appear in many applications, and the results obtained in this manuscript are applied to an example from magnetohydrodynamics. For a thorough account of this subject, the reader is referred to the monograph [16]. We will always assume the following condition is satisﬁed 1

Dom(|A| 2 ) ⊂ Dom(B ∗ ). This work was completed with the support of EPSRC grant no. EP/E037844/1.

(1.2)

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We denote by t the closure of the quadratic form of A which has domain 1 Dom(|A| 2 ). It follows from (1.2) and the assumption that A is bounded from below, that there exist constants a, b ≥ 0 such that B ∗ x2 ≤ at[x] + bx2

for all

1

x ∈ Dom(|A| 2 ).

(1.3)

We consider two classes of block operator matrices. If in addition to the condition (1.2), we have Dom(B) ⊂ Dom(C), and Dom(B) is a core for C, then the operator M0 is called top-dominant (or upper-dominant). A topdominant operator matrix is essentially self-adjoint, and the closure which we denote by M is given by x Dom(M) = : y ∈ Dom(C), x + (A − υI)−1 By ∈ Dom(A) (1.4) y x A(x + (A − υI)−1 By) − υ(A − υI)−1 By M = (1.5) y B ∗ x + Cy where υ < min σ(A) and is arbitrary (see [5, Section 4.2] and references 1 therein). If in addition to the condition (1.2), Dom(|C| 2 ) ⊂ Dom(B), then the operator M0 is called diagonally dominant. A diagonally dominant operator matrix is self-adjoint, and we therefore write M instead of M0 (see [8, Section 2]). Eigenvalues of operator matrices can often be characterised by variational principles (see for example [5, Theorem 2.1] and [8, Theorem 3.1]). Enclosures for eigenvalues can also be obtained in terms of the eigenvalues of A, the upper bound on the spectrum of C, and the constants a and b which satisfy (1.3) (see [8, Theorem 4.5]). In Sect. 2 we prove a relationship between σ(M) (the spectrum of M) and σ(A), σ(C), and the constants a and b. This leads to new enclosures for spectral points. Our approach is more general than the available variational principles; for example, we are not restricted to approximating only the discrete spectrum. In Sect. 3 we consider spectral subspaces L(α,∞) (M) for α ∈ R, that is, the range of the spectral projection associated to M and the interval (α, ∞). We are concerned with the existence of a so-called angular operator K : H1 → H2 with x : x ∈ Dom(K) . (1.6) L(α,∞) (M) = Kx If the representation (1.6) exists, the subspace L(α,∞) (M) is called a graph invariant subspace. Graph invariant subspaces and angular operators have been widely studied and the following list is by no means exhaustive [1,2,7, 9,11,13,15]. The case where A, B and C are bounded is considered in [9], where it is shown that if Δ is an interval in ρ(C) (the resolvent set of C) then the subspace LΔ (M) is graph invariant. In [1] the case where B is bounded and the diagonal entries are separated is considered. It is shown that for max σ(C) < α < min σ(A), the subspace L(α,∞) (M) admits the representation (1.6) with K ∈ B(H1 , H2 ). The top-dominant case with C ∈ B(H2 ) has been considered in [11]. The authors show that if there exists an α < min σ(A)

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such that C − αI − B ∗ (A − αI)−1 B 0, then L(α,∞) (M) admits the representation (1.6) with K ∈ B(H1 , H2 ). With a new approach, we prove the existence of graph invariant subspaces and bounded angular operators, our hypothesis is unrelated to those above, and is readily veriﬁed for a wide class of problems which includes any top-dominant or diagonally dominant matrix for which A has compact resolvent and C ∈ B(H2 ). If A has compact resolvent, then σ(M) ∩ (max σ(C), ∞) consists only of isolated eigenvalues of ﬁnite multiplicity. In Sect. 4 we consider basis properties of the ﬁrst component of eigenvectors corresponding eigenvalues which lie in the interval (max σ(C), ∞). From the existence of a bounded angular operator, one can deduce that the ﬁrst components of the corresponding eigenvectors form a Riesz basis for H1 (see [1, Theorem 3.5] and [11, Corollary 2.6]). Under our hypothesis we also obtain a Riesz basis, and even a Bari basis. However, our angular operator may not be everywhere deﬁned, in this case we ﬁnd the co-dimension of Dom(K) in terms of the so-called Schur complement. In the ﬁnal section we apply our results to a top-dominant operator matrix which arises in magnetohydrodynamics. 1.1. Operator Matrices and Schur Complements Associated with M is the ﬁrst Schur complement, which is a family of operators S(·) acting in H1 . When B is bounded, the ﬁrst Schur complement is given by S(λ) = A − λI − B(C − λI)−1 B ∗ ,

for λ ∈ ρ(C).

(1.7)

Evidently, S(λ) is deﬁned for all λ ∈ ρ(C), with Dom(S(λ)) = Dom(A), and S(λ) is self-adjoint for all λ ∈ ρ(C) ∩ R. The spectrum of S(λ) is given by σ(S) = {λ ∈ C : 0 ∈ σ(S(λ))}. It follows from the Schur factorisation that σ(S) ∩ ρ(C) = σ(M) ∩ ρ(C) (see [8, Section 4.1]). When B is unbounded, the operator (1.7) may not be densely deﬁned. However, the Schur complement can be deﬁned for any λ ∈ C with Re λ > max σ(C) using the following form 1 1 sλ (x, y) = (A−υI) 2 x, (A−υI) 2 y +(υ−λ) x, y − (C − λI)−1 B ∗ x, B ∗ y , 1

where Dom(sλ ) = Dom(|A| 2 ) and υ < min σ(A) is arbitrary. The Schur complement is then deﬁned using the ﬁrst representation theorem (for more details see [5, Section 4.2] and references therein). The spectra of the Schur complement and M coincide on {z ∈ C : Re z > max σ(C)}, however, if υ < min σ(A), Dom(A) ⊆ Dom(B ∗ ) and for every a > 0 there exists a b ≥ 0 such that B ∗ x ≤ a (A − υI)x, x + bx2

for all x ∈ Dom(A),

then the Schur complement can be deﬁned on ρ(C), and σ(S) ∩ ρ(C) = σ(M) ∩ ρ(C) (see [5, Section 4.2]). We will ﬁnd it useful to note that for any α ∈ ρ(M) for which S(α) is deﬁned, the restriction of (M − αI)−1 to H1 × Dom(F (α)) with F (α) = B(C − αI)−1 , is given by S(α)−1 −S(α)−1 F (α) . −(C − αI)−1 B ∗ S(α)−1 (C − αI)−1 + (C − αI)−1 B ∗ S(α)−1 F (α) (1.8)

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2. Resolvent Sets and Spectral Enclosures Let U be a subset of ρ(C) on which σ(S) and σ(M) coincide. If B is bounded, and λ ∈ σ(M)∩U , then for each ε > 0 there exists a normalised x ∈ Dom(A) such that Ax − λx − B(C − λI)−1 B ∗ x ≤ ε and therefore Ax − λx ≤ B(C − λI)−1 B ∗ x + ε B2 + ε. ≤ dist[λ, σ(C)] From the spectral theorem it follows that dist[λ, σ(A)] ≤

B2 . dist[λ, σ(C)]

The theorem below generalises this statement to the case where B is unbounded. We note that when B is bounded (1.3) holds with the constants a = 0 and b = B2 . Theorem 2.1. Let M be a top-dominant or diagonally dominant operator matrix and let a, b ≥ 0 satisfy (1.3). If λ ∈ σ(M) ∩ U and dist[λ, σ(C)] > a, then |aλ + b| dist[λ, σ(A)] ≤ . (2.1) dist[λ, σ(C)] − a Proof. Since λ ∈ / σ(C) and λ ∈ U , it follows that S(λ) is well deﬁned and 0 ∈ σ(S(λ)), that is, S(λ) does not have a bounded inverse. If λ ∈ σ(A) then the assertion is obvious. Suppose that λ ∈ / σ(A) and that (2.1) is false. With t the quadratic form associated to A, and using (1.3), we have for all 1 x ∈ Dom(|A| 2 ) | (C − λI)−1 B ∗ x, B ∗ x | ≤

at[x] b x, x B ∗ x2 ≤ + dist[λ, σ(C)] dist[λ, σ(C)] dist[λ, σ(C)] at[x] − aλ x, x aλ x, x + b x, x + = dist[λ, σ(C)] dist[λ, σ(C)] 1 1 a |A − λI| 2 x, |A − λI| 2 x ≤ dist[λ, σ(C)] 1 1 |aλ + b| |A − λI| 2 x, |A − λI| 2 x . + dist[λ, σ(A)]dist[λ, σ(C)]

That is,

1 1 | (C − λI)−1 B ∗ x, B ∗ x | ≤ δ |A − λI| 2 x, |A − λI| 2 x ,

where δ=

a |aλ + b| + . dist[λ, σ(C)] dist[λ, σ(A)]dist[λ, σ(C)]

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Since (2.1) is assumed to be false, we have δ < 1. By [6, Lemma VI.3.1] there exists an operator G ∈ B(H1 ) with G ≤ δ, such that for all x, y ∈ 1 Dom(|A| 2 ) we have 1 1 − (C − λI)−1 B ∗ x, B ∗ y = G|A − λI| 2 x, |A − λI| 2 y . Consider the operator T deﬁned as follows, 1

1 T = |A − λI| 2 (A − λI)|A − λI|−1 + G |A − λI| 2 1

1 = |A − λI| 2 I + G|A − λI|(A − λI)−1 (A − λI)|A − λI|−1 |A − λI| 2 , 1

note that Dom(T ) ⊆ Dom(|A| 2 ). Since G|A − λI|(A − λI)−1 ≤ δ|A − λI|(A − λI)−1 < 1, we deduce that (I + G|A − λI|(A − λI)−1 )−1 exists and is bounded. It follows that T −1 exists and is bounded, that is,

−1 1 1 T −1 = |A−λI|− 2 |A−λI|(A−λI)−1 I + G|A−λI|(A−λI)−1 |A−λI|− 2 . The Schur complement S(λ) is the unique operator which satisﬁes 1 Dom(S(λ)) ⊆ Dom(|A| 2 ) and S(λ)x, y = sλ (x, y) for all x ∈ Dom(S(λ)) 1 1 and y ∈ Dom(|A| 2 ). Also, Dom(T ) ⊆ Dom(|A| 2 ), and for all x ∈ Dom(T ) 1 and y ∈ Dom(|A| 2 ), we have 1 1 T x, y = |A − λI| 2 ((A − λI)|A − λI|−1 + G)|A − λI| 2 x, y 1 1 = ((A − λI)|A − λI|−1 + G)|A − λI| 2 x, |A − λI| 2 y 1 1 = (A − λI)|A − λI|−1 |A − λI| 2 x, |A − λI| 2 y 1 1 + G|A − λI| 2 x, |A − λI| 2 y 1 1 = (A − υI)|A − λI|−1 |A − λI| 2 x, |A − λI| 2 y 1 1 + G|A − λI| 2 x, |A − λI| 2 y 1 1 +(υ − λ) |A − λI|−1 |A − λI| 2 x, |A − λI| 2 y 1 1 = (A − υI) 2 x, (A − υI) 2 y + (υ − λ) x, y − (C − λI)−1 B ∗ x, B ∗ y = sλ (x, y), from which we deduce that T = S(λ), and therefore S(λ)−1 exists and is bounded. However, λ ∈ σ(M) ∩ U and therefore λ ∈ σ(S), the result follows from the contradiction. Corollary 2.2. Let c = max σ(C) and λ ∈ σ(M) ∩ (c + a, ∞), then dist[λ, σ(A)] ≤

aλ + b . λ−c−a

(2.2)

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If λ ∈ [μ, μ + r] where (μ, μ + 2r) ⊂ ρ(A) and μ ∈ σ(A), then λ ∈ [α− , α+ ] where 2 μ−c μ + c + 2a ± + a(a + c) + b. (2.3) α± = 2 2 If λ ∈ (μ − r, μ] where (μ − 2r, μ) ⊂ ρ(A), μ ∈ σ(A), and (μ − c)2 > 4aμ + 4b then λ ∈ / (β − , β + ) where 2 μ−c μ + c ± ± − (aμ + b). (2.4) β = 2 2 Proof. Note that aλ + b ≥ 0 whenever λ ≥ min σ(A), this follows from (1.3). If c < min σ(A) then (c, min σ(A)) ⊂ ρ(M) since sλ is a strictly positive form for λ ∈ (c, min σ(A)). Therefore (2.2) follows from (2.1). We now prove the second assertion. Using (2.2) we obtain aλ + b λ − μ = dist[λ, σ(A)] ≤ λ−c−a and therefore λ2 − (μ + c + 2a)λ + (c + a)μ − b ≤ 0. ±

−

(2.5)

+

From (2.5) we deduce that α ∈ R and λ ∈ [α , α ]. For the ﬁnal assertion, we note that dist[λ, σ(A)] = μ − λ, and obtain λ2 − (μ + c)λ + (c + a)μ + b ≥ 0, from which λ ∈ / (β − , β + ) follows.

Although c = max σ(C) may not be below min σess (M), the eigenvalues of M which are greater than c, but less than λe = min(σess (M)∩(c, ∞)), can often be approximated using a variational principle, which we now describe. For γ ∈ (c, ∞), let L(−∞,0) (S(γ)) be the spectral subspace of S(γ) corresponding to the interval (−∞, 0). We set κ(γ) = dim(L(−∞,0) (S(γ))) and assume there exists a γ ∈ (c, ∞) such that κ(γ) < ∞, then there exists an α such that (c, α) ⊂ ρ(M) (see [8, Theorem 3.1]). Set κ = κ(α) and let λ1 ≤ λ2 ≤ · · · ≤ λN , N ∈ N0 ∪ {∞}, be eigenvalues of M in the interval (c, λe ). Let μ1 ≤ μ2 ≤ · · · ≤ μM , M ∈ N0 ∪ {∞}, be the eigenvalues of A below σess (A), and set μk = min σess (A) for k > M . For n = 1, . . . , N , we have 2 μκ+n − c μκ+n + c + + aμκ+n + b (2.6) μκ+n ≤ λn ≤ 2 2 (see [8, Corollary 4.1 and Theorem 4.2]). The estimate (2.6) is particularly useful when A has compact resolvent. In this case, σess (M) ∩ (c, ∞) = ∅, σ(M) ∩ (c, ∞) consists of a sequence of eigenvalues λn → ∞, and κ(γ) < ∞ for all γ ∈ (c, ∞) (see [5, Theorem 4.5]). Corollary 2.2 is unlikely to oﬀer improvements to the estimate (2.6), however, we can obtain information about the resolvent set in the region (λe , ∞) as the corollary below shows. Also, in the theorem below, we are able to obtain spectral enclosures for spectral points in the interval (c, ∞), moreover, we are not restricted to

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approximating the discrete spectrum and we do not require the ﬁniteness of κ(γ). Corollary 2.3. Let a + c < μ1 , μ2 ∈ σ(A) and (μ1 , μ2 ) ⊂ ρ(A). If β2− < (μ1 + μ2 )/2 and α1+ < β2+ where

μ1 + c + 2a μ1 − c 2 ± ( ) + a(a + c) + b, α1± = 2 2

μ2 + c μ2 − c 2 ± ( ) − (aμ2 + b), β2± = 2 2 then μ1 ≤ α1+ < β2+ ≤ μ2 and (α1+ , β2+ ) ⊂ ρ(M). Proof. If μ1 > α1+ we have

2 μ1 − c μ1 − c −a>+ + a(a + c) + b, 2 2

then squaring both sides implies that aμ + b < 0 which contradicts (1.3). That β2+ ≤ μ2 follows immediately from the definition of β2+ . Let λ ∈ σ(M). If λ ∈ (μ1 , (μ1 +μ2 )/2], then λ < α1+ by Corollary 2.2. If λ ∈ ((μ1 +μ2 )/2, μ2 ) / (β2− , β2+ ). Suppose that then dist[λ, σ(A)] = μ2 − λ, and by Corollary 2.2 λ ∈ − λ ≤ β2 , then |μ1 − λ| < μ2 − λ and therefore dist[λ, σ(A)] < μ2 − λ; from the contradiction we deduce that λ ∈ (β2+ , μ2 ). Theorem 2.4. Let μ1 , μ2 , μ3 , μ4 ∈ σ(A), μ1 < μ2 ≤ μ3 < μ4 , and the pairs {μ1 , μ2 }, {μ3 , μ4 } satisfy the hypothesis of Corollary 2.3. With 2 μ2 − c μ2 + c + β2+ = − (aμ2 + b), 2 2 2 μ3 − c + c + 2a μ 3 + + + a(a + c) + b, α3 = 2 2 we have [β2+ , α3+ ] ∩ σ(M) = ∅, moreover, dim(L[β + ,α+ ] (M)) = dim(L[β + ,α+ ] 2 3 2 3 (A)). Proof. We consider the following family of operators A tB M0 (t) := for t ∈ [0, 1]. tB ∗ C

(2.7)

Denote the corresponding self-adjoint closures by M(t). Note that the matrix M0 (0) is diagonally dominant and therefore M(0) = M0 (0), moreover, we have dim(L[β + ,α+ ] (M(0))) = dim(L[β + ,α+ ] (A)). We now show that M(t) is 2

3

2

3

a holomorphic family. Let l = (α1+ + β2+ )/2 and r = (α3+ + β4+ )/2, then l, r ∈ ρ(A). For any t ∈ [0, 1] we have tB ∗ x2 ≤ B ∗ x2 ≤ at[x] + bx2

1

for all x ∈ Dom(|A| 2 ),

it therefore follows from Corollary 2.3 that l, r ∈ ρ(M(t)) for every t ∈ [0, 1]. We denote by S(λ, t) the Schur complement corresponding to the matrix M(t), that is, the operator corresponding to the form

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1 1 sλ,t (x, y) = (A−υI) 2 x, (A−υI) 2 y +(υ−λ) x, y −t2 (C−λI)−1 B ∗ x, B ∗ y , and the ﬁrst representation theorem. S(λ, 1) is a holomorphic family of type (B) with respect λ ∈ {z ∈ C : Re z > c} (see [6, Theorem VII.4.2]). Similarly, S(λ, t) is a holomorphic family of type (B) with respect to both variables λ ∈ {z ∈ C : Re z > c} and t ∈ R. Let (x, y)T , (u, v)T ∈ H1 ×Dom(B(C −αI)−1 ), then using (1.8) we have u x −1 (M(t) − lI) , y v −1 = S(l, t) x, u − t S(l, t)−1 B(C − lI)−1 y, u − t (C − lI)−1 B ∗ S(l, t)−1 x, v + (C − lI)−1 y, v + t2 (C − lI)−1 B ∗ S(l, t)−1 B(C − lI)−1 y, v . It follows from the definitions of top-dominant and diagonally dominant operators, that H1 × Dom(B(C − αI)−1 ) is dense in H1 × H2 . That M(t) is holomorphic on [0, 1] follows from [6, Theorem III.3.12 and Theorem VII.1.3] and the fact that S(l, t) is holomorphic on t ∈ R. Let Et ([β2+ , α3+ ]) be the spectral measure corresponding to the operator M(t) and the interval [β2+ , α3+ ]. For any t1 , t2 ∈ [0, 1] we have 1 (M(t1 ) − ζ)−1 − (M(t2 ) − ζ)−1 dζ Et1 − Et2 = − 2πi Γ

where Γ is a circle which passes through l and r, from which the result follows. To obtain accurate approximation of the spectrum, particularly of the discrete spectrum, a numerical method is likely to be necessary. In the region (λe , ∞) the traditional Galerkin (ﬁnite section) method is unreliable (see for example [10, Theorem 2.1]). There are several numerical methods available for locating spectral points in this region, for example, the second order relative spectrum and the method of Davies and Plum (see [4,10], respectively). Both of these methods can beneﬁt significantly from a priori information about the resolvent set and the dimension of spectral subspaces. The second order relative spectrum requires solving quadratic eigenvalue problems of the form P (M − z)2 |L u = 0 where P is the orthogonal projection onto a ﬁnite-dimensional subspace L (see [10]). To demonstrate, we suppose the hypothesis of Theorem 2.4 holds and that in addition dim(L[β + ,α+ ] (A)) = 1, 2 3 and therefore dim(L[β + ,α+ ] (M)) = 1. If we have a solution z to the quadratic 2 3 eigenvalue P (M−z)2 |L u = 0 for some subspace L, and such that z lies inside the open disc with centre (α1+ + β4+ )/2 and radius (β4− − α1+ )/2, then by [14, Theorem 2.2 and Remark 2.3] we obtain |Im z|2 |Im z|2 , Re z + = σ(M) ∩ [β2+ , α3+ ]. σ(M) ∩ Re z − + β4 − Re z Re z − α1+

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3. Graph Invariant Subspaces With E the spectral measure corresponding to M and α ∈ R, let L(α,∞) (M) := E((α, ∞))(H1 × H2 ). This section is concerned with the existence of angular operators corresponding to subspaces L(α,∞) (M), that is, operators Kα : H1 → H2 such that x (3.1) : x ∈ Dom(Kα ) . L(α,∞) (M) = Kα x Theorem 3.1. Let M be a top-dominant or diagonally dominant operator matrix and let C ∈ B(H2 ). If α ∈ R satisfies max σ(C) = c < α ∈ (ρ(M) ∩ ρ(A)) and δ=

|aα + b| 1 a + < , α − c dist[α, σ(A)](α − c) 2

(3.2)

then there exists a closed bounded operator Kα which satisfies (3.1). Proof. Similarly to the proof of Theorem 2.1 we have an operator G ∈ B(H1 ) such that S(α)−1 admits the following representation, −1 1

1 |A−αI|− 2 where G ≤ δ. S(α)−1 = |A−αI|− 2 (A−αI)|A−αI|−1 +G Note that (A − αI)|A − αI|−1 + G ∈ B(H1 ) and for all x ∈ H1 we have (A−αI)|A−αI|−1 x+Gx ≥ (A−αI)|A−αI|−1 x−Gx ≥ (1−δ)x, from which we obtain −1 1

1 S(α)−1 x, x = |A − αI|− 2 (A − αI)|A − αI|−1 + G |A − αI|− 2 x, x

−1 1 1 |A − αI|− 2 x, |A − αI|− 2 x = (A − αI)|A − αI|−1 + G −1

1 1 |A − αI|− 2 x|A − αI|− 2 x ≤ (A − αI)|A − αI|−1 + G −1

1 |A − αI|− 2 x2 ≤ (A − αI)|A − αI|−1 + G 1

≤ (1 − δ)−1 |A − αI|− 2 x2 . Let x ∈ H1 and y ∈ Dom(B(C − αI)−1 ). Using (1.8) we have x x (M − αI)−1 , y y −1 = S(α) x, x − S(α)−1 B(C − αI)−1 y, x − (C − αI)−1 B ∗ S(α)−1 x, y + (C − αI)−1 y, y + (C − αI)−1 B ∗ S(α)−1 B(C − αI)−1 y, y = S(α)−1 x, x − y, (C − αI)−1 B ∗ S(α)−1 x − (C − αI)−1 B ∗ S(α)−1 x, y + (C − αI)−1 y, y + S(α)−1 B(C − αI)−1 y, B(C − αI)−1 y , and for the last term on the right-hand side we have the upper bound 1 S(α)−1 B(C −αI)−1 y, B(C −αI)−1 y ≤ (1−δ)−1 |A−αI|− 2 B(C−αI)−1 y2 .

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1

The operator B ∗ |A − αI|− 2 is bounded by the closed graph theorem, and 1 therefore also the adjoint operator is bounded, in particular |A − αI|− 2 B is bounded. Similarly to the proof of Theorem 2.1 we have for any x ∈ H1 1 1 1 1 B ∗ |A − αI|− 2 x2 ≤ at[|A − αI|− 2 x] + b |A − αI|− 2 x, |A − αI|− 2 x 1

1

≤ at[|A − αI|− 2 x] − aα|A − αI|− 2 x2 + ≤ a x, x +

|aα+b| x, x dist[α, σ(A)]

|aα + b| x, x dist[α, σ(A)]

= δdist[α, σ(C)]x2 . 1

Therefore |A − αI|− 2 B2 ≤ δdist[α, σ(C)]. With E the spectral measure corresponding to C, we obtain x x −1 (M − αI) , y y −1 ≤ S(α) x, x − 2Re y, (C − αI)−1 B ∗ S(α)−1 x 1 1 |A − αI|− 2 B(C − αI)−1 y2 + (C − αI)−1 y, y + 1−δ ≤ S(α)−1 x, x − 2Re y, (C − αI)−1 B ∗ S(α)−1 x δdist[α, σ(C)] (C − αI)−1 y2 + (C − αI)−1 y, y + 1−δ = S(α)−1 x, x − 2Re y, (C − αI)−1 B ∗ S(α)−1 x δdist[α, σ(C)] 1 d Eμ y, y − (αI − C)−1 y, y + 1−δ (α − μ)2 R −1 −1 ∗ ≤ S(α) x, x − 2Re y, (C − αI) B S(α)−1 x 1 δ d Eμ y, y − (αI − C)−1 y, y + 1−δ (α − μ) R = S(α)−1 x, x − 2Re y, (C − αI)−1 B ∗ S(α)−1 x δ − 1 (αI − C)−1 y, y . + 1−δ We see that if x = 0 and y = 0, then the right-hand side is negative. Since Dom(B(C − αI)−1 ) is dense in H2 we deduce that L(α,∞) (M) is the graph of an operator, denote this operator by Kα . The subspace L(α,∞) (M) is closed, thus Kα is a closed operator. We suppose that Kα is unbounded. Therefore, there is a sequence xn → 0 with xn ∈ Dom(Kα ) and Kα xn → 1. Let x ∈ H1 , y ∈ Dom(B(C−αI)−1 ), and note that by the closed graph theorem (C − αI)−1 B ∗ S(α)−1 is bounded, then from the above inequality we have

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x x −1 (M − αI) , y y ≤ S(α)−1 x2 + 2(C − αI)−1 B ∗ S(α)−1 xy δ + − 1 (αI − C)−1 y, y . 1−δ Since Dom(B(C − αI)−1 ) is dense in H2 , for any y ∈ H2 there is a sequence yn → y with yn ∈ Dom(B(C − αI)−1 ). Evidently, the above inequality holds for all x ∈ H1 and for all y ∈ H2 . We obtain xn xn (M − αI)−1 , < 0 for all suﬃciently large n ∈ N. Kα xn Kα xn However, (xn , Kα xn )T ∈ L(α,∞) (M) for all n ∈ N and therefore xn xn −1 (M − αI) , ≥ 0 for all n ∈ N. Kα xn Kα xn From the contradiction we deduce that Kα is bounded.

Remark 3.2. If {μ1 , μ2 } is a pair which satisﬁes Corollary 2.3 then (α1+ + β2+ )/2 ∈ (ρ(M) ∩ ρ(A)) and a possible choice for α in (3.2) would be (α1+ + β2+ )/2. Also, if {μn , μn+1 } is a sequence of such pairs and |μj+1 − μj | → ∞, then aαj + b a + → 0. αj − c dist[αj , σ(A)](αj − c) For any normalised (x, y)T ∈ Dom(M), it follows from (1.4) and the 1 1 observation (A − υI)− 2 (A − υI)− 2 B = (A − υI)−1 B, that (A − υI)−1 By ∈ 1 1 1 Dom(|A| 2 ) and x ∈ Dom(|A| 2 ). Noting also that the adjoint of (A − υI)− 2 B 1 is given by B ∗ (A − υI)− 2 , and using (1.5) and (1.3), we have x x M , y y = (A − υ)(x + (A − υI)−1 By), x + υ x, x + B ∗ x, y + Cy, y 1 1 = (A − υ) 2 x, (A − υ) 2 x + υ x, x + B ∗ x, y + Cy, y 1 1 + (A − υ) 2 (A − υI)−1 By, (A − υ) 2 x 1 1 = t[x] + B ∗ x, y + Cy, y + (A − υI)− 2 By, (A − υ) 2 x = t[x] + B ∗ x, y + y, B ∗ x + Cy, y ≥ t[x] − 2B ∗ x + Cy, y ≥ t[x] − 2 at[x] + bx2 + Cy, y . If C ∈ B(H2 ) it follows that M is bounded from below and therefore form closable, the corresponding closed form we denote by a. Similarly, we see that

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Dom(a) = Dom(|A| 2 ) × H2 , and for any (x, y)T ∈ Dom(a) we have x a = t[x] + B ∗ x, y + y, B ∗ x + Cy, y . y

(3.3)

Corollary 3.3. Let M be a top-dominant or diagonally dominant operator matrix and let C ∈ B(H2 ). Suppose that for some c < α ∈ ρ(M) there exists a closed bounded operator Kα which satisfies (3.1). If dim(L(−∞,0) (S(γ))) < ∞ for some γ ∈ (c, ∞), and (c, α) ∩ σess (S) = ∅, then there exists a c˜ ∈ R with (c, c˜] ⊂ ρ(M), and a closed bounded operator Kc which satisfies (3.1), morec))) < ∞. over, codim(Dom(Kc )) = κ := dim(L(−∞,0) (S(˜ Proof. Let λ1 < · · · < λn be the eigenvalues of M which lie in the interval (c, α). Let x1 , . . . , xm be an orthonormal basis of Ker(S(λn )), and suppose ˜ ∈ Dom(Kα )\{0}. Set L = L(−∞,0] (S(λn )). that α1 x1 + · · · + αm xm = x 1 Note that L is a ﬁnite-dimensional subspace of Dom(|A| 2 ) and therefore ˜)T ∈ Dom(a). Using [8, Lemma 3.5], [5, proof of Theorem 2.1] and (˜ x, Kα x (3.3) we have 2 x ˜ x ˜ α
however, α > λn , and from the contradiction we deduce that L[λn ,∞) (M) is also the graph of an operator. This operator is a ﬁnite-dimensional extension of Kα , and therefore is also closed and bounded. We repeat this argument for each of the eigenvalues λ1 , . . . , λn which lie in the interval (c, α) and deduce that Kα has a ﬁnite-dimensional extension to a closed bounded operator Kc such that (3.1) holds. It follows that Dom(Kc ) is closed, and therefore codim(Dom(Kc )) is equal to dim(Dom(Kc )⊥ ) (see [6, Lemma III.1.40]). For proof of the existence of a c˜ with the stated properties see [8, Theorem c)) and for any 3.1]. Using [8, Lemma 3.4] we have for any x ∈ L(−∞,0) (S(˜ y ∈ H2 \{0} 2 x x ∗ ∗ a = t[x] + B x, y + y, B x + Cy, y ≤ c˜ y . (3.4) y c))), then there exists an x ∈ L(−∞,0) If codim(Dom(Kc )) < dim(L(−∞,0) (S(˜ (S(˜ c)) such that x ⊥ Dom(Kc )⊥ , that is, x ∈ Dom(Kc ). Therefore (x, Kc x)T ∈ Dom(a), (x, Kc x)T ∈ L(c,∞) (M), and 2 x c˜ →a Kc x yn Kc x

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contradicting (3.4), from which we deduce that codim(Dom(Kc )) ≥ κ. Let x ⊥ Dom(Kc ), then (x, 0)T ⊥ L(c,∞) (M), and using (1.8) x x 0 > (M − c˜I)−1 , = S(˜ c)−1 x, x , 0 0 from which we deduce that codim(Dom(Kc )) ≤ κ.

Theorem 3.4. Let M be a top-dominant or diagonally dominant operator matrix with A unbounded and C ∈ B(H2 ). If there exists an α ∈ (c, ∞)∩ρ(M) such that a/(α−c) < 1/2, then for sufficiently large β > c there exists a closed bounded operator Kβ which satisfies (3.1). Proof. Choose μ ∈ σ(A) ∩ (α, ∞), such that a |aα + b| 1 + < . α − c (μ − α)(α − c) 2

(3.5)

Let E be the spectral measure associated to A. We consider the operator matrix A + tE((−∞, μ)) B ˜ M0 := where t = μ − min σ(A). (3.6) B∗ C ˜ be the self-adjoint closure of (3.6). We have A + tE((−∞, μ)) ≥ μI > Let M ˜ is strictly αI > cI. It follows that the Schur complement associated to M ˜ Also note that min σ(M) ≤ positive on (c, μ), therefore (c, α] ⊂ ρ(M). ˜ and min σ(M), E((−∞, μ)) 0 ˜ M = M + tP, where P = and P = 1. 0 0 Denote by ˜t the form associated to the operator A + tE((−∞, μ)), then it is 1 1 clear that Dom(˜t) = Dom(t) = Dom(|A| 2 ). Moreover, for all x ∈ Dom(|A| 2 ) we have B ∗ x2 ≤ at[x] + bx2 ≤ a˜t[x] + bx2 , that is, the constants a, b ∈ R which satisfy (1.3), also satisfy (1.3) when we replace A with A + tE((−∞, μ)). We have A + tE((−∞, μ)) ≥ μI, therefore dist[α, σ(A + tE((−∞, μ)))] ≥ μ − α, and we obtain |aα + b| a |aα + b| 1 a + ≤ + < . α − c dist[α, σ(A + tE((−∞, μ)))](α − c) α − c (μ − α)(α − c) 2 Using Theorem 3.1 we deduce that there exists a closed bounded operator ˜ α which satisﬁes (3.1) for the subspace L(α,∞) (M). ˜ Since (c, α] ⊂ ρ(M), ˜ we K ˜ = L(α,∞) (M), ˜ that is, we have a closed bounded operator have L(c,∞) (M) ˜ ˜ Kc which satisﬁes (3.1) for the subspace L(c,∞) (M). ˜ respecLet E and E˜ be the spectral measures associated to M and M, tively. Let β > α, and Γ be a circle which passes through α and any α ˜ < min σ(M). For any normalised φ ∈ L(β,∞) (M) we have

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˜ ˜ E((−∞, α))φ, E((−∞, α))φ ˜ = E((−∞, α))φ, φ ˜ = [E((−∞, α)) − E((−∞, α))]φ, φ 1 ˜ − ζI)−1 − (M − ζI)−1 ]φ, φ dζ. =− [(M 2iπ

˜ E((−∞, α))φ2 =

Γ

˜ − ζI)−1 − (M − ζ)−1 = (M − ζI)−1 tP(M ˜ − ζI)−1 we have Since (M 1 ˜ − ζI)−1 φ, φ dζ ˜ [(M − ζI)−1 tP(M E(−∞, α))φ2 = − 2iπ Γ 1 ˜ − ζI)−1 φ, φ | dζ | [(M − ζI)−1 tP(M ≤ 2π Γ 1 ˜ − ζI)−1 φ, (M − ζI)−1 φ | dζ | tP(M = 2π Γ

t(α − α ˜) ˜ − zI)−1 max (M − wI)−1 φ. ≤ max (M z∈Γ w∈Γ 2 ˜ − zI)−1 , and note that Let M = maxz∈Γ (M ⎛ max (M − wI)−1 φ = max ⎝ w∈Γ

w∈Γ

⎛ ⎜ = max ⎝

R

∞

w∈Γ

≤

β

⎞ 12 1 d Eλ φ, φ ⎠ |λ − w|2 ⎞ 12 1 ⎟ d Eλ φ, φ ⎠ |λ − w|2

1 . β−α

We obtain

˜ E(−∞, α))φ ≤

t(α − α ˜ )M . 2(β − α)

Let 0 < ε < 1. For sufﬁciently large β ∈ R and any φ ∈ L(β,∞) (M) with ˜ φ = 1, we have E(−∞, α))φ ≤ ε. Suppose that φ is of the form 0 φ= and therefore y = 1. y We have

˜ E((α, ∞))φ =

x ˜ cx K

for some x ∈ H1 ,

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0 x ˜ y − K ˜ c x = [I − E((α, ∞))]φ ˜ = E((−∞, α))φ ≤ ε,

˜ c x = y−K ˜ c x ≤ y− K ˜ c x ≤ ε, from which we obtain x ≤ ε and 1−K ˜ therefore Kc ≥ (1 − ε)/ε. However, we may choose ε > 0 to be arbitrarily small. We deduce that for sufﬁciently large β ∈ R there exists a closed operator Kβ which satisﬁes (3.1) for the subspace L(β,∞) (M). A similar argument shows that Kβ is bounded. If dim(L(−∞,0) (S(γ))) < ∞ for some γ ∈ (c, ∞), and we can choose β in Theorem 3.4 with (c, β) ∩ σess (M) = ∅, then by Corollary 3.3 we can extend the operator Kβ to an operator Kc . In particular, we have the following corollary. Corollary 3.5. Let M be a top-dominant or diagonally dominant operator matrix where A has compact resolvent and C ∈ B(H2 ). There exists a c˜ ∈ R with (c, c˜] ⊂ ρ(M), and a closed bounded operator Kc which satisfies (3.1), c))) < ∞. moreover, codim(Dom(Kc )) = κ := dim(L(−∞,0) (S(˜ Proof. For a diagonally dominant or top-dominant matrix with C ∈ B(H2 ) and A having compact resolvent, we have σess (M) ∩ (c, ∞) = ∅ (see [5, Theorem 4.5]). The interval (c, ∞) contains a sequence of eigenvalues of ﬁnite multiplicity which accumulate at inﬁnity. The result follows from Theorem 3.4 and Corollary 3.3. The top-dominant case with C ∈ B(H2 ) has also been considered in [11]. We now demonstrate that our hypothesis includes operator matrices which cannot satisfy the hypothesis considered in [11]. The authors of [11] show that if there exists an α < min σ(A) such that C − αI − B ∗ (A − αI)−1 B 0, then L(α,∞) (M) is a graph invariant subspace, and the corresponding angular operator belongs to B(H1 , H2 ) (see [11, Theorem 2.5]). We suppose that the hypothesis of Corollary 3.5 is satisﬁed with κ = 0, and that min σ(A) < max σess (M). Suppose also that the hypothesis of [11, Theorem 2.5] is satisﬁed, that is, there exists an α < min σ(A) with C −αI −B ∗ (A − αI)−1 B 0. By [11, Theorem 2.5] this implies the existence of an operator Kα satisfying (3.1). Note that L(α,∞) (M) = L(α,c] (M) ⊕ L(c,∞) (M). The operator Kc implied by Corollary 3.5 has codim(Dom(Kc )) = κ < ∞, and is a restriction of the operator Kα . Since α < min σ(A) < max σess (M) ≤ c, we have dim(L(α,c] (M)) = ∞. Let x1 xκ+1 ,..., ∈ L(α,c] (M), Kα x1 Kα xκ+1 and

xi Kα xi

⊥

xj Kα xj

for i = j.

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The {x1 , . . . , xκ+1 } are a linearly independent set, but since codim(Dom(Kc )) = κ we have 0 = x = a1 x1 + · · · + aκ+1 xκ+1 ∈ Dom(Kc ) for some choice of a1 , . . . , aκ+1 , and therefore x x x ∈ L(α,c] (M) and = ∈ L(c,∞) (M), Kα x Kα x Kc x a contradiction.

4. Basis Properties In this section we derive basis properties for the ﬁrst components of the eigenvectors of M for which the corresponding eigenvalues lie in the interval (c, ∞). Corollary 4.1. Let M be a top-dominant or diagonally dominant operator matrix and let C ∈ B(H2 ). If A has compact resolvent, and {(xn , Kc xn )T }∞ n=1 are orthonormal eigenvectors corresponding to the eigenvalues of M which are greater than c, then the set {xn }∞ n=1 forms a Riesz basis for Dom(Kc ). Proof. The {(xn , Kc xn )T }∞ n=1 form an orthonormal basis for L(c,∞) (M), and for any x ∈ Dom(Kc ) we therefore have xn x x xn = βn , where βn = . Kc xn Kc xn Kc x Kc x Therefore x = βn xn , and the sequence {βn }∞ n=1 is unique because the T vectors (xn , Kc xn ) are a basis for L(c,∞) (M). We deduce that the {xn }∞ n=1 form a basis for Dom(Kc ). It remains to show that the {xn }∞ n=1 form a Riesz sequence. We have βn xn 2 + Kc |βn |2 , (4.1) x2 + Kc x2 = βn xn 2 = and therefore

|βn |2 ≤ (1 + Kc 2 )

βn xn 2 .

(4.2)

From (4.1) and (4.2) we have |βn |2 ≤ βn xn 2 ≤ |βn |2 , (1 + Kc 2 )−1 thus the {xn }∞ n=1 form a Riesz sequence, and therefore a Riesz basis for Dom(Kc ). Proposition 4.2. Let M be a top-dominant or diagonally dominant operator matrix. Let A have compact resolvent with eigenvalues {μn }∞ n=1 , dist[μn , σ(A)\{μn }] → ∞

as

n → ∞,

(4.3)

{λn }∞ n=1

and let C ∈ B(H2 ). Let be the eigenvalues of M in the interval (c, ∞). If E and Fn are the spectral measures corresponding to A and S(λn ), respectively, and γn = dist[λn , σ(M)\{λn }]/2, then we have E({μκ+n }) − Fn (Δn ) → 0

as

n → ∞,

where

Δn = (−γn , γn ). (4.4)

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Proof. We denote by Γn the circle with centre λn and radius γn . It follows from (2.6) that 1 ac + b − a2 +O (4.5) μκ+n ≤ λn ≤ μκ+n + a + μκ+n − c μ2κn (see [8, Corollary 4.4]). From (4.3) and (4.5) we deduce that dist[λn , σ(M)\ {λn }] → ∞ as n → ∞, therefore, for all sufﬁciently large n ∈ N, μκ+n is the only element from σ(A) which lies inside the circle Γn . For z ∈ Γn , we set δn (z) =

|az + b| a + , λn − c dist[z, σ(A)](λn − c)

also from (4.3) and (4.5) we deduce that δn := max δn (z) → 0 z∈Γn

as

n → ∞.

(4.6) 1

Similarly to the proof of Theorem 2.1, we have for all x ∈ Dom(|A| 2 ) 1 1 | (C − λn I)−1 B ∗ x, B ∗ x | ≤ δn (z) |A − zI| 2 x, |A − zI| 2 x . For all sufﬁciently large n ∈ N we have δn < 1, and by [6, Lemma VI.3.1] there exists a family of operators Gn (z) ∈ B(H1 ), parameterised by z ∈ Γn , with the property Gn (z) ≤ δn < 1, and such that 1 1 − (C − λn I)−1 B ∗ x, B ∗ y = Gn (z)|A − λI| 2 x, |A − λI| 2 y . Similarly to the proof of Theorem 2.1, we obtain the following 1

1 S(λn ) − (z − λn )I = |A − zI| 2 (A − zI)|A − zI|−1 + Gn (z) |A − zI| 2 , and we deduce that (z − λn ) ∈ ρ(S(λn )) for all z ∈ Γn . For any x, y ∈ H1 the inner product E({μκ+n })x − Fn (Δn )x, y is equal to 1 (A − ζI)−1 x − (S(λn ) − (ζ − λn )I)−1 x, y dζ. − 2iπ Γn

The inner product inside this integral equals

−1 I − I + |A − ζI|(A − ζI)−1 Gn (ζ) |A − ζI|(A − ζI)−1 |A 1 1 −ζI|− 2 x, |A − ζI|− 2 y , and using the Neumann series we see that this inner product is equal to m 1 1 |A−ζI|(ζI −A)−1 Gn (ζ) |A−ζI|(A−ζI)−1 |A−ζI|− 2 x, |A−ζI|− 2 y ,

∞

m=1

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and therefore | E({μκ+n })x − Fn (Δn )x, y | ∞ xy ≤ |A − ζI|(A − ζI)−1 Gn (ζ)m dζ 2πdist[Γn , σ(A)] m=1 Γn

≤ ≤

∞

|Γn |xy max Gn (z)m 2πdist[Γn , σ(A)] m=1 z∈Γn ∞ γn xy m δ . dist[Γn , σ(A)] m=1 n

It follows from (4.5) that γn /dist[Γn , σ(A)] → 1, and therefore with some M ∈ R independent of n ∈ N, we obtain | E({μκ+n })x − Fn (Δn )x, y | ≤ M

δn xy, 1 − δn

and the result follows from (4.6).

∞ Definition 4.3. Let {xn }∞ n=1 and {yn }n=1 be sequences in a Hilbert space ∞ ∞ with {yn }n=1 orthonormal. Then {xn }n=1 is a Bari sequence with respect to {yn }∞ n=1 if ∞

yn − xn 2 < ∞.

n=1 ∞ ∞ If {xn }∞ n=1 is a Bari sequence with respect to {yn }n=1 where {xn }n=1 and ∞ ∞ {yn }n=1 form a basis, then {xn }n=1 is a Bari basis with respect to {yn }∞ n=1 .

Corollary 4.4. Let M be a top-dominant or diagonally dominant operator matrix. Let A have compact resolvent with eigenvalues {μn }∞ n=1 , dist[μn , σ(A)\{μn }] → ∞, and let C ∈ B(H2 ). Suppose for some N ∈ N and with κ as in Corollary (3.5), the eigenvalues {μκ+n }∞ n=N are simple, and ∞

1 < ∞. (μn+1 − μn )2 n=1

(4.7)

T ∞ If {yn }∞ n=1 are orthonormal eigenvectors of A, and {(xn , Kc xn ) }n=1 are orthonormal eigenvectors corresponding to the eigenvalues of M which are ∞ greater than c, then {xn }∞ n=1 is a Bari sequence with respect to {yκ+n }n=1 .

Proof. From Proposition 4.2, we have E({μκ+n }) − Fn (Δn ) < 1 for all sufﬁciently large n ∈ N. We have xn ∈ Dom(S(λn )) and S(λn )xn = 0 (see [8, proof of Proposition 2.2] and [5, proof of Proposition 4.4]). Therefore xn ∈ Fn (Δn )H1 . Thus we have E({μκ+n })xn = 0 for all sufﬁciently large n ∈ N. For n ≥ N we have E({μκ+n })xn = xn , yκ+n yκ+n , and thus E({μκ+n })xn = yκ+n /E({μκ+n })xn . With the notation of Proposition 4.2, we obtain

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yκ+n − xn = E({μκ+n })(yκ+n − xn ) + (I − E({μκ+n }))(yκ+n − xn ) = E({μκ+n })xn /E({μκ+n })xn − E({μκ+n })xn +(I − E({μκ+n }))xn ≤ 2(I − E({μκ+n }))xn = 2(Fn (Δn ) − E({μκ+n }))xn 2M δn ≤ . 1 − δn ˜ ≥ N we have For some N ∞

yκ+n − xn 2 ≤

n=1

˜ −1 N

yκ+n − xn 2 + 4M 2

n=1

2 ∞ δn . 1 − δn

˜ n=N

That the second term on the right-hand side is ﬁnite follows from (4.5) and (4.7). Remark 4.5. If κ = 0 in Corollary 4.4 then the {xn }∞ n=1 is a Bari basis with . respect to {yn }∞ n=1

5. An Example from Magnetohydrodynamics The following operator appears in magnetohydrodynamics in the space L2ρ (0, 1) × L2ρ (0, 1) × L2ρ (0, 1), where L2ρ (0, 1) denotes the L2 -space with weight ρ, and D is the differential operator −id/dx. We consider the block operator matrix M0 given by

−1 ⎞ ⎛ −1 2 2 ρ Dρ(υa2 +υs2 )D+k ρ Dρυs2 +ig k υa (ρ−1 Dρ(υa2 +υs2 )+ig)k⊥ 2 2 ⎝ k⊥ (υa2 +υs2 )D−ig ⎠. k 2 υa2 +k⊥ υs k⊥ k υs2

2 2 2 2 k υs D−ig k⊥ k υs k υs 1,2 The operator M0 on the domain (Wρ2,2 (0, 1) ∩ W0,ρ (0, 1)) × Wρ1,2 (0, 1) × 1,2 Wρ (0, 1) is essentially self-adjoint (see [8, Example 4.5]). The eigenvalue problem Mu = λu describes the oscillations of a hot compressible gravitating plasma layer in an ambient magnetic ﬁeld (see [3] and references therein). The ﬁrst component of the vector u satisﬁes Dirichlet boundary conditions, ρ(x) the equilibrium density of the plasma, υa (x) the Alfv´en speed, υs (x) the sound speed, k⊥ (x) and k (x) are the coordinates of the wave vector with respect to the ﬁeld allied orthonormal bases, k(x)2 = k⊥ (x)2 + k (x)2 , and g is the gravitational constant. The essential spectrum of this operator is precisely the range of the functions υa2 k and υa2 υs2 k⊥ /(υa2 + υs2 ) (see [3, Section 5]). The operator M0 forms a top-dominant operator matrix, with 1,2 A := ρ−1 Dρ(υa2 + υs2 )D + k 2 υa2 : (Wρ2,2 (0, 1) ∩ W0,ρ (0, 1)) → L2ρ (0, 1),

and C :=

2 2 k 2 υa2 + k⊥ υs

k⊥ k υs2

k⊥ k υs2

k2 υs2

! : L2ρ (0, 1) × L2ρ (0, 1) → L2ρ (0, 1) × L2ρ (0, 1).

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We see that A is a Sturm–Liouville operator with Dirichlet boundary conditions. The operator C is self-adjoint and bounded, with " #

k 4 (υa2 + υs2 )2 k 2 (υa2 + υs2 ) 2 2 2 2 + − k k υa υs , c = max 2 4 and with a = max

"

2 (υa2 + υs2 )2 k⊥ + υs4 k2

#

υa2 + υs2

,

% $ g b = max max k 2 g 2 − (ρ((υa2 + υs2 )k⊥ + υs2 k )) − a min k 2 υa2 , 0 , ρ A and B ∗ satisfy (1.3) (see [8, Example 4.15]). If ρ and (υa2 + υs2 ) are positive and belong to C 2 ([0, 1]), then using the Liouville transform (see for example [12, Section 2.5.1]), we see that the eigenvalues of A are precisely those of a Sturm–Liouville operator in normal form with Dirichlet boundary conditions and bounded potential. Thus the eigenvalues of A satisfy dist[μn , σ(A)\{μn }] → ∞

and

∞

1 < ∞. (μ − μn )2 n+1 n=1

Theorem 5.1. Let ρ, (υa2 + υs2 ) ∈ C 2 ([0, 1]) and ρ > 0, υa2 + υs2 > 0 on [0, 1], and let M be the closure of M0 . There exists closed bounded operator Kc which satisfies (3.1), and codim(Dom(Kc )) = dim(L(−∞,0) (S(˜ c))) < ∞. If T ∞ {(xn , Kc xn ) }n=1 are orthonormal eigenvectors corresponding to the eigen∞ values {λn }∞ n=1 of M which are greater than c, then the set {xn }n=1 forms a ∞ Riesz basis for Dom(Kc ). Moreover, if {yn }n=1 are orthonormal eigenvectors ∞ of A, then {xn }∞ n=1 is a Bari sequence with respect to {yκ+n }n=1 .

References [1] Adamjan, A., Langer, H.: Spectral properties of a class of rational operator valued functions. J. Oper. Theory 33, 259–277 (1995) [2] Albeverio, S., Makarov, K., Motovilov, A.: Graph subspaces and the spectral shift function. Can. J. Math. 55(3), 449–503 (2003) [3] Atkinson, F., Langer, H., Mennicken, R., Shkalikov, A.: The essential spectrum of some matrix operators. Math. Nachr. 167, 5–20 (1994) [4] Davies, E.B., Plum, M.: Spectral pollution. IMA J. Numer. Anal. 24, 417– 438 (2004) [5] Eschwe, D., Langer, M.: Variational principles for eigenvalues of self-adjoint operator functions. Integral Equ. Oper. Theory 49, 287–321 (2004) [6] Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966) [7] Kostrykin, V., Makarov, K.A., Motovilov, A.K.: On the existence of solutions to the operator Riccati equation and the tan Θ theorem. Integral Equ. Oper. Theory 51, 121–140 (2005)

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[8] Kraus, M., Langer, M., Tretter, C.: Variational principles and eigenvalue estimates for unbounded block operator matrices and applications. J. Comput. Appl. Math. 171(1–2), 311–334 (2004) [9] Langer, H., Markus, A., Matsaev, V., Tretter, C.: Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements. J. Funct. Anal. 199(2), 427–451 (2003) [10] Levitin, M., Shargorodsky, E.: Spectral pollution and second order relative spectra for self-adjoint operators. IMA J. Numer. Anal. 24(3), 393–416 (2004) [11] Mennicken, R., Shkalikov, A.: Spectral decomposition of symmetric operator matrices. Math. Nachr. 179, 259–273 (1996) [12] Pryce, J.: Numerical Solution of Sturm-Liouville Problems. Oxford University Press, New York (1993) [13] Shkalikov, A.: Dissipative operators in a Krein space. Invariant subspaces and the properties of restrictions. Funct. Anal. Appl. 41(2), 154–167 (2007) [14] Strauss, M.: Quadratic projection methods for approximating the spectrum of self-adjoint operators. IMA J. Numer. Anal. (2010, in press) [15] Tretter, C.: Spectral Issues for Block Operator Matrices. Differential Equations and Mathematical Physics (Birmingham, AL, 1999). AMS/IP Stud. Adv. Math., vol. 16, pp. 407–423. Amer. Math. Soc., Providence (2000) [16] Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008) Michael Strauss Department of Mathematics University of Strathclyde 26 Richmond Street Glasgow, G1 1XH UK e-mail: [email protected] Received: August 29, 2009. Revised: December 9, 2009.

Integr. Equ. Oper. Theory 67 (2010), 279–300 DOI 10.1007/s00020-010-1781-z Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On Tractability and Ideal Problem in Non-Associative Operator Algebras Matej Breˇsar, Victor S. Shulman and Yuri V. Turovskii Abstract. The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojty´ nski’s problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given. Mathematics Subject Classification (2000). 17B05, 17B40, 17B60, 46H70, 47B07, 47B47. Keywords. Lie algebra, Jordan algebra, ideal, elementary operator, compact operator, invariant subspace, triangularizability, tractability.

1. Introduction This paper centers around the following problem by Wojty´ nski [18, Question 3]: Does every closed Lie algebra of compact quasinilpotent operators on a Banach space contain a non-trivial closed Lie ideal? At the moment we are not able to give a complete answer. We do give, however, several partial answers and, approaching the problem from different directions, we ﬁnd certain tools and introduce some concepts that might be of independent interest. The paper begins by introducing the concept of a tractable Lie algebra. It is deﬁned through a certain property of the associative algebra generated by all inner derivations of a Lie algebra in question. After considering this notion in Sect. 2 in a pure algebraic context and just from the point of view that is interesting in its own right, we present its connection to Wojty´ nski’s problem in Sect. 3. This section then contains some (partial) answers to this and to some related problems. In particular, it is shown that the Jordan algebra version of Wojty´ nski’s problem has a positive answer. In Sect. 4 we consider the The first author was supported by ARRS grant # P1-0288.

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situation where the algebra in question contains a non-zero ﬁnite rank operator; in this case the triangularization technique turns out to be useful. In Sect. 5 we consider the question on the existence of a not necessarily closed ideal. In particular we solve an algebraic version of Wojty´ nski’s problem: A Lie algebra of compact quasinilpotent operators is not simple, even without the assumption of closedness. Moreover, we prove that any inﬁnite-dimensional closed Lie or Jordan algebra of compact operators is not simple.

2. Tractable Lie Algebras Throughout, A will be an associative algebra over a ﬁeld F . For every a ∈ A we deﬁne La , Ra : A → A by La x = ax and Ra x = xa. Further, we write ad(a) = La − Ra . Let E(A) denote the algebra generated by all La and Rb , a, b ∈ A. The elements in E(A) are called elementary operators on A (in pure algebra E(A) is usually called the multiplication algebra of A, but having applications to the normed context in mind we will use a different terminology in E(A) are of the form E = and notation). Thus, the elements ◦ (A) we denote the ideal of E(A) La + Rb + i Lai Rbi , a, b, ai , bi ∈ A. By E consisting of elements of the form E = i Lai Rbi , ai , bi ∈ A. Of course, if A is unital, then E◦ (A) = E(A). But we are more interested in algebras without unity. Let L be a Lie subalgebra of A. We shall say that E ∈ E(A) is a Lie elementary operator on L if the restriction of E to L is a sum of products of operators of the form ad(a), a ∈ L. Clearly, L is invariant under E. We shall say that L is a tractable Lie algebra if there exists a Lie elementary operator on L which is not zero on L and coincides on L with some operator from E◦ (A). Of course, the tractability of L depends not only on L but also on A. But it shall always be clear from the context which algebra A we have in mind. We need some more notation. If L is a Lie subalgebra of A, then by L we denote the associative subalgebra of A generated by L. Further, we set ann(L) = {x ∈ L : xL = Lx = 0}. Note that ann(L) is an ideal of L. Let us ﬁrst record three simple observations. Remark 2.1. If L is a Lie subalgebra of an algebra A having an ideal J with the property [J, L] = J, and if J is tractable (at least with respect to A), then L is tractable. Indeed, let D be a non-zero Lie elementary operator on J which equals the restriction to J of an operator T ∈ E◦ (A). Then, for each a ∈ J, the operators D · ad(a) and T · ad(a) coincide on L. Moreover, the ﬁrst of them is a Lie elementary operator, while the second belongs to E◦ (A). So it sufﬁces to show that one can choose a in such a way that D · ad(a) = 0. But if this is impossible then D([J, L]) = 0. Since, by our assumptions, [J, L] = J, we get that D = 0, a contradiction.

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Remark 2.2. If L is noncommutative and A is unital, then L is automatically tractable. Indeed, taking any a ∈ L that does not lie in the center of L, we have ad(a) = La R1 − L1 Ra ∈ E◦ (A) and ad(a)L = 0. Using this, one can show that each simple ﬁnite-dimensional complex Lie algebra L of operators on a ﬁnite-dimensional space X is tractable (with respect to L). Indeed, it is well known [4, Theorem 3.7.8] that X = Y1 ⊕ Y2 ⊕ · · · ⊕ Yn , where all Yi are minimal (non-zero) subspaces invariant for L. Let us consider ﬁrst the case n = 1. In this case L is an irreducible algebra of operators; by Burnside’s Theorem, it coincides with the algebra B(X) of all operators on X. Hence L is unital and the claim follows from the preceding observation. Assume now that our claim is proved when the number of subspaces Yi is less than n. Let ϕi be the map which sends a ∈ L to a|Yi , and let Xi = j=i Yj . Setting Ai = {a ∈ L : a|Xi = 0}, we see that Ai is an ideal of L. If Ai = 0, for some i, then L is isomorphic to L|Xi ; by our assumption, L is tractable with respect to L. Thus we have to consider the case that Ai = 0 for each i. In this case ϕi (Ai ) is a non-zero ideal in ϕi (L). Since ϕi (L) is an algebra containing L|Yi , it is irreducible and equals B(Yi ) by Burnside’s Theorem. Hence it is simple and ϕi (Ai ) = B(Yi ); in particular, ϕi (Ai ) is unital. Thus there is ei ∈ L with ei |Yi = 1Yi , ei |Xi = 0. Clearly 1X = e1 + · · · + en ∈ L, so L is unital and L is tractable. Since [L, L] = L for a simple L, every Lie algebra of operators on a ﬁnite-dimensional space that have a simple Lie ideal is tractable by Remark 2.1. In particular, a semisimple Lie algebra of matrices is tractable. Remark 2.3. Let E be a Lie elementary operator on L. Thus we have (2.1) E= ad(ai1 ) ad(ai2 ) · · · ad(aini ), where aij ∈ L. We can rewrite E as E = Lu + Rv + where u=

ai1 ai2 · · · aini ,

v=

Lui Rvi ,

(2.2)

(−1)ni aini · · · ai2 ai1 ,

(2.3)

and ui , vi are elements in A that can also be expressed through the aij ’s. Hence we see that the following is true: if there exists an elementary Lie operator E on L of the form (2.1) such that EL = 0 and u and v from (2.3) lie in ann(L) (e.g., if both u and v are 0), then L is tractable. Let us show that the concept we introduced is not an empty one. Proposition 2.4. Let X = {x1 , x2 , . . .} be an infinite set, let A = F0 X be the non-unital free algebra on X, and let L be a Lie subalgebra of A generated by X. Then L is not tractable. Proof. We may regard elements in A as polynomials in noncommuting indeterminates xi , so we can deﬁne their degrees in the standard way. The Lie algebra L consists of polynomials that can be written as sums of what we will

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call Lie monomials—by these we mean scalar multiples of elements such as xi , [xi , xj ], [[xi , xj ], xk ], [[[xi , xj ], xk ], xl ], [[xi , xj ], [xk , xl ]], etc. If b1 , b2 are Lie monomials, so is [b1 , b2 ]; conversely, if b is a Lie monomial of degree ≥ 2 (i.e. b = λxi ), then b can be written as [b1 , b2 ] where b1 and b2 are Lie monomials whose degrees are smaller than the degree of b. Let E be a Lie elementary operator on L of the form (2.1). We can rewrite E according to (2.2). Suppose there exist wj , zj ∈ A such that E = Lu + Rv + Lui Rvi = Lwj Rzj . The set X is an inﬁnite one, so there exists r such that xr does not appear in the polynomials v, vi , zj . Considering Exr we thus have uxr = ui xr vi + wj xr zj − xr v. ai1 ai2 · · · aini = 0. But this is possible only if uxr = 0, and so u = Therefore, the proposition will be proved by showing that the following is true for all aij ∈ L: ad(ai1 ) ad(ai2 ) · · · ad(aini ) = 0. (2.4) ai1 ai2 · · · aini = 0 =⇒ Note that there is no loss of generality in assuming that all the aij ’s are Lie monomials; this is simply because every element in L is a sum of Lie monomials. We shall prove (2.4) by induction on the maximal degree d of the Lie monomials aij appearing in (2.4). If d = 1, that is, if each aij = λij xij for some λij ∈ C and xij ∈ X, then (2.4) trivially holds. Namely, different elements of the form xi1 xi2 · · · xin are linearly independent, and so the left hand side of (2.4) is 0 only in the trivial situation when the sum of the coefﬁcients at each xi1 xi2 · · · xin is equal to zero. This clearly forces the right hand side to be 0 too. Let d > 1. Each aij of degree d can be written as [bij , cij ] where bij and cij are Lie monomials of degree < d. Let air1 , · · · , airki , r1 < · · · < rki , be these elements. Then we have 0= ai1 · · · air1 · · · airki · · · aini = ai1 · · · [bir1 , cir1 ] · · · [birki , cirki ] · · · aini ai1 · · · bir1 cir1 · · · birki cirki · · · aini −ai1 · · · cir1 bir1 · · · birki cirki · · · aini = ± · · · + (−1)ki ai1 · · · cir1 bir1 · · · cirki birki · · · aini . We are now in a position to apply the induction assumption. Accordingly, 0= ad(ai1 ) · · · ad(bir1 )ad(cir1 ) · · · ad(birki )ad(cirki ) · · · ad(aini ) − ad(ai1 ) · · · ad(cir1 )ad(bir1 ) · · · ad(birki )ad(cirki ) · · · ad(aini )

± · · ·+(−1)ki ad(ai1 ) · · · ad(cir1 )ad(bir1 ) · · · ad(cirki)ad(birki ) · · · ad(aini) = ad(ai1 ) · · · [ad(bir1 ), ad(cir1 )] · · · [ad(birki ), ad(cirki )] · · · ad(aini ).

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However, [ad(bij ), ad(cij )] = ad(aij ) and so we have ad(aini ) = 0.

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ad(ai1 ) ad(ai2 ) · · ·

In the two propositions that follow we will present two types of examples of tractable Lie algebras. The ﬁrst one deals with Lie algebras that are in some sense close to associative ones. We say that a Lie subalgebra L of an associative algebra A is closed under triads if aba ∈ L for all a, b ∈ L. Besides the obvious example where L itself is an associative algebra, this also covers an important case where L is a Lie algebra of all skew-symmetric elements of an associative algebra A with involution ∗: L = {a ∈ A | a∗ = −a}. Proposition 2.5. Let L be a Lie subalgebra of an associative algebra A. If L is closed under triads, then L is tractable, unless acbab + babca = abacb + bcaba for all a, b, c ∈ L. Proof. Pick a, b ∈ L, and set E = ad(aba) ad(b) − ad(a) ad(bab). A straightforward computation shows that E can be represented as E = La Rbab + Lbab Ra − Laba Rb − Lb Raba . Accordingly, E is a Lie elementary operator on L which belongs to E0 (A). Thus L is tractable, unless Ec = acbab + babca − abacb − bcaba = 0 for every c ∈ L. The condition acbab+babca = abacb+bcaba for all a, b, c ∈ L means that L satisﬁes a very special polynomial identity. Using the theory of polynomial identities it would be possible to analyse this situation for some classes of Lie algebras; for example, if L = A is a prime algebra or if L is the set of skew-symmetric elements of a prime algebra A with involution, then using standard methods one could show that this polynomial identity can hold on L only in some exceptional cases. But this analysis would lead us afar from the scope of this paper. For the proof of the second proposition we need some auxiliary results. The ﬁrst one might be interesting in its own right. In particular it shows that every simple Lie algebra which is also a Lie ideal of an associative algebra, is, up to an isomorphism, equal to [B, B] where B is a simple associative algebra. This is kind of a converse to Herstein’s theorem stating that [B, B] is a simple Lie algebra if B is a simple associative algebra and [B, B] has trivial intersection with the center of R [3, Theorem 1.12]. In the proof we shall need another theorem by Herstein saying that a Lie ideal of a simple algebra B either contains [B, B] or is contained in the center of B [3, Theorem 1.3]. All these is true under the assumption that char(F ) = 2. Lemma 2.6. Let L be a Lie ideal of an associative algebra A. If L is simple (as a Lie algebra) and char(F ) = 2, then B = L/ann(L) is a simple associative algebra and L ∼ = L/ann(L) = [B, B]. Proof. Pick a ∈ L \ ann(L). We will prove the simplicity of B by showing that the ideal of L generated by a is equal to L.

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If a does not lie in the center Z of L, that is, if [a, L] = 0, then 0 = [a, l] ∈ L for some l ∈ L. Therefore the ideal of the Lie algebra L generated by [a, l] is equal to L. Consequently, L is contained in the ideal of L generated by a. Since the algebra L is generated by L, this ideal is equal to L. Assume therefore that a ∈ Z. We claim that there is x ∈ L such that ax ∈ / Z. Indeed, if ax was in Z for every x ∈ L, then we would have 0 = [ax, l] for every l ∈ L. Since a commutes with l, this implies that a[x, l] = 0. That is, a[L, L] = 0. However, [L, L] is an ideal of L, and certainly [L, L] = 0 since L is noncommutative. Therefore [L, L] = L. Thus, aL = 0, which clearly yields aL = 0, and so also La = 0. But this contradicts a ∈ / ann(L). This proves that ax ∈ / Z for some x ∈ L. But then, by what was proved in the preceding paragraph, the ideal of L generated by ax is equal to L. But then the ideal generated by a is also equal to L. Note that L ∩ ann(L) = 0 in view of the simplicity of L. Therefore the restriction of the quotient map x → x + ann(L) to L is an isomorphism between Lie algebras L and M = L/ann(L). Of course, M is a Lie ideal of B. It is clear that M is not contained in the center of B (since otherwise [L, L] would be contained in ann(L)). Therefore M ⊇ [B, B] by Herstein’s theorem [3, Theorem 1.3]. Since M is simple and [B, B] is obviously an ideal of M , it follows that M = [B, B]. The next lemma must be known, but we give the proof since it is rather short. We shall need it only for the case when B is a simple algebra, but in view of the proof it is more convenient to state it for prime algebras. Lemma 2.7. Let B be a noncommutative prime algebra. If b, c ∈ B are such that b[B, B]c = 0, then b = 0 or c = 0. Proof. Since non-zero elements in the center of a prime ring are not zero divisors, we may assume that at least one of b, c does not lie in the center. Without loss of generality we may assume that b is not in the center. We have b[x, y]c = 0 for all x, y ∈ B, that is, bxyc = byxc. In particular, bx(by)c = b2 yxc = b2 xyc. Thus, (bxb − b2 x)Bc = 0 for every x ∈ B. Since B is prime, this yields c = 0 or bxb = b2 x. If the latter occurs, then we have bxyb = b2 xy = bxby for all x, y ∈ B, i.e., bB[b, B] = 0. Therefore [b, B] = 0, that is, b lies in the center of B. Hence c = 0. Lemma 2.8. Let L be a Lie subalgebra of an associative algebra A. Suppose there exists a ∈ L such that an ∈ ann(L), an−1 Lan−1 = 0, and n does not divide char(F ). Then L is tractable. Proof. Note that the Lie elementary operator E = (ad(a))n coincides with an operator in E◦ (A) on L; namely, u and v from (2.3) lie in ann(L). Since Ran−2 E = −nLan−1 Ran−1 , E is not 0 on L. We now have enough information to prove our ﬁnal proposition in this section.

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Proposition 2.9. Let L be a Lie ideal of an associative algebra A over a field F with char(F ) = 0. Suppose that L is a simple Lie algebra and that L contains a non-zero nilpotent element. Then L is tractable. Proof. Since L contains non-zero nilpotents and since L ∩ ann(L) = 0 by simplicity of L, there exists a ∈ L and n ≥ 2 such that an ∈ ann(L) and / ann(L). an−1 ∈ Using the notation of Lemma 2.6 we set b = an−1 + ann(L) ∈ B. Since B is simple, we have b[B, B]b = 0 by Lemma 2.7. In view of Lemma 2.6 we then also have an−1 Lan−1 = 0. Therefore L is tractable by Lemma 2.8.

3. Weakly Tractable Subspaces of an Algebra and Topological Non-simplicity In what follows normed spaces and algebras are assumed to be complex. By an operator on a normed space we mean a bounded linear operator. Let L be a Lie algebra of compact quasinilpotent operators on a Banach space X, dim L > 1. Wojty´ nski’s problem (see Sect. 1) is to show that L has a non-trivial closed Lie ideal. The relation with the previous material is that if L is tractable then the answer is positive, as we shall see. Let us consider a more general situation. Let A be an algebra. If V is a subspace of A, let us denote by EV (A) the set of all operators T ∈ E(A) that preserve V : T V ⊂ V . Then EV (A) is a subalgebra of E(A). For example, if V is a Lie (Jordan) subalgebra of A then all operators ad(a) : x → ax − xa (respectively, pa : x → ax + xa), for a ∈ V , belong to EV (A). If V is a Lie (Jordan) ideal of A, then the same is true for all a ∈ A. Let V be a subspace of A. A subspace W of V is called El-stable with respect to V if all operators in EV (A) preserve W . In other words, W is El-stable if EV (A) ⊆ EW (A). We say that V is El-simple (with respect to A) if it does not have non-trivial El-stable subspaces. If A is normed, V is called topologically El-simple if it does not have non-trivial El-stable closed subspaces. Returning to the above examples we see that if V is a Lie subalgebra, a Jordan subalgebra or a Lie ideal of A, then all its El-stable subspaces with respect to A are, respectively, Lie ideals of V , Jordan ideals of V or Lie ideals of A. The converse is not true. The simplest example is the following. Let A be the algebra of all operators on a ﬁnite-dimensional space X over C. Then, as it is well known and easy to see, E(A) is the set of all operators on A. So EV (A) is the set of all operators leaving V invariant, whence V has no non-trivial El-stable subspaces. Now, if V is a non-simple Lie subalgebra of A, then V has non-trivial Lie ideals that are not El-stable. Therefore the statement that a Lie subalgebra, Jordan subalgebra, or Lie ideal V of a (normed) algebra A is not (topologically) El-simple, is a priori stronger than the statement that V is not a (topologically) simple Lie

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algebra, (topologically) simple Jordan algebra, or a minimal (closed) Lie ideal of A. Let E◦V (A) = E◦ (A) ∩ EV (A). A subspace V of an algebra A will be called weakly tractable (with respect to A) if E◦V (A)V = 0. It is clear that every tractable Lie algebra is weakly tractable (with respect to the same A). Let us show that Proposition 2.4 can be extended to weak tractability. Proposition 3.1. Let L and A be as in Proposition 2.4. Then L is not weakly tractable. Proof. In view of Proposition 2.4 it sufﬁces to show that every E ∈ EL (A) is a Lie elementary operator. Set E = k Lwk Rzk where wk , zk ∈ A∪{1}. We are assuming that Ef is a Lie polynomial (i.e., a sum of Lie monomials) whenever f ∈ A is a Lie polynomial. Without loss of generality we may assume that none of the polynomials wk , zk involves x1 . Since x1 is a Lie polynomial, it fol lows that Ex1 = wk x1 zk is a Lie polynomial,which is linear in x1 . Therefore there exists a Lie elementary operator D = ad(xi1 )ad(xi2 ) . . . ad(xini ), xij = x1 , such that Ex1 = Dx1 ; indeed, this follows easily from ad([u, v]) = [ad(u), ad(v)]. So we have [xi1 , [xi2 , [. . . , [xini , x1 ] . . .]]]. wk x1 zk = Now, this is the identity in the free algebra, and so we may replace x1 by any other element. Therefore, wk gzk = [xi1 , [xi2 , [. . . , [xini , g] . . .]]] l

holds for every g ∈ A. That is, Eg = Dg for every g ∈ A, i.e., E = D.

Note that if A is an algebra of compact operators, then the ideal E◦ (A) of E(A) consists of compact operators [17]. Proposition 3.2. Let V be a closed subspace in a closed algebra A of compact quasinilpotent operators, and let dim(V ) > 1. If V is weakly tractable with respect to A, then V is not topologically El-simple with respect to A. Proof. By [15, Lemma 5.10], all elementary operators on A are quasinilpotent. Let M be the algebra of the restrictions of all operators in EV (A) to V . Since E(A) consists of quasinilpotents, the same is true for M. Since all operators in E◦ (A) are compact, it follows from our assumptions that M contains a non-zero compact operator. It can be easily seen from Lomonosov’s Lemma [9] that if an algebra of quasinilpotent operators contains a non-zero compact operator then it has a non-trivial invariant subspace. Clearly a subspace invariant under M is an El-stable subspace of V . Corollary 3.3. Let L be a closed non-one-dimensional Lie algebra L of compact quasinilpotent operators on a Banach space X. If L is weakly tractable with respect to the closure of L, then L is not topologically simple. Proof. Let A be the closed subalgebra in B(X) generated by L. By [14, Corollary 11.6], it consists of compact quasinilpotent operators. Since L is weakly tractable, the condition E◦L (A)L = 0 holds. By Proposition 3.2,

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L is not a topologically El-simple subspace of A. Since each El-stable subspace of L is an ideal of L, we conclude that L is not topologically simple as a Lie algebra. Corollary 3.4. A closed non-one-dimensional Jordan algebra J of compact quasinilpotent operators on a Banach space X is not topologically simple. Proof. Let A be the closed algebra generated by J. It follows easily from [5, Corollary 11.1], that A consists of compact quasinilpotent operators. For each a ∈ A, let pa be the operator on A deﬁned as above. Then pa ∈ EJ (A) for each a ∈ J. It follows that wa = 12 (p2a − pa2 ) belongs to EJ (A) for a ∈ J. Since wa (x) = axa, we have that wa ∈ E◦ (A). If wa (J) = 0 for some a ∈ J, then, by Proposition 3.2, J has a non-trivial El-stable closed subspace, which is clearly a Jordan ideal. Suppose that wa (J) = 0 for all a ∈ J. Then (La Rb + Lb Ra )J = (wa+b − wa − wb )(J) = 0 for every a, b ∈ J. Hence (ab + ba)c + c(ab + ba) = 0 for all a, b, c ∈ J. Let Ib be the closed ideal of J generated by an element b in J. If ab + ba = 0, for some a, b ∈ J, then Iab+ba is a required ideal. If, however, ab + ba = 0 for all a, b ∈ J, then Ib = Cb is a required ideal for any non-zero b. For each algebra A, set Z2 (A) = {x ∈ A : a2 xa = axa2 for all a ∈ A}. Corollary 3.5. Let A be a closed algebra of compact quasinilpotent operators. Then each minimal closed Jordan ideal of A is one-dimensional, and each minimal closed Lie ideal of A is either one-dimensional or contained in Z2 (A). Proof. Suppose that J is a minimal closed Jordan ideal of A with dim(J) > 1. Then pa ∈ EJ (A) for each a ∈ A, and we obtain as above, in terms of the proof of Corollary 3.4, that wa (J) = 0 for all a ∈ A. If ab + ba = 0, for some a ∈ A, b ∈ J, then the Jordan ideal Iab+ba of A is one-dimensional, a contradiction. If, however, ab + ba = 0 for all a ∈ A, b ∈ J, then Ib = Cb is a Jordan ideal of A, for any non-zero b ∈ J, a contradiction. Suppose that V is a minimal closed Lie ideal in A with dim(V ) > 1. Since V is a Lie ideal, each operator Ta = (ad(a))3 − ad(a3 ) leaves it invariant: Ta ∈ EV (A) for a ∈ A. If Ta V = 0 then a2 xa = axa2 for all x ∈ V . Thus if V is not contained in Z2 (A) then there is a ∈ A with Ta V = 0. Since Ta ∈ E◦ (A), it follows from Proposition 3.2 that V is not topologically El-simple. Hence V is not a minimal closed Lie ideal of A. Let us call a normed algebra bicompact if all operators La Rb , a, b ∈ A, are compact. Proposition 3.2 and Corollary 3.5 can be easily extended to the case that A is an arbitrary bicompact Jacobson-radical Banach algebra. The following example shows that a bicompact radical Banach algebra can have a minimal closed Lie ideal of inﬁnite dimension. Example. It was shown in [2] that for any quasinilpotent operator T ∈ B(X) there is an algebraic norm ||| · ||| majorizing the operator norm on the algebra A(T ) generated by T , such that the completion B of A(T ) with respect to

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||| · ||| is a bicompact radical Banach algebra. Choosing for T a quasinilpotent operator without invariant subspaces (its existence was proved in [11]) we obtain that there is a bicompact radical Banach algebra B with a nontrivial topologically irreducible representation π on a Banach space X. Let A = B ⊕ X with multiplication (a ⊕ x)(b ⊕ y) = ab ⊕ π(a)y. Then A is clearly a bicompact radical Banach algebra. The subspace J = {0} ⊕ X is a Lie ideal of A and it is easy to check that it does not contain smaller closed Lie ideals. Let WOT denote the weak operator topology on B(X). Let us say that a set U of operators is weakly ﬁnitely generated if there is a ﬁnite subset W of U such that the WOT-closed algebra generated by W contains U. For a closed Lie subalgebra L of a Banach algebra A, let K2 (L) be the set of all a ∈ L such that La2 + Ra2 is compact as an operator from L into L. Then K2 (L) is invariant under the set exp(ad(L)) of inner automorphisms of L. Indeed, if Q is the unit ball of L, ϕt = exp(t ad(b)) for b ∈ L and every t ∈ R, then ϕt is the restriction to L of the automorphism Lexp(tb) Rexp(−tb) = exp(Ltb ) exp(R−tb ) of the closed algebra generated by L, and Lϕt (a)2 + Rϕt (a)2 Q = Lexp(tb) Rexp(−tb) (La2 + Ra2 ) ϕ−t (Q). for each a ∈ K2 (L). As ϕ−t (Q) is a bounded subset of L, the set (La2 + Ra2 )ϕ−t (Q) is precompact, whence Lϕt (a)2 + Rϕt (a)2 is compact as an operator from L into L. Theorem 3.6. Let L be a closed non-one-dimensional Lie algebra of compact quasinilpotent operators, and let N be a set in K2 (L) invariant under exp(ad(L)). If N is non-zero and weakly finitely generated, then L is not topologically simple. Proof. Let A be the closure of L. By [14, Corolary 11.6], A consists of compact quasinilpotent operators. By [15, Lemma 5.10], E(A) consists of quasinilpotents. For a ∈ N , clearly ad(a)2 = La2 + Ra2 − 2La Ra is a compact operator from L into L, therefore it is a compact operator on L. If ad(a)2 is non-zero on L then EL (A) has in L a non-trivial closed invariant subspace by Lomonosov’s Lemma [9]. This subspace is a closed ideal of L. So it sufﬁces to consider the case that ad(a)2 = 0 for all a ∈ N . Let K be a ﬁnite subset of N weakly generating N . By Zelmanov’s theorem [20, Theorem 1], the elements with the property ad(a)2 = 0 generate a locally nilpotent subalgebra of L. Thus the Lie algebra generated by K is nilpotent, hence its center is non-zero. So there is a non-zero element b ∈ L commuting with K. It follows that b commutes with the WOT-closed algebra generated by K, hence with N . The closed linear span I of N is an ideal of L. Indeed, for each b ∈ L, the automorphisms ϕt = exp(t ad(b)) of L preserve N. So these automorphisms preserve I, whence 1 ad(b)(x) = lim (ϕt (x) − x) ∈ I t→0 t for each x ∈ I.

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Hence if L is topologically simple then I = L. It follows that b commutes with all elements of L, and so Cb is a closed ideal of L, a contradiction. An element a of a normed algebra A is called completely continuous if La and Ra are compact operators on A. Corollary 3.7. Let L be a closed non-one-dimensional Lie algebra of compact quasinilpotent operators, N1 = {a ∈ L : a2 = 0}, N2 = {a ∈ L : a2 ∈ ann(L)}, let N3 be the set of all a ∈ L such that a2 commutes with each element of L, N4 the set of all a ∈ L such that a2 is completely continuous element of L, and let N5 be the set of all a ∈ L such that La2 and Ra2 are compact as operators from L into L. If Ni is non-zero and weakly finitely generated for some i, then L is not topologically simple. Proof. All sets are invariant under exp(ad(L)). Clearly N1 ⊂ N2 ⊂ N4 ⊂ N5 ⊂ K2 (L), and N3 ⊂ N4 by [1, Theorem 1]. So the statement follows by Theorem 3.6. The last result of the section establishes the topological non-simplicity in a quite special situation. In the next section it will be used to establish the algebraic non-simplicity in a very general case (see Theorem 5.1). Recall that for each subspace Y ⊂ X and for each operator a leaving Y invariant, one can consider the restriction a|Y of a to Y and the quotient operator a|(X/Y ) on X/Y which sends x + Y to ax + Y . We denote a|Y by πY (a) and a|(X/Y ) by πX/Y (a). The maps a → πY (a) and a → πX/Y (a) are representations of the algebra of all operators leaving Y invariant; they are called the restriction representation and quotient representation deﬁned by Y . Theorem 3.8. Let L be an infinite-dimensional Banach Lie algebra. Assume that there is a bounded non-zero representation h of L by finite rank operators on a normed space X. Then L is not topologically simple. If moreover h is injective, then L has a proper closed ideal of finite codimension. Proof. Without loss of generality, one may assume that X is complete and h is injective. Every x ∈ X deﬁnes an operator Ux : L −→ X by Ux (a) = h(a)x. The space U of operators Ux is locally ﬁnite-dimensional. Indeed, Ua = h(a)X is ﬁnite-dimensional for every a ∈ L. By a theorem of Livshits [8], there are a ﬁnite-dimensional space U1 of operators from L to X and a ﬁnite-dimensional subspace W of X such that U ⊂ U1 + UW , where UW is the space of all (bounded) operators from L to X with ranges in W . It follows that dim(U/U ∩ UW ) < ∞. Let X0 = {x ∈ X : Ux ∈ UW }. Then X0 is closed in X and has ﬁnite codimension in X. It is clear that h(L)X0 ⊂ W . Let Y = {x ∈ X : dim(h(L)x) < ∞}. It is a subspace of X. As h(L)h(a)x ⊂ h(L)x + h(a)h(L)x for every a ∈ L and x ∈ X, we see that h(L)h(a)x is of ﬁnite dimension for every x ∈ Y , i.e., Y is invariant for h(L). As Y contains X0 , it is a closed subspace of ﬁnite codimension in X. The quotient representation πX/Y of h(L) is of ﬁnite rank, so its kernel is of ﬁnite

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codimension. If πX/Y = 0, then {a ∈ L : πX/Y (h(a)) = 0} is a proper closed ideal of ﬁnite codimension in L. Assume that πX/Y = 0. Since πY (h(L)) is locally ﬁnite-dimensional, it follows as above from the theorem of Livshits that there is a ﬁnite-dimensional subspace Z of Y such that dim(πY (h(L))/πY (h(LZ ))) < ∞, where LZ is the space of all a ∈ L such that h(a)Y ⊂ Z. Clearly LZ is a closed subalgebra of ﬁnite codimension in L. By [7, Theorem 7.1], if LZ = L then L has a proper closed ideal of ﬁnite codimension. So we may assume that LZ = L, that is πY /Z = 0. Then Z is an invariant subspace for h(L). The restriction representation πZ of h(L) is of ﬁnite rank, so its kernel is of ﬁnite codimension. If πZ = 0, then {a ∈ L : πZ (h(a)) = 0} is a proper closed ideal of ﬁnite codimension in L. Assume that πZ = 0. As πX/Y = πY /Z = πZ = 0, then h(a)h(b)h(c) = 0 for every a, b, c ∈ L. Hence [[L, L], L] = 0 and any closed subspace of L containing [L, L] is an ideal of L. If [L, L] is dense in L, then L is commutative, and every closed subspace of L is an ideal of L. In any case, L has a proper closed ideal of ﬁnite codimension.

4. Around the Triangularizability A set Γ of closed subspaces of a normed space X is called a closed subspace chain if it is linearly ordered by inclusion. A closed subspace chain is called complete if it contains the intersection and the closure of the sum of subspaces of each of its subchains. The gaps of a complete closed subspace chain Γ are deﬁned as pairs (Z, Y ) ∈ Γ × Γ with Z =⊂ Y such that there are no subspaces in Γ intermediate between Z and Y . The quotients Y /Z for such pairs are called the gap-quotients of Γ. If Γ consists of invariant subspaces for a set M of operators on X, every gap (Z, Y ) induces a representation πY /Z of M by operators on Y /Z given by πY /Z (a)(y + Z) = ay + Z, which is called a gap-representation of M with respect to Γ. A closed subspace chain is called a maximal closed subspace chain if it is not a subchain of a larger chain. This is equivalent to the condition that all its gap-quotients (if they exist) are one-dimensional. Complete chains without gap-quotients are called continuous. A set M of operators on X is called triangularizable if there is a maximal closed subspace chain consisting of invariant subspaces for M . It was proved by Ringrose [12] (see also [10, Theorem 5.12]) that if a complete closed subspace chain Γ in a Banach space X (including 0 and X of course) consists of subspaces invariant for a compact operator a then Sp∗ (a) = ∪Sp∗ (πZ/Y (a)) where the union is over all gaps Y ⊂ Z of Γ, and Sp∗ denotes the non-zero part of spectrum: Sp∗ (a) = Sp(a)\{0}. In particular, if Γ is maximal then a is quasinilpotent if and only if πZ/Y (a) = 0 for all gaps. It follows that if a Lie or Jordan algebra J is triangularizable then the set of all quasinilpotent compact operators in J is a closed ideal of J.

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We will apply the following analogue of the Ringrose Theorem: Lemma 4.1. Let Γ be a maximal closed subspace chain in a normed space X. A finite rank operator a that leaves invariant all subspaces in Γ is nilpotent if and only if it belongs to the kernels of all gap-representations of Γ. The non-trivial part of the proof is contained in the proof of Lemma 5.5 below, and we omit it here. Theorem 4.2. Let J be a non-one-dimensional, Lie or Jordan, algebra of operators on a normed space X. If J contains a non-zero finite rank operator and if J is triangularizable, then J is not topologically simple. Proof. Assuming the contrary, we need only to consider the case when the ideal I of all ﬁnite rank operators in J is dense in J. Let Γ be a maximal closed subspace chain consisting of invariant subspaces for J, and let, as usual, πY be the restriction representation of J on Y , for every Y ∈ Γ. Assume ﬁrstly that πY is non-zero for every non-zero Y ∈ Γ. Let Γa = {Y ∩ aX : Y ∈ Γ} for any a ∈ J. If a is a ﬁnite rank operator in J, then Γa consists of a ﬁnite number of ﬁnite-dimensional subspaces invariant for a. Let (0, Z) be the gap of Γa , and let Y be a subspace in Γ such that Y ∩ aX = 0. As aY = aY ∩ aX ⊂ Y ∩ aX = 0, we obtain that πY (a) = 0. If Y = 0 then πY = 0 and the kernel of πY is a non-trivial ideal of J, a contradiction. If there are no non-zero Y ∈ Γ with Y ∩ aX = 0, then every non-zero subspace in Γ contains Z. This implies that there is a gap-quotient of Γ containing Z. As gap-quotients of Γ are one-dimensional, Z ∈ Γ and Z is one-dimensional. When J is a Lie algebra, πZ ([b, c]) = 0 for every b, c ∈ J. As the kernel of πZ is zero, J is commutative and is a sum of one-dimensional ideals, a contradiction. Consider the case when J is a Jordan algebra. If I contains a nonzero nilpotent operator, one may assume, using Lemma 4.1, that J has a dense ideal I0 of nilpotent ﬁnite rank operators. Then πZ vanishes on I0 and therefore on J, a contradiction. Therefore I has no non-zero nilpotent operators. As the kernel of πZ is zero and πZ ([[b, c], d]) = 0, we obtain that [[b, c], d] = 0, for every b, c, d ∈ J. As the algebra generated by a ∈ I is a commutative semisimple ﬁnite-dimensional algebra, it is a ﬁnite direct sum of simple algebras each of which is isomorphic to the ﬁeld. So a is a ﬁnite linear combination λi pi of orthogonal projections that also belong to I. It is easy to see from [[pi , b], pi ] = 0 that each pi commutes with every b ∈ J. Hence J is a semisimple commutative associative algebra. If I contains at least two orthogonal non-zero projections p and q, then J has non-trivial ideals pJ and qJ, otherwise I is one-dimensional and J = I. In any case, we have a contradiction. Now consider the general case when πY = 0 for some non-zero Y ∈ Γ. Then there is a largest subspace W ∈ Γ such that πW = 0 and πY = 0 for every Y ∈ Γ properly contaning W . As πX = 0, then W = X. Let πX/W be the quotient representation of J on X/W , and let π = πX/W . Note that π(J)

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is a triangularizable set of operators on X/W , Λ = {Y /W : W ⊂ Y ∈ Γ} is a maximal closed subspace chain consisting of invariant subspaces for π(J). Assume ﬁrst that π = 0. As the kernel of π is zero, π(J) is non-onedimensional. Let πY /W be the restriction representation of π(J) on Y /W for every Y /W ∈ Λ. If πY /W = 0 for every non-zero Y /W ∈ Λ, then the above argument for πY /W instead of πY leads to a contradiction. So there is a non-zero Y /W ∈ Λ such that πY /W = 0. Then aY ⊂ W and aW = 0 for every a ∈ J, whence πY (a)πY (b) = 0 for every a, b ∈ J. This implies that πY (ab ± ba) = 0 for all a, b ∈ J. As πY = 0 and its kernel is trivial, every one-dimensional subspace of J is an ideal of J, a contradiction. At last, assume that π = 0. The above argument for πX/W instead of πY /W shows that J is a sum of one-dimensional ideals of J, a contradiction. This shows that J is not topologically simple. For an operator a acting on a non-complete normed space we use the term quasinilpotent if the equality limn→∞ an 1/n = 0 holds. Corollary 4.3. Let J be a non-one-dimensional Jordan algebra of quasinilpotent operators on a normed space X. If J contains a non-zero finite rank operator, then J is not topologically simple. Proof. Without loss of generality, one may assume that X is complete and that J consists of quasinilpotent compact operators. By [5, Corollary 11.1], J is triangularizable. Therefore J is not topologically simple by Theorem 4.2. The result cannot be considered as a consequence of Theorem 3.4 because we do not assume that J is closed. The following result was proved in [19, Theorem 6] under additional conditions of separability and closedness for Lie algebras of quasinilpotent compact operators. A normed Lie algebra L is called Engel if ad(a) is a quasinilpotent operator on L for every a ∈ L. Corollary 4.4. Let L be a non-one-dimensional Engel Lie algebra of operators on a normed space. If L contains a non-zero finite rank operator then L is not topologically simple. Proof. Assuming the contrary, we have that the ideal of ﬁnite rank operators in L is dense in L. Without loss of generality, one can assume that X is a Banach space. Since L consists of compact operators, it is triangularizable by [14, Corollary 11.5]. By Theorem 4.2, L is not topologically simple, a contradiction. Let h : L −→ M be a homomorphism of Lie algebras. Then h · adL (a) = adM (h(a)) · h for every a ∈ L. Suppose that M is normed and h(L) is dense in M . Then adM (h(a)) = 0 if and only if a ∈ ker(h · adL ). If adL (a) is of ﬁnite rank, then, under the same conditions on h, adM (h(a)) is of ﬁnite rank. We say that a normed Lie algebra L is semi-Engel if there are an Engel normed Lie algebra M and a bounded injective homomorphism h : L −→ M

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with dense range. It is clear that every Engel normed Lie algebra is semi-Engel. Note also that every Lie algebra of quasinilpotent operators on a normed space is an Engel normed algebra. Corollary 4.5. Let L be a non-one-dimensional semi-Engel normed Lie algebra. If ad(a) is of finite rank for some non-zero a ∈ L, then L has a non-trivial closed ideal. Proof. Let M be an Engel normed Lie algebra and h : L −→ M be a bounded injective homomorphism with dense range. Let N = ad(M ). It is a normed Lie algebra of quasinilpotent operators on the completion of M which contains a ﬁnite rank operator adM (h(a)). If adL = 0 then L is commutative and is a sum of one-dimensional ideals. If adL = 0 and either dim(N ) < 1 or adM (h(a)) = 0, then ker(adL ) is a non-trivial ideal of L. Assume now that ker(adL ) = 0. Then dim(N ) > 1 and adM (h(a)) = 0, and N has a non-trivial closed ideal I by Corollary 4.4. Then {b ∈ L : adM (h(b)) ∈ I} is a non-trivial closed ideal of L. Let M be a set of operators on a normed space X. We say that M is almost triangularizable if there is a complete chain (including 0 and X) of closed subspaces invariant for M that admits only ﬁnite-dimensional gapquotients (if any exist of course). Corollary 4.6. Let J be an infinite-dimensional, Lie or Jordan, algebra of operators on a normed space X. If J contains a non-zero finite rank operator and if J is almost triangularizable, then J is not topologically simple. Proof. Assume, to the contrary, that J is topologically simple and let Γ be a maximal chain of closed J-invariant subspaces with only ﬁnite-dimensional gaps. Since kernels of non-zero ﬁnite rank representations of J are non-trivial ideals, one may assume that all gap-representations of J with respect to Γ are zero. So gap-quotients of Γ are one-dimensional, and J is in fact triangularizable. By Theorem 4.2, J is not topologically simple. A normed space X is called an operator range if there is a bounded operator T from a Banach space Y onto X. Theorem 4.7. Let J be a normed, Lie or Jordan, algebra of finite rank operators on an infinite-dimensional Banach space X. If J is an operator range then J has a non-trivial invariant closed subspace of finite dimension or codimension. Proof. Let J = T V for some bounded operator from a Banach space V . Assume that J has no non-trivial invariant closed subspaces of ﬁnite dimension or codimension. Every x ∈ X deﬁnes a bounded operator Sx : V −→ X by Sx (v) = (T v)x. The space U of operators Sx is locally ﬁnite-dimensional. Indeed, Uv = (T v)X is ﬁnite-dimensional for every v ∈ V . By theorem of Livshits [8], there are a ﬁnite-dimensional space U1 of operators from V to X and a ﬁnite-dimensional subspace W of X such that U ⊂ U1 + UW , where UW is the space of all bounded operators from V to X with ranges in W . It follows that dim(U/U ∩ UW ) < ∞. Let X0 = {x ∈

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X : Sx ∈ UW }. Then X0 is closed in X and has ﬁnite codimension in X. If a = T v and x ∈ X0 then ax = (T v)x = Sx (v) ∈ W , so that JX0 ⊂ W . Let Y = {x ∈ X : dim(Jx) < ∞}. It is a subspace of X. As Jax ⊂ Jx + aJx for every a ∈ J and x ∈ X, we see that Jax is of ﬁnite dimension for every x ∈ Y , i.e., Y is invariant for J. As Y contains X0 , it is a closed subspace of ﬁnite codimension in X. As J has no non-trivial invariant closed subspaces of ﬁnite codimension, Y = X. Then J is locally ﬁnite-dimensional. It follows as above from the theorem of Livshits that there is a ﬁnite-dimensional subspace Z of X such that dim(J/JZ ) < ∞, where JZ is the space of all a ∈ J such that aX ⊂ Z. Clearly JZ is a closed subalgebra of ﬁnite codimension in J. Then there is a ﬁnite-dimensional subspace J0 such that J = J0 +JZ . As J0 consists of ﬁnite rank operators, J0 X is ﬁnite-dimensional. As JX ⊂ J0 X + JZ X ⊂ J0 X + Z, we obtain that J0 X + Z is a ﬁnite-dimensional subspace invariant for J. As J has no non-trivial invariant closed subspaces of ﬁnite dimension, JX = 0, a contradiction. Corollary 4.8. Let J be a normed, Lie or Jordan, algebra of finite rank operators on an infinite-dimensional Banach space X. If J is an operator range then J is almost triangularizable. Proof. Let Γ be a maximal closed subspace chain consisting of invariant subspaces for J. Assume, to the contrary, that there is an inﬁnite-dimensional gap-quotient Y /Z of Γ. Let π be the quotient representation of J corresponding to Y /Z. Then π(J) = πT (V ) is an operator range. By Theorem 4.7, π(J) is reducible. This implies that there is an invariant closed subspace for J between Z and Y , a contradiction. So J is almost triangularizable. The following result is a Jordan algebra analogue of Theorem 3.8. Corollary 4.9. Let J be an infinite-dimensional Banach Jordan algebra. Assume that there is a bounded non-zero representation h of J by finite rank operators on a normed space X. Then J is not topologically simple. Proof. Without loss of generality, one may assume that X is complete. Assume, to the contrary, that J is topologically simple. Then h is injective. As h(J) is an inﬁnite-dimensional Jordan algebra of ﬁnite rank operators and an operator range, then it is almost triangularizable by Corollary 4.8. Then h(J) is not topologically simple by Corollary 4.6. Therefore J is not topologically simple, a contradiction. For Lie algebras of operators, Theorem 4.7 allows us to obtain the following statement of independent interest. Corollary 4.10. Let L be a Lie algebra of operators on an infinite-dimensional Banach space X, and let J be the set of all finite rank operators in L. If J is a non-zero operator range, then L has a non-trivial invariant closed subspace. Proof. If J is ﬁnite-dimensional, then the statement follows from [15, Theorem 4.33]. If J is inﬁnite-dimensional, then, taking into account that J has a

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non-trivial invariant subspace of ﬁnite dimension or codimension by Theorem 4.7, the statement follows by [6, Theorems 6.4 and 7.1]. If L consists of compact operators then the condition J = 0 also implies by [13, Theorem 2] that L has a non-trivial invariant closed subspace.

5. Non-simplicity of Lie and Jordan Algebras In this section we present results on the existence of ideals (non-necessarily closed) in Lie or Jordan algebras of operators which act on normed spaces or on vector spaces without topology. For the sake of simplicity, vector spaces and algebras are considered over algebraically closed ﬁelds of characteristic 0. A natural algebraic analog of Wojty´ nski’s problem would be the question: is every Lie algebra of nilpotent operators non-simple? But this question has a negative answer, as the following example shows. Example. Let A be a simple nil-algebra. Such exists by the celebrated result of Smoktunowicz [16]. By [3, Theorem 1.12], L = [A, A] is a simple Lie algebra. If a ∈ L, then a is nilpotent, an = 0, which yields ad(a)2n−1 = 0. Thus, there exist simple Lie algebras L such that ad(a) is nilpotent for all a ∈ L. The adjoint representation a → ad(a) of such an algebra is injective, so ad(L) is a simple Lie algebra of nilpotent operators. To obtain positive results we impose the conditions that some operators in algebras in question are of ﬁnite rank or compact. Theorem 5.1. Let L be an infinite-dimensional Banach Lie algebra such that ad(a) has at most countable spectrum for each a ∈ L. If h(L) contains a non-zero compact operator, for some bounded representation h on a normed space, then L is not simple. Proof. Assuming the contrary, we need only to consider the case when h(L) consists of compact operators on a Banach space and h is injective. Let M be the closure of h(L). Assume ﬁrst that there is an element a ∈ L whose ad(a) has a non-zero spectrum. Then there is an isolated non-zero point λ in the spectrum. As h · adL (b) = adM (h(b)) · h for every b ∈ L, we obtain, for Riesz projections pλ (adL (a)) and pλ (adM (h(a))) corresponding to λ, that −1 h · pλ (adL (a)) = (2πi) h · (μ − adL (a))−1 dμ Ω

⎛ ⎞ −1 = (2πi)−1 ⎝ (μ − adM (h(a))) dμ⎠ · h Ω

= pλ (adM (h(a))) · h,

(5.1)

where Ω is an admissible contour in C enclosing the point λ. As pλ (adL (a)) = 0, pλ (adM (h(a))) is not zero on h(L). Since the range of pλ (adM (h(a))) consists of ﬁnite rank operators by [15, Lemma 3.12], h(L) contains non-zero ﬁnite rank operators. Then L is not simple by Theorem 3.8, a contradiction.

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Therefore one may assume that L is Engel. If adM (h(a)) is not quasinilpotent, it follows from (5.1) that pλ (adM (h(a))) = 0 on h(L) for each isolated point λ = 0 in the spectrum of adM (h(a)). Since h(L) is dense in M , we obtain that pλ (adM (h(a))) = 0, a contradiction. So h(L) is also Engel and is not simple by Theorem 5.3. This implies that L is also not simple, a contradiction. Corollary 5.2. Let J be an infinite-dimensional closed, Lie or Jordan, algebra of operators on a Banach space. If J contains a non-zero compact operator, then J is not simple. Proof. The case of Lie algebras immediately follows from Theorem 5.1. Let J be Jordan and, without loss of generality, consist of compact operators. If J has ﬁnite rank operators, then one may assume that J consists of ﬁnite rank operators, and the assertion follows from Corollary 4.9. Otherwise no operator in J has Riesz projections corresponding to nonzero points of its spectrum. Therefore J consists of quasinilpotent compact operators, and the assertion follows by Corollary 3.4. Theorem 5.3. Let L be a non-one-dimensional Engel Lie algebra of operators on a normed space X. If L contains a non-zero compact operator, then L is not simple. Proof. Assume, to the contrary, that L is simple. Without loss of generality, one may of course assume that X is complete. As the set I = {a ∈ L : a is compact} is an ideal of L, it coincides with L. So L is an Engel Lie algebra of compact operators on a Banach space. Let A be the closed algebra generated by L. By [14, Corollary 11.5], A is commutative modulo the Jacobson radical. Then J = {a ∈ L : a is quasinilpotent} is an ideal of L which contains [L, L]. If J = 0 then L is commutative and is a sum of one-dimensional ideals. So L = J is a Lie algebra of quasinilpotent compact operators. By [15, Lemma 5.10], E(A) consists of quasinilpotent operators. Then the algebra B generated by ad(L) consists of quasinilpotent operators. As Ba does not contain a for every non-zero a ∈ L, either Ba or the one-dimensional subspace generated by a is a non-trivial ideal of L, a contradiction. Note that this result cannot be considered as a special case of Corollary 5.2 because we do not assume the completeness of L. We need an algebraic version of triangularizability. For this we introduce the notions of complete subspace chain, maximal subspace chain, continuous subspace chain in vector spaces which differ from the corresponding notions for normed spaces only by the absence of the word “closed” in the definitions. Gaps and gap-representations are deﬁned as above. A set M of operators on a vector space X is called strictly triangularizable if there is a maximal subspace chain consisting of invariant subspaces for M . The following theorem supplies us with examples of strictly triangularizable sets of operators.

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Theorem 5.4. Any Lie or Jordan algebra J of nilpotent finite rank operators generates an algebra of nilpotent operators, and every algebra of nilpotent operators is strictly triangularizable. Proof. Let A be the algebra generated by J. Every element a of A is a polynomial in some ﬁnite set K of elements of L. As every ﬁnite set of ﬁnite rank operators is locally ﬁnite in the sense that it generates a ﬁnite-dimensional algebra, the algebra B generates by K is ﬁnite-dimensional. Note that the Lie or Jordan algebra I generated by K is in B ∩ J and consists therefore of nilpotents. Taking a regular representation πl : b → bx of B on the vector space X = B, we have that πl (I) consists of nilpotent operators on a ﬁnite-dimensional space. By [4, Theorem 2.2.1], πl (B) consists of nilpotent operators on X. Then an+1 = πl (a)n a = 0 for some n, i.e. a is a nilpotent operator. This means that A is an algebra of nilpotent operators. Let Γ be a maximal chain of invariant subspaces for an algebra A of nilpotent operators. If Γ is not a maximal subspace chain, then there is a non-one-dimensional gap-quotient W = Y /Z. It is clear that B = πY /Z (A) is an non-zero algebra of nilpotent operators. As x ∈ / Bx for every non-zero x ∈ W , either Bx or {y ∈ W : By = 0} is a non-trivial invariant subspace for B. This means that there is an intermediate invariant subspace for A between Z and Y , a contradiction. This theorem shows that if A is a simple nil-algebra then E(A) is not a nil-algebra since it has no invariant subspaces. On the other hand, considering the regular representations, it follows from the theorem that every simple nil-algebra has a maximal subspace chain consisting of left ideals and a maximal subspace chain consisting of right ideals. The following lemma works in the pure algebraic setting as well as, with a minor modiﬁcation, for operators on a normed space (we have mentioned this in the previous section). Lemma 5.5. Let Γ be a complete subspace chain containing 0 and X, and a be a finite rank operator on X leaving subspaces of Γ invariant. If all gap-representations with respect to Γ vanish on a, then a is nilpotent. Proof. Let X0 = aX and let Λ = {Y ∩ X0 : Y ∈ Γ}. Since X0 is ﬁnite-dimensional, Λ is a ﬁnite chain of subspaces 0 = Z0 Z1 · · · Zm = X0 that are invariant for a. Let W2j = ∩{Y ∈ Γ : Zj ⊂ Y } for j > 0. Then Zj = ∩{Y ∩ X0 : Zj ⊂ Y ∈ Γ} = W2j ∩ X0 . For every Y ∈ Γ with Y W2j , we have that Y ∩ X0 Zj . If Zj−1 Y ∩ X0 then there is an intermediate subspace of Λ between Zj−1 and Zj , a contradiction. So Y ∩ X0 ⊂ Zj−1 . Thus for every Y ∈ Γ with W2j−2 ⊂ Y W2j for j > 1, we have that aY ⊂ Y ∩ aX = Y ∩ X0 = Zj−1 = W2j−2 ∩ X0 ⊂ W2j−2 .

(5.2)

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{Y ∈ Γ : Y W2j } for j > 0, and let W0 = 0 and Let W2j−1 be W2m+1 = X. Then W2j−1 ∈ Γ and it follows from (5.2) that aW2j−1 ⊂ W2j−2 . If W2j−1 = W2j then aW2j ⊂ W2j−2 , otherwise (W2j−1 , W2j ) is a gap of Γ, whence aW2j ⊂ W2j−1 by assumption. Taking into account that aW2m+1 ⊂ W2m , we obtain that aWj ⊂ Wj−1 for every j > 0. Hence a2m+1 X = a2m+1 W2m+1 ⊂ a2m W2m ⊂ · · · ⊂ aW1 ⊂ W0 = 0. In particular, Lemma 5.5 says that if Γ is continuous, a is automatically a nilpotent operator. On the other hand, it is clear that if Γ is maximal then every one of its gap-representation (being of rank one) vanishes on nilpotent operators leaving subspaces of Γ invariant. The following reﬂects this fact and is a sort of a converse assertion to Theorem 5.4. Theorem 5.6. Let J be a strictly triangularizable, Lie or Jordan, algebra of operators. Then the set of all nilpotent finite rank operators in J is an ideal of J. Proof. Apply Lemma 5.5.

The restriction on ranges in the theorem is essential. It is not difﬁcult to construct two nilpotent operators a, b in a (strictly) triangularizable algebra such that a + b is not nilpotent. The simplest example is following. Let Γ be a maximal continuous subspace chain in a vector space X. Then Γ(2) = {Y ⊕ Y : Y ∈ Γ} is a continuous chain of subspaces for X ⊕ X. Let a, b be deﬁned by a(x, y) = (0, x) and b(x, y) = (y, 0), for all (x, y) ∈ X ⊕ X. Then a and b are nilpotents, subspaces of Γ(2) are invariant for a and b, but (a + b)2 is the identity operator. In fact the Ringrose result can be transferred to the algebraic setting in full generality: for every ﬁnite rank operator a leaving the subspaces of a maximal (subspace or closed subspace) chain Γ invariant, the set {π(a)} for dim X < ∞ or {π(a)} ∪ {0} for dim X = ∞ is the spectrum of a, where π runs over gap-representations with respect to Γ. This is probably well known but we could not ﬁnd a reference. A Lie algebra L is called nil-Engel if ad(a) is a nilpotent operator, for every a ∈ L. Corollary 5.7. Let L be a non-one-dimensional nil-Engel Lie algebra. If ad(a) is of finite rank for some non-zero a ∈ L, then L is not simple. Proof. Let M = ad(L). The set I = {b ∈ L : ad(b) is of ﬁnite rank} is a nonzero ideal of L. So, one may assume that I = L. Then M is a Lie algebra of nilpotent ﬁnite rank operators. By Theorem 5.4, it has a non-trivial invariant subspace which is an ideal of L. Now we list several results which are algebraic analogs of the results in Sect. 4. Theorem 5.8. Let J be a non-one-dimensional, Lie or Jordan, algebra of operators on a vector space X. If J contains a non-zero finite rank operator and if J is strictly triangularizable, then J is not simple.

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Proof. Similar to the proof of Theorem 4.2 with using Lemma 5.5 instead of Lemma 4.1 and using Theorem 5.6. The following assertion is an algebraic analog of Corollary 4.3. Corollary 5.9. Let J be a non-one-dimensional Jordan algebra of nilpotent operators. If J contains a non-zero finite rank operator, then J is not simple. Proof. If J is simple, then it consists of nilpotent ﬁnite rank operators and is strictly triangularizable by Theorem 5.4. By Theorem 5.8, J is not simple, a contradiction. So J is not simple. Let M be a set of operators. We say that M is strictly almost triangularizable if there is a maximal chain of subspaces invariant for M that admits only ﬁnite-dimensional gap-quotients. Corollary 5.10. Let J be a infinite-dimensional, Lie or Jordan, algebra of operators on a vector space X. If J contains a non-zero finite rank operator and if J is strictly almost triangularizable, then J is not simple. Proof. Similar to the proof of Corollary 4.6 with using Theorem 5.8 instead of Theorem 4.2. Acknowledgements ˇ The authors are grateful to Yuri Bahturin and Peter Semrl for helpful conversations, and to the referee for useful remarks.

References [1] Bonsall, F.F.: Compact linear operators from an algebraic standpoint. Glasgow Math. J. 8, 41–49 (1967) [2] Bonsall, F.F.: Operators that act compactly on an algebra of operators. Bull. Lond. Math. Soc. 1, 163–170 (1969) [3] Herstein, I.N.: Topics in Ring Theory. The University of Chicago Press, Chicago (1969) [4] Jacobson, N.: Lie Algebras. Interscience, New York (1962) [5] Kennedy, M., Shulman, V., Turovskii, Y.V.: Invariant subspaces of subgraded Lie algebras of compact operators. Int. Eqs. Oper. Theory 63, 47–93 (2009) [6] Kissin, E., Shulman, V., Turovskii, Y.V.: Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals. J. Funct. Anal. 256, 323–351 (2009) [7] Kissin, E., Shulman, V., Turovskii, Y.V.: Banach Lie algebras with Lie subalgebras of finite codimension have Lie ideals. J. Lond. Math. Soc. 80, 603–626 (2009) [8] Livshits, L.: Locally finite-dimensional sets of operators. Proc. Am. Math. Soc. 119, 165–169 (1993) [9] Lomonosov, V.I.: On invariant subspaces of operators commuting with compact operators. Funct. Anal. Appl. 7, 213–214 (1973) [10] Radjavi, H., Rosenthal, P.: Invariant Subspaces. Springer, Berlin (1973)

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[11] Read, C.J.: Quasinilpotent operators and the invariant subspace problem. J. Lond. Math. Soc. 56, 595–606 (1997) [12] Ringrose, J.R.: Superdiagonal forms for compact linear operators. Proc. Lond. Math. Soc. 12, 367–384 (1962) [13] Shulman, V., Turovskii, Y.V.: Solvable Lie algebras of operators have invariant subspaces. Spectral Evol. Prob. (Simferopol, Crimea, Ukraine) 9, 38–44 (1999) [14] Shulman, V., Turovskii, Y.V.: Joint spectral radius, operator semigroups and a problem of W. Wojty´ nski. J. Funct. Anal. 177, 383–441 (2000) [15] Shulman, V., Turovskii, Y.V.: Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action. J. Funct. Anal. 223, 425–508 (2005) [16] Smoktunowicz, A.: A simple nil ring exists. Comm. Algebra 30, 27–59 (2002) [17] Vala, K.: On compact sets of compact operators. Ann. Acad. Sci. Fenn. Ser. A I 351, 1–8 (1964) [18] Wojty´ nski, W.: Engel’s theorem for nilpotent Lie algebras of Hilbert-Schmidt operators. Bull. Acad. Polon. Sci. 24, 797–801 (1976) [19] Wojty´ nski, W.: Banach-Lie algebras of compact operators. Stud. Math. 59, 263– 273 (1977) [20] Zelmanov, E.: Absolute divisors of zero in Jordan pairs and Lie algebras. Matem. Sbornik 112, 611–629 (1980) (in Russian) Matej Breˇsar Faculty of Mathematics and Physics University of Ljubljana Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics, University of Maribor Maribor, Slovenia e-mail: [email protected] Victor S. Shulman Department of Mathematics Vologda State Technical University Vologda, Russia e-mail: shulman [email protected] Yuri V. Turovskii Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan Baku, Azerbaijan e-mail: [email protected] Received: August 29, 2009. Revised: December 27, 2009.

Integr. Equ. Oper. Theory 67 (2010), 301–326 DOI 10.1007/s00020-010-1758-y Published online June 3, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

The Lower Weyl Spectrum of a Positive Operator Egor A. Alekhno Abstract. For the lower Weyl spectrum − σw (T ) = σ(T − K), 0≤K∈K(E)≤T

where T is a positive operator on a Banach lattice E, the conditions − − (T ) = σw (T ∗ ) holds, are established. In parfor which the equality σw ticular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ∞ such that the spectral radius − − (T )\(σf (T ) ∪ σw (T ∗ )), where σf (T ) is the Fredholm specr(T ) ∈ σw trum, is given. The conditions which guarantee the order continuity of the residue T−1 of the resolvent R(., T ) of an order continuous operator T ≥ 0 at r(T ) ∈ / σf (T ), are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, r(T ) ∈ / σf (T ), can not have the trivial band En∼ of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from Eσ∼ ⊗F , where F is a Banach lattice, Eσ∼ is a disjoint complement of the band En∼ of E ∗ . Mathematics Subject Classification (2000). Primary 47B65, 47A10, 47A11, 47A58; Secondary 46B42, 47B37, 47B07. Keywords. Positive operator, essential spectra, residue of resolvent, order continuity, r-norm, o-weak compactness, space ∞ .

1. Introduction and Preliminaries This paper is a continuation of research which was begun by the author in notes [6,7] and devoted to special subsets of the spectrum of a positive operator T on a Banach lattice E. For terminology, notions, and properties on the theory of Banach lattices and operators on them not explained or proved in this note, we refer to [1,8]; see also [15,16]. Throughout the note, unless otherwise stated, Banach lattices E and F will be assumed to be inﬁnite dimensional and an operator T from E into F (or into E) will be assumed linear and (norm) bounded.

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In Sects. 3–6 where the spectral properties are considered, spaces will be assumed complex, and in Sect. 7 spaces will be assumed real. By the term operator, we mean a linear operator. Let Z be a Banach space, T be a bounded operator on Z. We denote by N (T ) and R(T ) the null space and the range of T , respectively. That is, N (T ) = {z ∈ Z : T z = 0}, R(T ) = {T z : z ∈ Z}. An operator T is said to be Fredholm ([1, p. 156], [3, p. 33]) if the dimension of the null space N (T ) and the dimension of the quotient space Z/R(T ) are both ﬁnite. As usual, the spectrum of an operator T on Z will be denoted by σ(T ). The Fredholm spectrum ([1, p. 299], [3, p. 41]) of an operator T is the set σf (T ) = {λ ∈ C : λ − T is not a Fredholm operator on Z}, and the Weyl spectrum ([1, p. 312], [3, pp. 133, 135]) of an operator T is the set σw (T ) = σ(T + K), K∈K(Z)

where K(Z) is the set of all compact operators on Z. In the case, when T is a positive operator on a Banach lattice E, the lower Weyl spectrum [6] of an operator T is the set − σw (T ) = σ(T − K). 0≤K∈K(E)≤T

Clearly, the inclusions − (T ) ⊆ σ(T ) σf (T ) ⊆ σw (T ) ⊆ σw

(1.1)

hold. In particular, if E is an inﬁnite dimensional Banach lattice, then − (T ) = ∅. In [7] the example of an operator T ≥ 0 for which all incluσw sions of (1.1) are proper, was given. This paper is devoted to investigate some properties of the lower Weyl − (T ) of a positive operator T on a Banach lattice E and a probspectrum σw lems which are related to this. Before the statement of the main results, recall some definitions and notations in Riesz spaces and Banach lattices which will be used further on. Let E be a (Archimedean) Riesz space. The cone of all positive elements of E is denoted by E + , i.e., E + = {x ∈ E : x ≥ 0}. The band B of E is called a projection band ([8, p. 32], [16, pp. 61, 135]) whenever B ⊕ B d = E. For a Banach lattice E the Lorenz seminorm (see [13]) on E is deﬁned by the formula xL = inf {sup zα E : 0 ≤ zα ↑ |x|}. α

(1.2)

In a real Riesz space E a net xα is said to be order convergent [8, p. 30] to o x ∈ E, xα −→ x, whenever there exists a net yα satisfying |xα − x| ≤ yα ↓ 0. An operator T : E → F , where E and F are real Riesz spaces, is said to be a regular operator [8, p. 10] whenever it can be written as a difference of two positive operators, and is said to be order continuous [8, p. 42]

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o

whenever xα −→ 0 in E implies T xα −→ 0 in F . The collection of all regular operators and all order continuous operators from E into F will be denoted by Lr (E, F ) and Ln (E, F ), respectively. For an operator T : E → F we say that its modulus |T | exists [8, p. 9] whenever the supremum |T | := T ∨ (−T ) exists in the canonical order of the space of all linear maps from E into F . Obviously, if an operator T possesses a modulus, then T is regular. In the case, when E and F are complex Riesz spaces, an operator T from E into F is called regular if its real and imaginary parts are both regular; similarly for an order continuous operator. Moreover, in the case of a complex Riesz space o E the relation xα −→ x means that xα and x belong to the underlying real Riesz space ER of E and xα is order convergent to x in ER . Every operator T ∈ Ln (E, F ) [2] and likewise every operator T ∈ Lr (E, F ) is order bounded, that is, mapping order bounded subsets of E onto order bounded subsets of F . If E and F are (real or complex) Banach lattices, then [1, p. 22] every order bounded operator T from E into F is bounded. Therefore, the inclusions Ln (E, F ) ⊆ L(E, F ) and Lr (E, F ) ⊆ L(E, F ) hold, where, of course, L(E, F ) is the space of all bounded operators between E and F ; in particular, every positive operator T from E into F is bounded. The operator T : E → F acting from a Banach lattice E to a Banach lattice F is said to be an o-weakly compact [8, p. 310] whenever T maps order bounded subsets of E onto relatively weakly compact subsets of F . Clearly, every weakly compact operator and so every compact operator, is o-weakly compact. For an operator T ∈ Lr (E, F ), where E and F are real Banach lattices, its r-norm ([9], [15, p. 27]) is deﬁned by T r = inf {S : 0 ≤ S ∈ L(E, F ), |T x| ≤ S|x|, x ∈ E}. Under the r-norm the space Lr (E, F ) is a Banach space. If |T | exists its r-norm is T r = |T |. In the general case, the inequality T r ≥ T is valid. An operator T ∈ Lr (E, F ) is called r-compact [9] if it can be approximated in the r-norm by an operators of ﬁnite-rank. Every r-compact operator T possesses the modulus |T | and |T | is r-compact [9]. Thus, the space Kr (E, F ) of all r-compact operators is a Banach lattice under the r-norm and the ordering induced by the canonical order of L(E, F ). For a (real or complex) Riesz space E the order dual of E is deﬁned by E ∼ = Lr (E, R) and the order continuous dual of E is deﬁned by En∼ = Ln (E, R) [1, p. 21]. Through (En∼ )◦ will be denoted the polar of the band En∼ with respect to the dual system E, E ∗ , that is, (En∼ )◦ = {x ∈ E : x∗ x = 0 for all x∗ ∈ En∼ }. The band of all functionals in the Riesz space E ∼ that are disjoint from the band En∼ will be denoted by Eσ∼ . A Banach lattice E has order continuous norm [1, §2.3] if En∼ = E ∗ ; equivalently, E is an ideal of E ∗∗ . The Banach lattice E is called a KB-space ∗∗ [8, pp. 225–226] if (E ∗ )∼ n = E; equivalently, E is a band of E .

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We now state for convenience the following result [7] which has a speciﬁc importance below. Theorem 1.1. Let T be a positive operator on a Banach lattice E such that the spectral radius r(T ) ∈ / σf (T ) and there exists a net of a compact operators Kα satisfying 0 ≤ Kα x ↑ T x for all x ≥ 0. If T is order continuous and the − / σw (T ). order continuous dual En∼ separates the points of E, then r(T ) ∈

2. The Statement of the Main Results It is well known that for an arbitrary operator T on a Banach space Z the equality σw (T ) = σw (T ∗ ) holds, where T ∗ is the adjoint of T . In Sect. 3 the − (T ), will be question when an analogue holds for the lower Weyl spectrum σw discussed. The main result of this section is the next theorem. Theorem 2.1. Each of the following conditions ensures that for a positive − − operator T on a Banach lattice E the equality σw (T ) = σw (T ∗ ) holds: (a) (b) (c) (d)

− (T ) is valid (in particular, σw (T ) = σ(T )); The equality σw (T ) = σw − − (T ∗∗ ) is valid; The equality σw (T ) = σw The Banach lattice E has order continuous norm; The operator T is order continuous and there exists a Banach lattice F such that E = F ∗ and F = En∼ , moreover T ∗ (Eσ∼ ) ⊆ Eσ∼ .

− − (T ) = σw (T ∗ ) does not hold for an arbitrary Banach The equality σw lattice E. Namely, Sect. 4 will be devoted the construction of a weakly com− − (T ) = σw (T ∗ ) (see, in particular, pact positive operator T on ∞ for which σw Example 4.6). In Sect. 5, using this example, we will show that in Theorem 1.1 the assumption about the order continuity of T is essential (see Example 5.1). The assumption in Theorem 1.1 that En∼ separates E, will also be discussed. How important it is, is closely connected with the search conditions which guarantee the order continuity of the residue T−1 of the resolvent R(., T ) of an order continuous positive operator T at the point r(T ). This is the aim of Sect. 6. The main result of this section is the next theorem.

Theorem 2.2. Let T be a positive order continuous operator on a Banach lattice E, moreover r(T ) ∈ / σf (T ). Then each of the following conditions ensures that the residue T−1 of the resolvent R(., T ) at r(T ) is order continuous: (a) The operator T is o-weakly compact; (b) The band (En∼ )◦ is a projection band and Lorenz seminorm (1.2) on (En∼ )◦ is a norm. Moreover, it turns out that the problem of the order continuity of the residue T−1 of R(., T ) at r(T ) leads to a study of compact order continuous operators on spaces with trivial order continuous dual, and is connected with the approximation problem. This is a treatment of Sect. 7. We give here the basic result of it.

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Theorem 2.3. Let E and F be two Banach lattices, and let T : E → F be a non-zero order continuous operator. Then T ∈ / Eσ∼ ⊗ F , where the closure in Lr (E, F ) with the r-norm. In particular, if En∼ = {0}, then T is not r-compact. − − 3. When Does the Equality σw (T ) = σw (T ∗ ) Hold? − − First note that the inclusion σw (T ∗ ) ⊆ σw (T ) always holds. Indeed, if λ ∈ / − / σ(T −K) for some K ∈ K(E), 0 ≤ K ≤ T . So λ ∈ / σ(T ∗ −K ∗ ) σw (T ), then λ ∈ − / σw (T ∗ ). Below the conditions when the equaland 0 ≤ K ∗ ≤ T ∗ , that is, λ ∈ − − ∗ ity σw (T ) = σw (T ) holds, will be proved (see Theorem 2.1 above). For a Banach space Z, jZ will denote the natural embedding jZ : Z → Z ∗∗ . We shall identity jZ (Z) with the space Z without any further explanations. When we do so, the identiﬁcation will be clear from the context. The following lemma is known. We include here a short proof for the sake of completeness, and because the construction of the required operator is important later on.

Lemma 3.1. Let Z be a Banach space and an operator T ∈ L(Z ∗ ). The following assertions are equivalent: (a) The subspace Z of Z ∗∗ is T ∗ -invariant; (b) There exists a unique operator S ∈ L(Z) such that S ∗ = T ; (c) The operator T is σ(Z ∗ , Z)-continuous. In particular, S ∈ K(Z) iﬀ T ∈ K(Z ∗ ). If Z is a Banach lattice, then S ≥ 0 iﬀ T ≥ 0. Proof. (a) ⇒ (b) For an arbitrary element y ∈ Z there exists a unique element x ∈ Z such that jZ (x) = T ∗ (jZ (y)). ∗

(3.1)

∗

Put Sy = x. Fix z ∈ Z and z ∈ Z . The relations (S ∗ z ∗ )z = z ∗ (Sz) = jZ (Sz)z ∗ = (T ∗ (jZ (z)))z ∗ = jZ (z)T z ∗ = (T z ∗ )z hold, hence S ∗ = T . (b) ⇒ (c) If a net zα∗

σ(Z ∗ ,Z)

−→ 0, then we have (T zα∗ )z = zα∗ (Sz) → 0 for

σ(Z ∗ ,Z)

an arbitrary z ∈ Z, so T zα∗ −→ 0.

σ(Z ∗ ,Z)

(c) ⇒ (a) Fix an element z ∈ Z. Let zα∗ −→ 0. From the relations (T ∗ jZ (z))zα∗ = (T zα∗ )z → 0, it follows that the functional T ∗ jZ (z) is σ(Z ∗ , Z)-continuous, whence T ∗ jZ (z) ∈ Z. The last assertions follow at once from the equality S ∗ = T . Lemma 3.2. Let a Banach E be the direct sum of projection bands Bi , lattice n operator on E such that is, the equality E = i=1 Bi holds. If T is a positive n − − that Bi is T -invariant for all i = 1, . . . , n, then σw (T ) = i=1 σw (Ti ), where Ti is the restriction of T to Bi .

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− (Ti ) for all i, then there exist an operators Ki ∈ K(Bi ) Proof. If λ ∈ / σw / σ(Ti − Ki ). The operator satisfying the relations 0 ≤ Ki ≤ Ti and λ ∈ n n / σ(T − K) = i=1 σ(Ti − Ki ), K = i=1 Ki is compact, 0 ≤ K ≤ T and λ ∈ − (T ). so λ ∈ / σw − (T ), then λ ∈ / σ(T − K), where K ∈ K(E) For the converse, if λ ∈ / σw and 0 ≤ K ≤ T . Bands Bi are K-invariant therefore, K has a representation n / σ(Ti − Ki ) for K = i=1 Ki with nKi ∈−K(Bi ) and 0 ≤ Ki ≤ Ti . Clearly, λ ∈ all i, whence λ ∈ / i=1 σw (Ti ).

We proceed now to the proof of Theorem 2.1 (see Sect. 2) which collects − (T ) = the necessary conditions guaranteeing the validity of the equality σw − ∗ σw (T ). Proof of Theorem 2.1. (a) The desired equality follows from the relations − − − σw (T ∗ ) ⊆ σw (T ) = σw (T ) = σw (T ∗ ) ⊆ σw (T ∗ ).

(b) Sufﬁciently to observe that − − − − (T ∗∗ ) ⊆ σw (T ∗ ) ⊆ σw (T ) = σw (T ∗∗ ). σw − (T ∗ ). There exists a compact operator K on E ∗ such that (c) Let λ ∈ / σw / σ(T ∗ − K). Clearly, 0 ≤ K ∗ ≤ T ∗∗ and E is 0 ≤ K ≤ T ∗ and λ ∈ ∗ K -invariant as E is a T ∗∗ -invariant ideal of E ∗∗ . By Lemma 3.1, there exists a compact operator S on E satisfying 0 ≤ S ≤ T and S ∗ = K. − (T ). This implies Finally, the operator λ−(T −S) is invertible, so λ ∈ / σw − − ∗ σw (T ) ⊆ σw (T ), as required. (d) First of all we remark that F = (F ∗ )∼ n . In particular, F has order continuous norm. From the order continuity of the operator T , we have T ∗ (F ) ⊆ F . Deﬁne an operator T as the restriction of T ∗ to En∼ = jF (F ). Since Eσ∼ is T ∗ -invariant, then Lemma 3.2 implies − − (T ) ⊆ σw (T ∗ ). σw

(3.2)

By Lemma 3.1, there exists a positive operator S on F satisfying the following equalities S ∗ = T and jF (Sx) = T ∗ jF (x) for all x ∈ F (see (3.1)). So the restriction of T to F coincides with S, then it is easy to − − (S) = σw (T ). Using the assertion (c) and (3.2), we have see that σw − − − − − − (T ) = σw (S ∗ ) = σw (S) = σw (T ) ⊆ σw (T ∗ ) ⊆ σw (T ), σw − − (T ) = σw (T ∗ ). hence σw

− (T ) = In the next section an example of an operator T such that σw will be given (Example 4.6). The condition (b) of the previous theorem implies the necessity of the − − (T ) and σw (T ∗∗ ). Recall that for an operstudy of the connection between σw ator T ∈ L(Z), where Z is a Banach space, ρ∞ (T ) denotes the unbounded component in C of the resolvent set ρ(T ) of T . − (T ∗ ), σw

Theorem 3.3. Let T be a positive operator on a Banach lattice E having order continuous norm. Then: − − (T ) implies r(T ) ∈ σw (T ∗∗ ); (a) r(T ) ∈ σw

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(b) If for every K ∈ K(E ∗∗ ), 0 ≤ K ≤ T ∗∗ , the equality ρ(T ∗∗ − K) = ρ∞ (T ∗∗ − K) holds, then

− σw (T )

=

(3.3)

− σw (T ∗∗ ).

− (T ∗∗ ), then for some K ∈ K(E ∗∗ ), 0 ≤ K ≤ T ∗∗ , Proof. (a) If r(T ) ∈ / σw the operator r(T ) − (T ∗∗ − K) is invertible. Therefore, r(T ) > r(T ∗∗ − K) ≥ r(T − K|E ), where K|E is the restriction of K to E, that is, − (T ). r(T ) ∈ / σw − (b) If λ ∈ / σw (T ∗∗ ), then for some K ∈ K(E ∗∗ ), 0 ≤ K ≤ T ∗∗ , the relations [1, p. 256]

λ ∈ ρ(T ∗∗ − K) = ρ∞ (T ∗∗ − K) ⊆ ρ(T − K|E ) − hold, whence λ ∈ / σw (T ).

The equality (3.3) holds for every bounded operator T on a Banach space Z with the spectrum of T is at most countable. Namely, in this case ρ(T + S) = ρ∞ (T + S)

(3.4)

for every S ∈ I(Z), where I(Z) is the set of all inessential operators on Z [3, §7.1; in particular, p. 379], that is, I(Z) is a collection of an operators S ∈ L(Z) satisfying T + S is a Fredholm operator on Z whenever T is a Fredholm operator on Z. The inclusion ([1, p. 162], [3, p. 371]) K(Z) ⊆ I(Z) is valid. For the proof of (3.4) it is enough to show that for every operator S ∈ I(Z) the spectrum of T + S is at most countable. Indeed, in this case there is a path joining an arbitrary point of ρ(T + S) with a point of the circle {λ : |λ| = r(T + S)} and lying inside of ρ(T + S), whence the equality ρ(T + S) = ρ∞ (T + S) follows. So ﬁx λ ∈ σ(T + S)\σ(T ). The / σf (T + S). There is a path lying equality σf (T ) = σf (T + S) implies λ ∈ outside σf (T + S) and joining λ with some point ξ ∈ ρ(T + S), hence [1, p. 300] λ is an isolated point of σ(T + S) and so of σ(T + S)\σ(T ). Therefore, σ(T + S)\σ(T ) and so σ(T + S) is at most countable. In the case T ∈ Ln (E), where E is a Banach lattice, the band En∼ is ∗ T -invariant. Denote the restriction of T ∗ to En∼ by T . The proof of the following assertion is analogous to the part (b) of Theorem 3.3: If ρ(T ∗ − K) = − − (T ) ⊆ σw (T ∗ ). ρ∞ (T ∗ − K) for every 0 ≤ K ∈ K(E ∗ ) ≤ T ∗ , then σw The proof of the following assertion is quite similar to that of the part (a) of Theorem 3.3: If 0 ≤ T ∈ L(E), a closed ideal A of E is T -invariant, − then the inclusion r(T ) ∈ σw (T |A ), where T |A is the restriction of T to A, − implies r(T ) ∈ σw (T ). We shall close this section with few remarks dealing with Lozanovsky’s spectrum [6] σ(T − Q) σl (T ) = 0≤Q≤T Q≤K∈K(E)

of a positive operator T on a Banach lattice E. The conditions when σw (T ) ⊆ σl (T ) holds, are given in [7]. Again σl (T ∗ ) ⊆ σl (T ) is valid. The following theorem is similar to Theorem 2.1.

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Theorem 3.4. Each of the following conditions ensures that for a positive operator T on a Banach lattice E the equality σl (T ) = σl (T ∗ ) holds: (a) The equality σl (T ) = σl (T ∗∗ ) is valid; (b) E and E ∗ are atomic with order continuous norms; (c) E is a KB-space. Proof. (a) The proof is analogous to the proof of the part (b) of Theorem 2.1. (b) The inequalities 0 ≤ Q ≤ K, where K is a compact operator either on − E or on E ∗ , imply [17] the compactness of Q, hence σl (T ) = σw (T ) and − − (T ∗ ). By the part (c) of Theorem 2.1, we have σw (T ) = σl (T ∗ ) = σw − (T ∗ ), so σl (T ) = σl (T ∗ ). σw (c) The Banach lattice E is a band of E ∗∗ as E is a KB-space. Deﬁne / the real positive order projection from E ∗∗ onto jE (E) by PE . Let λ ∈ σl (T ∗ ). Then the operator λ − (T ∗ − Q) is invertible, where 0 ≤ Q ≤ T ∗ , Q ≤ K ∈ K(E ∗ ). The space E = jE (E) is Q∗ -invariant. By Lemma 3.1, there exists an operator Q0 on E satisfying Q∗0 = Q, 0 ≤ Q0 ≤ T

(3.5)

and jE (Q0 y) = Q∗ jE (y) for all y ∈ E (see (3.1)). The space E is also PE K ∗ -invariant. There exists K0 ∈ K(E) such that jE (K0 y) = PE K ∗ (jE (y)) for all y ∈ E. Then for x ∈ E + we have jE (Q0 x) = Q∗ (jE (x)) = PE Q∗ (jE (x)) ≤ PE K ∗ (jE (x)) = jE (K0 x), hence Q0 ≤ K0 ∈ K(E).

(3.6)

Thus, according to the invertibility of the operator λ − (T − Q0 ) and the relations (3.5) and (3.6), we have λ ∈ / σl (T ), so σl (T ) = σl (T ∗ ). In the proof of the part (c) of the previous theorem the existence of (positive) projection from E ∗∗ onto the ideal E was only used. In fact, this implies [14] that E is a KB-space. Remark also that E ∗ has order continuous norm iﬀ E ∗ is a KB-space. In particular, if E ∗∗ is a Banach lattice with an order continuous norm then E is a KB-space (see [8, p. 225]). It is not known if the equality σl (T ) = σl (T ∗ ) holds for an arbitrary positive operator T on a Banach lattice E.

4. An Example of an Operator T Such That − − σw (T ) = σw (T ∗ ) − − If a Banach lattice E has order continuous norm, then σw (T ) = σw (T ∗ ) (Theorem 2.1, (c)). The main example of a Banach lattice which does not have order continuous norm, is the space ∞ of all bounded sequences with the sup norm. Below the example of a weakly compact positive operator T − − (T ) = σw (T ∗ ), will be obtained. on the space ∞ such that σw

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First of all we recall some definitions and results about ∞ . The space ∞ is an AM -space with a unit e = (1, 1, . . .), so by Kakutani–Bohnenblust– Krein–Krein theorem [8, p. 194], ∞ is lattice isometric onto a space C(K) of all continuous functions on some Hausdorﬀ compact topological space K; ˘ in fact, K is homeomorphic to the Stone-Cech compactification βN of the set of natural numbers N. The Banach lattice ∗∞ can be identiﬁed with the direct sum of 1 and s∞ = {x∗ ∈ ∗∞ : x∗ (c0 ) = {0}}, where c0 is the space ∼ s of all sequences converging to zero, by this (∞ )∼ n = 1 and (∞ )σ = ∞ ; in s particular, 1 ⊥ ∞ . It is easily to see that an operator T on ∞ is norm bounded iﬀ T has a representation T x = (x∗1 x, x∗2 x, . . .), where x∗n ∈ ∗∞ and supn x∗n < ∞. In this case the order continuity of T is equivalent to the condition x∗n ∈ 1 for all n. The following result gives the conditions of the compactness of an operator T . Lemma 4.1. Let T x = (x∗1 x, x∗2 x, . . .) be a bounded operator on ∞ , x∗n ∈ ∗∞ . Then: (a) An operator T is compact iﬀ the set {x∗n }∞ n=1 is relatively norm compact in ∗∞ ; (b) An operator T is weakly compact iﬀ the set {x∗n }∞ n=1 is relatively weakly compact in ∗∞ . Proof. (a) Necessity. The set T U , where U is the closed unit ball of ∞ , is relatively norm compact. Therefore [11, p. 260], for an arbitrary m > 0 there exist a disjoint partition of the set of natural numbers N = i=1 Ni and elements ni ∈ Ni such that supn∈Ni |(T x)ni − (T x)n | ≤ for all x ∈ U and i = 1, . . . , m. So supn∈Ni |x∗ni x−x∗n x| ≤ , whence supn∈Ni x∗ni − m ∗ x∗n ≤ for i = 1, . . . , m, that is, {x∗n }∞ n=1 ⊆ i=1 B(xni , ), where ∗ ∗ B(xni , ) is the closed ball centered at xni with radius . The last inclusion means that the set {x∗n }∞ n=1 is totally bounded, so is relatively norm compact. The sufﬁciency contains in [5] (the proof of Theorem 2). For the sake of completeness we include the proof. Let the sequence xk ∈ ∞ , xk ≤ M , M > 0. By passing to a subsequence if needed, we can assume that limk→∞ x∗n xk = zn for all n. The sequence T xk converges to the element z = (z1 , z2 , . . .) ∈ ∞ in the norm. In fact, assuming by way of contradiction and passing to one more subsequence if necessary, we can ﬁnd a subsequence nk of N such that |x∗nk xk − znk | ≥ 1 > 0 for all k,

x∗nk

→

x∗0

in the norm of lim

k→∞

x∗0 xk

∗∞

(4.1)

and limk→∞ znk = z0 . Then

= z0 .

(4.2)

2 Indeed, ﬁx 2 > 0 and choose k1 such that x∗0 − x∗nk ≤ 3M , |z0 − 1 2 2 ∗ znk1 | ≤ 3 . There exists k2 such that |xnk xk − znk1 | ≤ 3 for each 1 k ≥ k2 . So |x∗0 xk − z0 | ≤ 2 for k ≥ k2 , that is, (4.2) holds. Finally, limk→∞ (x∗nk xk − znk ) = 0, contrary to (4.1).

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(b) The operator T on ∞ is weakly compact iﬀ [8, pp. 335, 347] T is a Dunford–Pettis operator, that is, T carries weakly convergent sequences σ(∞ ,∗ )

onto norm convergent sequences. Let xk −→∞ 0. The sequence T xk → 0 in the norm of ∞ iﬀ limk→∞ x∗n xk = 0 uniformly for n. By Grothendieck theorem [16, p. 126], the letter is equivalent to the asser tion that {x∗n }∞ n=1 is relatively weakly compact. Lemma 4.2. Let T x = (x∗1 x, x∗2 x, . . .) be a positive operator on the space ∞ , moreover T ≤ 1, x∗n ∈ s∞ and limn→∞ x∗n = 1. Then T k = 1 for all k; in particular, the spectral radius r(T ) = 1. Proof. The equalities T = T e and x∗n = x∗n e imply 1 ≥ T = supn x∗n ≥ 1, hence T = 1. So T k ≤ 1 for all k, that is, 0 ≤ T k e ≤ e.

(4.3)

Next, if x = (x1 , x2 , . . .) ∈ ∞ , 0 ≤ x ≤ e and limn→∞ xn = 1, then limn→∞ (T x)n = 1. Actually, we have (T x)n = x∗n x = x∗n e − x∗n (e − x) = x∗n → 1 as e − x ∈ c0 . From the last relations, the elementary induction and the inequalities (4.3), it is easy to see that limn→∞ (T k e)n = 1 for all k. Hence, T k = 1. Now by Gelfand formula [1, p. 243], the equality r(T ) = 1 is obvious. On the other hand, the space ∗∞ is an AL-space, so by Kakutani– Bohnenblust–Nakano theorem [8, p. 192], there exists a lattice isometry Φ∗∞ from ∗∞ onto the space of all integrable functions L1 (Ω∞ , μ∞ ), moreover the measure μ∞ is not σ-ﬁnite and is not purely atomic (if the functional x∗ ∈ ∗∞ is a generalized limit, then [4] the restriction of μ∞ to the support of the function Φ∗∞ x∗ is a non-atomic measure). By this, the band Φ∗∞ 1 is a L1 -space associated with an atomic measure. To continue our discussion, we need the following construction. Let (Ω, Σ, μ) be an arbitrary non-atomic probability measure space. Deﬁne the function r0 ≡ 1. There exist disjoint measurable sets A11 and A12 such that Ω = A11 ∪ A12 and μ(A11 ) = μ(A12 ) = 12 . Put r1 = χA12 − χA11 . Sets Ani , 1 ≤ i ≤ 2n , and the sequence rn , n ∈ N, will be constructed by induction. Assume that sets Ani , 1 ≤ i ≤ 2n , with the properties n

Ani ∩ Anj = ∅, i = j, Ω = 2n

2 i=1

Ani , μ(Ani ) =

1 2n

i

and the function rn = i=1 (−1) χAni have been constructed. Next, there exist sets An+1,i , 1 ≤ i ≤ 2n+1 , such that An+1,i ∩ An+1,j = ∅, i = j, Ω =

n+1 2

i=1

An+1,i , μ(An+1,i ) =

1 , 2n+1

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moreover if i is odd, then An+1,i ∪ An+1,i+1 = An, i+1 . Now the function rn+1 2 2n+1 i is deﬁned by rn+1 = i=1 (−1) χAn+1,i . Any sequence constructed as above is called a sequence of Rademacher functions (see [1, pp. 496–497]). Clearly, |rn | = 1. The relation rn

σ(L1 ,L∞ )

−→

0 is valid.

2n−1 Lemma 4.3. Let Bn = i=1 An,2i−1 , where sets Ani were deﬁned above. m 1 the equality μ( Then for every subsequence B n i i=1 Bni ) = 1 − 2m holds for ∞ all m. In particular, μ( i=1 Bni ) = 1. Proof. It sufﬁces to establish that the set Bn1 ∪ · · · ∪ Bnm can be written as a union of 2nm −2nm −m (different) sets of the form Anm ,i . To verify this, we use induction on m. For m = 1 the set Bn1 consists of 2n1 −1 = 2n1 −2n1 −1 sets of the form An1 ,i . For the induction step, suppose that the statement is true for some m. Then for every k the set Bn1 ∪· · ·∪Bnm consists of 2nm +k −2nm −m+k sets of the form Anm +k,i and the set Ω\(Bn1 ∪ · · · ∪ Bnm ) consists of 2nm −m sets of the form Anm ,i . Therefore, the set Bnm +k \(Bn1 ∪ · · · ∪ Bnm ) can be written as a union of 2nm −m+k−1 sets of the form Anm +k,i , so the set Bn1 ∪· · ·∪Bnm ∪Bnm +k consists of 2nm +k −2nm −m+k +2nm −m+k−1 sets of the form Anm +k,i . Taking k = nm+1 −nm , we obtain that the set Bn1 ∪· · ·∪Bnm+1 can be written as a union of 2nm+1 − 2nm+1 −m + 2nm+1 −m−1 = 2nm+1 − 2nm+1 −(m+1) sets of the form Anm+1 ,i , as desired.

If E is an ideal of the space of measurable functions L0 (ν), then for an arbitrary measurable set A the order projection on the space E is deﬁned by PA x = χA x, x ∈ E. Lemma 4.4. Let E be a Banach function space associated with L0 (ν). If the sequence zn ∈ E + , moreover zn → z = 0 in E, then there exist a subsequence znk , a set A with ν(A) > 0 and a number a > 0 such that znk ≥ aχA for all k. Proof. There exist a set B, ν(B) > 0, and a number a > 0 satisfying PB z ≥ ∞ 2aχB . Indeed, sets Bi = {s : z(s) ≥ 1i } have the property i=1 Bi = {s : z(s) > 0}, hence ν(Bi0 ) > 0 for some i0 as z = 0. Putting B = Bi0 , a = 2i10 , we obtain PB z ≥ 2aχB . The sequence zn has [1, p. 195] a subsequence znk which is relatively uniformly convergent to z, that is, there exists a positive function u ∈ L0 (ν) such that for each > 0 the inequality |znk −z| ≤ u holds for all k ≥ k . For some A ⊆ B, ν(A) > 0, the function PA u is bounded. Pick 1 > 0 such that the inequality 1 PA u ≤ aχA is valid. Then 1 PA u ≥ |PA znk − PA z| ≥ PA z − PA znk for k ≥ k1 , whence znk ≥ PA znk ≥ 2aχA − 1 PA u ≥ 2aχA − aχA = aχA , as claimed.

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Lemma 4.5. Let rn be a sequence of Rademacher functions on a space Ω with a non-atomic probability measure μ deﬁned above, and let E be a Banach function space associated with L0 (μ). If a sequence zn ∈ E such that the set {zn }∞ n=1 is relatively norm compact in E and the inequalities r0 +rn ≥ zn ≥ 0 hold in L0 (μ) for every n ∈ N, then zn → 0 in E. Proof. The sequence of Rademacher functions rn is equal rn = 2n 2n−1 i i=1 (−1) χAni , where sets Ani are deﬁned above, Bn = i=1 An,2i−1 . Obviously, the relations 0 ≤ PBn zn ≤ PBn (r0 + rn ) = 0 hold in L0 (μ) thus, PBn zn = 0

(4.4)

for all n. Assume that some subsequence znk of zn satisﬁes znk → z = 0 in E. By Lemma 4.4, we can suppose the existence of a set A, μ(A) > 0, and a number a > 0 for which the inequality znk ≥ aχA holds for all k. From (4.4), we get PBnk χA = χA∩Bnk = 0, that is, μ(A ∩ Bnk ) = 0. This gives ∞ μ(A ∩ k=1 Bnk ) = 0. By virtue of Lemma 4.3, we have μ(A) = 0, a contradiction. So z = 0. Therefore, the zero function is the only accumulation point of the set {zn }∞ n=1 in E, whence zn → 0 in E. Now we are ready to give an example of a positive operator T such that − − σw (T ) = σw (T ∗ ). Example 4.6 (a weakly compact positive operator T on the space ∞ satis− − (T ) = σw (T ∗ )). fying σw Step 1. The construction of T . Let Φ∗∞ be a lattice isometry from ∗∞ onto L1 (Ω∞ , μ∞ ). Fix an arbitrary measurable set A, μ∞ (A) > 0, such that the restriction μA of the measure μ∞ to A is the non-atomic measure. We can assume that μ∞ (A) = 1. χA . Consider an arbitrary Rademacher sequence rn , n ∈ N, Put x∗0 = Φ−1 ∗ ∞ supported by A with the measure μA . In the case of the necessity, we can consider the sequence rn as a sequence in L1 (Ω∞ , μ∞ ). Let x∗n = x∗0 + Φ−1 ∗ rn . σ(L1 ,L∗ )

∞

σ(∗ ,∗∗ )

∞ ∞ x∗0 . Clearly, x∗n ∈ s∞ for n ≥ 0. The relation rn −→ 1 0 implies x∗n −→ ∗ ∗ Consider the operator T on ∞ deﬁned by T x = (x1 x, x2 x, . . .). Then T = 1 and T ≥ 0 as x∗n = 1 and x∗n ≥ 0 for all n. By Lemma 4.1, (b), the operator T is weakly compact. Furthermore, the relation T (c0 ) = {0} implies

T ∗ (∗∞ ) ⊆ s∞ .

(4.5)

σf (T ) = σw (T ) = {0}, σ(T ) = {0, 1}

(4.6)

Step 2. The equalities

are valid. Since T is weakly compact, by Dunford–Pettis theorem [8, p. 337], the operator T 2 is compact, so T is a Riesz operator, i.e., σf (T ) = σw (T ) = {0}. In particular, every non-zero point λ ∈ σ(T ) is an eigenvalue of T . The equality T e = e implies {0, 1} ⊆ σ(T ). Fix a non-zero λ ∈ σ(T ). From the above,

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it follows that T x = λx for some x = 0. The equalities x∗0 x = limn→∞ x∗n x = λ limn→∞ xn give 1 lim xn = x∗0 x. (4.7) n→∞ λ So λxn = x∗n x = λ1 x∗0 x, hence xn = λ12 x∗0 x for all n. In particular, x∗0 x = 0. Using (4.7) once more, we have λ1 x∗0 x = λ12 x∗0 x therefore, λ = 1. Consequently, σ(T ) = {0, 1}. Remark also that we have established the following equality for the null space of the operator I − T N (I − T ) = {ae : a ∈ C}.

(4.8)

The proof of the equality N (I − T ∗ ) = {ax∗0 : a ∈ C}

(4.9)

is similar. Step 3. Let K be a compact operator on the space ∞ presenting in the form Kx = (z1∗ x, z2∗ x, . . .), zn∗ ∈ ∗∞ , and satisfying the inequalities 0 ≤ K ≤ T . Then zn∗ → 0 in ∗∞ . Indeed, the inequalities 0 ≤ K ≤ T imply 0 ≤ zn∗ ≤ x∗n . On the other hand, according to Lemma 4.1, (a), the set {zn∗ }∞ n=1 is relatively norm compact in ∗∞ , hence 0 ≤ Φ∗∞ zn∗ ≤ χA + rn and {Φ∗∞ zn∗ }∞ n=1 is relatively norm compact in L1 (μA ). Using Lemma 4.5, we conclude that Φ∗∞ zn∗ → 0 in L1 (μA ) thus, zn∗ → 0 in ∗∞ . Step 4. The equality − σw (T ) = {0, 1}

(4.10)

is valid. − − The inclusions σf (T ) ⊆ σw (T ) ⊆ σ(T ) and Step 2 give {0} ⊆ σw (T ) ⊆ − {0, 1}. We will show that 1 ∈ σw (T ). In fact, let 0 ≤ K ≤ T hold, where K ∈ K(∞ ), Kx = (z1∗ x, z2∗ x, . . .), zn∗ ∈ ∗∞ . The relations T −K ≤ T = 1 are valid and functionals x∗n − zn∗ belong to s∞ . Using Step 3, we have zn∗ → 0 in ∗∞ therefore, x∗n −zn∗ = (x∗n −zn∗ )e = 1−zn∗ e → 1 as n → ∞. This implies, − (T ). via Lemma 4.2, that r(T − K) = 1, so 1 ∈ σ(T − K). Hence, 1 ∈ σw

Step 5. The spectral radius r(T ) = 1 is a simply pole of the resolvent R(., T ), moreover for the residue T−1 of R(., T ) at the point λ = 1 satisﬁes T−1 = x∗0 ⊗ e. According to Step 2, we have σf (T ) = {0}. This guarantees that the point λ = 1 is a pole of R(., T ) and T−1 is a ﬁnite-rank operator. For every element x ∈ ∞ the equalities T 2 x = T (x∗1 x, x∗2 x, . . .) = (x∗0 x)e hold as limn→∞ x∗n x = x∗0 x. Hence, if (I − T )2 x = 0, then x − 2T x + T 2 x = 0 or xn − 2x∗n x + x∗0 x = 0 for all n. Therefore, limn→∞ xn = x∗0 x, so x∗n x = x∗0 x. 2 This implies xn = x∗0 x. We get N ((I − T ) ) = {ae : a ∈ C}. Now a glance 2 at (4.8) yields N (I − T ) = N ((I − T ) ), so α(I − T ) = 1, where α(I − T ) is the ascent of I − T . It follows [1, p. 267] that the point λ = 1 is a simply pole of R(., T ). Using (4.8), (4.9) and the equalities [1, pp. 266, 268]

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∗ ) = N (I − T ∗ ), we obtain a representation of R(T−1 ) = N (I − T ) and R(T−1 the operator T−1 in the form T−1 = x∗0 ⊗ e, as desired. ∗∗ ∗∗ Step 6. Let x∗∗ 0 be a positive functional on ∞ such that x0 (1 ) = {0} and ∗∗ ∗∗ ∗ x0 ≤ 1. Then for the operator K = x0 ⊗ x0 the inequalities 0 ≤ K ≤ T ∗ hold.

Indeed, if 0 ≤ x∗ ∈ 1 , then Kx∗ = 0 ≤ T ∗ x∗ . Let 0 ≤ x∗ ∈ s∞ . The equality T ∗ x∗ = x∗ x∗0 ∗

is valid, so Kx =

∗ ∗ (x∗∗ 0 x )x0

∗

≤ x

x∗0

(4.11)

∗ ∗

=T x .

∗∗ ∗∗ Step 7. Let x∗∗ 0 be a positive functional on ∞ such that x0 (1 ) = {0} and ∗∗ ∗∗ ∗ ∗ x0 ≤ 1, moreover x0 x0 = 1. Then for the operator K = x∗∗ 0 ⊗ x0 the ∗ equality r(T − K) = 0 holds.

Assume by way of contradiction that λ = r(T ∗ − K) > 0. The number λ is an eigenvalue of the operator T ∗ − K as σf (T ∗ − K) = σf (T ) = {0}. So T ∗ x∗ − Kx∗ = λx∗

(4.12)

∗ ∗ for some positive functional x∗ ∈ ∗∞ , x∗ = 1. The equality T−1 T ∗ = T−1 yields ∗ ∗ ∗ ∗ ∗ λT−1 x∗ = T−1 T ∗ x∗ − T−1 Kx∗ = T−1 x∗ − T−1 Kx∗ .

Using Step 5, we have ∗ ∗ ∗ ∗ (1 − λ)x∗0 = (1 − λ)T−1 x∗ = T−1 Kx∗ = ((Kx∗ )e)x∗0 = (x∗∗ 0 x )x0 .

Hence, ∗ λ = 1 − x∗∗ 0 x .

(4.13) ∗

According to (4.5) and (4.12), we have x ∈

s∞ .

It follows from (4.11)

that ∗ ∗ ∗∗ ∗ ∗ λx∗ = x∗ x∗0 − Kx∗ = x∗ x∗0 − (x∗∗ 0 x )x0 = (1 − x0 x )x0 . ∗ ∗∗ ∗ The equality (4.13) implies x∗ = x∗0 . So λ = 1 − x∗∗ 0 x = 1 − x0 x0 = 0, which ∗ is impossible. Therefore, r(T − K) = 0. − Step 8. The equality σw (T ∗ ) = {0} is valid.

There exists a positive functional x∗∗ 0 satisfying all conditions of Step 7. In fact, consider the functional x∗ on L1 (Ω∞ , μ∞ ) deﬁned by x∗ x = x dμ∞ . A

∗ Then the functional x∗∗ x∗ satisﬁes the desired conditions. Using 0 = Φ∗ ∞ Steps 6 and 7, we get the relations 0 ≤ K ≤ T ∗ and r(T ∗ − K) = 0, where ∗ − / σw (T ∗ ). On the other hand, K = x∗∗ 0 ⊗ x0 , hence 1 ∈ − − {0} = σw (T ∗ ) ⊆ σw (T ∗ ) ⊆ σw (T ) = {0, 1}. − (T ∗ ) = {0}. Thus, σw − − Finally, according to Steps 4 and 8 the relation σw (T ) = σw (T ∗ ) holds.

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Nevertheless, remark that for the operator T from the previous example the equalities σl (T ) = σl (T ∗ ) = {0} satisfy. Indeed, it sufﬁces to observe that the operator T is dominated by the rank-one operator 2x∗0 ⊗ e (we use the notations from Example 4.6). It is easy to see that the operator T from Example 4.6 is not order continuous. In general, if T is an arbitrary bounded operator on ∞ , then the order continuity of T is equivalent to the fact that the subspace 1 of ∗∞ is T ∗ -invariant. If s∞ is also T ∗ -invariant (equivalently, c0 is T -invariant) − − (T ) = σw (T ∗ ). It and T is positive, then, using Theorem 2.1, (d), we have σw − − ∗ is not known if the equality σw (T ) = σw (T ) holds for an arbitrary positive order continuous operator T on a Banach lattice E.

5. The Order Continuity is Important in Theorem 1.1! The objective of this section is to show the essentiality of the assumption about the order continuity of the operator T in Theorem 1.1 (see Sect. 1). Example 5.1 We will use the notations and results from Example 4.6. The positive operator T x = (x∗1 x, x∗2 x, . . .) acts on ∞ . The order continuous dual / (∞ )∼ n = 1 separates ∞ . According to (4.6) and (4.10), we have r(T ) = 1 ∈ − (T ). Obviously, the sequence Kn x = (x∗1 x, . . . , x∗n x, 0, 0, . . .) σf (T ) and 1 ∈ σw of positive compact operators satisﬁes Kn ↑ T (such sequence exists for each positive operator on a Banach function space associated with a σ-ﬁnite atomic measure). It is not known if the assumption in Theorem 1.1 that En∼ separates the points of E, is essential. The main tool of the proof of Theorem 1.1 (see [7], the proof of Theorem 20) is the following theorem about the Frobenius normal form [7]. Let B be a projection band of E. Through PB will be denoted the order projection onto B, i.e., if x = x1 +x2 , where x1 ∈ B, x2 ∈ B d , then PB x = x1 . Put TB = PB T PB and denote the restriction of TB to B by TB . Recall that in a Dedekind complete Riesz space every band is a projection band [8, p. 33]. Theorem 5.2 Let E be a Dedekind complete Banach lattice such that the order continuous dual En∼ separates the points of E. Let T be a positive order continuous operator on E, moreover r(T ) ∈ / σf (T ). Then there exist T -invariant bands Bi , E = Bn ⊃ Bn−1 ⊃ · · · ⊃ B1 ⊃ B0 = {0}, ) = r(T ) holds for some i = 1, . . . , n, then such that if the equality r(TBi ∩Bi−1 d the operator TB ∩B d is band irreducible. i

i−1

In fact (see [7, Sect. 2.2]), the assertion that En∼ separates E, is only necessary as the condition which guarantees the order continuity of the residue T−1 of R(., T ) at r(T ) (see Theorem 2.2 above). Thus, the question arises naturally: Does the order continuity of the operator T imply the order continuity of the residue T−1 in the general case? The aﬃrmative answer will main

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the validity of Theorem 5.2 and so Theorem 1.1 by the additional assumption of the Dedekind completeness without the condition that En∼ separates E. In the following section we will discuss this question in detail.

6. The Order Continuity of the Residue The main task of this section is to discuss the conditions of the order continuity of the residue T−1 of the resolvent R(., T ) at r(T ). Recall that if G is some set of linear functionals on Y and H ⊆ Z, where Y and Z are vector spaces, then k

yi ⊗ zi : yi ∈ G, zi ∈ H, i = 1, . . . , k, k ∈ N . G⊗H = i=1

In particular, if Y and Z are Banach spaces, then Y ∗ ⊗ Z is the set of all ﬁnite-rank bounded operators from Y into Z. Let T be a positive operator on some Banach lattice E. If the spectral radius r(T ) is a pole of the resolvent R(., T ), then R(., T ) has the Laurent expansion R(λ, T ) =

1 1 T−m + · · · + T−1 + T0 + (λ − r(T ))T1 + · · · , (λ − r(T ))m λ − r(T ) (6.1)

around r(T ), where m is the order of the pole of R(., T ) at r(T ). Mention / σf (T ) that T−m ≥ 0 and all operators Ti are real. The spectral radius r(T ) ∈ iﬀ [1, pp. 300–302] r(T ) > 0, the point r(T ) is a pole of R(., T ) and the residue T−1 ∈ E ∗ ⊗ E; by this T−i ∈ E ∗ ⊗ E, i = 1, . . . , m. Next, if an operator T ≥ 0 is order continuous, then [1, p. 256] R(λ, T ) is also order continuous for each λ > r(T ). Nevertheless, this fact and the relation (we assume that r(T ) is a pole of R(., T ) of the order m) m

T−m = lim (λ − r(T )) R(λ, T ) λ↓r(T )

(6.2)

do not imply the order continuity of the operator T−m in general. Actually, there exists [13] the example of a sequence of order continuous positive operators which converge in norm to a positive operator which is not order continuous (even σ-order continuous). Here the following result holds (see [13], Theorem 2.16, where the given fact was established for rather other class than the class of order continuous operators, while in the our case the proof of it is analogous and will be omitted). Lemma 6.1 Let E and F be Banach lattices and suppose that the Lorenz seminorm (1.2) on F is a norm, i.e., for x ∈ F the equality xL = 0 implies x = 0 (in particular, this is true when Fn∼ separates F or F is an AM -space with a unit). If a sequence Sk ∈ Ln (E, F ) converges in L(E, F ) to S ≥ 0, then S ∈ Ln (E, F ). Therefore, if 0 ≤ T ∈ Ln (E), the Lorenz seminorm on E is a norm and for R(., T ) the expansion (6.1) holds, then, using (6.2), we have T−m ∈ Ln (E). When E is a Dedekind complete AM -space with a unit, for the proof

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sufﬁciently to observe that [1, pp. 96–97] the space Ln (E) is a band of the Banach lattice L(E) and so is closed (for the operator norm). In fact, in this case it is easy to see from the equalities Ti = lim (λ − r(T ))−i R(λ, T ) − λ↓r(T )

1 i−1 Ti−1 , m T−m − · · · − (λ − r(T )) (λ − r(T ))

(6.3)

i ≥ −(m − 1), that the relations Ti ∈ Ln (E) hold. Lemma 6.2 Let E and F be two Riesz spaces, moreover En∼ = {0}. Then the equality (E ∼ ⊗ F ) ∩ Ln (E, F ) = {0} is valid. Proof. It sufﬁces to consider the case when an operator K ∈ (E ∼ ⊗ F ) ∩ k Ln (E, F ) has a representation K = i=1 x∗i ⊗ xi with functionals x∗i ∈ E ∼ real and elements x1 , . . . , xk in the underlying real Riesz space FR of F linearly independent. Let a net zα ↓ 0 in E. The net x∗i zα converges to some number ai , i = 1, . . . , k, as every functional x∗ ∈ E ∼ has the decomposio k tion x∗ = (x∗ )+ − (x∗ )− . Then Kzα −→ i=1 ai xi . Taking into account the k order continuity of K, the last relation yields i=1 ai xi = 0. Hence, ai = 0. Thus, x∗i are order continuous. Therefore, x∗i = 0 for all i. So K = 0, as required. The following result which at once follows from the previous lemma, gives a necessary condition of the order continuity of the residue. Theorem 6.3 Let T be a positive operator on a Banach lattice E, r(T ) ∈ / σf (T ). If there is at least one an order continuous operator among of the operators T−m , . . . , T−1 in the expansion (6.1), then En∼ = {0}. We start our discussion on a sufﬁcient conditions of the order continuity of T−1 with the next auxiliary results. Lemma 6.4 Let E be a Riesz space, let E0 be a ﬁnite dimensional vector subspace of the underlying real Riesz space of E, and let Γ be a vector subspace of the space of linear functionals on E separating the points of E. If a net σ(E,Γ)

o

zα ∈ E0 and zα −→ 0, then zα −→ 0 in E. Proof. The collection of restrictions of functionals from Γ to E0 will be denoted by Γ0 . Clearly, Γ0 separates E0 , so the topology σ(E0 , Γ0 ) is well deﬁned and it coincides with every Hausdorﬀ linear topology on E0 . In particular, it coincides nwith the topology generated by the norm z = max1≤i≤n |bi |, en is a basis of E0 . Then zα → 0 or where z = i=1 bi ei and e1 , . . . , n max1≤i≤n |bαi | → 0, where zα = i=1 bαi ei . So, maxβ≥α zβ ↓α 0, hence n o |zα | ≤ (maxβ≥α zβ ) i=1 |ei | ↓α 0, that is, zα −→ 0. The following lemma is similar to Lemma 6.1. Lemma 6.5 Let E and F be Banach lattices, moreover Fn∼ separates F . If a sequence Sk ∈ Ln (E, F ) converges in L(E, F ) to S ∈ E ∗ ⊗ F , then S ∈ Ln (E, F ).

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Proof. Let a non-zero functional x∗ ∈ Fn∼ . By Lemma 6.4, it is enough to o show that if a net xα −→ 0 in E, then x∗ (Sxα ) → 0. There exists a net zα ↓ 0 satisfying |xα | ≤ zα . Fix > 0 and an index α0 . Pick k0 with S − Sk0 ≤ ∗ 2 x∗

zα0 (we can assume zα0 = 0) and α1 ≥ α0 with |x (Sk0 xα )| ≤ 2 for α ≥ α1 . Then |x∗ (Sxα )| ≤ |x∗ ((S − Sk0 )xα )| + |x∗ (Sk0 xα )| ≤ + |x∗ (Sk0 xα )| ≤ 2 for α ≥ α1 , as claimed. Lemma 6.6 If a net zα in a Banach lattice E is relatively weakly compact set o

and zα −→ 0, then zα

σ(E,E ∗ )

−→ 0.

Proof. Pick a net yα with |zα | ≤ yα ↓ 0. Let z be a weak cluster point of the set {zα }, that is [11, p. 29], for each σ(E, E ∗ )-neighbourhood U of the point z and each α0 there exists α ≥ α0 such that zα ∈ U . Fix β. For α ≥ β we have z − yβ ≤ z − yα ≤ z − zα . For an arbitrary x∗ ∈ (E ∗ )+ and > 0 pick α ≥ β with x∗ (z − zα ) ≤ . Hence x∗ (z − yβ ) ≤ , so z ≤ yβ for all β. Consequently, z ≤ 0. Analogously, −z ≤ 0. Finally, z = 0. We obtain that the zero is only a weak cluster point of {zα }. Fix x∗ ∈ E ∗ . If the net x∗ zα does not converge to zero, then for every α there exists βα ≥ α such that |x∗ zβα | ≥ > 0.

(6.4)

The set {zβα } is a net. Indeed, for indexes βα1 , . . . , βαn pick α0 satisfying α0 ≥ βαi for all i = 1, . . . , n. Then βα0 ≥ βαi . Therefore [11, p. 29], the net {zβα } has a weak cluster point z . Obviously, z is also a weak cluster point of {zα }. So, as showed above, z = 0, which is impossible in view of (6.4). Thus, limα x∗ zα = 0. Lemma 6.7 Let T : E → F be an o-weakly compact order continuous operator between two Banach lattices. Then the inclusion R(T ∗ ) ⊆ En∼ is valid. In particular, for E = F and λ = 0 the inclusion N ∞ (λ − T ∗ ) ⊆ En∼ ∞ ∞ ∗ holds, where N (λ − T ) = k=1 N ((λ − T ∗ )k ). o

Proof. We may suppose that the operator T is real. Consider a net xα −→ 0 in E. Assume xα is order bounded. By the order continuity of T , we have o T xα −→ 0. On the other hand, by the o-weakly compactness of T , the set σ(F,F ∗ )

{T xα } is relatively weakly compact. Using Lemma 6.6, we obtain T xα −→ 0, hence x∗ (T xα ) → 0 for all x∗ ∈ F ∗ . So the relation T ∗ x∗ ∈ En∼ holds, as desired. In the case, when E = F and λ = 0, it sufﬁces to observe that [3, p. 3] the inclusion N ∞ (λ − T ∗ ) ⊆ R(T ∗ ) is valid. Corollary 6.8 Suppose that there exists a non-zero o-weakly compact order continuous operator T : E → F , where E and F are Banach lattice. Then En∼ = {0}.

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Lemma 6.9 Let E be a Banach lattice, T be a positive operator on E, moreover r(T ) ∈ / σf (T ). If B is a T -invariant projection band, then in a sufﬁciently small deleted neighbourhood of r(T ) we have PB R(λ, T )PB d = R(λ, TB )PB T PB d R(λ, TB d ). Proof. The relation [10] r(T ) ∈ / σf (TB ) ∪ σf (TB d ) is true. Thus, in some sufﬁciently small deleted neighbourhood U of r(T ) the operators R(., TB ) and R(., TB d ) are well deﬁned. The band B is R(λ, T )-invariant for λ ∈ U . Then for λ ∈ U we have (λ − TB )PB R(λ, T )PB d (λ − TB d ) = PB (λ − T PB )R(λ, T )(λ − PB d T )PB d = PB (λ − T + T PB d )R(λ, T )(λ − PB d T )PB d = PB (I + T PB d R(λ, T ))(λ − PB d T )PB d = PB (λ − PB d T )PB d + PB T PB d R(λ, T )(λ − T + PB T )PB d = PB T PB d (I + R(λ, T )PB T )PB d = PB T PB d , and the proof is ﬁnished.

Our purpose here is to establish Theorem 2.2 (see Sect. 2) giving the necessary conditions of the order continuity of the residue. ∗ Proof of Theorem 2.2. (a) The equality R(T−1 ) = N ((r(T )−T ∗ )m ) is valid, where m is the order of the pole of R(., T ) at r(T ). By Lemma 6.7, k ∗ ) ⊆ En∼ . The operator T−1 has a representation T−1 = i=1 x∗i ⊗ R(T−1 xi with elements x1 , . . . , xk linearly independent. Pick functionals k ∗ ∗ z1∗ , . . . , zk∗ with zj∗ xi = δji , i, j = 1, . . . , k. Since T−1 zj = ( i=1 xi ⊗ x∗i )zj∗ = x∗j , we have x∗j ∈ En∼ . Therefore, T−1 ∈ Ln (E). (b) Step 1. The case (En∼ )◦ = {0}, that is, the band En∼ separates E. In this case the validity of the given assertion was mentioned in [7]. The proof will be derived here because it was omitted in [7]. Moreover, for the completeness and by the reason of a significance of this assertion, we will give the proof in two different ways. The ﬁrst way. Since r(T ) ∈ / σf (T ), the operators T−m , . . . , T−1 in (6.1) are of ﬁnite rank. Now the desired assertion follows at once from the relations (6.2), (6.3) and Lemma 6.5. The second way. The idea of the proof is borrowed from [12], Propositions 4, 5, where the analogous statement was proved for the case σf (T ) = {0}. Let T be the restriction of T ∗ to En∼ . Then r(T ) = / σf (T ). If m be the order of the pole of R(., T ) at r(T ), then r(T ) ∈

N ((r(T ) − T )m ) ⊆ N ((r(T ) − T ∗ )m ), dim N ((r(T ) − T ∗ )m ) = dim N ((r(T ) − T )m ) < ∞.

(6.5)

Next, let T be the restriction of (T )∗ to (En∼ )∼ n . The Banach lattice E . If j is this natural one-to-one can be considered as a subspace of (En∼ )∼ n n embedding, then the equality jn (T x) = T (jn (x)) holds. So N ((r(T ) − T )m ) ⊆ N ((r(T ) − T )m ) ⊆ N ((r(T ) − (T )∗ )m ).

(6.6)

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Using dim N ((r(T ) − (T )∗ )m ) = dim N ((r(T ) − T )m ) < ∞ and the relations (6.5) and (6.6), we have ∗ ) = N ((r(T ) − T ∗ )m ) = N ((r(T ) − T )m ) ⊆ En∼ . R(T−1

Consequently, T−1 ∈ Ln (E). Step 2. The general case. The band (En∼ )◦ is T -invariant. Put B = (En∼ )◦ . The relation E = B d ⊕ B is valid. The operators TB and TB are order continuous and [7] r(TB ) = r(TB ) holds. The band B is T−1 -invariant, whence PB d T−1 PB = 0.

(6.7)

We wish to show that r(TB ) < r(T ) holds. To see this, let r(TB ) = r(T ). Then [10] r(TB ) ∈ / σf (TB ), so r(TB ) ∈ / σf (TB ). The Lorenz seminorm on the band B is a norm. By remarks after Lemma 6.1, the nonzero ﬁnite-rank operator (TB )−mB = (TB − r(TB ))mB −1 (TB )−1 , where (TB )−1 and mB is the residue and the order of the pole of R(., TB ) at r(TB ), respectively, is order continuous. According to the relation Bn∼ = {0} and Lemma 6.2, which is impossible. Thus, r(TB ) < r(T ). So [7] (T−1 )B = 0,

(6.8)

moreover r(TB d ) = r(TB d ) = r(T ). Since the band (B d )∼ n separates d B , using Step 1, we get the order continuity of the residue (TB d )−1 of R(., TB d ) at r(TB d ). If (TB d )−1 is the residue of R(., TB d ) at r(TB d ), then the band B d is (TB d )−1 -invariant and [7] the restriction of (TB d )−1 to B d coincides with (TB d )−1 . Therefore, from [7] again, the operator (T−1 )B d = (TB d )−1 ∈ Ln (E),

(6.9)

so the operators (TB d )−i = (TB d − r(TB d ))i−1 (TB d )−1 , i ≥ 1, are also order continuity. Using Lemma 6.9, we have PB R(λ, T )PB d = R(λ, TB )PB T PB d R(λ, TB d ), where functions R(λ, TB ) and R(λ, TB d ) are analytic on some deleted neighbourhood of the point r(T ) and have representations R(λ, TB ) =

∞

(−1)i (λ − r(T ))i R(r(T ), TB )i+1 ,

i=0

R(λ, TB d ) =

1 1 (T d )−1 (T d )−m + · · · + (λ − r(T ))m B λ − r(T ) B + (TB d )0 + · · ·

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(here m is the order of the pole of R(., TB d ) at r(T )). Using the order continuity of R(r(T ), TB ) and of (TB d )−i , i = 1, . . . , m, we get PB T−1 PB d ∈ Ln (E). So according (6.7), (6.8) and (6.9) T−1 = (T−1 )B + PB T−1 PB d + PB d T−1 PB + (T−1 )B d = PB T−1 PB d + (T−1 )B d ∈ Ln (E), and the proof is ﬁnished.

In the case of (a) the previous theorem is true if instead of r(T ) an / σf (T ), is arbitrary non-zero isolated point λ0 of the spectrum σ(T ), λ0 ∈ considered. In the general case it is not known if the order continuity of a positive operator T , r(T ) ∈ / σf (T ), implies the order continuity of the residue T−1 of R(., T ) at r(T ). Moreover, the author does not know an example of a Banach lattice E such that the band (En∼ )◦ is not a projection band. If 0 ≤ T ∈ L(E), where E is a Banach lattice, and r(T ) ∈ / σf (T ), then the residue T−1 of R(., T ) at r(T ) is a non-zero ﬁnite-rank operator. If, in addition to this, En∼ = {0} (most important examples of Banach lattices satisfying the property En∼ = {0} are an AM -space C[0, 1] and its a Dedekind completion), then by Lemma 6.2, the operator T−1 can not be order continuous. Thus, a question arises naturally: Can a positive order continuous operator T , r(T ) ∈ / σf (T ), on a Banach lattice E with the property En∼ = {0} exist? The aﬃrmative answer to this question means that the order continuity of the operator T does not imply the order continuity of the residue T−1 of R(., T ) at r(T ). As it see from the proof of Theorem 2.2 (see the case of (b), Step 2), the negative answer should have meant the order continuity of T−1 if (En∼ )◦ is a projection band. By Corollary 6.8, such operator T can not be o-weakly compact, so it can not be compact. In particular, there exists no a non-zero order continuity compact operator on C[0, 1]. In the next section this result will be derived (see Corollaries 7.8, 7.9 below) from other a more general result (see Theorem 2.3 above) which allows looking at the reason of the absence of a non-zero order continuous compact operator on C[0, 1] in a new fashion, namely with the point of view of the approximation problem. Remark also that the conditions 0 ≤ T ∈ Ln (C[0, 1]) imply r(T ) ∈ σf (T ) as the Lorenz seminorm on C[0, 1] is a norm.

7. Compact Order Continuous Operators Let E and F be two Banach lattices. If a Banach lattice F has the approximation property (in particular, is an AM -space), then every compact operator K : E → F is the limit in the operator norm of a sequence of ﬁnite-rank operators. On the other hand, every operator S ∈ E ∗ ⊗ F , where E and F are arbitrary Banach lattices, has a decomposition S = S1 + S2 with S1 ∈ En∼ ⊗ F ⊆ Ln (E, F ) and S2 ∈ Eσ∼ ⊗ F , moreover En∼ ⊗ F ⊥ Eσ∼ ⊗ F in L(E, F ) (see Lemma 7.3). Below the conditions when an order continuous operator T : E → F can be approximated by an operators from Eσ∼ ⊗ F , will be considered.

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Lemma 7.1 Let T and K be an operators from a Riesz space E into a Riesz space F (not necessarily Dedekind complete) such that T ∈ Ln (E, F ) and K ∈ (Eσ∼ )+ ⊗ F + . If an operator S : E → F satisﬁes T + S ≥ K and S ≥ 0, then S ≥ K. In particular, if the modulus |T − K| exists, then |T − K| ≥ K. k Proof. The operator K has a representation K = i=1 x∗i ⊗ xi , x∗i ∈ (Eσ∼ )+ , k x∗i ⊥ En∼ is valid. Fix > 0 and xi ∈ F + . Obviously, the relation i=1 k + z ∈ E . The equality [8, p. 46] inf{supα i=1 x∗i zα : 0 ≤ zα ↑ z} = 0 holds. k Consequently, there exists a net zα , 0 ≤ zα ↑ z, satisfying i=1 x∗i zα ≤ for all α. Then T (z − zα ) ≥ K(z − zα ) − S(z − zα ) =

k

i=1

(x∗i (z−zα ))xi −S(z−zα ) ≥

k

(x∗i z − )xi −Sz = Kz−

i=1

k

xi − Sz.

i=1

k Using z − zα ↓ 0 and T ∈ Ln (E, F ), we infer 0 ≥ Kz − i=1 xi − Sz. Since is arbitrary, this implies 0 ≥ Kz − Sz. Hence, S ≥ K, and we are done. If |T − K| exists, then the inequality |T − K| ≥ K − T implies T + |T − K| ≥ K, whence |T − K| ≥ K. Recall that a locally convex-solid topology on a Riesz space F is a locally convex topology generated by a family of lattice seminorms {pi : i ∈ A} on F , that is, seminorms having the property: |x| ≤ |y| in F implies pi (x) ≤ pi (y) for every i ∈ A (for detail, see [8, §11]). Lemma 7.2 Let 0 ≤ T ∈ Ln (E, F ), where E and F are Riesz spaces. Then for every element z ∈ E + satisfying T z > 0 there exist no a collection of an operators Ki ∈ (Eσ∼ )+ ⊗ F + , i ∈ A, and a locally convex-solid topology τz on F such that zero belongs to the τz -closures of the set {|T − Ki |z : i ∈ A}. Proof. Assuming by way of contradiction, we ﬁnd a net Kα ∈ (Eσ∼ )+ ⊗ F + and a locally convex-solid topology τz on F such that moduli |T − Kα | exist and τ

z 0. |T − Kα |z −→

(7.1) τz

The inequality |T z − Kα z| ≤ |T − Kα |z implies |T z − Kα z| −→ 0, hence τ

z Kα z −→ T z.

(7.2)

On the other hand, Lemma 7.1 guarantees |T − Kα | ≥ Kα , so |T − Kα |z − Kα z ≥ 0. Since the cone F + is τz -closed, it follows from the relations (7.1) and (7.2) that −T z ≥ 0. This implies T z ≤ 0, which is a contradiction. Clearly, if an operator K ∈ E ∗ ⊗ F , where E and F are Banach lattices, then the modulus of K exists and is r-compact (see Sect. 1). In fact, |K| belongs to the closure of (E ∗ )+ ⊗ F + in Lr (E, F ) with the r-norm [16, pp. 253–254, the proof of Theorem IV.4.6]. Lemma 7.3 Let E and F be two Banach lattices. If K1 ∈ En∼ ⊗ F and K2 ∈ Eσ∼ ⊗ F , then inf {|K1 |, |K2 |} = 0 in an ordered vector space L(E, F ).

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Proof. From the preceding discussion there exist moduli |K1 | and |K2 |. k Obviously, if K1 = i=1 x∗i ⊗ xi with x∗i ∈ En∼ , xi ∈ F , then |K1 | ≤

k

|x∗i ⊗ xi | =

i=1

k

k

|x∗i | ⊗ |xi | ≤

i=1

k

k

|x∗i | ⊗

i=1

k

|xi | = x∗ ⊗ x,

i=1

where x∗ = i=1 |x∗i | ∈ (En∼ )+ , x = i=1 |xi |. Analogously, |K2 | ≤ y ∗ ⊗ y, y ∗ ∈ (Eσ∼ )+ , y ∈ F + . Now remain to notice that (x∗ ⊗ x) ∧ (y ∗ ⊗ y) ≤ (x∗ ⊗ (x + y))∧(y ∗ ⊗ (x + y)) = (x∗ ∧ y ∗ )⊗(x + y) = 0 as x∗ ⊥ y ∗ .

Lemma 7.4 Let K ∈ Eσ∼ ⊗ F , where E and F are two Banach lattices. Then we have |K| ∈ (Eσ∼ )+ ⊗ F + , where the closure in Lr (E, F ) with the r-norm. Proof. There exists a sequence Kn ∈ (E ∗ )+ ⊗ F + converging to |K| in the r-norm. For an arbitrary n the operator Kn has a decomposition Kn = Kn1 + Kn2 with Kn1 ∈ (En∼ )+ ⊗ F + and Kn2 ∈ (Eσ∼ )+ ⊗ F + . Consider the band BEσ∼ ⊗F generated by the set Eσ∼ ⊗ F of Kr (E, F ). By the previous lemma, Kn1 ⊥ BEσ∼ ⊗F in Kr (E, F ). Obviously, |K| ∈ BEσ∼ ⊗F , so Kn1 ⊥ Kn2 − |K| in Kr (E, F ). Then |Kn1 | + |Kn2 − |K|| = |Kn1 + Kn2 − |K|| = |Kn − |K|| → 0 in Kr (E, F ). The inequality |Kn1 | + |Kn2 − |K|| ≥ |Kn2 − |K|| implies Kn2 → |K| in the r-norm, as claimed. Lemma 7.5 Let E and F be two Riesz spaces. If an operator T ∈ Ln (E, F ) possesses a modulus |T |, moreover for every x ∈ E + the equality |T |x = sup {T y : |y| ≤ x}

(7.3)

holds, then also |T | ∈ Ln (E, F ). For the case of a Dedekind complete Riesz space F the proof of this assertion can be found in [15, pp. 29–30, Proposition 1.3.9] (see also [8, p. 43, Theorem 4.3]). In the our case the proof of it is analogous, but for the sake of completeness we include the proof. Proof. Consider a net xα ↓ 0 in E. Let |T |xα ≥ z ≥ 0 in F . Fix an index β. For every |y| ≤ xβ and α ≥ β the inequality |y − (y + ∧ xα − y − ∧ xα )| ≤ xβ − xα holds. Therefore, T y + z − |T |xβ ≤ |T (y + ∧ xα )| + |T (y − ∧ xα )|. Hence, using the relations y + ∧ xα ↓ 0, y − ∧ xα ↓ 0 and the order continuity of T , we obtain T y + z ≤ |T |xβ . It follows from (7.3) that z ≤ 0. Now we are in a position to prove Theorem 2.3 (see Sect. 2).

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Proof of Theorem 2.3. Assume by way of contradiction that the operator T ∈ Lr (E, F ) and there exists a sequence Kn ∈ Eσ∼ ⊗ F which is convergent to T in the r-norm. Then the operator T is r-compact. In particular [9], the modulus of T exists and the equality (7.3) is valid. By Lemma 7.5, |T | is order continuous. Next, |Kn | → |T | in the r-norm. By Lemma 7.4, we can assume that Kn ∈ (Eσ∼ )+ ⊗ F + . Obviously, |Kn − |T ||z → 0 in F for each z ∈ E + . According to Lemma 7.2, we have |T |z = 0 and hence T = 0, a contradiction. Lemma 7.6 For a Banach lattice E and an AM -space F the next statements hold: (a) If a sequence Kn ∈ E ∗ ⊗ F converges in L(E, F ) to an operator T , then T ∈ Lr (E, F ) and Kn → T in the r-norm; (b) The equality K(E, F ) = Kr (E, F ) is valid. Proof. (a) Since F is an AM -space, by Krengel theorem [8, p. 271], the space K(E, F ) of all compact operators from E into F is a Banach lattice under the r-norm. The sequence Kn is a · r -Cauchy sequence. Indeed, Kn − Km r = |Kn − Km | = |Kn − Km |∗∗ .

(7.4)

∗

Next, the relation Kn − Km ∈ E ⊗ F implies [16, p. 296] ∗∗ |. |Kn − Km |∗∗ = |Kn∗∗ − Km

(7.5)

∗∗

The space F is [8, pp. 188, 193] a Dedekind complete AM -space with a unit, so [1, p. 96] ∗∗ ∗∗ | = Kn∗∗ − Km = Kn − Km . |Kn∗∗ − Km

Therefore, using (7.4) and (7.5), we have Kn −Km r = Kn −Km → 0 as n, m → ∞. Thus, Kn converges in the r-norm. Obviously, Kn → T in the r-norm. (b) By Grothendieck’s results ([16, p. 239], [1, pp. 125–129]), an AM -space F has the approximation property, that is, every operator K ∈ K(E, F ) can be approximated in the operator norm by an operators of ﬁnite rank. It remains to use of (a). Now we are ready to derive a number of corollaries of Theorem 2.3. Corollary 7.7 Let E and F be two Banach lattices, moreover F is an AM / Eσ∼ ⊗ F , where space. If a non-zero operator T belongs to Ln (E, F ), then T ∈ the closure in L(E, F ) with the operator norm. Proof. If there exists a sequence Kn ∈ Eσ∼ ⊗ F which is convergent to the operator T in L(E, F ), then by previous lemma, Kn → T in the r-norm. It is a contradiction in view of Theorem 2.3. Corollary 7.8 If E and F are Banach lattices, F is an AM -space, En∼ = {0}, then K(E, F ) ∩ Ln (E, F ) = {0}. Corollary 7.9 If E is an AM -space with En∼ = {0} (for example, E = C[0, 1]), then K(E) ∩ Ln (E) = {0}.

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In the case of a Dedekind complete Banach lattice F the space Lr (E, F ) is a Banach lattice under the r-norm [8, p. 248], so the band Ln (E, F ) of order continuous operators is closed in the r-norm, that is, the relations Sk ∈ Ln (E, F ), Sk → S in the r-norm, imply S ∈ Ln (E, F ). The next theorem improves this fact and Theorem 2.3. Theorem 7.10 Let E and F be two Banach lattices with F Dedekind complete, and let Si ∈ Ln (E, F ) and Ki ∈ Eσ∼ ⊗ F be two arbitrary collections of an operators, i ∈ A. If inf i∈A Si − Ki r = 0, then inf i∈A Ki r = 0. Proof. Clearly, inf i∈A |Si | − |Ki |r = 0. Fix > 0. Using Lemma 7.4, we ﬁnd an operators Qi ∈ (Eσ∼ )+ ⊗ F + such that inf |Si | − Qi r = 0, sup |Ki | − Qi r ≤ .

i∈A

i∈A

(7.6)

By Lemma 7.1, ||Si | − Qi | ≥ Qi , it follows from (7.6) that inf i∈A Qi r = 0. Using (7.6) once more, we have Ki r ≤ + Qi r for all i, so inf i∈A Ki r ≤ . Letting ↓ 0 yields inf i∈A Ki r = 0, as desired. Acknowledgements The author would like to express a deep thank to the referee for many helpful suggestions that has led to a better version of the paper. The author also thanks Prof. I. Labuda for useful remarks.

References [1] Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. In: Graduate Studies in Mathematics, vol. 50 (2002) [2] Abramovich, Y.A., Sirotkin, G.: On order convergence of nets. Positivity 9(3), 287–292 (2005) [3] Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer, Dordrecht (2004) [4] Alekhno, E.A.: Some special properties of Mazurs’ functionals, I. Trans. Math. Inst. Nats. Akad. Navuk Belarusi 12(1), 17–20 (2004) (Russian) [5] Alekhno, E.A.: Some properties of the weak topology in the space L∞ . Vestsi Nats. Akad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk (3), 31–37 (2006) (Russian) [6] Alekhno, E.A.: Some properties of essential spectra of a positive operator. Positivity 11(3), 375–386 (2007) [7] Alekhno, E.A.: Some properties of essential spectra of a positive operator, II. Positivity 13(1), 3–20 (2009) [8] Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, New York (1985) [9] Arendt, W.: On the o-spectrum of regular operators and the spectrum of measures. Math. Z. 178(2), 271–287 (1981) [10] Caselles, V.: On the peripheral spectrum of positive operators. Isr. J. Math. 58(2), 144–160 (1987)

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[11] Dunford, N., Schwartz, J.T.: Linear Operators. Part 1: General Theory. Wiley, New York (1958) [12] Grobler, J.J., Reinecke, C.J.: On principal T -bands in a Banach lattice. Integral Equ. Oper. Theory 28(4), 444–465 (1997) [13] Kitover, A.K., Wickstead, A.W.: Operator norm limits of order continuous operators. Positivity 9(2), 341–355 (2005) [14] Lozanovsky, G.Y.: On projections on some Banach lattices. Mat. Zametki 4(1), 41–44 (1968) (Russian) [15] Meyer-Nieberg, P.: Banach Lattices. Springer-Verlag, Berlin (1991) [16] Schaefer, H.H.: Banach Lattices and Positive Operators. Springer-Verlag, Berlin (1974) [17] Wickstead, A.W.: Converses for the Dodds-Fremlin and Kalton-Saab theorems. Math. Proc. Cambr. Philos. Soc. 120(1), 175–179 (1996) Egor A. Alekhno Faculty of Mechanics and Mathematics Belarussian State University Minsk, Belarus e-mail: [email protected] Received: February 12, 2009. Revised: October 17, 2009.

Integr. Equ. Oper. Theory 67 (2010), 327–339 DOI 10.1007/s00020-010-1779-6 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

A Note on an Operator-Theoretic Approach to Classic Boundary Value Problems for Harmonic and Analytic Functions in Complex Plane Domains Vladimir Ryzhov Abstract. A general spectral boundary value problems framework is utilized to restate Poincar´e, Hilbert, and Riemann problems for harmonic and analytic functions in an abstract operator-theoretic setting. Mathematics Subject Classification (2000). 30E25, 35J25, 47F05. Keywords. Spectral boundary value problems, M-operators, analytic functions, harmonic functions, Riemann–Hilbert problem, Poincar´e problem.

Introduction The last several years have witnessed an increased interest revealed by the mathematical community toward the abstract operator-theoretic methods in applications to spectral boundary value problems for differential operators and operator matrices. It is sufﬁcient to point out numerous recently published papers [1–5,8–11,16–20,25–28] along with their extensive bibliographies in order to appreciate the potential and vitality of emerging concepts and developing approaches. The general theory has been successfully applied to boundary value problems for general elliptic partial differential operators of even order in bounded Lipschitz domains, for nonselfadjoint (2 × 2)-block operator matriust operators, for additive ces acting in L2 (0, 1) × L2 (0, 1) known as Hain-L¨ perturbations of multiplication operators and some other cases inspired by the theory of elliptic partial differential operators. The presented paper is an attempt to embrace the study of boundary value problems of complex analysis by a general operator theoretic framework.

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We follow the line of reasoning developed in [26–28] and hope to demonstrate the utility of the abstract technique by reformulating classic problems of Poincar´e, Hilbert, and Riemann for harmonic and analytic functions in bounded simply connected and sufﬁciently smooth domains of the complex plane. Keeping this goal in mind, no attempt is made to report any function analytic results on solvability and respective properties of solutions of these problems. For the comprehensive treatment (at least in the classical setting) the interested reader is referred to the authoritative resources [6,14,22,23], where all the details can be found. The paper consists of two sections. After recollecting relevant definitions and statements from [26–28] we apply obtained results to the Laplace operator on the plane domain. Then, by appropriate choice of boundary conditions, we arrive at the standard statements of three aforementioned problems of complex analysis. As usual, R, C are the sets of real and complex numbers. For two separable Hilbert spaces H1 , H2 and a linear operator A acting from H1 to H2 the notation A : H1 → H2 means that A is deﬁned everywhere in H1 and bounded. The domain, range, and kernel of A are D(A), R(A), and ker(A), respectively. The writing A : f → g for f ∈ D(A) is equivalent to Af = g. The symbol ρ(A) is used for the resolvent set of A. If A : H → H and λ ∈ C, then the inclusion λ−1 ∈ ρ(A) means that the operator I − λA is boundedly invertible, i. e. the inverse (I − λA)−1 exists and is bounded in H. When discussing function theoretic concepts, the Lebesgue measure is assumed.

1. Spectral Boundary Value Problems 1.1. Spaces and Operators Let H be a Hilbert space and T : H → H be a bounded linear operator. Assume ker(T ) = {0} and denote A0 the left inverse of T so that A0 T f = f,

f ∈H

Note that A0 with the domain D(A0 ) = R(T ) needs not be bounded, closed or even densely deﬁned. Let E be another Hilbert space and Π : E → H be a linear mapping with Ker(Π) = {0} satisfying R(T ) ∩ R(Π) = {0} ˙ It follows that the linear set R(T ) + R(Π) is in fact the direct sum R(T )+ R(Π). ˙ Introduce the linear operator A in H with the domain D(A) := R(T )+ R(Π) by

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f ∈ H, ϕ ∈ E

Obviously, Ker(A) = R(Π),

R(A) = H,

A0 = A|R(T )

Since Ker(Π) = {0}, there exists the left inverse γ0 of Π such that Ker(γ0 ) = {0} and γ0 Πϕ = ϕ,

ϕ∈E

We extend the operator γ0 from its domain D(γ0 ) = R(Π) to the linear map Γ0 deﬁned on D(A) by Γ0 : T f + Πϕ → ϕ,

f ∈ H, ϕ ∈ E

It is clear that Ker(Γ0 ) = R(T ),

R(Γ0 ) = E,

γ0 = Γ0 |R(Π)

1.2. Spectral Boundary Value Problem The spectral boundary value problem for an unknown u ∈ D(A) is deﬁned for f ∈ H, ϕ ∈ E by the system of two equations (A − λI)u = f (1.2.1) Γ0 u = ϕ where λ ∈ C is the spectral parameter. Since T : H → H, the bounded inverse (I − λT )−1 exists for any λ in a small neighborhood of λ = 0. To justify the terminology, we note that in the applications given below the ﬁrst equation (1.2.1) is realized as the “main” equation for the operator A deﬁned in a bounded domain of the complex plane, whereas the equality Γ0 u = ϕ plays the role of a boundary condition. The operator Γ0 is interpreted as a “boundary map” deﬁned on D(A) with values in the “boundary space” E. Lemma 1.1. Suppose λ−1 ∈ ρ(T ). Then Ker(A − λI) = R((I − λT )−1 Π) Proof. Let u ∈ D(A). ˙ Since u ∈ D(A) = R(T )+R(Π) there exist f ∈ H and ϕ ∈ E such that u = T f + Πϕ. Then (A − λI)u = (A − λI)(T f + Πϕ) = f − λ(T f + Πϕ) = (I − λT )f − λΠϕ Assuming (A − λI)u = 0 and λ−1 ∈ ρ(T ) we obtain f = λ(I − λT )−1 Πϕ. Substitution into u = T f + Πϕ yields u = λT (I − λT )−1 Πϕ + Πϕ = [I + λT (I − λT )−1 ]Πϕ = (I − λT )−1 Πϕ To prove the inverse, put v = (I − λT )−1 Πϕ with some ϕ ∈ E and observe that due to the equality AT = I we have (A − λI)(I − λT )−1 = (A − λI)[I + λT (I − λT )−1 ] = A − λI + λ(A − λI)T (I − λT )−1 = A − λI + λ(I − λT )(I − λT )−1 = A

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Therefore, (A − λI)v = (A − λI)(I − λT )−1 Πϕ = AΠϕ = 0 since Ker(A) = R(Π), which completes the proof.

The following theorem describes solutions of (1.2.1) for λ−1 ∈ ρ(T ). Theorem 1.2. If λ−1 ∈ ρ(T ), then the problem (1.2.1) is uniquely solvable for any f ∈ H, ϕ ∈ E with the solution −1 f + (I − λT )−1 Πϕ uf,ϕ λ = T (I − λT )

(1.2.2)

Proof. The uniqueness of solution follows from the standard arguments. Namely, if u1 , u2 ∈ D(A) are two solutions, then for their difference u0 = u1 − u2 = Af0 + Πϕ0 with some f0 ∈ H, ϕ0 ∈ E we have (A − λI)u0 = 0 and Γ0 u0 = 0. Since Ker(Γ0 ) = R(T ) and Γ0 Π = I, the second identity gives ϕ0 = 0. Then the ﬁrst identity yields 0 = (A − λI)T f0 = (I − λT )f0 . The equality f0 = 0 follows now from the assumption λ−1 ∈ ρ(T ). Let us verify the representation (1.2.2). According to Lemma 1.1 the term (I − λT )−1 Πϕ belongs to Ker(A − λI). Thus we have −1 f = (A − λI)(I −λT )−1 T f = AT f = f (A − λI)uf,g λ = (A − λI)T (I −λT )

deﬁned by (1.2.2) due to the The condition Γ0 u = ϕ is fulﬁlled for uf,ϕ λ obvious calculation −1 Πϕ = Γ0 [I + λT (I − λT )−1 ]Πϕ = Γ0 Πϕ = ϕ Γ0 uf,ϕ λ = Γ0 (I − λT )

where we used the equalities Ker(Γ0 ) = R(T ) and Γ0 Π = I. The proof is complete. 1.3. M-operator Let Λ be a linear operator in E deﬁned on the domain D(Λ) ⊂ E. Intro˙ duce the linear map Γ1 on D(Γ1 ) = R(T )+ΠD(Λ) ⊂ D(A) with the range R(Γ1 ) ⊂ E by Γ1 : T f + Πϕ → Π∗ f + Λϕ,

f ∈ H, ϕ ∈ D(Λ)

Obviously, Γ1 T = Π∗ ,

Γ1 Π = Λ

(1.3.1)

Note that Γ1 T is bounded as an adjoint to the bounded operator Π. In applications below Γ1 is realized as the “second boundary operator” complementary to the map Γ0 . Definition 1.3. The M-operator is an operator-valued function M (λ) of the spectral parameter λ defined on D(Λ) for λ−1 ∈ ρ(T ) by the equality M (λ)Γ0 uλ = Γ1 uλ ,

uλ ∈ Ker(A − λI) ∩ D(Γ1 )

To check correctness of this definition, assume uλ ∈ Ker(A−λI)∩D(Γ1 ) and λ−1 ∈ ρ(T ). Then according to Lemma 1.1 uλ = (I − λT )−1 Πϕ where Γ0 uλ = ϕ with some ϕ ∈ D(Λ). Therefore, Γ0 uλ = 0 means ϕ = 0, which in turn implies uλ = 0 and Γ1 uλ = 0.

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Theorem 1.4. For λ−1 ∈ ρ(T ) M (λ) = Γ1 (I − λT )−1 Π = Λ + λΠ∗ (I − λT )−1 Π,

D(M (λ)) = D(Λ)

Proof. Due to Lemma 1.1, any uλ ∈ Ker(A − λI) has the form uλ = (I − λT )−1 Πϕ with some ϕ ∈ E. If ϕ ∈ D(Λ) then uλ ∈ D(Γ1 ) and Γ1 uλ = Γ1 (I − λT )−1 Πϕ = Γ1 [I + λT (I − λT )−1 ]Πϕ = [Γ1 Π + λΓ1 T (I − λT )−1 Π]ϕ = [Λ + λΠ∗ (I − λT )−1 Π]Γ0 uλ where we used the equalities Γ1 Π = Λ, Γ1 T = Π∗ , and Γ0 uλ = ϕ.

Corollary 1.5. Values of the analytic operator-function M (λ) − M (0), λ−1 ∈ ρ(T ) are bounded operators in E. 1.4. Boundary Conditions Let β0 , β1 be two linear operators, such that D(Λ) ⊂ D(β0 ) and β1 : E → E. Consider the following spectral boundary value problem for u ∈ D(Γ1 ) ⊂ D(A) (A − λI)u = f (1.4.1) (β0 Γ0 + β1 Γ1 )u = ϕ where f ∈ H, ϕ ∈ E and λ ∈ C is the spectral parameter. Theorem 1.6. Assume λ−1 ∈ ρ(T ) is such that the equation [β0 + β1 M (λ)] ψ = g

(1.4.2)

for an unknown ψ ∈ E is uniquely solvable for any g ∈ E. Then the boundary value problem (1.4.1) has a unique solution uf,ϕ ∈ λ D(A) given by −1 f + (I − λT )−1 ΠΨf,ϕ uf,ϕ λ = T (I − λT ) λ

where

Ψf,ϕ λ

(1.4.3)

∈ E solves (1.4.2) with g = ϕ − β1 Π∗ (I − λT )−1 f

(1.4.4)

Remark 1.7. Formally, the left hand side of (1.4.2) is meaningful only for ψ ∈ D(M (z)) = D(Λ). However, the domain of β0 + β1 M (z) can be wider than D(Λ), for example, if the operator sum β0 + β1 Λ is bounded. We take such possibilities into consideration and the general solution to (1.4.2) is sought in the whole space E. Proof. Due to Lemma 1.1, the second term in (1.4.3) belongs to Ker(A − λI). Therefore, −1 f =f (A − λI)uf,ϕ λ = (A − λI)T (I − λT )

Thus the element (1.4.3) solves the ﬁrst equation in (1.4.1). Let us verify that the second equation in (1.4.1) is also satisﬁed. To that end we need to calculate the sum (β0 Γ0 + β1 Γ1 )uf,ϕ where uf,ϕ is deﬁned λ λ

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f,ϕ by (1.4.3). Assuming for the moment that Ψf,ϕ λ ∈ D(Λ) so that uλ ∈ D(Γ1 ), we have according to the properties of Γ0 , Γ1 , and Theorem 1.4 f,ϕ −1 Γ0 uf,ϕ f + (I − λT )−1 ΠΨf,ϕ λ = Γ0 [T (I − λT ) λ ] = Ψλ −1 ∗ −1 Γ1 uf,ϕ f +(I −λT )−1 ΠΨf,ϕ f + M (λ)Ψf,ϕ λ = Γ1 [T (I −λT ) λ ] = Π (I −λT ) λ

Since Ψf,ϕ satisﬁes (1.4.2), (1.4.4), we have λ f,ϕ ∗ −1 (β0 Γ0 + β1 Γ1 )uf,ϕ f + M (λ)Ψf,ϕ λ = β0 Ψλ + β1 [Π (I − λT ) λ ] ∗ −1 = (β0 + β1 M (λ))Ψf,ϕ f λ + β1 Π (I − λT )

= ϕ − β1 Π∗ (I − λT )−1 f + β1 Π∗ (I − λT )−1 f = ϕ as required. Now the condition uf,ϕ ∈ D(Γ1 ) can be relaxed by treating the λ expression β0 Γ0 +β1 Γ1 as an operator sum initially deﬁned on D(Γ1 ) and then extended to its maximal domain contained in D(A) ⊂ E. is a solution to the system (A − Calculations above show that uf,ϕ λ λI)u = f , Γ0 u = Ψf,ϕ . According to the uniqueness part of Theorem 1.2, λ this solution is unique if two equalities f = 0 and Ψf,ϕ = 0 imply ϕ = 0. λ In turn, this implication follows from the unique solvability of (1.4.2). The proof is complete. 1.5. Operator Node In this subsection we discuss connections between the spectral boundary value problems (1.2.1), (1.4.1) and the theory of open systems, thereby translating the setting of previous sections into alternative, in some sense more intuitive terms. The reader is referred to the books [12,13,24,29] for the background information on the linear systems theory. The collection {T, Π, Λ; H, E} of two Hilbert spaces and three operators introduced above, deﬁnes the block operator matrix acting in the space H ⊕E and often called the operator node T Π M= (1.5.1) Π∗ Λ deﬁned as follows: The The node M is associated with an open system M are identiﬁed with H, state and the input–output spaces of the system M are realized as elements of H and are E respectively. The inner states of M governed by the equation (A−λI)u = 0 with λ ∈ C. In other words, the inner states are vectors from Ker(A − λI). Elements of E represent external data by the control sent to the input and read from the output of the system M −1 and observation processes. For λ ∈ ρ(T ) and ϕ ∈ E, the control process is −1 Πϕ ∈ Ker(A − λI). given as the input-state mapping ϕ → uϕ λ = (I − λT ) The state-output mapping representing the observation process is deﬁned ϕ ϕ as uϕ λ → Γ1 uλ assuming uλ ∈ D(Γ1 ), or equivalently, ϕ ∈ D(Λ). In this model the transfer function that maps inputs into outputs coincide with the M-operator M (λ) : ϕ → Γ1 uϕ λ . The map Λ is called the feedthrough operator. The role of Λ becomes clear if we note that for λ = 0 the input-output mapping reduces to the correspondence ϕ → Λϕ.

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The condition (β0 Γ0 + β1 Γ1 )u = ϕ can be interpreted as a description by “mixing” its inputs and outputs into a of the system obtained from M new input deﬁned by the operator sum β0 Γ0 + β1 Γ1 . The second term represents the feedback procedure that sends the original output Γ1 uϕ λ , modiﬁed along the way by the operator β1 , back to the system’s input. Thus state vectors of this new system are elements u ∈ Ker(A − λI) that satisfy the equation (β0 Γ0 + β1 Γ1 )u = ϕ, where ϕ is the input. In a similar way, with a suitable choice of operators α0 , α1 , the output can be redeﬁned as the sum ϕ ϕ (α0 Γ0 + α1 Γ1 )uϕ λ , where (A − λ)uλ = 0 and Γ0 uλ = ϕ is the input of system A combination of these two “mixing” operations leads to the system with M. ϕ the output (α0 Γ0 + α1 Γ1 )uϕ λ where uλ ∈ Ker(A − λI) is the state satisfying the condition (β0 Γ0 + β1 Γ1 )u = ϕ, and ϕ is considered as the input. The is called the fractional linear transformation of M. It is resulting system N not difﬁcult to see that the mapping N (λ) : (β0 + β1 M (λ))ϕ → (α0 + α1 M (λ))ϕ Here ϕ ∈ D(Λ) is regarded as a parameter. If is the transfer function of N. the operator sum β0 + β1 M (λ) is boundedly invertible, then N (λ) can be written as a linear operator N (λ) = (α0 + α1 M (λ))(β0 + β1 M (λ))−1 In general, if β0 + β1 M (λ) is not invertible, N (λ) is a multi-valued map, or in other terminology, a linear relation on the Hilbert space E ⊕ E. Trivial inputs satisfying (β0 + β1 M (λ))ϕ = 0 correspond to the inner states that always exist and produce non-trivial output regardless of the input applied to the system. is obtained Expression for the feedthrough operator Θ of the system N by setting λ = 0, Θ : (β0 + β1 Λ)ϕ → (α0 + α1 Λ)ϕ Assuming (β0 + β1 Λ) is invertible, Θ = (α0 + α1 Λ)(β0 + β1 Λ)−1 . Existence of both factors, as well as existence of their product here and in the formula for N (λ) above, requires further justiﬁcation, especially in cases where participating operators are unbounded. The detailed discussion of relevant issues in the setting of abstract boundary value problems can be found in [28]. A brief illustration of these concepts is given below using the Hilbert boundary value problem for analytic functions as an example.

2. Applications Let D ⊂ C be a bounded simply connected domain of the complex plane C with smooth boundary ∂D. Let us deﬁne the main and the boundary Hilbert spaces to be H = L2 (D), E = L2 (∂D). It is well known that the inhomogeneous boundary value problem for the Dirichlet Laplacian in H Δu = f,

u|∂D = 0

(2.0.2)

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is uniquely solvable for any f ∈ H. Let T : H → H be the corresponding solution operator T : f → u acting in L2 (D). The range R(T ) consists of all functions from the Sobolev class W22 (D) vanishing on the boundary [7]. Therefore, R(T ) is dense in L2 (D). Following the general schema, we deﬁne Π : L2 (∂D) → L2 (D) to be the solution operator for the problem Δu = 0,

u|∂D = ϕ

(2.0.3)

where ϕ ∈ L2 (∂D). Clearly, Π is the operator of harmonic continuation of functions deﬁned on ∂D into the domain D. It is an integral operator with the kernel expressed in terms of Green’s function of the domain D. If uϕ is a solution to (2.0.3) corresponding to ϕ ∈ L2 (∂D), then the element ϕ is uniquely (in the sense of L2 (∂D)) recovered from uϕ by the boundary trace mapping γ0 : u → u|∂D . Thus, γ0 Π = IE . The solution of homogeneous problem Δu = 0 with the condition u|∂D = 0 is trivial and therefore the equality R(T ) ∩ R(Π) = {0} holds. Deﬁne the ˙ operator A as the Laplacian with the dense domain D(A) = R(T )+R(Π) and introduce Γ0 on the domain D(Γ0 ) = D(A) as the trace operator γ0 extended by the null mapping on the set R(T ) = D(A)\R(Π). Denote A0 the restriction of A to R(T ). Trace properties of functions from the Sobolev class W22 (D) imply that A0 is in fact the Dirichlet Laplacian on D(A0 ) = R(T ) and A0 T = I. ∂u Let Γ1 : u → ∂n be the trace of the outer normal derivative of ∂D u ∈ D(A) deﬁned on the dense set of sufﬁciently smooth functions in the and application of closure of D. The integral representation for T = A−1 0 the Fubini theorem show that Γ1 T = Π∗ : H → E, as prescribed by (1.3.1), see [28]. All components of the operator node M from (1.5.1) are now completely determined except for the parameter Λ deﬁned on the domain D(Λ) ⊂ L2 (∂D). Below we give three definitions of Λ resulting in three boundary value problems for harmonic and analytic functions in D. We are concerned with Eq. (2.0.3) and for simplicity’s sake, the only case of the system (1.4.1) with λ = 0 is discussed. Relevant results for the spectral problem with any λ ∈ C easily follow from the abstract considerations carried out in Sect. 1. 2.1. Poincar´e Problem Definitions of operators Γ0 and Γ1 given above suggest the “natural” choice for Λ. Since Γ1 maps a smooth function deﬁned in D to the trace of its normal derivative on ∂D and Π is the operator of harmonic continuation, we have for a smooth function ϕ on ∂D ∂uϕ 0 Γ1 Π : ϕ → ∂n ∂D where uϕ 0 is the solution to Au = 0, satisfying the boundary condition u|∂D = ϕ. Operator Ω := Γ1 Π is called the Dirichlet-to-Neumann map for the Laplacian Δ in D. It is known that Ω is selfadjoint as an operator in L2 (∂D) deﬁned on the Sobolev class W21 (∂D). Put Λ = Ω with the domain D(Λ) = W21 (∂D).

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According to Theorem 1.6, for two mappings β0 : W21 (∂D) → L2 (∂D) and β1 : L2 (∂D) → L2 (∂D), and any function g ∈ L2 (∂D) the solvability of system Au = 0,

(β0 Γ0 + β1 Γ1 )u = g

(2.1.1)

is equivalent to the solvability of (β0 + β1 Ω)ϕ = g.

(2.1.2)

Let β˜0 , β˜1 , γ˜ be three complex valued measurable functions on ∂D. Deﬁne operators β0 and β1 by dϕ + γ˜ ϕ β1 : ϕ → β˜1 ϕ (2.1.3) β0 : ϕ → β˜0 ds d is the operator of (generalized) differentiation in L2 (∂D). For a where ds sufﬁciently smooth ϕ the harmonic function uϕ = Πϕ is continuously differϕ entiable in the closure D, and the trace of its tangential derivative ∂u ∂τ on the boundary ∂D satisﬁes ∂uϕ dϕ = ∂τ ds ∂D

Thus, the boundary condition in (2.1.1) is meaningful at least for harmonic functions u ∈ L2 (D) with boundary values from W21 (∂D). Solvability of (2.1.1) with β0 , β1 deﬁned in (2.1.3) therefore is determined by the solvability of d + β˜1 Ω + γ˜ ϕ = g, g ∈ L2 (∂D) β˜0 ds for ϕ ∈ W21 (∂D). Since Ωϕ =

∂uϕ ∂n ∂D ,

this condition can be rendered as ∂u ∂u β˜0 + β˜1 + γ˜ u|∂D = g, g ∈ L2 (∂D) ∂τ ∂D ∂n ∂D

(2.1.4)

for the unknown function u harmonic in D. When β˜0 , β˜1 , γ˜ , and g are sufﬁciently regular and real valued, the problem (2.1.4) reduces to the classical Poincar´e’s problem for harmonic functions [23]. 2.2. Hilbert Problem Hilbert problem in the domain D consists in seeking an analytic function w = u + iv deﬁned in D with the real and imaginary parts u, v satisfying the following condition on the boundary ∂D a(s)u(s) + b(s)v(s) = g(s),

(2.2.1)

with real valued functions a, b, and g. For simplicity, we consider the case when D is the unit disc D = {z ∈ C | |z| < 1} in the complex plane with the boundary T = {z ∈ C | |z| = 1}. In order to apply the general schema we need to recall some properties of Hilbert transform H acting in L2 (T), see [15,21]. The operator H is bounded in L2 (T) and for the real valued ϕ ∈ L2 (T), the sum ϕ + iHϕ is a boundary value of the function w = u + iv analytic in D. In other words, if w = u + iv is analytic in D, where u, v are

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harmonic real valued functions such that the trace ϕ = u|T is in L2 (T) , then ϕ = v|T is also in L2 (T) and ϕ = Hϕ. The function ϕ is called the harmonic conjugate of ϕ. Deﬁne Λ to be the Hilbert transform, Λ = H. Then the boundary condition (1.4.1) results in the equation (β0 + β1 H)ϕ = g that can be rewritten as β0 ϕ + β1 ϕ =g

(2.2.2)

Let β0 : ϕ → aϕ, β0 : ϕ → bϕ be two multiplication operators by measurable functions a, b on T. Under additional assumption that a, b, ϕ, and g are real valued, the condition (2.2.2) corresponds to the Hilbert problem (2.2.1) for the unknown function w = u + iv analytic in D. If ϕ ∈ L2 (T) solves the equation a(s)ϕ(s) + b(s)(Hϕ)(s) = g(s),

(2.2.3)

for almost all s ∈ T, then the solution to (2.2.1) is w = u + iv with the real and imaginary parts u = Πϕ and v = ΠHϕ. In the language of open systems theory, Eq. (2.2.2) can be regarded as a corresponding to the operator node (1.5.1) redeﬁned input of the system M with Λ = H. To make this point clear, consider the left hand side of (2.2.2) and with β0 = 1, with β0 = 1, β1 = i as the input of the new system N Then the feedthrough operator of N is the map β1 = −i as the output of N. Θ : (I + iH)ϕ → (I − iH)ϕ,

ϕ ∈ L2 (T)

which cannot be written in the form Θ = (I −iH)(I +iH)−1 because I +iH is not boundedly invertible. Property H2 = −I of the Hilbert transform yields (I + iH)(I − iH) = 0 and therefore the set Ker(I + iH) is not trivial. In fact Ker(I + iH) and R(I − iH) coincide. Thus, the mapping Θ is a linear relation on L2 (T) ⊕ L2 (T). However, its restriction to the set (I + iH)(L2 (T)) ⊕ {0} where (L2 (T)) denotes all real valued functions from L2 (T), deﬁnes an operator θ = (I −iH)(I +iH)−1 that maps boundary values of functions w = u+iv analytic in D with u|T ∈ (L2 (T)) to the boundary values of complex conjugate function w ¯ = u − iv. Note that the operator θ is not linear over the ﬁeld of complex numbers because aθw = θaw for w ∈ D(θ) and a ∈ C, unless a is a real number. 2.3. Riemann Problem The Riemann problem for analytic functions is another case that can be studied by means of the general theory exposed in Sect. 1. Let D be the simply connected bounded domain D ⊂ C with regular boundary ∂D and B, g be measurable complex valued functions on ∂D. A pair of functions Φ± is a solution to the corresponding Riemann problem if Φ+ is analytic in D, Φ− is analytic in C\D, non-tangential boundary values of Φ± on the contour ∂D exist almost everywhere, and Φ+ (s) − B(s)Φ− (s) = g(s),

a.e. s ∈ ∂D

(2.3.1)

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Note that all considerations carried out in the beginning of this section for the Laplacian remain fully applicable, because Φ± are linear combinations of harmonic functions deﬁned on their respective domains. Let S be the Cauchy singular integral operator on the contour ∂D deﬁned for ϕ ∈ L2 (∂D) by 1 ϕ(t)dt ϕ(t)dt 1 = lim S : ϕ → Φ(s) = πi t−s πi ε↓0 t−s |t−s|>ε

∂D

2

Operator S is bounded in L (∂D). For notational convenience, denote D+ = D and D− = C\D. Two functions ϕ(t)dt 1 , z ∈ D± Φ± (z) = πi t−z ∂D

where ϕ ∈ L2 (∂D) are analytic and possess non-tangential boundary values almost everywhere on the contour ∂D lim

z→s, z∈D ±

Φ± (z) = Φ± (s),

a.e. s ∈ ∂D

The Sokhotzki–Plemelj formulae [6,14,23] Φ+ (s) = ϕ(s) + Φ(s),

Φ− (s) = −ϕ(s) + Φ(s)

a.e. s ∈ ∂D

(2.3.2)

and boundedness of S show that Φ± ∈ L2 (∂D). Introduce two multiplication operators β0 : ϕ → a(s)ϕ(s), β1 : ϕ → b(s)ϕ(s), where a, b are measurable functions of s ∈ ∂D. Then the choice Λ : ϕ → Sϕ and the boundary condition from (1.4.1) leads to the equation for unknown ϕ ∈ L2 (∂D) a(s)ϕ(s) + b(s)(Sϕ)(s) = g(s)

(2.3.3)

Put a = A + B, b = A − B with some measurable functions A, B deﬁned on ∂D. Then for Φ = Sϕ, the left hand side of (2.3.3) takes the form aϕ + bSϕ = (A + B)ϕ + (A − B)Sϕ = A (ϕ + Φ) − B (−ϕ + Φ) Due to (2.3.2), Eq. (2.3.3) becomes A(s)Φ+ (s) − B(s)Φ− (s) = g(s),

s ∈ ∂D

For A(s) = 1 we arrive at the Riemann boundary value problem (2.3.1). Other types of boundary value problems can be described by Eq. (2.3.3) if we continue to treat a and b as linear operators. For example, let τ : ϕ(s) → ϕ(α(s)), s ∈ ∂D be the composition operator where α(s) is an arbitrary oneto-one mapping of the contour ∂D onto itself with continuous derivative α (s) = 0. The choice a = Aτ + B, b = Aτ − B where two multiplication operators A, B are as above, results in the so-called Riemann boundary value problem with shift, see [14] for details, A(s)Φ+ [α(s)] − B(s)Φ− (s) = g(s) Note in conclusion that the case λ = 0 of the general spectral problem (1.4.1) appears to be irrelevant for the study of analytic functions in the paper’s context. However, the spectral theory approach may prove beneﬁcial in the

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study of boundary value problems for the ﬁrst-order differential operators of complex analysis, most notably, Cauchy–Riemann and Beltrami operators on domains (for example, see [6] for their definitions).

References [1] Alpay D., Behrndt, J.: Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators, 26 p. Preprint (2008). arXiv:0807. 0095v1 [math.FA] [2] Amrein, W.O., Pearson, D.B.: M -operators: a generalisation of Weyl–Titchmarsh theory. J. Comput. Appl. Math. 171(1–2), 1–26 (2004) [3] Ashbaugh M.S., Gesztesy F., Mitrea M., Teschl G.: Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains. Preprint (2009). arXiv:0907.1442v1 [math.SP] 60 p. [4] Ashbaugh, M.S., Gesztesy, F., Mitrea, M., Shterenberg, R., Teschl, G.: The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem, 16 p. Preprint (2009). arXiv:0907.1439v1 [math.SP] [5] Behrndt, J., Langer, M.: Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal. 243(2), 536–565 (2007) [6] Begehr, H.G.W.: Complex analytic methods for partial differential equations. An introductory text. World Scientiﬁc, River Edge (1994) [7] Browder, F.E.: On the spectral theory of elliptic differential operators. I. Math. Ann. 142, 22–130 (1961) [8] Brown, B.M., Grubb, G., Wood, I.G.: M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr. 282(3), 314–347 (2009) [9] Brown, B.M., Hinchcliﬀe, J., Marletta, M., Naboko, S., Wood, I.: The abstract Titchmarsh–Weyl M-function for adjoint operator pairs and its relation to the spectrum. Integral Equ. Oper. Theory 63(3), 297–320 (2009) [10] Brown, B.M., Marletta, M.: Spectral inclusion and spectral exacteness for PDEs on exterior domains. IMA J. Numer. Anal. 24, 21–43 (2004) [11] Brown, B.M., Marletta, M., Naboko, S., Wood, I.: Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. Lond. Math. Soc. 2(77), 700–718 (2008) [12] Corless, M.J., Frazho, A.E.: Linear systems and control. An operator perspective. Marcel Dekker, Inc., New York (2003) [13] Curtain, R.F., Zwart, H.J.: An Introduction to Inﬁnite-Dimensional Linear Systems Theory. Springer, Berlin (1995) [14] Gakhov, F.D.: Boundary Value Problems. Pergamon Press, Oxford (1966) [15] Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981) [16] Gesztesy F., Mitrea M.: Self-Adjoint Extensions of the Laplacian and KreinType Resolvent Formulas in Nonsmooth Domains, 67 p. Preprint (2009). arXiv:0907.1750v1 [math.AP] [17] Gesztesy, F., Mitrea, M.: Generalized Robin Boundary Conditions, Robin-toDirichlet maps, and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains, 50 p. Preprint (2008). arXiv:0803.3179v2 [math.AP]

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[18] Gesztesy, F., Mitrea, M.: Robin-to-Robin Maps and Krein-type resolvent formulas for Schrodinger operators on bounded Lipschitz domains, 23 p. Preprint (2008) arXiv:0803.3072v2 [math.AP] [19] Gesztesy, F., Mitrea, M., Zinchenko, M.: Variations on a theme of Jost and Pais. J. Funct. Anal. 253(2), 399–448 (2007) [20] Gesztesy, F., Mitrea, M., Zinchenko, M.: Multi-dimensional versions of a determinant formula due to Jost and Pais. Rep. Math. Phys. 59(3), 365–377 (2007) [21] Koosis, P.: Introduction to Hp Spaces, 2nd edn. University Press, Cambridge (1998) [22] Mikhlin, S.G, Pr¨ ossdorf, S.: Singular Integral Operators. Springer, Berlin (1986) [23] Muskhelishvili, N.I.: Singular Integral Equations. Boundary problems of function theory and their application to mathematical physics. Noordhoﬀ International Publishing, Leyden (1977) [24] Partington, J.R.: Linear operators and linear systems. An analytical approach to control theory. Cambridge University Press, Cambridge (2004) [25] Posilicano, A.: Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Methods Funct. Anal. Topol. 10(2), 57–63 (2004) [26] Ryzhov, V.: A general boundary value problem and its Weyl function. Opuscula Math. 27, 305–331 (2007) [27] Ryzhov, V.: Weyl–Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control. Complex Anal. Oper. Theory 3, 289–322 (2009) [28] Ryzhov, V.: Spectral Boundary Value Problems and their Linear Operators, 38 p. Preprint (2009). arXiv:0904.0276v1 [math-ph] [29] Staffans, O.J.: Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005) Vladimir Ryzhov Mathematics, Statistics and Physics Unit University of British Columbia Okanagan 3333 University Way Kelowna BC V1V 1V7 Canada e-mail: [email protected] Received: August 30, 2009. Revised: November 21, 2009.

Integr. Equ. Oper. Theory 67 (2010), 341–364 DOI 10.1007/s00020-010-1782-y Published online April 14, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On the H¨ ormander Classes of Bilinear Pseudodifferential Operators ´ ad B´enyi, Diego Maldonado, Virginia Naibo Arp´ and Rodolfo H. Torres Abstract. Bilinear pseudodifferential operators with symbols in the bilinear analog of all the H¨ ormander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calder´ on–Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well. Mathematics Subject Classification (2000). Primary 35S05, 47G30; Secondary 42B15, 42B20. Keywords. Bilinear pseudodifferential operators, bilinear H¨ ormander classes, symbolic calculus, Calder´ on–Zygmund theory.

1. Introduction Many linear operators encountered in analysis are best understood when represented as singular integral operators in the space domain, while others are better treated as pseudo-differential operators in the frequency domain. In some particular situations both representations are readily available. In many others, however, one of them is only given abstractly, and through the existence of a distributional kernel or symbol which is hard or impossible to compute. Both representations have proved to be tremendously useful. The representation of operators as pseudodifferential ones usually yields simple L2 estimates, explicit formulas for the calculus of transposes and composition, D. Maldonado was partially supported by the NSF under Grant DMS 0901587. R. H. Torres was supported in part by NSF under Grant DMS 0800492 and a General Research Fund allocation of the University of Kansas.

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and invariance properties under change of coordinates in smooth situations. As it is known, this makes pseudodifferential operators an invaluable tool in the study of partial differential equations and they are employed to construct parametrices and study regularity properties of solutions. The integral representation on the other hand, is often best suited for other Lp estimates and motivates or indicates what results should hold in other metric and measure theoretic situations where the Fourier transform is no longer available. This has found numerous applications in complex analysis, operator theory, and also in problems in partial differential equations where the domains or functions involved have a minimum amount of regularity. Work on singular integral and pseudodifferential operators started with explicit classical examples and was then directed to attack speciﬁc applications in other areas. Switching back and forth, many eﬀorts were also oriented to the understanding of naturally appearing technical questions and to the testing of the full power of new techniques as they developed. In fact, sometimes the technical analytic tools studied preceded the applications in which they were much later used. The Calder´ on–Zygmund theory and the related real variable techniques played a tremendous role in all these accomplishments. This is explained in detail from both the historical and technical points of view in, for example, the book by Stein [28]. The study of bilinear operators within harmonic analysis is following a similar path. The ﬁrst systematic treatment of bilinear singular integrals and pseudodifferential operators in the early work of Coifman and Meyer [13,15] originated from speciﬁc problems about Calder´ on’s commutators, and soon lead to the study of general boundedness properties of pseudodifferential operators [14]. Later on, the work of Lacey and Thiele [22,23] on a speciﬁc singular integral operator (the bilinear Hilbert transform which also goes back to Calder´ on) and the new techniques developed by them immediately suggested the study of other bilinear operators and the need to understand and characterize their boundedness and computational properties. See for example [3,16,17,25], to name a few works in this direction. The development of the symbolic calculus for bilinear pseudodifferential operators started in B´enyi and Torres [7] and was continued in B´enyi et al. [5]. Other results speciﬁc to bilinear pseudodifferential operators were obtained in B´enyi et al. [2,4,6,8] and, much recently, in Bernicot [9] and Bernicot and Torres [10]. As in the linear case, many of the results obtained were motivated too by the Calder´ on–Zygmund theory and its bilinear counterpart as developed in Grafakos and Torres [18]; see also [12,19,24]. The literature is by now vast, see [27] for further references. We want to contribute with this article to the understanding of the properties of all the bilinear analogs of the linear H¨ ormander classes of pseudodifm (see the next ferential operators. These bilinear classes are denoted by BSρ,δ section for technical definitions). Only some particular cases of them have been studied before; mainly the cases when ρ = 1 or when ρ = δ = 0. The symbolic calculus has only been developed for the case ρ = 1 and δ = 0. Our goal is to complete the symbolic calculus for all the possible (and meaningful) values of δ and ρ.

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We could quote from the introduction of H¨ ormander’s work [20]: “In this work the use of Fourier transformations has been emphasized; as a result no singular integral operators are apparent...”, but we are clearly guided by previous works that relate, in the case of operators of order zero, to bilinear Calder´on–Zygmund singular integrals. In fact, the existence of calculus 0 , δ < 1, gives an alternative way to prove the boundedness of such for BS1,δ operators in the optimal range of Lp spaces directly from the multilinear T 1-Theorem in [18]. That is, without using Littlewood–Paley arguments as in the already cited monograph [14] or the work [2]. While the composition of pseudodifferential operators (with linear ones) forces one to study different classes of operators introduced in [5], previous results in the subject left some level of uncertainty about whether the computation of transposes could still be accomplished within some other bilinear H¨ormander classes. The forerunner work [7] dealt mainly with the class 0 and a significant part of the proofs given in [7] relied on the so-called BS1,0 Peetre inequality which does not go through in general for other values of ρ = 1, δ = 0, m = 0. We resolve this problem in the present article using ideas inspired in part by some computations in Kumano-go [21], and developing the calculus of transposes for all the bilinear H¨ ormander classes for which such calculus is possible. The excluded classes are the ones for which ρ = δ = 1. This restriction is really necessary as proved in [7]. In fact, as the linear class 0 0 , the class BS1,1 is forbidden in the sense that two related pathologies S1,1 occur: it is not closed under transposition and it contains operators which fail to be bounded on product of Lp spaces, even though the associated kernels for operators in this class are of bilinear Calder´ on–Zygmund type. The analogy between results in the linear and multilinear situations is in general only a guide to what could be expected to transfer from one context to the other. Some multilinear results arise as natural counterparts to linear ones, but often the techniques employed need to be substantially sharpened or replaced by new ones. It is actually far more complicated to prove the existence of calculus in the bilinear case than in the linear one. Some properties of the symbols of bilinear pseudodifferential operators on Rn can be guessed from those of linear operators in R2n . Though some of our computations are reminiscent of those for linear pseudodifferential operators or Fourier integral operators, the calculus of transposes for bilinear operators does not follow from the linear results by doubling the number of dimensions. Boundedness results cannot be obtained in this fashion either. The essential obstruction is the fact that the integral of a function of two n-dimensional variables (x, y) ∈ R2n yields no information about the (n-dimensional) integral of its restriction to the diagonal (x, x), x ∈ Rn . On the other hand, a few point-wise estimates can be obtained in a more direct way from the linear case using the method of doubling the dimensions. For example, it is useful to establish ﬁrst precise point-wise estimates on the bilinear kernels associated to the operators in various H¨ ormander classes. We are able to derive them ´ from the linear ones investigated by Alvarez and Hounie [1]. It is interesting too that some results do not extend to the multilinear context. A notorious example is the Calder´ on–Vaillancourt result in [11] for

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0 0 the L2 boundedness of the class S0,0 . One may expect the class BS0,0 to map, 2 2 1 say, L × L into L , but this fails unless additional conditions on the symbol are imposed; see [8]. Such class only maps into an optimal modulation space (M 1,∞ ) which is larger than L1 . See also [4] and [6] for more details. Similarly (and using duality and the existence of the calculus for transposes), it is nat0 , with 0 < δ < ρ < 1 maps L2 × L∞ into ural to ask whether the class BSρ,δ 2 0 maps L2 into L2 L – recall that H¨ormander’s results in [20] give that Sρ,δ for the same range of ρ and δ. Alternatively, it may only be possible to obtain the boundedness from L2 × X into L2 , where X is a space smaller than L∞ . Though we do not know the answer to the former questions, we give a result in the direction of the latter. We also obtain some other new boundedness properties involving Sobolev spaces which take the form of fractional Leibniz’ rules. In the next section we present the technical definitions and the precise statements of our results about symbolic calculus. Sections 3 and 4 contain the proofs of those results. Section 5 contains the results about the point-wise estimates for the kernels, while Section 6 has the boundedness results alluded to before.

2. Symbolic Calculus in the Bilinear H¨ ormander Classes We start by recalling the linear pseudodifferential operators in the H¨ ormander m . These are operators of the form classes Sρ,δ Tσ (f )(x) = σ(x, ξ)f(ξ)eix·ξ dξ Rn

where the symbol σ satisﬁes the estimates |∂xα ∂ξβ σ(x, ξ)| ≤ Cαβ (1 + |ξ|)m+δ|α|−ρ|β| , for all x, ξ ∈ Rn , all multi-indices α, β, and some positive constants Cαβ . These operators are a priori deﬁned for appropriate test functions. In this article we study the natural bilinear analog σ(x, ξ, η)f(ξ) g (η)eix·(ξ+η) dξdη, Tσ (f, g)(x) = Rn Rn

where the symbol σ satisﬁes now the estimates |∂xα ∂ξβ ∂ηγ σ(x, ξ, η)| ≤ Cαβγ (1 + |ξ| + |η|)m+δ|α|−ρ(|β|+|γ|) , n

(2.1)

also for all x, ξ, η ∈ R , all multi-indices α, β, γ and some positive constants m (Rn ), or Cαβγ . The class of all symbols satisfying (2.1) is denoted by BSρ,δ m simply BSρ,δ when it is clear from the context to which space the variables x, ξ, η belong to. The transposes of such operators are deﬁned as usual by the duality relations T (f, g), h = T ∗1 (h, g), f = T ∗2 (f, h), g.

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We will write σ∼

∞

σj

j=0

if there is a non-increasing sequence mN −∞ such that σ−

N −1

mN σj ∈ BSρ,δ ,

j=0

for all N > 0. The spaces of test functions that we will use will be the space Cc∞ of inﬁnitely differentiable functions with compact support or the Schwartz space S. When given their usual topologies, their duals are D and S , the spaces of distributions and of tempered distributions, respectively. We will also consider Ccs , s ∈ N, the space of functions with compact support and continuous derivatives up to order s; W s,2 , the Sobolev space of functions having derivatives in L2 up to order s; and W0s,∞ , the completion of Ccs with respect to the norm sup|γ|≤s Dγ g L∞ . Unless speciﬁed otherwise, the underlying space will be assumed to be Rn . We develop a symbolic calculus for bilinear pseudodifferential operators m , m ∈ R, 0 ≤ δ ≤ with symbols in all the bilinear H¨ ormander classes BSρ,δ ρ ≤ 1, δ < 1. Our ﬁrst two theorems state that the H¨ ormander classes are closed under transposition and that the symbols of the transposed operators have appropriate asymptotic expansions. Theorem 2.1 (Invariance under transposition). Assume that 0 ≤ δ ≤ ρ ≤ 1, m m δ < 1, and σ ∈ BSρ,δ . Then, for j = 1, 2, Tσ∗j = Tσ∗j , where σ ∗j ∈ BSρ,δ . m , then Theorem 2.2 (Asymptotic expansion). If 0 ≤ δ < ρ ≤ 1 and σ ∈ BSρ,δ ∗1 ∗2 σ and σ have asymptotic expansions i|α| ∂xα ∂ξα (σ(x, −ξ − η, η)) σ ∗1 ∼ α! α

and σ ∗2 ∼

i|α| α

α!

∂xα ∂ηα (σ(x, ξ, −ξ − η)).

More precisely, if N ∈ N then i|α| m+(δ−ρ)N ∂ α ∂ α (σ(x, −ξ − η, η)) ∈ BSρ,δ σ ∗1 − α! x ξ

(2.2)

|α|
and σ ∗2 −

i|α| m+(δ−ρ)N ∂ α ∂ α (σ(x, ξ, −ξ − η)) ∈ BSρ,δ . α! x η

(2.3)

|α|
In relation to asymptotic expansions we also prove the following two theorems.

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m

Theorem 2.3. Assume that aj ∈ BSρ,δj , j ≥ 0 and mj −∞ as j → ∞. ∞ m0 Then, there exists a ∈ BSρ,δ such that a ∼ j=0 aj . Moreover, if ∞ m b ∈ BSρ,δ = ∪ BSρ,δ and b ∼ m

∞

aj ,

j=0

then −∞ m = ∩ BSρ,δ . a − b ∈ BSρ,δ m

m BSρ,δj , j

Theorem 2.4. Assume that aj ∈ a ∈ C ∞ (Rn × Rn × Rn ) be such that

≥ 0 and mj −∞ as j → ∞. Let

|∂xα ∂ξβ ∂ηγ a(x, ξ, η)| ≤ Cαβγ (1 + |ξ| + |η|)μ ,

(2.4)

for some positive constants Cαβγ and μ = μ(α, β, γ). If there exist μN → ∞ such that |a(x, ξ, η) −

N

aj (x, ξ, η)| ≤ CN (1 + |ξ| + |η|)−μN ,

(2.5)

j=0 m0 and a ∼ then a ∈ BSρ,δ

∞

aj .

j=0

The continuity on Lebesgue spaces of bilinear pseudodifferential oper0 ators with symbols in the class BS1,δ with 0 ≤ δ < 1 has been intensely addressed in the literature. It is nowadays a well-known fact that the bilinear 0 kernels associated to bilinear operators with symbols in BS1,δ , 0 ≤ δ < 1, are bilinear Calder´ on–Zygmund operators in the sense of Grafakos and Torres [18]. Recall the following result ([18, Corollary 1]) which is an application of the bilinear T 1-Theorem therein: 0 If T and its transposes, T ∗1 and T ∗2 , have symbols in BS1,1 , then they p q r can be extended as bounded operators from L × L into L for 1 < p, q < ∞ and 1/p + 1/q = 1/r. 0 0 ⊂ BS1,1 , we can As mentioned in the introduction, and since BS1,δ directly combine this result with Theorem 2.1 to recover the following optimal version of a known fact. 0 Corollary 2.5. If σ is a symbol in BS1,δ , 0 ≤ δ < 1, then Tσ has a bounded p q r extension from L × L into L , for all 1 < p, q < ∞, 1/p + 1/q = 1/r.

3. Proofs of Theorems 2.1 and 2.2 In the following, we assume that the symbol σ has compact support (in all three variables x, ξ, and η) so that the calculations in the proofs of Theorems 2.1 and 2.2 are properly justiﬁed. All estimates are obtained with constants independent of the support of σ and an approximation argument can be used to obtain the results for symbols that do not have compact support; see [7] for further details regarding such an approximation argument.

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We restrict the proofs of Theorems 2.1 and 2.2 to the ﬁrst transpose of Tσ (j = 1). As in [7], we rewrite Tσ∗1 as a compound operator. We have ∗1 T (h, g)(x) = c(y, ξ, η)h(y) g (η)e−i(y−x)·ξ eix·η dξdηdy, y

η

ξ

where c(y, ξ, η) = σ(y, −ξ − η, η). Straightforward calculations show that c satisﬁes the same differential inequalities as σ. Indeed, by the Leibniz rule we can write γ1 γ2 |∂yα ∂ξβ ∂ηγ c(y, ξ, η)| |∂yα ∂ξβ ∂η2 ∂η3 σ(y, −ξ − η, η)| |γ1 |+|γ2 |=|γ|

(1 + |ξ + η| + |η|)m+δ|α|−ρ(|β|+|γ1 |+|γ2 |) (3.1)

(1 + |ξ| + |η|)m+δ|α|−ρ(|β|+|γ|) , with constants of the form |γ1 |+|γ2 |=|γ| Cαβγj 2m+δ|α|+ρ(|β|+|γ|) . By appropriately changing variables of integration, the symbol of T ∗1 is given in terms of c by the following expression: a(x, ξ, η) = c(x + y, z + ξ, η)e−iz·y dydz. (3.2) Proof of Theorem 2.1. We will use the representation (3.2) of a to show that m a ∈ BSρ,δ . By (3.1) and since ∂xα ∂ξβ ∂ηγ c(x + y, z + ξ, η)e−iz·y dydz, ∂xα ∂ξβ ∂ηγ a(x, ξ, η) = it is enough to work with α = β = γ = 0. Our techniques are inspired in part by ideas in [21, Lemma 2.4, page 69]. In the following, ﬁx ξ ∈ Rn , η ∈ Rn , and set A := 1 + |ξ| + |η|. We have to prove that |a(x, ξ, η)| Am with a constant independent of the support of σ. Let l0 ∈ N, 2l0 > n. Writing 2

e−iz·y = (1 + A2δ |y| )−l0 (1 + A2δ (−Δz ))l0 e−iz·y , integration by parts gives a(x, ξ, η) =

q(x, y, z, ξ, η)e−iz·y dydz,

(3.3)

where q(x, y, z, ξ, η) =

(1 + A2δ (−Δz ))l0 c(x + y, z + ξ, η) 2

(1 + A2δ |y| )l0

We now estimate (−Δy )l q for l ∈ N.

.

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(−Δy )l q =

∂yβ

|α|=2lα

2δ

(1 + A

IEOT

|α|=2lα

β≤α

Cα ∂yα q(x, y, z, ξ, η) =

i even

2 −l0

|y| )

∂yα−β

i even

Cαβ ×

(1 + A (−Δz )) c(x + y, z + ξ, η) . (3.4) 2δ

l0

Note that

−l0

β

2 2 .

∂y (1 + A2δ |y| )−l0 ≤ Cβl0 Aδ|β| 1 + A2δ |y|

(3.5)

Moreover, if Pl0 = {γ = (γ1 , . . . , γn ) : γi even and |γ| = 2j, j = 0, . . . , l0 }, then (1 + A2δ (−Δz ))l0 c(x + y, z + ξ, η) = Cγ Aδ|γ| ∂ξγ c(x + y, z + ξ, η), γ∈Pl0

and therefore

|∂yα−β (1 + A2δ (−Δz ))l0 c(x + y, z + ξ, η) | ≤ Cγαβ Aδ|γ| (1 + |z + ξ| + |η|)m+δ(|α|−|β|)−ρ|γ| .

(3.6)

γ∈Pl0

From (3.4), (3.5) and (3.6), we get −l0

(−Δy )l q 1 + A2δ |y|2 Cαβl0 Aδ|β| |α|=2l β≤α αi even

⎛

×⎝

⎞

Cγαβ Aδ|γ| (1 + |z + ξ| + |η|)m+δ(|α|−|β|)−ρ|γ| ⎠ .

(3.7)

γ∈Pl0

Deﬁne the sets Ω1 = {z : |z| ≤

Aδ 2 },

We then have

Ω2 = {z :

Aδ 2

··· +

a(x, ξ, η) =

≤ |z| ≤

Ω1 y

Ω3 = {z : |z| ≥

A 2 }.

··· +

Ω2 y

A 2 },

· · · := I1 + I2 + I3 . Ω3 y

Note that 1 3 A ≤ 1 + |z + ξ| + |η| ≤ A, 2 2

z ∈ Ω1 ∪ Ω2 .

(3.8)

and 1 + |z + ξ| + |η| ≤ A + |z| ≤ 3 |z| ,

z ∈ Ω3 .

(3.9)

Estimation For I1 . The estimate (3.7) with l = 0, (3.8), and δ − ρ ≤ 0 give, for z ∈ Ω1 , 2 Cγ Aδ|γ| (1 + |z + ξ| + |η|)m−ρ|γ| |q| ≤ (1 + A2δ |y| )−l0 γ∈Pl0 2δ

≤ (1 + A

2 −l0

|y| )

γ∈Pl0 2 −l0

(1 + A2δ |y| )

Am .

Cγ Am+(δ−ρ)|γ|

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Therefore, since 2l0 > n,

|I1 | Am

1 (1 +

Ω1 y

A2δ

2 |y| )l0

dydz ∼ Am .

Estimation For I2 . Integration by parts gives

q(x, y, z, ξ, η)e−iz·y dy =

y

=

1 2l0

|z|

q(x, y, z, ξ, η)(−Δy )l0 e−iz·y dy

y

1 2l0

|z|

(−Δy )l0 (q(x, y, z, ξ, η))e−iz·y dy.

y

Using (3.7) with l = l0 , (3.8), and δ − ρ ≤ 0 we get, for z ∈ Ω2 ,

(−Δy )l0 q

−l0 2 ≤ 1+A2δ |y| Cαβl0 Aδ|β| cγαβ Aδ|γ| Am+δ(|α|−|β|)−ρ|γ| γ∈Pl0

|α|=2l0 β≤α αi even

−l0 2 ≤ 1 + A2δ |y| Cαβl0 cγαβ Am+δ|α|+(δ−ρ)|γ| γ∈Pl0

|α|=2l0 β≤α αi even

Am+2l0 δ 1+

A2δ

2

|y|

l 0 .

Recalling that 2l0 > n, we get |I2 | ≤

1

2l0

Ω2

|z|

y

Am+2l0 δ−δn

Am+2l0 δ

l dydz 2 0 1 + A2δ |y| −2l |z| 0 dz ∼ Am . δ

|z|≥ A2

Estimation For I3 . Let l ∈ N to be chosen later. Again, integration by parts gives

−iz·y

q(x, y, z, ξ, η)e

dy =

y

=

1

2l

|z|

1

y

2l

|z|

q(x, y, z, ξ, η)(−Δy )l e−iz·y dy

y

(−Δy )l (q(x, y, z, ξ, η))e−iz·y dy.

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Using (3.7) and (3.9), and deﬁning m+ = max(0, m), we get, for z ∈ Ω3 ,

(−Δy )l q

−l0 2 1 + A2δ |y| Cαβl0 Aδ|β| ⎛ ×⎝

|α|=2l β≤α αi even

⎞

Cγαβ Aδ|γ| (1 + |z + ξ| + |η|)m+δ(|α|−|β|)−ρ|γ| ⎠

γ∈Pl0 2

1+A2δ |y|

−l0

Cαβl0

δ(|β|+|γ|)

Cγ,α,β |z|

m+ +δ(|α|−|β|)

|z|

γ∈Pl0

|α|=2l β≤α αi even

−l0 2 m +δ(2l+2l0 ) 1 + A2δ |y| |z| + . We then have

1

|I3 |

2l

Ω3

|z|

2

1 + A2δ |y|

−l0

y m+ +2l0 δ+2l(δ−1)

|z|

∼ |z|≥ A 2

∼ A−δn

m+ +δ(2l+2l0 )

|z|

dz

2

1 + A2δ |y|

dydz −l0

dy

y

m+ +2l0 δ+2l(δ−1)

|z|

dz.

|z|≥ A 2

We now choose l ∈ N sufﬁciently large so that m+ + 2l0 δ + 2l(δ − 1) < −n

and

− δn + m+ + 2l0 δ + 2l(δ − 1) + n < m.

The existence of such an l is guaranteed by the condition 0 ≤ δ < 1. Finally, |I3 | A−δn+m+ +2l0 δ+2l(δ−1)+n ≤ Am . Proof of Theorem 2.2. As in the proof of Theorem 2.1 we use the representation (3.2) for the symbol of T ∗1 . Deﬁne i|α| α α 1 ∂x ∂ξ c(x, ξ, η) = aα (x, ξ, η) := ∂ξα c(x + y, ξ, η) e−iz·y z α dydz. α! α! m+(δ−ρ)|α|

By the estimates (3.1), aα ∈ BSρ,δ with constants independent of the support of σ. We will show that

α1 α2 α3

(3.10)

∂x ∂ξ ∂η (a(x, ξ, η) − |α|
≤ C(1 + |ξ| + |η|)m+(δ−ρ)N +δ|α1 |−ρ(|α2 |+|α3 |) , where C = Cα1 α2 α3 N is independent of the support of σ. This then shows m+(δ−ρ)N that a − |α|
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Taylor’s theorem 1 aα (x, ξ, η) = a(x, ξ, η)− ∂ξα c(x+y, ξ +tz, η) z α e−iz·y dydz, α! |α|
|α|=N

where t ∈ (0, 1) and t = t(x, y, ξ, z, η). Note that, because of the estimates (3.1), it is enough to prove (3.10) for α1 = α2 = α3 = 0. Inequality (3.10) follows from computations similar to the ones in Theorem 2.1. We include them here for the reader’s convenience and for completeness. Fix N ∈ N0 and a multiindex α with |α| = N. For l0 ∈ N, integration by parts gives ∂ξα c(x + y, ξ + tz, η) z α e−iz·y dydz (3.11) Iα := = q(x, y, z, ξ, η)e−iz·y dydz, where q(x, y, z, ξ, η) =

(1 + A2δ (−Δz ))l0 (∂ξα c(x + y, ξ + tz, η) z α ) 2

(1 + A2δ |y| )l0

.

We now estimate (−Δy )l q for l ∈ N. Cν ∂yν q(x, y, z, ξ, η) (−Δy )l q = |ν|=2l νi even

=

2 Cνβ ∂yβ (1 + A2δ |y| )−l0

|ν|=2l νi even β≤ν

× ∂yν−β (1 + A2δ (−Δz ))l0 (∂ξα c(x + y, ξ + tz, η) z α ) .

(3.12)

As before, if Pl0 = {γ = (γ1 , . . . , γn ) : γi even and |γ| = 2j, j = 1, . . . , l0 }, then (1 + A2δ (−Δz ))l0 (∂ξα c(x + y, ξ + tz, η) z α ) Cγ Aδ|γ| ∂zγ (∂ξα c(x + y, ξ + tz, η) z α ) = γ∈Pl0

=

Cγω Aδ|γ| ∂zω z α (∂ξα+γ−ω c)(x + y, ξ + tz, η) t|γ|−|ω| ,

γ∈Pl0 ω≤γ,ω≤α

and therefore |∂yν−β (1 + A2δ (−Δz ))l0 (∂ξα c(x + y, ξ + tz, η) z α ) | N −|ω| Cγωνβ Aδ|γ| |z| (1 + |ξ + tz| + |η|)m+δ(|ν|−|β|)−ρ(N +|γ|−|ω|) . ≤ γ∈Pl0 ω≤γ,ω≤α

(3.13)

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From (3.12), (3.5) and (3.13), we get −l0

(−Δy )l q 1 + A2δ |y|2 N −|ω| × Cγωνβl0 Aδ(|β|+|γ|) |z| |ν|=2l,νi even β≤ν,γ∈Pl0 ω≤γ,ω≤α

×(1 + |ξ + tz| + |η|)m+δ(|ν|−|β|)−ρ(N +|γ|−|ω|) .

(3.14)

Letting again Ω1 = {z : |z| ≤ we have

Aδ 2 },

Ω2 = {z :

Ω1 y

≤ |z| ≤

··· +

Iα =

Aδ 2

Ω3 = {z : |z| ≥

A 2 },

··· +

Ω2 y

A 2 },

· · · := I1 + I2 + I3 . Ω3 y

Note that 1 3 A ≤ 1 + |ξ + tz| + |η| ≤ A, 2 2

z ∈ Ω1 ∪ Ω2 , t ∈ (0, 1),

(3.15)

and 1 + |ξ + tz| + |η| ≤ A + |z| ≤ 3 |z| ,

z ∈ Ω3 , t ∈ (0, 1).

(3.16)

The estimates in (3.10) for α1 = α2 = α3 = 0 follow if we prove that |Ii | Am+(δ−ρ)N ,

i = 1, 2, 3.

Estimation For I1 . The estimate (3.14) with l = 0, (3.15), and δ − ρ < 0 give, for z ∈ Ω1 , 2 |q| ≤ (1 + A2δ |y| )−l0 Cγωl0 Am+(δ−ρ)(|γ|+N −|ω|) γ∈Pl0 ω≤γ,ω≤α

Am+(δ−ρ)N 2

(1 + A2δ |y| )l0

.

Therefore, if we choose 2l0 > n, we get 1 m+(δ−ρ)N |I1 | A dydz ∼ Am+(δ−ρ)N . 2 2δ (1 + A |y| )l0 Ω1 y

Estimation For I2 . Integration by parts gives 1 −iz·y q(x, y, z, ξ, η)e dy = q(x, y, z, ξ, η)(−Δy )l0 e−iz·y dy 2l |z| 0 y y 1 = (−Δy )l0 (q(x, y, z, ξ, η))e−iz·y dy. 2l0 |z| y

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Using (3.14) with l = l0 , (3.15), and δ − ρ < 0 we get, for z ∈ Ω2 ,

(−Δy )l0 q

−l0 2 N −|ω| ≤ 1+A2δ |y| Cγωνβl0 Aδ|γ| |z| Am+δ2l0 −ρ(N +|γ|−|ω|) . |ν|=2l0 ,νi even β≤ν,γ∈Pl0 ω≤γ,ω≤α

Choosing l0 such that 2l0 > N + n,

|I2 | ≤ A−nδ

Cγωνβl0 Aδ|γ| Am+δ2l0 −ρ(N +|γ|−|ω|)

|ν|=2l0 ,νi even β≤ν,γ∈Pl0

|z|N −|ω|−2l0 dz δ

|z|≥ A2

ω≤γ,ω≤α

Am+(δ−ρ)N .

Estimation For I3 . Let l ∈ N to be chosen later. Again, integration by parts gives 1 q(x, y, z, ξ, η)e−iz·y dy = q(x, y, z, ξ, η)(−Δy )l e−iz·y dy 2l |z| y y 1 = (−Δy )l (q(x, y, z, ξ, η))e−iz·y dy. 2l |z| y

Using (3.14) and (3.16), and deﬁning m+ = max(0, m), we get, for z ∈ Ω3 , −l0

δ(|β|+2l0 ) N −|ω|

(−Δy )l q 1 + A2δ |y|2 Cγωνβl0 |z| |z| |ν|=2l,νi even β≤ν,γ∈Pl0 ω≤γ,ω≤α m+ +δ(2l−|β|)

× |z| We then have |I3 | A−δn

|ν|=2l,νi even β≤ν,γ∈Pl0 ω≤γ,ω≤α

m+ +2l(δ−1)+2l0 δ+N

|z|

Cγωνβl0

dz.

|z|≥ A 2

Choosing l ∈ N sufﬁciently large so that m+ + 2l(δ − 1) + 2l0 δ + N < −n and −δn + m+ + 2l(δ − 1) + 2l0 δ + N + n < m + (δ − ρ)N, we obtain |I3 | A−δn+m+ +2l(δ−1)+2l0 δ+N +n ≤ Am+(δ−ρ)N .

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4. Proofs of Theorems 2.3 and 2.4 Proof of Theorem 2.3. The second part of the statement is immediate if we write for all N > 0, ⎛ ⎞ ⎛ ⎞ N −1 N −1 a − b = ⎝a − aj ⎠ − ⎝b − aj ⎠ ∈ BS mN , ρ,δ

j=0

j=0

and recall that mN −∞ as N → ∞. For the proof of the ﬁrst part of Theorem 2.3 we proceed by explicitly constructing a. Let ψ ∈ Cc∞ (Rn × Rn ) such that 0 ≤ ψ ≤ 1, ψ(ξ, η) = 0 on {(ξ, η) : |ξ| + |η| ≤ 1} and ψ(ξ, η) = 1 on {(ξ, η) : |ξ| + |η| ≥ 2}. Deﬁne a(x, ξ, η) =

∞

ψ( j ξ, j η)aj (x, ξ, η),

j=0

where j 0 as j → ∞ is an appropriately chosen sequence of numbers in m0 . The choice of this sequence will be made explicit (0, 1) so that a ∈ BSρ,δ below. For each ﬁxed ∈ (0, 1) we have (a) ψ( ξ, η) = 0 for |ξ| + |η| ≤ 1/ ; (b) ψ( ξ, η) = 1 for |ξ| + |η| ≥ 2/ ; (c) For all β, γ such that |β| + |γ| ≥ 1, ∂ξβ ∂ηγ ψ( ξ, η) = 0 for |ξ| + |η| ≤ 1/ or |ξ| + |η| ≥ 2/ ; (d) |∂ξβ ∂ηγ ψ( ξ, η)| ≤ cβγ |β|+|γ| . In particular, because of (a) and (b) (that is, we only care about pairs (ξ, η) such that 1/ < |ξ|+|η| < 2/ ), (d) is equivalent to (e) |∂ξβ ∂ηγ ψ( ξ, η)| ≤ cβγ (1 + |ξ| + |η|)−|β|−|γ| . This in turn is equivalent to saying that the family {ψ( ξ, η)}0< <1 0 represents a bounded set in BS1,0 (endowed with the topology induced by appropriate semi-norms; see [7]). m Based on the estimate (e) and aj ∈ BSρ,δj , we can control each of the terms in the sum that deﬁnes a. By using Leibniz’ rule, we immediately obtain |∂ξβ ∂ηγ ψ( j ξ, j η)aj (x, ξ, η)| ≤ Cj,αβγ (1 + |ξ| + |η|)mj +δ|α|−ρ(|β|+|γ|) .

(4.1)

−j

Let us now select j such that Cj,αβγ j ≤ 2 for all |α + β + γ| ≤ j. Due to (b) and (4.1), we can therefore write, for all |α + β + γ| ≤ j, |∂ξβ ∂ηγ ψ( j ξ, j η)aj (x, ξ, η)| ≤ 2−j (1 + |ξ| + |η|)mj +1+δ|α|−ρ(|β|+|γ|) .

(4.2)

Now, for a ﬁxed (x, ξ, η), the sum deﬁning a(x, ξ, η) is ﬁnite. Indeed, by (a), if inﬁnitely many terms corresponding to a subsequence (jk ) are non-zero, we necessarily have |ξ| + |η| > 1/ jk → ∞ as k → ∞, a contradiction. In particular, we also have a ∈ C ∞ (Rn × Rn × Rn ). Fix then a triple of multi-indices (α, β, γ) and let J > 0 be such that |α + β + γ| ≤ J. We split a = S1 (a) + S2 (a),

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where S1 (a) =

J−1

ψ( j ξ, j η)aj (x, ξ, η)

j=0

and S2 (a) =

∞

ψ( j ξ, j η)aj (x, ξ, η).

j=J m

Since S1 (a) is a ﬁnite sum and each of its terms belongs to BSρ,δj ⊂ m0 m0 BSρ,δ , we infer that S1 (a) ∈ BSρ,δ . To estimate S2 (a), recall ﬁrst that for all j ≥ J, mj + 1 ≤ mJ + 1 ≤ m0 , thus, by using (4.2), we get that ⎞ ⎛ ∞ 2−j ⎠ (1 + |ξ| + |η|)m0 +δ|α|−ρ(|β|+|γ|) , |∂xα ∂ξβ ∂ηγ S2 (a)| ≤ ⎝ j=J m0 . which implies that S2 (a) ∈ BSρ,δ m0 Thus, we conclude that a ∈ BSρ,δ . We ﬁnally arrive to the asymptotic expansion of a. We have

a−

N −1 j=0

aj =

N −1

(ψ( j ξ, j η) − 1)aj +

j=0

∞

ψ( j ξ, j η)aj .

j=N

In the ﬁrst sum, because of (b), we only care about |ξ| + |η| < 2/ N −1 , and therefore we can achieve whatever decay we wish. For the second sum, we mN . The proof is proceed exactly as above to show that it belongs to BSρ,δ complete. Proof of Theorem 2.4. Note that, by Theorem 2.3, we know that there exists m0 such that some b ∈ BSρ,δ b∼

∞

aj .

j=0

Therefore, it will be sufﬁcient to show that a − b ∈ BS −∞ . We start by noticing that |a(x, ξ, η) − b(x, ξ, η)| ≤ |a(x, ξ, η) −

N −1

aj (x, ξ, η)| + |b(x, ξ, η) −

j=0

N −1

aj (x, ξ, η)|

j=0

≤ CN (1 + |ξ| + |η|)−μN −1 + C˜N (1 + |ξ| + |η|)mN ≤ cN (1 + |ξ| + |η|)−N , because both −μN −1 and mN converge to −∞ as N → ∞. To estimate the derivatives of the difference a − b we will employ the following useful result; see the book by Taylor [26, p.41]:

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If K1 , K2 are two compact sets such that K1 ⊂ K2◦ ⊂ K2 and u ∈ Cc2 (Rn ), then sup |Dα u(z)| sup |u(z)| sup |Dα u(z)|. |α|=1

z∈K1

z∈K2

|α|≤2

z∈K2

This is an immediate consequence of the estimate of a ﬁrst order partial differential operator in terms of its second order partial differential operator:

∂u/∂xj 2L∞ u L∞ ∂ 2 u/∂x2j L∞ ,

u ∈ Cc2 (Rn ).

Let then K be a compact set such that x ∈ K. Set K1 = K × {0} × {0}, and let K2 be a compact neighborhood of K1 . For ﬁxed ξ, η, deﬁne Fξ,η (x, ζ, ζ ) = a(x, ξ + ζ, η + ζ ) − b(x, ξ + ζ, η + ζ ). We can write sup |∇x,ξ,η (a − b)(x, ξ, η)|2 =

x∈K

sup (x,ζ,ζ )∈K2

|Fξ,η (x, ζ, ζ )|

sup (x,ζ,ζ )∈K1

|α|≤2

|∇(x,ζ,ζ ) Fξ,η (x, ζ, ζ )|2

sup (x,ζ,ζ )∈K2

α |Dx,ζ,ζ (a − b)(x, ξ + ζ, η + ζ )|

≤ CN sup (1 + |ξ + ζ|+|η + ζ |)−N (1+|ξ + ζ|+|η + ζ |)max(μ,m0 +2(δ−ρ)) . (x,ζ,ζ )

Since we have the freedom of choosing the compact neighborhood K2 , we can assume that on it |ζ| ≤ 1/3, |ζ | ≤ 1/3. Then, by the triangle inequality, we have 1 1 + |ξ + ζ| + |η + ζ | ≥ (1 + |ξ| + |η|), 3 and therefore we get, for all N > 0, the estimate |∂x ∂ξ ∂η (a − b)(x, ξ, η)| ≤ CN 3N 2max(μ,m0 +2(δ−ρ)) (1 + |ξ| + |η|)−N . Analogously, we will be able to control all the derivatives of a − b by |∂xα ∂ξβ ∂ηγ (a − b)(x, ξ, η)| ≤ CN (α, β, γ)(1 + |ξ| + |η|)−N . This proves that a − b ∈ BS −∞ and the proof is complete.

5. Pointwise Kernel Estimates In this section we will describe decay/blow-up estimates for bilinear kernels associated to pseudodifferential operators. These estimates can be summarized as follows. m , 0 < ρ ≤ 1, 0 ≤ δ < 1, m ∈ R, and let k(x, y, z) Theorem 5.1. Let p ∈ BSρ,δ denote the distributional kernel of associated bilinear pseudodifferential operator Tp . Let Z+ denote the set of non-negative integers and for x, y, z ∈ Rn , set

S(x, y, z) = |x − y| + |x − z| + |y − z|.

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Given α, β, γ ∈ Zn+ , there exists N0 ∈ Z+ such that for each N ≥ N0 , sup (x,y,z):S(x,y,z)>0

•

357

S(x, y, z)N |Dxα Dyβ Dzγ k(x, y, z)| < ∞

Suppose that p has compact support in (ξ, η) uniformly in x. Then k is smooth, and given α, β, γ ∈ Zn+ and N ∈ Z+ , there exists C > 0 such that for all x, y, z ∈ Rn with S(x, y, z) > 0 |Dxα Dyβ Dzγ k(x, y, z)| ≤ C(1 + S(x, y, z))−N .

• •

Suppose that m + M + 2n < 0 for some M ∈ Z+ . Then k is a bounded continuous function with bounded continuous derivatives of order ≤ M . Suppose that m + M + 2n = 0 for some M ∈ Z+ . Then there exists a constant C > 0 such that for all x, y, z ∈ Rn with S(x, y, z) > 0, sup |α+β+γ|=M

•

|Dxα Dyβ Dzγ k(x, y, z)| ≤ C| log |S(x, y, z)||.

Suppose that m+M +2n > 0 for some M ∈ Z+ . Then, given α, β, γ ∈ Zn+ , there exists a positive constant C such that for all x, y, z ∈ Rn with S(x, y, z) > 0, sup |α+β+γ|=M

|∂xα ∂yβ ∂zγ k(x, y, z)| ≤ CS(x, y, z)−(m+M +2n)/ρ .

Proof. Given a bilinear symbol p(x, ξ, η) with x, ξ, η ∈ Rn , set X = (x1 , x2 ) ∈ R2n , ζ = (ξ, η) ∈ R2n and deﬁne the linear symbol P in R2n as x1 + x2 P (X, ζ) = p , ξ, η . 2 m It follows easily that if p ∈ BSρ,δ (Rn ), for some m ∈ R and ρ, δ ∈ [0, 1], then m (R2n ). Indeed, given α = (α1 , α2 ), β = (β1 , β2 ) ∈ Z2n P ∈ Sρ,δ +

|α +α |

1

1 2 x1 + x2

α β

α1 +α2 β1 β2 , ξ, η

(∂x ∂ξ ∂η p)

∂X ∂ζ P (X, ζ) =

2 2 |α1 +α2 | 1 ≤ Cα,β (1 + |ξ| + |η|)m−ρ|β1 +β2 |+δ|α1 +α2 | 2 |α1 +α2 | 1 = Cα,β (1 + |ζ|)m−ρ|β|+δ|α| . 2

In R2n , for the associated linear operator TP we now have TP (F )(X) = P (X, ζ)eiX·ζ F(ζ)dζ R2n

=

p Rn Rn

x1 + x2 , ξ, η eix1 ·ξ eix2 ·η F(ξ, η) dξdη, 2

and also

TP (F )(X) =

K(X, Y )F (Y )dY, R2n

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where K(X, Y ) = F2n (P (X, ·))(Y − X),

X, Y ∈ R2n ,

and F2n denotes the Fourier transform in R2n . Next, for the bilinear symbol p we write Tp (f, g)(x) = p(x, ξ, η)eix·(ξ+η) fˆ(ξ)ˆ g (η)dξdη Rn Rn

=

p(x, ξ, η)eix·ξ eix·η f ⊗ g(ξ, η)dξdη

Rn Rn

= TP (f ⊗ g)(x, x) = K((x, x), (y, z))(f ⊗ g)(y, z)dydz Rn Rn

=

K((x, x), (y, z))f (y)g(z)dydz. Rn

Rn

Therefore, the distributional bilinear kernel k(x, y, z) of the bilinear operator Tp is given by k(x, y, z) = K((x, x), (y, z)),

x, y, z ∈ Rn ,

where K(X, Y ) is the distributional linear kernel associated to the linear operator TP in R2n . Finally, the pointwise estimates for linear kernels associm (R2n ) in [1, Theorem 1.1] imply the desired pointwise ated to symbols in Sρ,δ estimates for the bilinear kernel k(x, y, z).

6. An L2 × W0s,∞ → L2 Boundedness Property 0 If σ ∈ BSρ,δ , by freezing g, Tσ (·, g) can be regarded as a linear pseudodifferential operator (with symbol depending on g), that is, Tσ (f, g)(x) = σg (x, ξ)fˆ(ξ)eiξx dξ, ξ

where

σg (x, ξ) =

σ(x, ξ, η)ˆ g (η)eiηx dη.

η 2

Moreover, the well-known L boundedness of a linear pseudodifferential oper0 , 0 ≤ δ ≤ ρ ≤ 1, δ < 1, there exist constants C0 ator asserts that if τ ∈ Sρ,δ and k ∈ N (independent of τ ) such that

Tτ (u) L2 ≤ C0 |τ |k u L2 , where

u ∈ S(Rn ),

|τ |k = max sup ∂xα ∂ξβ τ (x, ξ) (1 + |ξ|)−δ|α|+ρ|β| . |α|,|β|≤k x, ξ

In fact, k can be taken equal to [n/2] + 1, see [14, p. 30].

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0 , 0 ≤ δ ≤ ρ ≤ 1, δ < 1. Then Theorem 6.1. Let σ ∈ BSρ,δ

Tσ : L2 × W0s,∞ → L2 , where s is any integer satisfying [n/2] + 1 + n. 1−δ 0 Moreover, if g ∈ Ccs (Rn ) then σg ∈ Sρ,δ , and s>

(6.1)

|σg |[n/2]+1 g W0s,∞ := sup Dγ g L∞ |γ|≤s

with a constant depending only on the

0 BSρ,δ -norm

of σ up to order n + 2.

Remark 6.2. Note that Theorem 6.1 includes the case 0 ≤ ρ = δ < 1 and, in particular, the case ρ = δ = 0, where, as pointed out in the introduction, the mapping from L2 (Rn ) × L∞ (Rn ) into L2 (Rn ) fails. 0 Proof. Let g ∈ Ccs (Rn ) and σ ∈ BSρ,δ , 0 ≤ δ ≤ ρ ≤ 1, δ < 1. We assume σ has compact support so all calculations below can be justiﬁed. However, all constants are independent of the support of σ and a limiting argument proves the result when σ does not have compact support (see [7]). Fix multiindices α and β such that |α| , |β| ≤ [n/2] + 1. Deﬁne A = 1 + |ξ| and let l0 ∈ N (to be chosen later) and Pl0 = {γ = (γ1 , . . . , γn ) : γi is even and |γ| = 2j, j = 0, . . . , l0 }. We have,

∂xα ∂ξβ σg (x, ξ) =

cγ,α

γ≤α

=

=

z

y

cγ,α,θ

γ≤α,θ∈Pl0

=

z

y

(1 + A2δ (−Δz ))l0 (∂xγ ∂ξβ σ(x, ξ, z)z α−γ )

cγ,α

γ≤α

z

∂xγ ∂ξβ σ(x, ξ, z)z α−γ eizy g(x − y) dydz

y

Aδ|θ| ∂zθ (∂xγ ∂ξβ σ(x, ξ, z)z α−γ )

cγ,α,θ,ω

γ≤α,θ∈Pl0

eizy g(x − y) dydz (1 + A2δ |y|2 )l0

eizy g(x − y) dydz (1 + A2δ |y|2 )l0

Aδ|θ| ∂xγ ∂ξβ ∂zθ−ω σ(x, ξ, z)z α−γ−ω

z y

eizy g(x − y) dydz. (1 + A2δ |y|2 )l0

ω≤min{θ,α−γ}

Fix γ ≤ α, θ ∈ Pl0 , ω ≤ min{θ, α − γ} and set eizy g(x − y) p(x, ξ) = Aδ|θ| ∂xγ ∂ξβ ∂zθ−ω σ(x, ξ, z)z α−γ−ω dydz. 2 (1 + A2δ |y| )l0 z

y

Deﬁne the sets Ω1 = {z : |z| ≤

Aδ 2 },

We then have

Ω2 = {z : ··· +

Ω1 y

≤ |z| ≤

p(x, ξ) =

Aδ 2

A 2 },

Ω3 = {z : |z| ≥

··· +

Ω2 y

· · · := I1 + I2 + I3 . Ω3 y

Note that A ≤ 1 + |ξ| + |z| ≤ 2A,

z ∈ Ω1 ∪ Ω2

A 2 }.

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and that |z| ≤ 1 + |ξ| + |z| ≤ 2 |z| ,

z ∈ Ω3

Estimation For I1 . Choose l0 such that 2l0 > n. Then |α|−|γ|−|ω| |z| |I1 | Aδ|θ| g L∞ (1+|ξ|+|z|)δ|γ|−ρ(|β|+|θ|−|ω|) dzdy 2 (1 + A2δ |y| )l0 y |z|≤ Aδ 2

δ|θ|

∼A

δ|γ|−ρ(|β|+|θ|−|ω|)

g L∞ A

A−δn Aδ(|α|−|γ|−|ω|+n)

= A(δ−ρ)(|θ|−|w|) Aδ|α|−ρ|β| g L∞ ≤ Aδ|α|−ρ|β| g L∞ . Note that in the last inequality we have used that δ ≤ ρ and |θ| − |w| ≥ 0. Estimation For I2 . Let l ∈ N to be chosen later. We have

g(x − y) (−Δy )l (eizy ) dydz 2 (1 + A2δ |y| )l0 |z|2l Ω2 y z α−γ−ω γ β θ−ω g(x − y) l = Aδ|θ| ∂ ∂ ∂ σ(x, ξ, z) (−Δ ) eizy dy dz. y x z ξ (1 + A2δ |y|2 )l0 |z|2l

I2 =

Aδ|θ| ∂xγ ∂ξβ ∂zθ−ω σ(x, ξ, z)z α−γ−ω

y

Ω2

Now,

g(x − y)

l

(−Δy ) 2

(1 + A2δ |y| )l0

2 −l0 ν 2δ μ−ν

=

Cνμ ∂y ((1 + A |y| ) )∂y g(x − y)

|μ|=2l

μi even,ν≤μ 2 ≤ Cνμl0 Dμ−ν g L∞ Aδ|ν| (1 + A2δ |y| )−l0 . |μ|=2l μi even,ν≤μ

Therefore,

|z α−γ−ω |

γ β θ−ω

Aδ|θ| |I2 | ≤ ∂ ∂ σ(x, ξ, z)

∂

x z ξ 2l |z| Ω2

≤

Cνμl0 Dμ−ν g L∞

|μ|=2l μi even,ν≤μ

×Aδ|ν| A−δn dz |α|−|γ|−|ω|−2l |z| Cνμl0 Dμ−ν g L∞ Aδ(|θ|+|ν|−n) |μ|=2l μi even,ν≤μ

Ω2

×(A + |z|)δ|γ|−ρ(|β|+|θ|−|ω|) dz ≤ Aδ(|θ|+2l−n) g W 2l,∞ Aδ|γ|−ρ(|β|+|θ|−|ω|) Aδ(|α|−|γ|−|ω|−2l+n) , 0

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where we have used that A + |z| ∼ A on Ω2 and l has been chosen so that |α| − |γ| − |ω| − 2l + n < 0.

(6.2)

Thus we obtain, |I2 | ≤ C Aδ(|θ|+2l−n) g W 2l,∞ Aδ|γ|−ρ(|β|+|θ|−|ω|) Aδ(|α|−|γ|−|ω|−2l+n) 0

= C Aδ|α|−ρ|β| A(δ−ρ)(|θ|−|w|) g W 2l,∞ 0

δ|α|−ρ|β|

≤ CA

g W 2l,∞ , 0

since (δ − ρ)(|θ| − |w|) ≤ 0. Estimation For I3 . Here we impose some extra conditions to l above. As in the estimation for B2 we have |I3 | ≤ Cνμl0 Dμ−ν g L∞ Aδ(|θ|+|ν|−n) |μ|=2l μi even,ν≤μ

×

|α|−|γ|−|ω|−2l

|z|

(A + |z|)δ|γ|−ρ(|β|+|θ|−|ω|) dz.

Ω3

Using that A + |z| ∼ |z| in Ω3 we have |α|−|γ|−|ω|−2l+δ|γ|−ρ(|β|+|θ|−|ω|) δ(|θ|+2l−n) |z| dz. |I3 | ≤ C g W 2l,∞ A 0

Ω3

Now, let us choose l as in the estimation of I2 and satisfying |α| − |γ| − |ω| − 2l + δ |γ| − ρ(|β| + |θ| − |ω|) + n < 0. For example, any choice of l such that 2l > |α| + n satisﬁes the inequality above. Then, |I3 | ≤ C Aδ(|θ|+2l−n) A|α|−|γ|−|ω|−2l+δ|γ|−ρ(|β|+|θ|−|ω|)+n g W 2l,∞ 0

= C Aδ|γ|−ρ|β| A(δ−1)(2l−n)+|α|−|γ| A(δ−ρ)|θ| A(ρ−1)|ω| g W 2l,∞ . 0

Since δ < 1, we can choose l sufﬁciently large so that (δ − 1)(2l − n) + |α| − |γ| ≤ 0

(6.3)

and since δ − ρ ≤ 0, ρ − 1 ≤ 0 and |γ| ≤ |α| we obtain |I3 | ≤ C Aδ|α|−ρ|β| g W 2l,∞ . 0

Finally, by choosing l = s/2, with s as in (6.1), we guarantee that both conditions (6.2) and (6.3) are satisﬁed, and the proof is complete. m Corollary 6.3. Let m ≥ 0 and σ ∈ BSρ,δ , 0 ≤ δ ≤ ρ ≤ 1, δ < 1. Then, if s is any integer satisfying (6.1), the following fractional Leibniz rule-type inequality holds true

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Tσ (f, g) L2 ≤ C f W m,2 g W0s,∞ + f W0s,∞ g W m,2 ,

f, g ∈ Cc∞ (Rn ). (6.4)

Proof. Corollary 6.3 follows from Theorem 6.1 and composition with Bessel potentials of order m, along the lines of Theorem 2 in [5]. We only need to m and φ is a C ∞ function on R such that 0 ≤ φ ≤ 1, notice that, if σ ∈ BSρ,δ supp(φ) ⊂ [−2, 2] and φ(r) + φ(1/r) = 1 on [0, ∞), then, the symbols σ1 and σ2 deﬁned by 1 + |ξ|2 σ1 (x, ξ, η) = σ(x, ξ, η)φ (1 + |ξ|2 )−m/2 1 + |η|2 and

σ2 (x, ξ, η) = σ(x, ξ, η)φ

1 + |η|2 1 + |ξ|2

(1 + |η|2 )−m/2

0 satisfy σ1 , σ2 ∈ BSρ,δ , and the corresponding operators Tσ , Tσ1 , and Tσ2 are related through

Tσ (f, g) = Tσ1 (J m f, g) + Tσ2 (f, J m g), where J m denotes the linear Fourier multiplier with symbol (1+|ξ|2 )m/2 .

Acknowledgments The authors would like to thank Kasso Okoudjou for useful discussions regarding the results presented here and to the anonymous referee for his/her comments and suggestions.

References ´ [1] Alvarez, L., Hounie, J.: Estimates for the kernel and continuity properties of pseudo-differential operators. Ark. Mat. 28, 1–22 (1990) ´ Bilinear pseudodifferential operators with forbidden symbols on [2] B´enyi, A.: Lipschitz and Besov spaces. J. Math. Anal. Appl. 284, 97–103 (2003) [3] B´enyi, A., Demeter, C., Nahmod, A., Thiele, C.M., Torres, R.H., Villarroya, P.: Modulation invariant bilinear T(1) theorem. J. Anal. Math. 109, 279–352 (2009) ´ Gr¨ [4] B´enyi, A., ochenig, K., Heil, C., Okoudjou, K.A.: Modulation spaces and a class of bounded multilinear pseudodifferential operators. J. Oper. Theory 54, 301–313 (2005) ´ Nahmod, A., Torres, R.H.: Sobolev space estimates and symbolic cal[5] B´enyi, A., culus for bilinear pseudodifferential operators. J. Geom. Anal. 16(3), 431–453 (2006) ´ Okoudjou, K.: Modulation space estimates for multilinear pseudo[6] B´enyi, A., differential operators. Stud. Math. 172, 169–180 (2006) ´ Torres, R.H.: Symbolic calculus and the transposes of bilinear [7] B´enyi, A., pseudodifferential operators. Comm. Partial Differ. Equ. 28, 1161–1181 (2003)

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´ Torres, R.: Almost orthogonality and a class of bounded bilinear [8] B´enyi, A., pseudodifferential operators. Math. Res. Lett. 11, 1–12 (2004) [9] Bernicot, F.: Local estimates and global continuities in Lebesgue spaces for bilinear operators. Anal. PDE 1, 1–27 (2008) [10] Bernicot, F., Torres R.H.: Sobolev space estimates for a class of bilinear pseudodifferential operators lacking symbolic calculus (2010, in preparation) [11] Calder´ on, A., Vaillancourt, R.: On the boundedness of pseudo-differential operators. J. Math. Soc. Jpn. 23, 374–378 (1971) [12] Christ, M., Journ´e, J-L.: Polynomial growth estimates for multilinear singular integral operators. Acta Math. 159, 51–80 (1987) [13] Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975) [14] Coifman R.R., Meyer Y.: Au-del` a des op´erateurs pseudo-diﬀ´erentiels, 2nd edn, vol. 57. Ast`erisque (1978) [15] Coifman, R.R., Meyer, Y.: Commutateurs d’int´egrales singuli`ers et op´erateurs multilin´eaires. Ann. Inst. Fourier Grenoble 28, 177–202 (1978) [16] Gilbert, J., Nahmod, A.: Boundedness of bilinear operators with non-smooth symbols. Math. Res. Lett. 7, 767–778 (2000) [17] Grafakos, L., Li, X.: The disc as a bilinear multiplier. Am. J. Math. 128, 91–119 (2006) [18] Grafakos, L., Torres, R.H.: Multilinear Calder´ on-Zygmund theory. Adv. Math. 165, 124–164 (2002) [19] Kenig, C., Stein, E.M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999) [20] H¨ ormander, L.: Pseudo-differential operators and hypoelliptic equations. Singular integrals. In: Proceedings of Symposium in Pure Mathematics X, 138–183 (1966). American Mathematical Society, Providence (1967) [21] Kumano-go, H.: Pseudodifferential Operators. MIT Press, Cambridge (1981) [22] Lacey, M., Thiele, C.: Lp bounds for the bilinear Hilbert transform, 2 < p < ∞. Ann. Math. 146, 693–724 (1997) [23] Lacey, M., Thiele, C.: On Calder´ on’s conjecture. Ann. Math. 149, 475– 496 (1999) [24] Maldonado, D., Naibo, V.: Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity. J. Fourier Anal. Appl. 15(2), 218–261 (2009) [25] Muscalu, C., Tao, T., Thiele, C.: Multilinear operators given by singular multipliers. J. Am. Math. Soc. 15, 469–496 (2002) [26] Taylor, M.: Pseudodifferential operators. Princeton University Press, New Jersey (1981) [27] Torres, R.H.: Multilinear singular integral operators with variable coefﬁcients. Rev. Union Mat. Argentina 50, 157–174 (2009) [28] Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

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´ ad B´enyi Arp´ Department of Mathematics Western Washington University 516 High St Bellingham, WA 98225, USA e-mail: [email protected] Diego Maldonado and Virginia Naibo Department of Mathematics Kansas State University 138 Cardwell Hall Manhattan, KS 66506, USA e-mail: [email protected]; [email protected] Rodolfo H. Torres Department of Mathematics University of Kansas Lawrence, KS 66045, USA e-mail: [email protected] Received: September 17, 2009. Revised: January 2, 2010.

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Integr. Equ. Oper. Theory 67 (2010), 365–375 DOI 10.1007/s00020-010-1786-7 Published online March 20, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Closed Range Property for Holomorphic Semi-Fredholm Functions J¨org Eschmeier and Dominik Faas Abstract. Given Banach spaces X and Y , we show that, for each operator-valued analytic map α ∈ O(D, L(Y, X)) satisfying the finiteness condition dim(X/α(z)Y ) < ∞ pointwise on an open set D in Cn , α the induced multiplication operator O(U, Y ) −→ O(U, X) has closed range on each Stein open set U ⊂ D. As an application we deduce that the generalized range R∞ (T ) = k≥1 |α|=k T α X of a commuting mul tioperator T ∈ L(X)n with dim(X/ n i=1 Ti X) < ∞ can be represented as a suitable spectral subspace. Mathematics Subject Classification (2000). Primary 47A13, 32C35; Secondary 47A11, 47A53. Keywords. Closed range property, coherent sheaves, Fredholm theory.

0. Introduction A classical result in several complex variables going back to H. Cartan says that, over a Stein open set U ⊂ Cn , each ﬁnitely generated ideal (g1 , . . . , gr ) in r O(U ) is closed. rIn this setting, the operator-valued map αg : U → L(C , C), αg (z)(xi ) = i=1 gi (z)xi , is analytic and the above cited result precisely means that the image of the induced multiplication operator αg : O(U, Cr ) → O(U, C), (αg f )(z) = αg (z)f (z) is closed. To indicate a proof, let us observe that, more generally, every anaCr α lytic operator-valued map α ∈ O(U, L(Cr , Cs )) induces a morphism OU → s C Cs between coherent analytic sheaves. But then the image sheaf Imα ⊂ OU OU is a coherent subsheaf and r

α

C 0 → ker α → OU −→ Imα → 0

becomes an exact sequence of coherent analytic sheaves on U . Since coherent sheaves are acyclic on Stein open sets in Cn , the induced sequence of

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section spaces over U remains exact. By Cartan’s Abgeschlossenheitssatz (cf. Sect. V.4 in [7]) the subspace r s ) = Γ(U, Imα) ⊂ Γ(U, OU ) = O(U, Cs ) αO(U, Cr ) = αΓ(U, OU

is closed in the usual Fr´echet space topology. Let X, Y be complex Banach spaces. In the present note we show that, for each Stein open set U ⊂ Cn and each operator-valued map α in O(U, L(Y, X)) with the property that dim X/α(z)Y < ∞ for every z ∈ U , the induced multiplication operator α : O(U, Y ) → O(U, X),

(αf )(z) = α(z)f (z)

has closed range. To prove this result we use methods from Markoe [12] and X Leiterer [10] to show that in the above situation the image sheaf Imα ⊂ OU is a Banach coherent subsheaf in the sense of Leiterer [10]. But then α

Y 0 → ker α → OU −→ Imα → 0

becomes an exact sequence of Banach coherent sheaves. Since Banach coherent sheaves are acyclic on Stein open sets in Cn , we obtain as in the ﬁnitedimensional case the closedness of the subspace Y X ) = Γ(U, Imα) ⊂ Γ(U, OU ) = O(U, X). αO(U, Y ) = αΓ(U, OU

For single variable operator-valued functions of the form α(z) = z − T , where T ∈ L(X) is a bounded operator on a Banach space, closed range theorems of the above type are known and have been applied in the local spectral theory of Banach-space operators (see [2,13,14]). For instance, for every operator satisfying the ﬁniteness condition dim(X/T X) < ∞, one can show that the generalized range R∞ (T ) = k≥1 T k X of T has a representation of the form R∞ (T ) = XT (C\U ), where U is a suitable open zero neighbourhood in C. As an application of our results we show in the ﬁnal part of the paper that an analogous formula holds for commuting multioperators.

1. Main Result Recall that an analytic sheaf F on an analytic space (X, OX ) is said to be an analytic Fr´echet sheaf if all section spaces F(U ) (U ⊂ X open) are Fr´echet O(U )-modules and the restriction maps F(U ) → F(V ),

s → s|V

(U, V ⊂ X open with V ⊂ U )

are continuous. A continuous morphism ϕ : F → G between analytic Fr´echet sheaves is a sheaf homomorphism such that for every open set U ⊂ X the induced section map ϕ : Γ (U, F) → Γ (U, G) is continuous. An analytic Fr´echet sheaf F on X is called Banach coherent (in the sense of Leiterer [10]) if, for each point x ∈ X and each integer n ≥ 0, there are an open

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neighbourhood U ⊂ X of x and a Banach-free resolution for F|U of length n, that is, an exact sequence of continuous morphisms OEn |U −→ · · · −→ OE1 |U −→ OE0 |U −→ F|U −→ 0, where E0 , . . . , En are suitable Banach spaces. Examples of Banach coherent analytic Fr´echet sheaves are coherent analytic sheaves and sheaves of the E , where E is a Banach space. The reader will ﬁnd more details in form OX Chapter 4 of [4]. In the following let X, Y be complex Banach spaces and let α : D → L(Y, X) be a holomorphic map on a ﬁxed open set D ⊂ Cn . We start by considering some special cases of our main result. In particular we prove a local version which shows that the range of α : O(U, Y ) → O(U, X) is closed if U ⊂ D is a small Stein open neighbourhood of a point z ∈ D such that dim (X/α(z)Y ) < ∞. First we consider the case that the Banach space X on the right-hand side is ﬁnite dimensional, i.e., we have X = Cp for some p ∈ N. Lemma 1.1. If X = Cp for some p ∈ N and D is Stein, then αO(D, Y ) ⊂ O(D, Cp ) is a closed subspace. p

Y C Proof. By a result of Markoe [12] (Proposition 5) the image sheaf αOD ⊂ OD is a coherent subsheaf. Therefore the second and the third sheaf in the short exact sequence α

Y Y −→ αOD −→ 0 0 −→ ker α → OD

of analytic Fr´echet sheaves are Banach coherent in the sense of Leiterer [10] (cf. Section 4.5 in [4]). But then also the ﬁrst sheaf ker α is Banach coherent (Proposition 4.5.7 in [4]). Since Banach coherent analytic Fr´echet sheaves are quasi-coherent and quasi-coherent analytic Fr´echet sheaves are acyclic on Stein open subsets (Theorem 4.5.2 and Proposition 4.3.3 (b) in [4]), the induced sequence of section spaces α

Y Y ) −→ Γ(D, αOD ) −→ 0 0 −→ Γ(D, ker α) → Γ(D, OD

remains exact. By the ‘Abgeschlossenheitssatz’ for coherent sheaves (Chapter V §6.4 in [7]) it follows that p

Y Y C ) = Γ(D, αOD ) ⊂ Γ(D, OD ) = O(D, Cp ) αO(D, Y ) = αΓ(D, OD

is a closed subspace.

Now we consider the case that X is an arbitrary Banach space, but we assume in addition that the kernel of the operator α(z) is continuously projected in Y . Lemma 1.2. Let 0 ∈ D and dim (X/α(0)Y ) < ∞. If ker α(0) ⊂ Y is continuously projected, then there is an open set V ⊂ Cn with 0 ∈ V ⊂ D such that αO(U, Y ) ⊂ O(U, X) is closed for every Stein open subset U ⊂ V .

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Proof. We use a construction due to Markoe [12] to reduce the assertion to the case where X is ﬁnite dimensional. Deﬁne T = α(0) ∈ L(Y, X) and choose closed subspaces M ⊂ Y and N ⊂ X such that Y = M ⊕ ker T

and

X = T Y ⊕ N.

Note that N is ﬁnite dimensional. Let us write iN : N → X for the inclusion mapping. For z ∈ D, the operator α(z) : M ⊕ ker T → T Y ⊕ N possesses a matrix representation of the form a(z) b(z) α(z) = c(z) d(z) with suitable analytic operator-valued functions a, b, c, d. Since both the maps a(0) ∈ L(M, T Y ) and (α(0), iN ) ∈ L(M ⊕N, X) are invertible, we can choose an open neighbourhood V ⊂ D of 0 in such a way that a(z) ∈ L(M, T Y ) and (α(z), iN ) ∈ L(M ⊕ N, X) are invertible for every z ∈ V . Fix an open subset U ⊂ V . For the holomorphic operator-valued function u : U → L (ker T, N ) ,

u(z) = d(z) − c(z)a(z)−1 b(z),

it was shown by Eschmeier [3] that the identity uO(U, ker T ) = O(U, N ) ∩ αO(U, Y ) holds. Therefore the map ϕU : O(U, N )/uO(U, ker T ) → O(U, X)/αO(U, Y ),

[f ] → [f ]

is a well-deﬁned one-to-one linear map which is continuous if both sides are equipped with their canonical quotient topologies. To show that ϕU is in fact a homoeomorphism, we construct a continuous right inverse. For this purpose, we ﬁrst observe that by construction the operators (α(z),iN )−1 r(z) = rY (z), rN (z) : X −→ M ⊕ N → Y ⊕ N depend analytically on z ∈ V and satisfy α(z)rY (z)x + rN (z)x = x (x ∈ X, z ∈ V ). Thus for f ∈ O(U, X) we have f = αrY f + rN f. In particular we ﬁnd that rN αO(U, Y ) ⊂ O(U, N ) ∩ αO(U, Y ) = u O(U, ker T ). Hence we obtain a well-deﬁned continuous linear mapping ΨU : O(U, X)/αO(U, Y ) → O(U, N )/uO(U, ker T ),

[f ] → [rN f ].

Since f − rN f = αrY f ∈ αO(U, Y ) for all f ∈ O(U, X), it follows that ΨU is a right inverse for ϕU . Hence both maps are topological isomorphisms.

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To complete the proof it sufﬁces to observe that by Lemma 1.1 the range space of ΨU is Hausdorﬀ, whenever U ⊂ V is a Stein open set. Our next aim is to show that Lemma 1.2 remains true without the condition that ker α(0) ⊂ Y is continuously projected. To prove this we use an idea due to Kaballo [8]. Theorem 1.3. (Closed-Range Theorem, local version) Suppose that 0 ∈ D and that dim (X/α(0)Y ) < ∞. Then there is an open set V ⊂ Cn with 0 ∈ V ⊂ D such that αO(U, Y ) ⊂ O(U, X) is closed for every Stein open subset U ⊂ V . Proof. By shrinking D if necessary, we may suppose that dim (X/α(z)Y ) < ∞ for all z ∈ D. As shown by Kaballo [8] (see 1.1, 1.2 and 1.3) there is a diagram Z @

ϕ

1 (A)

α0 (z)

α(z) ˜

@ R @ - 1 (B)

ρ

π

? Y

α(z)

? - X

with surjective continuous linear operators π, ρ, ϕ and holomorphic functions α ˜ : D → L(Z, 1 (B)) and α0 : D → L(1 (A), 1 (B)) such that ker α ˜ (0) is continuously projected, dim 1 (B)/˜ α(z)Z < ∞ for every z ∈ D and such that the intertwining relations π ◦ α0 (z) = α(z) ◦ ρ

and π ◦ α ˜ (z) = α(z) ◦ ρ ◦ ϕ

hold for all z ∈ D. More explicitly (see also [8]) the space Z can be chosen as the direct sum Z = 1 (A) ⊕ ker π and the mappings ϕ and α(z) ˜ can be deﬁned by ϕ(x, y) = x and α(z)(x, ˜ y) = α0 (z)x + y. We consider now an arbitrary open set U ⊂ D. Then πU : O(U, 1 (B)) → O(U, X),

f → πf

is a surjective continuous linear operator between Fr´echet spaces (see e.g. Appendix 1 in [4]). We claim that −1 ˜ O(U, Z). πU αO(U, Y ) = α To check this, note ﬁrst that π α ˜ f = α (ρϕf ) for f ∈ O(U, Z). To prove the opposite inclusion, consider a function f ∈ O(U, 1 (B)) such that πU (f ) = αg

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for some g ∈ O(U, Y ). Using the surjectivity of ρ we can choose a function h ∈ O(U, 1 (A)) with g = ρh. It follows that πf = αρh = πα0 h and hence that f = α0 h + (f − α0 h) = α ˜ (h, f − α0 h) ∈ α ˜ O(U, Z). The formula just proved implies that the map πU induces a well-deﬁned linear isomorphism πˆU : O(U, 1 (B))/˜ αO(U, Z) −→ O(U, X)/αO(U, Y ),

[f ] → [πu f ] .

Obviously this map is continuous if both sides are equipped with their quotient topologies. We will show that also the inverse of πˆU is continuous. Since πU : O(U, 1 (B)) → O(U, X) is a surjective continuous linear operator between Fr´echet spaces, there exists a continuous, not necessarily linear, right inverse rU : O(U, X) → O(U, 1 (B)) for πU (Satz 7.1 in [11]). For f, g ∈ O(U, X) with f − g ∈ αO(U, Y ), we have πU (rU f − rU g) = f − g ∈ αO(U, Y ). But then rU f − rU g ∈ α ˜ O(U, Z). Hence rU induces a well-deﬁned continuous mapping rˆU : O(U, X)/αO(U, Y ) → O(U, 1 (B))/˜ αO(U, Z),

[f ] → [rU f ].

By construction rˆU is the inverse of πˆU (in particular, rˆU is linear) and hence πˆU is a topological isomorphism. Since by Lemma 1.2 there is an open neighbourhood V ⊂ D of 0 such that α ˜ O(U, Z) ⊂ O(U, 1 (B)) is closed for every Stein open subset U ⊂ V , the assertion follows.

Let X, Y be Banach spaces and let α ∈ O (D, L(Y, X)) be a holomorphic map on an open neighbourhood D of 0 in Cn such that dim (X/α(0)Y ) < ∞. Corollary 1.4. There exist an open neighbourhood V ⊂ D of 0, a Banach space E, a finite-dimensional subspace N ⊂ X and a holomorphic mapping u : V → L(E, N ) such that ρU : O(U, N )/uO(U, E) → O(U, X)/αO(U, Y ),

[f ] → [f ]

are well-defined topological isomorphisms for each open subset U ⊂ V . Proof. We assume that dim (X/α(z)Y ) < ∞ for all z ∈ D and keep the notation from the proof of Theorem 1.3. As shown there, for each open subset U ⊂ D, the mapping πˆU : O(U, 1 (B))/˜ αO(U, Z) −→ O(U, X)/αO(U, Y ),

[f ] → [πU f ]

is a well-deﬁned topological isomorphism. By the proof of Lemma 1.2 there are an open neighbourhood V ⊂ D of 0, a Banach space E, a ﬁnite-dimensional subspace N ⊂ 1 (B) and a holomorphic mapping u : V → L(E, N )

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such that, for every open subset U ⊂ V , the mapping ϕU : O(U, N )/uO(U, E) → O(U, 1 (B))/˜ αO(U, Z),

[f ] → [f ]

is a well-deﬁned topological isomorphism. Note that σU : O(U, N )/uO(U, E) → O(U, πN )/(πu)O(U, E),

[f ] → [πf ]

and ρU : O(U, πN )/(πu)O(U, E) → O(U, X)/αO(U, Y ),

[f ] → [f ]

are well-deﬁned continuous linear maps with ρU ◦ σU = πˆU ◦ ϕU for every open subset U ⊂ V . Since the maps σU are surjective, it follows easily, that the maps ρU are topological isomorphisms. In [3] a version of Corollary 1.4 was used to show that on the open subset D0 = {z ∈ D; dim (X/α(z)Y ) < ∞} ⊂ D X Y the quotient sheaf OD /αOD is coherent. Indeed Corollary 1.4 implies that 0 0 locally on D0 there are sheaf isomorphisms of the form ∼

ρV : OVN /uOVE −→ OVX /αOVY ,

[(f, U )z ] → [(f, U )z ] ,

where N ⊂ X is a ﬁnite-dimensional subspace, E is a suitable Banach space and u : V → L(E, N ) is an analytic map. Since by a result of Markoe (see [12], Proposition 5) the quotient sheaf on the left is coherent and since coherence X Y /αOD follows. is a local property, the coherence of the analytic sheaf OD 0 0 It is well known (cf. Section 4.1 in [4]) that the section spaces of a coherent analytic sheaf carry a canonical nuclear Fr´echet space topology which, for the sheaf OU , is given by the identiﬁcation Γ (U, OU ) ∼ = O(U ). Morphisms of coherent sheaves induce continuous linear maps between section spaces. Hence, in the situation explained above, we obtain induced topological isomorphisms ∼ (U ⊂ V open) Γ U, OVN /uOVE −→ Γ U, OVX /αOVY between nuclear Fr´echet spaces. The quotient map OVN → OVN /uOVE induces continuous linear maps (U ⊂ V open). O(U, N )/uO(U, E) −→ Γ U, OVN /uOVE Using the commutativity of the diagrams ∼ - O(U, X) Γ U, OVX

- O(U, X)/αO(U, Y ) (ρU )−1

? ∼ Γ U, OVX /αOVY

Γ U, OVN /uOVE

? O(U, N )/uO(U, E)

qU we deduce that the section map Γ U, OVX −→ Γ U, OVX /αOVY is continuous for every open U ⊂ V .

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From now on let us make the assumption that dim (X/α(z)Y ) < ∞ for Y all z ∈ D. By the above cited result of Markoe [12] the image sheaf αOD is coherent when X is ﬁnite dimensional. More generally, for an arbitrary Y is still Banach coherent. Banach space X the image sheaf αOD Corollary 1.5. Let α ∈ O (D, L(Y, X)) be a holomorphic operator-valued function on an open set D ⊂ Cn such that dim (X/α(z)Y ) < ∞ for all z ∈ D. Then the quotient map q

X X Y −→ OD /αOD OD

is a continuous morphism between Banach coherent analytic Fr´echet sheaves. Y is a Banach coherent analytic Fr´echet In particular, the image sheaf αOD sheaf. Proof. Let U ⊂ D be an arbitrary open subset. By the remarks following Corollary 1.4 we can choose an open cover U = k∈N Uk such that all the section maps qUk X X Y (k ∈ N) Γ Uk , OD −→ Γ Uk , OD /αOD are continuous. A straightforward application of the closed graph theorem for Fr´echet spaces shows that also the section map qU X X Y Γ U, OD −→ Γ U, OD /αOD X X Y is continuous. Thus we have shown that q : OD −→ OD /αOD is a continuous morphism between analytic Fr´echet sheaves. For each open set U ⊂ D, the space Y = ker qU Γ U, αOD X . Thereis a Fr´echet space as a closed subspace of the Fr´echet space Γ U, OD Y fore the image sheaf αOD is an analytic Fr´echet sheaf and Y X → OD 0 −→ αOD

q

X Y −→ OD /αOD −→ 0

is an exact sequence of continuous morphisms between analytic Fr´echet sheaves. Since the second and the third sheaf in this sequence are Banach coherent, by the result of Leiterer (Proposition 4.5.7 in [4]) used before, the same is true for the ﬁrst sheaf. This observation completes the proof. Since Banach coherent analytic sheaves are acyclic on Stein open sets, Corollary 1.5 allows us to deduce our main result by a repetition of the arguments that lead to a proof in the case of ﬁnite-dimensional image spaces. Theorem 1.6. (Closed-Range Theorem) Let X, Y be Banach spaces. Suppose that α ∈ O (D, L(Y, X)) is an analytic operator-valued map on an open set D ⊂ Cn such that dim (X/α(z)Y ) < ∞ for all z ∈ D. Then αO(U, Y ) ⊂ O(U, X) is closed for every Stein open subset U ⊂ D.

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Proof. By Corollary 1.5 the second and the third sheaf in the exact sequence of analytic Fr´echet sheaves α

Y Y −→ αOD −→ 0 0 −→ ker α → OD

are Banach coherent. The result of Leiterer cited above (Proposition 4.5.7 in Y [4]) implies that the same is true for the subsheaf ker α ⊂ OD . Since Banach coherent analytic sheaves are acyclic on Stein open subsets (Theorem 4.5.2 and Proposition 4.3.3 (b) in [4]), the induced sequence of section spaces α

Y Y ) −→ Γ(U, αOD ) −→ 0 0 −→ Γ(U, ker α) → Γ(U, OD

remains exact on each Stein open subset U ⊂ D. Hence Y Y X αO(U, Y ) = αΓ(U, OD ) = Γ(U, αOD ) ⊂ Γ(U, OD ) = O(U, X)

is a closed subspace for each Stein open subset U ⊂ D.

2. An Application Let T = (T1 , . . . , Tn ) ∈ L(X)n be a commuting tuple ofbounded linear n operators on a complex Banach space X such that dim (X/ i=1 Ti X) < ∞. Then there is an open neigbourhood D of the origin z = 0 ∈ Cn such that n

(zi − Ti )X < ∞ (z ∈ D), dim X/ i=1

and Theorem 1.6 applied to the analytic operator-valued map αT : D → L(X n , X), αT (z) = (z1 − T1 , . . . , zn − Tn ) n shows that i=1 (zi − Ti )O(U, X) ⊂ O(U, X) is closed for each Stein open set U ⊂ D. Hence the spectral subspaces n

n XT (C \ U ) = X ∩ (zi − Ti )O(U, X) ⊂ X i=1

are closed for every Stein open set U ⊂ D. In particular, the space ∞

X∞ = XT (Cn \U ) = XT Cn \ B k1 (0) ⊂ X U ∈U (0)open

k=1

is a countable union of closed linear subspaces of X. By Corollary 1.2 in X Xn /αT OD is a coherent analytic sheaf. Hence [3], the quotient sheaf H = OD its stalk at z = 0 is a ﬁnitely generated module over the local Noetherian ring O0 . The spaces Mk = |α|=k T α X (k ∈ N) form a decreasing sequence of ﬁnite-codimensional closed subspaces of X. Following the notation used in the one-variable case, we write R∞ (T ) = k≥1 Mk . Let m ⊂ O0 be the maximal ideal of O0 . Then in [5], Lemma 3.22 (see also [6], Section 1.3 for the Hilbert-space case), it was shown that the map Φ : X → H0 , x → [x], induces vector-space isomorphisms Φk : X/Mk → H0 /mk H0 ,

x + Mk → [x] + mk H0

(k ≥ 1).

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By Krull’s intersection theorem (cf. [15, Theorem 4.19]), applied to the Noetherian O0 -module H0 , the right vertical map in the commuting diagram X qX ∞ k=1

Φ

- H0 qH

?

X/Mk

(Φk )k≥1

-

∞

?

H0 /mk H0 ,

k=1

is injective. Here the vertical maps are deﬁned by qX (x) = (x + Mk )k≥1 and qH (h) = h + mk H0 k≥1 . Since the lower horizontal map is an isomorphism, we obtain that X∞ = Ker Φ = Ker qX = R∞ (T ) is a closed subspace of X. By Baire’s category theorem there is a natural ∞ n number k ≥ 1 such that R (T ) = XT C \B k1 (0) . Thus we have obtained the following result. Theorem 2.1. Let T = (T1 , . . . , Tn ) ∈ L(X)n be a commuting tuple of bounded n operators on a Banach space X such that dim (X/ i=1 Ti X) < ∞. Then there is an open neigbourhood U of the origin z = 0 ∈ Cn such that n

∞ n X R (T ) = XT (C \ U ) = x ∈ X; x ∈ (zi − Ti )O0 . i=1

For a single operator T ∈ L(X) with ∞ dim X/T X < ∞, the preceding representation of the space R∞ (T ) = k=1 T k X is well known (see for instance [9], Proposition 3.7.2). As an elementary application one obtains that the restriction of T to the invariant subspace R∞ (T ) = is surjective. In [6] X. Fang asked whether, n more generally, in the setting of Theorem 2.1 the identity R∞ (T ) = i=1 Ti R∞ (T ) holds. Since in the multivariable case there are examples of commuting tuples T ∈ L(X)n which possess closed spectral subspaces XT (F ) such that the surjectivity spectrum of T |XT (F ) is not contained in F (see [1]), this question remains open at this time.

References [1] Eschmeier, J.: Are commuting systems of decomposable operators decomposable? J. Oper. Theory 12, 213–219 (1984) [2] Eschmeier, J.: On the essential spectrum of Banach space operators. Proc. Edinb. Math. Soc. 43, 511–528 (2000) [3] Eschmeier, J.: Samuel multiplicity for several commuting operators. J. Oper. Theory 60, 399–414 (2008) [4] Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. London Mathematical Society Monographs, New Series, vol. 10, Clarendon Press, Oxford (1996)

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[5] Faas, D.: Zur Darstellungs- und Spektraltheorie f¨ ur nichtvertauschende Operatortupel, Dissertation, Saarbr¨ ucken, 2008 [6] Fang, X.: The Fredholm index of a pair of commuting operators II. J. Funct. Anal. 256, 1669–1692 (2009) [7] Grauert, H., Remmert, R.: Theorie der Steinschen R¨ aume, Grundlehren der mathematischen Wissenschaften. vol. 227, Springer, Berlin (1977) [8] Kaballo, W.: Holomorphe Semi-Fredholmfunktionen ohne komplementierte Kerne bzw. Bilder. Math. Nachrichten 91, 327–335 (1979) [9] Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs, vol. 20, The Clarendon Press, New York (2000) [10] Leiterer, J.: Banach coherent analytic Fr´echet sheaves. Math. Nachrichten 85, 91–109 (1978) [11] Mantlik, F.: Parameterabh¨ angige lineare Gleichungen in Banach- und in Fr´echetr¨ aumen, Dissertation, Universit¨ at Dortmund, 1988 [12] Markoe, A.: Analytic families of differential complexes. J. Funct. Anal. 9, 181– 188 (1972) [13] Miller, T.L., Miller, V.G., Neumann, M.M.: The Kato-type spectrum and local spectral theory. Czech. Math. J. 57, 831–842 (2007) [14] Miller, T.L., M¨ uller, V.: The closed range property for Banach space operators. Glasgow Math. J. 50, 17–26 (2008) [15] Northcott, D.G.: Lessons on Rings, Modules and Multiplicities. Cambridge University Press, London (1968) J¨ org Eschmeier and Dominik Faas Fachrichtung Mathematik Universit¨ at des Saarlandes Postfach 151150 66041 Saarbr¨ ucken Germany e-mail: [email protected]; [email protected] Received: September 24, 2009. Revised: December 16, 2009.

Integr. Equ. Oper. Theory 67 (2010), 377–424 DOI 10.1007/s00020-010-1787-6 Published online April 2, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On Passive and Conservative State/Signal Systems Mikael Kurula Abstract. This article is devoted to a study of continuous-time passive and conservative systems within the state/signal framework. The main idea of the state/signal approach is to not a priori distinguish between inputs and outputs, but rather to combine these two into a single external signal. The so-called node space is introduced as the product of two copies of the state space of the system and one copy of the space where the external signals of the system live. This node space is equipped with a sesquilinear product that makes it a Kre˘ın space. A generating subspace is deﬁned as a closed subspace of the node space, and the node space determines the trajectories of a state/signal system. One of the main results of this article is that a subspace of the node space generates a passive state/signal system if and only if it is a maximally nonnegative subspace of the node space and it satisﬁes a certain nondegeneracy condition. In this case the generating subspace can be interpreted as the graph of a scattering-passive input/state/output system node. Mathematics Subject Classification (2000). Primary 47A48, 93C25; Secondary 47N70, 46C20. Keywords. State/signal, input/state/output, inﬁnite-dimensional system, linear system, passive, conservative, scattering.

1. Introduction In this paper we study continuous-time passive and conservative linear systems within the so-called state/signal framework. This framework allows us to treat inputs and outputs on an equal basis. Indeed, a limitation of the input/state/output approach to systems theory is that inputs and outputs are considered to be ideal. When systems are interconnected, however, every input also acts as an output and vice versa, because the subsystems will always inﬂuence each other mutually. The state/signal formulation is useful for instance for modelling interconnections where a partial collapse of the This research was supported by the Academy of Finland, project number 201016 and the Finnish Graduate School in Mathematical Analysis and its Applications.

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state space takes place. This situation, which is not covered by the standard feedback theory, see [26], is illustrated in Example 4.9, where we model the interconnection of two capacitors in parallel. State/signal systems in continuous time were introduced in [14] and in the present article we continue the development of their theory. The ﬁrst steps in the direction of conservative continuous-time state/signal theory were taken already in Ball and Staffans [9], and Malinen and Staffans [18]. The theory of discrete-time state/signal systems has been developed by Arov and Staffans in [4–8]. For an overview how the state/signal theory uniﬁes the theories of different types of passive discrete-time systems; see [27]. The present article mainly gives continuous-time analogues of some of the results in [5]. The state/signal framework is similar to the behavioural theory developed by Jan Willems and his coauthors and followers; see [23] for a good introduction. A main difference between these two formulations is that the system state plays an important role in the state/signal setting, whereas in the behavioural formulation this seems not to be the case. Another main diﬀerence is that the behavioural approach handles partial diﬀerential equation models by viewing the spatial variables as extra time variables, which are not distinguished from the actual time variable, but in the state/signal framework one keeps time one-dimensional. As a consequence of this diﬀerence, the so-called n − D behavioural systems have a ﬁnite-dimensional state space and state/signal systems have inﬁnite-dimensional state spaces. The state/signal theory is much more developed than the corresponding behavioural theory for partial diﬀerential equation systems. Here we use mainly energy-based methods, whereas the ﬁnite-dimensional behavioural theory is built using algebraic tools. Another approach to modelling conservative systems, which is closely related to the state/signal approach, uses the concepts of port-Hamiltonian systems and Dirac structures. Van der Schaft and Maschke are two of the main authors in this ﬁeld, which originates from the modelling of conservative physical systems which are often nonlinear, see [15,20,21,30]. Although some work has been done to extend the port-Hamiltonian system approach to distributed-parameter systems, most of the theory still concerns ﬁnitedimensional systems. In the port-Hamiltonian approach the existence of solutions is often motivated by physical arguments, but in this article we are also interested in mathematical proofs for existence of system trajectories. The theory of boundary triplets and their application to solving boundary control problems is now classical; see [13]. The concept of boundary triplets has been generalised to that of boundary relations by a group of authors in papers such as [10] and [12]. This work is also closely connected to the state/signal theory. Passive inﬁnite-dimensional input/state/output systems in continuous time have previously been studied a.o. in [1–3,17–19,24,25,29,31]. We now proceed to describe the contents of the paper in more detail. To avoid unnecessary repetition we introduce the abbreviations i/s/o for input/state/output, i/o for input/output and s/s for state/signal.

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A common model for a linear continuous-time time-invariant system is x(t) ˙ A B x(t) (1.1) Σ: = , t > 0, x(0) = x0 . y(t) C D u(t) ∂ Here A, B, C and D are linear operators, and x˙ = ∂t x. The function x is called the state trajectory, u the input signal, y the output signal, and together they form an input/state/output (i/s/o) trajectory (u, x, y) of Σ. The state trajectory x takes values in the state space X , the input u lives in the input space U, and the output y in the output space Y. For the moment the operA B ] is assumed to map [ X ] continuously into X , but we will soon ator [ C Y D U A B ] by an unbounded operator node, which we deﬁne in Definition replace [ C D 2.4, in order to allow a larger set of applications. A system reminiscent of (1.1) is commonly known as an i/s/o system. We formalise the idea of equal treatment of inputs and outputs in (1.1) by considering the input space U and the output space Y to be closed subspaces of a combined external signal space: W := U Y. We can always {0} Y . achieve this by setting W := Y U and identifying Y = {0} and U = U

Then we rewrite the system (1.1) in graph form to get rid of the explicit input u(t) and output y(t): ⎡ ⎤ x(t) ˙ ⎣ x(t) ⎦ ∈ V, t > 0, x(0) = x0 , where (1.2) w(t) ⎧⎡ ⎫ ⎤ ⎬ z ⎨ z A B x V = ⎣ x ⎦ = y C D u ⎭ ⎩ u+y ⎡ ⎤ A B ⎦ X . 0 = ⎣ 1X (1.3) U C D + 1U We call a system of the type (1.2) a differential s/s model of the system Σ in (1.1). We now return to the general inﬁnite-dimensional setting. In the study of passive systems it is natural to require W to be a Kre˘ın space, as we will see later. Some theory of Kre˘ın spaces and brief definitions of the function spaces which we need throughout this article can be found in the appendix. The following definition should be compared to (1.2). Definition 1.1. Let I be a subinterval of R with positive length, let X be a Hilbert space, and let W be a Kre˘ın space. Let V be a closed subspace of X X W

.

The space V(I) of classical trajectories on I generated by V consists of x(t) ˙ 1 C (I;X ) x all pairs [ w ] ∈ C(I;W , such that x(t) ∈ V for all interior points t in I. We abbreviate V := V[0, ∞).

w(t)

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If x, ˙ x and w are all continuous on I, and

x(t) ˙ x(t) w(t)

∈ V for all internal

points t of I, then the inclusion holds at every t ∈ I, with one-sided derivatives at any end-points of I. Thus we often replace the statement that the inclusion holds for all internal points t of I by the shorter statement that it holds for all t ∈ I. The state space X in Definition 1.1 represents the internal memory of the system, whereas the external signal space is used to interconnect the s/s system to the outside world. The external signal space can be decomposed into a direct-sum decomposition W = U Y of an input space U and an output space Y in various ways and different decompositions yield different A B ] used to describe V i/s/o representations (1.1). Indeed, the operator [ C D in (1.3) corresponds to the given decomposition W = U Y. The different i/s/o representations can have different properties; they can for instance be passive with respect to different supply rates. More precisely, a direct-sum decomposition of W = U Y forms an admissible i/o pair (U, Y) for the subspace V if there exists an operator node A&B , which we will deﬁne precisely in Definition 2.4, with input space U, C&D state space X and output space Y, such that V can be written as the graph A&B in the following way: of C&D ⎡ ⎢ V =⎣

A&B

⎤

⎥ 1X 0 ⎦ Dom C&D + 0 1U

A&B C&D

.

(1.4)

Denote the projection of W onto U along Y by PUY and the complementary projection by PYU . Then admissibility of the i/o pair (U, Y) yields the folx ]∈ lowing i/s/o representation: a pair [ w generated by V if and only if

x(t) ˙ PYU w(t)

=

A&B C&D

C 1 (R+ ;X ) C(R+ ;W)

x(t) , PUY w(t)

is a classical trajectory

t ≥ 0,

A&B where C&D is the operator node in (1.4). This is a very general representation of a linear time-invariant system with input signal PUY w, state trajectory x and output signal PYU w. Operator node representations are studied in more detail in Sect. 2. X We call any subspace V of the triple X a generating subspace, meaning W that it generates some space V of classical trajectories. Let us now impose some additional structure on a generating subspace in order to make it a state/signal node. Definition X 1.2. Let X be a Hilbert space and W a Kre˘ın Space, and let V ⊂ X . We say that (V ; X , W) is an ordinary state/signal node (shortly W s/s node) if V has the following properties:

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The space V is closed in the norm ⎡ ⎤ z ⎣ x ⎦ = z2 + x2 + w2 , X X W w where · W denotes an arbitrary admissible norm on W; see Definition A.3. z The space V has the property 0 ∈ V =⇒ z = 0. 0

There exists some T > 0 such that ⎡ ⎤ z0 x ⎣ ⎦ ∀ x0 ∈ V ∃ ∈ V[0, T ] : w w0

⎡

⎤ ⎡ ⎤ x(0) ˙ z0 ⎣ x(0) ⎦ = ⎣ x0 ⎦ . w0 w(0)

Comparing condition (ii) to Definition 1.1, we see that (ii) means that the derivative of the current state is uniquely determined by the current state and the current external signals at any given time. The condition in fact says that we have chosen an appropriate state space, as we will see in Proposition 4.7. Condition (iii) implies that every classical trajectory on an arbitrary interval [0, T ] can be extended in the forward-time direction to a trajectory on R+ . According to the following definition, a s/s system is essentially a collection of trajectories generated by a s/s node, and as such it is a very general object. Definition 1.3. Let (V ; X , W) be a s/s node and I a subinterval of R with positive length. The space W(I) trajectories generated by V on I is the of generalised C(I;X ) x ] ∈ W(I) if and only closure of V(I) in L2 (I;W) . By this we mean that [ w loc C(I;X ) xn x as if there exists a sequence of [ w ] ∈ V(I) that tends to [ ] in 2 w n L (I;W) loc

n → ∞. We abbreviate W[0, ∞) by W. The triple Σs/s = (W; X , W) is called the state/signal system (s/s system) induced by (V ; X , W).

Any system can intuitively be called passive if it lacks internal energy sources. We now describe how this translates to s/s systems. The energy stored in state x0 ∈ X is given by the norm of x0 squared: x0 2X = (x0 , x0 )X and similarly we let the inner product on W describe how the trajectories generated by V exchange power with the surroundings via the external signal w, so that [w(t), w(t)]W measures the amount of energy absorbed from the surroundings per time unit at time t. This energy exchange is inherently indefinite, because energy can ﬂow in both directions. Therefore we need to allow W to have an indefinite inner product, i.e., we need to let W be a Kre˘ın space. We conclude that a passive s/s system should have the following property: for every generalised (or equivalently for every classical) trajectory, the energy stored in the state should at all times be at most the energy of the

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initial state plus the total energy absorbed from the surroundings, i.e.,

x ∀ w

t ∈W:

x(t)2X

≤

x(0)2X

+

[w(s), w(s)]W ds,

t ≥ 0. (1.5)

0

It follows from the proof of Proposition 4.3 that (1.5) is equivalent to the statement x ∀ ∈ V : (x(t), ˙ x(t))X + (x(t), x(t)) ˙ t ≥ 0. (1.6) X ≤ [w(t), w(t)]W , w Noting that

x(t), ˙ x(t) X + x(t), x(t) ˙ = X

∂ x2X (t), ∂t

(1.7)

we can interpret (1.6) as a statement that the change of energy stored in the state at no time instance exceeds the power input from the outside world. We need to consider classical trajectories in (1.6), because the state part of a generalised trajectory is in general not differentiable. Motivated by this discussion we now introduce a so-called power product in order to be able to measure the amount of energy dissipated by a given trajectory at a given time. Definition 1.4. Let X Let be a Hilbert space with inner product (·, ·)X and let W be a Kre˘ın space with indefinite inner product [·, ·]W . The (contin X

uous-time) node space is K := X equipped with the sesquilinear power W product ⎡⎡ 1 ⎤ ⎡ 2 ⎤⎤ z z ⎣⎣ x1 ⎦ , ⎣ x2 ⎦⎦ := [w1 , w2 ]W − (z 1 , x2 )X − (x1 , z 2 )X . (1.8) w1 w2 K

We prove that K is a Kre˘ın space in Proposition A.2. The power product x in (1.8) can be interpreted in the following way. Let [ w ] ∈ V be a classical trajectory, let t ≥ 0 and denote ⎡⎡ ⎤ ⎡ ⎤⎤ x(t) ˙ x(t) ˙ p(t) := ⎣⎣ x(t) ⎦ , ⎣ x(t) ⎦⎦ . w(t) w(t) K x ] dissipates energy at a rate of p(t) per time If p(t) > 0, then the trajectory [ w x ] accumulates energy at a rate of |p(t)| per unit at time t. If p(t) < 0, then [ w x ] preserves energy at time t. As a remark, time unit and if p(t) = 0 then [ w in Sect. 3 we introduce s/s systems systems whose trajectories evolve with time going in the backwards direction. A trajectory of such a time-reﬂected s/s system dissipates energy at time t if p(t) < 0, because a unit of time is negative in this case. It turns out that it is natural to deﬁne the dual of the s/s node (V ; X , W) by (V [⊥] ; X , W), where the orthogonal companion V [⊥] of V in K is the space

V [⊥] := {k ∈ K | ∀v ∈ V :

[v, k]K = 0}

(1.9)

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of vectors that are [·, ·]K -orthogonal to V . See Sect. 3 for a more detailed exposition on the dual of a s/s node. A s/s node (V ; X , W) is passive if V is a maximally nonnegative subspace of the Kre˘ın space K. This means that [v, v]K ≥ 0 for all v ∈ V and that V has no proper extension which preserves this property. The maximality requirement is related to the fact that also the dual should be passive in the time-reﬂected sense that V [⊥] ≤ 0, i.e., [v, v]K ≤ 0 for all v ∈ V [⊥] . A s/s node (V ; X , W) is conservative if V = V [⊥] , which means that all trajectories of the primal node as well as all those of its dual preserve the energy at all times. Sect. 4 contains general results on passive and conservative s/s systems. Theorem 4.5 yields that every triple ; X , W), where V is a maximally z (V nonnegative subspace of K, such that 0 ∈ V =⇒ z = 0, is a passive s/s 0 node. The theorem moreover says that every fundamental decomposition, as deﬁned in Definition A.1, of the external signal space W induces an admissible i/o pair for a passive s/s node. Operator node representations arising from fundamental decompositions of the external signal space W are called scattering representations, because every scattering representation of a passive or conservative s/s node is a scattering passive or scattering conservative i/s/o system node, respectively; see Definition 5.4. We study scattering representations and their connection to passivity of s/s nodes in Section 5. There we also clarify how passivity relates to the notion of L2 -well-posedness, which was introduced for s/s systems in [14]. If we drop the maximality assumption on the generating subspace V , i.e., we only assume that V ≥ 0, then by the text preceding Example 5.3 there is no guarantee of the existence of a scattering representation. The particular citations to auxiliary results that we make in this article are chosen because they are formulated suitably for our needs. Many of these results have been formulated earlier in other contexts. The book [26] collects much of the background that we need and for simplicity we often cite results from this book. The reader may consult this source for further references to the original versions of the various results. The author is very grateful to Olof Staffans for his generous help with this article.

2. Operator Node Representation of State/Signal Nodes We begin this section by listing some useful properties of the classical trajectories generated by a s/s node. Thereafter we move on to introduce operator nodes and study how these can be used to represent s/s nodes. In order to proceed in this way we need to introduce some operators for manipulating trajectories. We denote the operator which shifts its argument function to the left by an amount c ∈ R by τ c , so that (τ c f )(t) = f (t + c) for all t such that t + c ∈ Dom (f ). The operator that restricts its argument function f to I ⊂ Dom (f ) is denoted by ρI , so that (ρI f )(t) = f (t) for t ∈ I.

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By c we denote the concatenation operator at c ∈ R, i.e.: f (t), t < c, t ∈ Dom (f ) (f c g)(t) = . g(t), t ≥ c, t ∈ Dom (g) Lemma 2.1. Let (V ; X , W) be a s/s node with classical trajectories V. Then the following claims are valid: (i) For all −∞ < a < b < ∞ and c ∈ R: V[a, b] = τ c V[a + c, b + c] (ii)

and

V[a, ∞) = τ c V[a + c, ∞).

For all positive-length subintervals I of the interval I we have ρI V(I) ⊂ V(I ) and, moreover, ∀b ∈ (a, b] : ρ[a,b ] V[a, b] = V[a, b ]

and

∀b > a : ρ[a,b ] V[a, ∞) = V[a, b ]. (iii)

(iv)

x1 ] ∈ V[a, c] and Let −∞ < a < c < b < ∞ and assume that [ w 1 x2 x1 x2 c [ w2 ] ∈ V[a, b] if and only if x1 (c) = x2 (c) [ w2 ] ∈ V[c, b]. Then [ w1 ] and w1 (c) = w2 (c). For all T > 0 we have ⎫ ⎧⎡ ⎤ ˙ ⎬ ⎨ x(0) x ∈ V[0, T ] . (2.1) V = ⎣ x(0) ⎦ w ⎭ ⎩ w(0)

This claim is also valid for T = ∞ in the sense that it remains true if we replace V[0, T ] by V. (v) The spaces V[0, T ], 0 < T ≤ ∞, are uniquely determined by V and vice versa. (vi) Property (iii) of Definition 1.2 holds for some T > 0 if and only if it holds for all T> 0. C 1 (R+ ;X ) x x (vii) A pair [ w ] ∈ C(R+ ;W) lies in V if and only if ρ[0,T ] [ w ] ∈ V[0, T ] for all T > 0. Proof. All of these claims were proved in [14, Sect. 2], except for claim (vii), x x ] ∈ V then ρ[0,T ] [ w which we now prove. If [ w ] ∈ V[0, T] for all T > 0 by C 1 (R+ ;X ) + C(R ;W) x(t ˙ 0) that x(t0 ) w(t0 )

x claim (ii). Conversely assume only that [ w ]∈

x . If [ w ] ∈ V, then

there by Definition 1.1 exists a t0 > 0 such

∈ V . This implies

that

x ρ[0,t0 ] [ w ]

∈ V[0, t0 ].

Property (iv) says that for every vector v0 ∈ V , we can ﬁnd a trajectory x [w ]

on an interval of arbitrary length, such that the initial value

x(0) ˙ x(0) w(0)

of

the trajectory is the given vector v0 . A consequence of this result is given in claim (ii), which says that classical trajectories can always be restricted and extended in the forward-time direction to arbitrary intervals. Although most of the results in this paper are given for the interval [0, T ], they can be generalised immediately to all intervals [a, b], a < b, or [a, ∞), because of claim (ii) and the shift invariance property in claim (i).

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An operator node is a relatively complicated mathematical object and we therefore need to recall some terminology before we can deﬁne it properly. Definition 2.2. Let A be a closed operator on the Banach space X . The resolvent set Res (A) of A is the set of all λ ∈ C such that λ − A maps Dom (A) one-to-one onto X . Fix α ∈ Res (A), assume that X1 := Dom (A) is dense in X , and equip X1 with the norm x1 := (α − A)xX . Denote by X−1 the completion of X with respect to the norm x−1 := (α − A)−1 xX . This norm is weaker than the norm · X , because x−1 ≤ (α − A)−1 xX for all x ∈ X . The spaces X1 and X−1 deﬁned above satisfy X1 ⊂ X ⊂ X−1 with dense and continuous embeddings. This construction is sometimes referred to as “rigging”. Different choices of α ∈ Res (A) give rise to the same triple (X1 , X , X−1 ) of spaces, because although the norms on X1 and X−1 depend on α, all the norms on X1 (all the norms on X−1 ) are equivalent. The norm on X1 is also equivalent to the graph norm (2.6) of A. If X is a Hilbert space, then, so are X1 and X−1 . The triple (X1 , X , X−1 ) is also called the Gelfand triple induced by A, see e.g. [26, Sect. 3.6] for more details. Assume that β ∈ Res (A). Then β − A maps X1 = Dom (A) isomorphically onto X . The operator A can also be considered as a continuous operator which maps the dense subspace X1 of X into X−1 and we denote the unique continuous extension of A to an operator X → X−1 by A|X . Then the operator β − A|X maps X isomorphically onto X−1 and (β − A|X )−1 is the continuous extension of (β − A)−1 from X to X−1 . Definition 2.3. Let X be a Banach space. A family t → At , t ≥ 0, of bounded linear operators on X is a semigroup on X if A0 = 1 and As+t = As At for all s, t ≥ 0. The semigroup is strongly continuous, or shorter C0 , if limt→0+ At x0 = x0 for all x0 ∈ X . A C0 semigroup A is a contraction semigroup if the norm of At as an operator on X satisﬁes At ≤ 1 for all t ≥ 0. The generator A : X ⊃ Dom (A) → X of A is the (in general unbounded) linear operator deﬁned by 1 Ax0 := lim+ (At x0 − x0 ) t→0 t

(2.2)

with Dom (A) consisting of those x0 ∈ X for which the limit (2.2) exists in X. The generator A of a C0 semigroup on X is closed and Dom (A) is dense in X ; see [22, Theorem 1.2.7]. Moreover, according to [22, Theorem 1.2.6], a C0 semigroup A is uniquely determined by its generator A in the following way. For every x0 ∈ Dom (A), the function x : t → At x0 , t ≥ 0, is the unique continuously differentiable solution of the initial value problem x(t) ˙ = Ax(t) t ≥ 0, x(0) = x0 . The operators At , t ≥ 0, are then extended by continuity to all of X . It may therefore be said that A generates A. From [26, Theorem 3.2.9(i)] we know that Res (A) = ∅ for every C0 -semigroup generator.

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The following definition is essentially Definition 4.7.2 of [26], and we use the notation of [26] for the operator A&B. This notation should be interpreted as described in item (iii) below. Definition 2.4. By an i/s/o-operator node (shortly operator node) on the triple (U, X , Y) of Banach spaces we mean a linear operator A&B X A&B X : ⊃ Dom → C&D U C&D Y with the following properties: A&B is closed. (i) The operator C&D A&B , deﬁned by (ii) The so-called main operator A : Dom (A) → X of C&D x x A&B Ax = A&B on Dom (A) = x ∈ X ∈ Dom (, 2.3) 0 0 C&D has domain dense in X and nonempty resolvent set. The operator A&B can be extended to an operator A|X B that maps [ X U ] continuously A&B into X−1 . satisﬁes the condition (iv) The domain of C&D A&B x X A|X x + Bu ∈ X . Dom = ∈ C&D u U A&B is called an i/s/o system node if its main An operator node C&D operator A generates a C0 semigroup. The operator node is a time-reflected i/s/o system node if −A generates a C0 semigroup, and in this case we say that A generates a C0 semigroup in backward time. A&B The triple (u, x, y) is said to be a classical i/s/o of C&D if trajectory x(t) ˙ x(t) + 1 + + A&B u ∈ C(R ; U), x ∈ C (R ; X ), y ∈ C(R ; Y), and y(t) = C&D u(t) for all t ≥ 0. (iii)

We return to time reﬂection and motivate the choice of the term “timereﬂected i/s/o system node” in the next section. The following definition says that admissibility of a given i/o pair for a generating subspace V means that V can be written as the graph of an operator node. Definition 2.5. Let V ⊂ K and W = U Y. We say that (U, Y) is an admissible i/o pair of V if there exists an A&B on (U, X , Y), such that operator node C&D ⎡ ⎤ A&B A&B ⎦ Dom 1X 0 V =⎣ . (2.4) C&D C&D + 0 1U A&B In thiscase we call C&D an operator node representation of V and write A&B ; X , U, Y . Vop = C&D A&B ; X , U, Y , then we call = If (V ; X , W) is a s/s node and V op C&D A&B C&D an operator node representation of both the s/s node (V ; X , W) and of the s/s system Σ that the node generates.

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An i/o pair (U, Y) is admissible for the system Σ if it is admissible for at least one of its generating s/s nodes. An operator node representation is in a sense an input/state/output representation of a s/s node, but for clarity we call it an operator node representation, because the term i/s/o representation was given a different meaning in [14, Deﬁnition 4.5]. Note that (V ; X , W) is not necessarily a s/s node even if V ⊂ K has an admissible i/o pair, because condition (iii) of Definition 1.2 might be violated. We proceed to investigate this issue. It follows from conditions (iii) and (iv) of Definition 2.4 that thetop A&B to half A&B of an operator node is a closed operator from Dom C&D X ; see the proof of [26, Lemma 4.3.10]. Therefore the domain of an operator node is a Banach space with the graph norm of A&B: # 2 x x + x2 + u2 . A&B = (2.5) ! " X U u u A&B X Dom C&D

A&B

The closedness of C&D implies that the operator node is continuous with respect to this norm. If X is a Hilbertspace and U has a Hilbert-space A&B determines a Hilbert-space inner topology, then the norm of Dom C&D product by polarisation. By taking u = 0 in (2.5) we obtain the following graph norm of A for Dom (A): 2 xDom(A) = AxX + x2X . (2.6) This norm makes Dom (A) a Banach space, and if X is a Hilbert space, then the norm deﬁnes a Hilbert-space inner product on Dom (A). The operator A is trivially continuous with respect to this norm. Lemma 2.6. Let A be a densely defined operator on the Hilbert space X such that Res (A) = ∅. The homogeneous Cauchy problem x(t) ˙ = Ax(t),

t > 0,

x(0) = x0 ,

(2.7)

has a unique solution x ∈ C 1 (R+ ; X ) for every initial value x0 ∈ Dom (A) if and only if A generates a C0 semigroup on X . A proof can be found for instance in [22, Theorem 4.1.3]. The lemma allows us to explain the difference between operator nodes and i/s/o system nodes, i.e. the existence of a semigroup, in terms of existence and uniqueness of classical trajectories. See Definition A.10 for a description of the function 1 (I; X ). space Hloc Proposition2.7. Let I = [a, b] or I = [a, ∞), where a < b, let V ⊂ K, and A&B let Vop = C&D ; X , U, Y be an operator node representation. Then the following claims are all true: A&B is a system node, then the triple (V ; X , W) is a s/s node with (i) If C&D admissible i/o pair (U, Y).

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1 (I; X ) and u ∈ L2loc (I; U), Assume that the pair [ ux ], where x ∈ Hloc x(t) + Bu(t) in X−1 almost everywhere satisfies the equation x(t) ˙ = A| XA&B if and only if x ∈ C 1 (I; X ) and on I. Then [ ux ] ∈ C I; Dom C&D u ∈ C(I; U). x(t) for all t ∈ I. If these conditions all hold, then x(t) ˙ = A&B u(t) A&B x x A pair [ w ] lies in V(I) if and only if PUY w ∈ C I; Dom C&D and x(t) x(t) ˙ A&B f or all t ∈ I. (2.8) = C&D PYU w(t) PUY w(t) x ]∈ For all xa ∈ Dom (A) there exists a unique classical trajectory [ w Y V(I), such that x(a) = xa and PU w = 0, if and only if A generates a A&B is a system node. C0 semigroup on X . In this case C&D

Proof. Claim (ii) was shown to hold as a part of the proof of [14, Lemma 5.7]. We now show how claim (iii) follows almost directly from Definition 1.1 A&B x and claim (ii). Assume that PUY w ∈ C I; Dom C&D and that (2.8)

U holds. Then PY w ∈ C(I; Y) by [14, Lemma 5.6] and, moreover, (2.4) yields x(t) ˙ x ] ∈ V(I). Conversely assume that that x(t) ∈ V for all t ∈ I. Thus [ w w(t) x(t) ˙ x [w ] ∈ V(I), so that x(t) ∈ V for all t ∈ I. According to Definition 1.1, we w(t)

Y have x ∈ C 1 (I; X ) and w ∈ C(I; W), and then x(t) PU w ∈ C(I; U) because the ˙ projection is continuous. The inclusion x(t) ∈ V again means that (2.8) w(t) x A&B by claim (ii). holds for all t ∈ I and thus PUY w ∈ C I; Dom C&D We use Definition 1.2 to prove claim (i). The graph V in (2.4) of any z

operator node is closed by Definition 2.4, and moreover, 0 ∈ V =⇒ 0 z0 z0 0 x z = A&B [ 0 ] = 0. Let now w0 ∈ V be arbitrary, so that PYU w0 = 0 A&B x0 x Y PU w0 by (2.4). We need to construct a classical trajectory [ w ] ∈ V, C&D x(0) ˙ z0 x0 . According to [26, Lemma 4.7.8], we can deﬁne such that x(0) = w 0 w(0) 1 + C (R ;X ) u(t) := PUY w0 for t ≥ 0 and let [ xy ] be the unique solution in C(R+ ;Y) A&B x(t) x(t) ˙ x of the equation y(t) = C&D u(t) , t ≥ 0, x(0) = x0 , so that [ u ] lies A&B + x by claim (ii). Then [ u+y ] ∈ V by claim (iii) and in C R ; Dom C&D x(0) ˙ z0 x0 x(0) by construction. = w u(0)+y(0)

0

We split the proof of claim (iv) into two parts: one for the case I = R+ and one for the case I = [0, b], where b > 0. It is sufﬁcient to consider these two cases, because we can assume that a = 0 without loss of generality, due to Lemma 2.1(i) and the shift invariance of the equation x(t) ˙ = Ax(t).

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Part 1: Assume that I = R+ . The operator A is densely deﬁned in X and Res (A) is nonempty by Definition 2.4. We may thus use Lemma 2.6. x Fix x0 ∈ Dom (A) arbitrarily. If [ w ] ∈ V and PUY w = 0, then x ∈ 1 + C (R ; X ) and x(t) ˙ = Ax(t) for all t ≥ 0 by claim (iii). if x ∈ Conversely, A&B ˙ = Ax(t) for all t ≥ 0, then [ x0 ] ∈ C R+ ; Dom C&D C 1 (R+ ; X ) and x(t) x for all t ≥ 0 we then get [ w ] ∈ V, by claim (ii). Deﬁning w(t) := C&D x(t) 0 Y according to claim (iii), and PU w = 0 by construction. The above argument shows that x ∈ C 1 (R+ ; X ) satisﬁes x(t) ˙ = Ax(t), x ] ∈ V. This implies that t ≥ 0, if and only if there exists a w such that [ w x ] ∈ V, such that x(0) = x0 for every x0 ∈ Dom (A), there exists a unique [ w Y and PU w = 0, if and only if the equation x(t) ˙ = Ax(t), t ≥ 0 and x(0) = x0 , has a unique continuously differentiable solution for all x0 ∈ Dom (A). By Lemma 2.6, this holds if and only if A generates a C0 semigroup on X . Regarding A&B the last claim of (iv), if A generates a C0 semigroup, then the is a system node by Definition 2.4. This ﬁnishes the proof operator C&D of claim (iv) for the case I = [a, ∞). Part 2 : Now assume that I = [0, b], where b > 0. We reduce this case to to xb ] ∈ V[0, b], such the case I = R+ by proving that there exists a unique [ w b Y x ] ∈ V, that xb (0) = x0 and PU wb = 0 if and only if there exists a unique [ w Y such that x(0) = x0 and PU w = 0. x ] ∈ V, First assume that there for all x0 ∈ Dom (A) exists a unique [ w Y such that x(0) = x0 and PU w = 0. Then ﬁx x0 ∈ Dom (A) arbitrarily and let xb x x [w ] ∈ V satisfy x(0) = x0 and PUY w = 0. By Lemma 2.1(ii), [ w ] := ρ[0,b] [ w ] b Y lies in V[0, b], and trivially xb (0) = x0 and PU wb = 0. In order to prove xc uniqueness, we let also [ w ] ∈ V[0, b] with xc (0) = x0 and PUY wc = 0. Then c By the assumption at the beginning xc (b) ∈ Dom (A) by claim (iii) and x$ (2.3). of this paragraph, there exists a w ∈ V such that $(0) = xc (b) and PUY w $= $ x% x xc −b x $ ∈V ] τ 0. Claims (i) and (iii) of Lemma 2.1 yields that w% := [ w b c w $ with x %(0) = x0 and PUY w % = 0. We also assumed that the initial state ξ(0) uniquely determines a trajectory [ ωξ ] ∈ V with zero input: PUY ω = 0. This x% xb xc x x implies that w% = [ w ] and therefore we obtain [ w ] = ρ[0,b] [ w ] = [w ]. We c b xb have shown that there for every x0 ∈ Dom (A) exists a unique [ wb ] ∈ V[0, b], such that xb (0) = x0 and PUY wb = 0. Now conversely assume that there for every x0 ∈ Dom (A) exists a xb unique [ w ] ∈ V[0, b], such that xb (0) = x0 and PUY wb = 0. In order to prove b x that there for all x0 ∈ Dom (A) exists a unique [ w ] ∈ V, such that x(0) = x0 Y and PU w = 0, we ﬁrst ﬁx x0 ∈ Dom (A) arbitrarily. By assumption we can xn ] ∈ V[0, b], such that x1 (0) = x0 and xn+1 (0) = xn (b), ﬁnd a sequence [ w n xn Y PU wn = 0 for all n ≥ 1, since [ w ] ∈ V[0, b] and PUY wn = 0 imply that n xn (b) ∈ Dom (A). By claims (i), (iii) and (vii) of Lemma 2.1 we have that

x w

:=

x1 w1

b τ

−b

x2 w2

2b τ −2b . . . ∈ V

and obviously x(0) = x1 (0) = x0 and PUY w = 0.

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x% w %

∈ V with x(0) = x0 and PUY w = 0, then % x %n (n−1)b x := ρ[0,b] τ ∈ V[0, b], n ≥ 1, w %n w %

according to claims (i) and (ii) of Lemma 2.1. Moreover, x %1 (0) = x1 (0) and PUY w %n = 0 for all n ≥ 1. By assumption this implies that x %1 = x1 . Using xn x %n = [w ] for all n ≥ 1 and thus we induction over n, we obtain that w n %n x% x arrive at w% = [ w ]. The proof is now done.

We end this section by describing in which sense this article also covers boundary control. The following definition is [17, Deﬁnition 1.1]. Definition 2.8. A triple (L, K, G) is a boundary i/s/o node on the triple (U, X , Y) of Banach spaces if it has the following properties: (i) The linear operators G have the same domain Z. L, K and (ii)

The operator

L K G

:Z→

X Y U

is closed.

(iii) The operator G is surjective and has kernel N (G) dense in X . (iv) The operator A := L|N (G) has a nonempty resolvent set. If all of these conditions hold and A generates a C0 semigroup on X , then the boundary i/s/o node is internally well-posed. If the conditions (i–iv) hold and −A generates a C0 semigroup, then the boundary i/s/o node is internally well-posed in backward time. Recall that if A generates a C0 semigroup on X , then Dom (A) = N (G) is dense in X and the resolvent set is of A is nonempty. In this case (L, K, G) satisﬁes conditions (iii) and (iv) of Definition 2.8 if G is surjective. If (L, K, G) is a boundary i/s/o node on (U, X , Y) then we, according to [17, Theorem 2.3], always obtain an operator node on (U, X , Y) by deﬁning A&B A&B L ] 1X −1 on Dom := [ K = Ran 1GX . In this case the C&D C&D G operator node representation ⎤ ⎡ A&B ⎦ 1X 0 (2.9) V =⎣ Dom (S) C&D + 0 1U can be written as

⎤ L V = ⎣ 1X ⎦ Dom (L). K +G ⎡

This representation is formally independent of the i/o pair (U, Y), but note that conditions (iii) and (iv) in Definition 2.8 still depend on the choice of i/o pair. Example 2.9. Equip C with the usual Hilbert-space inner product u1 , u2 C = u1 u2 and let X := L2 (R+ ; C). Set Z := H 1 (R+ ; C). The elements of Z are continuous and we may therefore deﬁne the point-evaluation operator at 0 on Z by δ0 , so that δ0 x = x(0) for all x ∈ Z.

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∂ − ∂z In this example we show that Ξ := with domain Z is a boundδ0 ary i/s/o node both on (C, L2 (R+ ; C), {0}), i.e., with input space U = C and output space Y = {0}, and ({0}, L2 (R+ ; C), C), i.e. U = {0} and Y = C, by checking the conditions listed in Definition 2.8. In the ﬁrst case G = δ0 and K = 0, and in the second case G = 0 and K = δ0 . We prove that the ﬁrst of these two boundary i/s/o nodes is internally well-posed (in forward time) and that the second one is internally well-posed in backward time. Condition (i) of Definition 2.8 is met by Ξ according to the definition of Ξ. We proceed to verify that Ξ satisﬁes condition (iii) in the case G = δ0 . Note therefore that the space of functions x ∈ C ∞ (R+ ; C), such that x(0) = 0, is dense in Z and that every such function lies in N (G). Moreover, G is sura ∈ H 1 (R+ ; C) and jective, because for every a ∈ C, the function xa := 1+z xa (0) = a. Condition (iii) is trivial in the case G = 0. We now prove that Ξ satisﬁes condition (iv), beginning with the case where G = 0. According to [26, Example 2.3.2, 3.2.3 and 3.3.1], the operator ∂ 2 + + and it generates the incoming ∂z : Z → L (R ; C) has resolvent set C 2 + left-shift C0 semigroup τ+ on L (R ; C): x ∈ L2 (R+ ; C), t ≥ 0, for almost all z ≥ 0. ∂ Consequently, the resolvent set of − ∂z is C− . Z We next look at the case where G = δ0 . Example 3.5.11(ii) of [26] ∂ : Z ∩ N (δ0 ) → L2 (R+ ; C) generates the adjoint yields that the operator − ∂z ∗ semigroup τ+ of τ+ . This adjoint semigroup is the outgoing right-shift C0 semigroup on L2 (R+ ; C): ∗ t x(z − t), t ≥ 0, z ≥ t, for almost all z ≥ 0. (τ+ ) x (z) = 0, t ≥ 0, z < t, ∂ ∂ is therefore the adjoint of ∂z , cf. [26, Theorem The operator − ∂z N (δ0 ) Z 3.5.6(v)]. Thus the resolvent set of this operator is also C+ , because α ∈ Res (A) if and only if α ∈ Res (A∗ ). Condition (ii) is satisﬁed by Ξ both when G = 0 and when G = δ0 . ∂ with domain Z is nonempty, Indeed, by the above, the resolvent set of − ∂z and therefore this operator is closed. Moreover, δ0 is continuous with respect ∂ , which is the standard Sobolev norm on H 1 (R+ ; C), to the graph norm of − ∂z and therefore Ξ∂ is a closed operator. − ∂z Thus is a boundary i/s/o node with both i/o pairs (C, {0}) and δ0 ({0}, C), i.e., with both G = δ0 , K = 0 and G = 0, K = δ0 , respectively. In the ﬁrst case, the boundary i/s/o node is internally well-posed, because it has ∗ . The boundary i/s/o node in the second case is internally the semigroup τ+ ∂ generates τ+ . If the second boundwell-posed in backward time since ∂z Z ary i/s/o node were to be internally well-posed also in forward time, then the ∂ would have to contain some right-half-plane; see [26, resolvent set of − ∂z Z Theorem 3.2.9(i)]. This is clearly not the case because by the above we know t x)(z) = x(z + t), (τ+

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− that this resolvent set equals C . The same argument can be used to show ∂ that ∂z Z∩N (δ ) does not generate a C0 semigroup on X . 0 According to [17, Theorem 2.3], both −1 2 + ∂ L (R ; C) 1 S := − : ⊃ Dom (S) → L2 (R+ ; C) C ∂z δ0 with domain δ10 Z and its so-called flow inverse 2 + ∂ × L (R ; C) − ∂z × 2 + S := , : L (R ; C) ⊃ Dom S → C δ0

Dom (S × ) = Z, are therefore operator nodes. Summarising the example, we have established that S and −S × are i/s/o system nodes, that −S and S × are time-reﬂected i/s/o system nodes, and that all of these four operators are operator nodes. (The sign of δ0 is unimportant.) Scattering- and impedance-conservative boundary i/s/o nodes are studied in [17] and [18]. In [16], Malinen studies ﬁve examples of boundary control systems after developing the tools necessary for this task. Some results applicable to interconnection of impedance-conservative boundary control systems are given in [15]. We return to ﬂow inversion at the end of Sect. 4.

3. Time-Reflected and Dual State/Signal Nodes The dynamics of the systems we have considered so far evolve with increasing time. The intuitive idea of a time-reﬂected s/s node, which we now introduce, is a s/s node whose trajectories evolve in backward time. Later in this section we study state/signal duals. Time-reﬂected and dual s/s nodes generalise the corresponding notions given for i/s/o systems in [28]. Definition 3.1. Let V ⊂ K and T < 0. We call (V ; X , W) a time-reflected s/s node if V has the following properties: (i) (ii) (iii)

V closed, z is 0 ∈ V =⇒ z = 0 and 0

x for all v0 ∈ V there exists a [ w ] ∈ V[T, 0] such that

x(0) ˙ x(0) w(0)

= v0 .

Comparing Definitions 3.1 and 1.2, we see that the difference between ordinary and time-reﬂected s/s nodes is that time-reﬂected s/s nodes are initialised at the right endpoint of the time interval, t = 0 in this case, and evolve in backward time. This determines the time direction of the s/s node. Note, however, that the generating subspaces V in Definition 1.1 have no x ] time direction, because we only require that the generated trajectories [ w should be smooth enough and satisfy the appropriate time interval.

x(t) ˙ x(t) w(t)

∈ V for all internal points of

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In order to be able to formulate the next proposition, we need to introduce the time-reflection operator R, which reﬂects its argument function about zero, so that (Rf )(t) = f (−t), for −t ∈ Dom (f ). We also denote R− = (−∞, 0]. Proposition 3.2. Let T > 0, let V ⊂ K and let I be a positive-length subinter −1 0 0 val of R. Define the so-called time-reﬂection V ← of V by V ← := 0 1 0 V 0 0 1 and denote the space of classical trajectories generated by V ← on I by V← (I). x x x ] ∈ V[0, T ] if and only if R [ w ] ∈ V← [−T, 0], and [ w ] ∈ V if Then [ w x and only if R [ w ] ∈ V← (R− ). The triple (V ← ; X , W) is a time-reflected s/s node if and only if (V ; X , W) is an ordinary s/s node and vice versa. We have ⎤ ⎡ A&B A&B ⎦ ⎣ 1 0 ⇐⇒ V = Dom C&D C&D + 0 1 ⎤ (3.1) ⎡ −A&B A&B ← ⎦ 1 0 V =⎣ Dom C&D 0 1 C&D + and the sets of admissible i/o pairs for V and V ← coincide. x Proof. By Definition 1.1, [w ] ∈ V[0, T ] if and only if x ∈ C 1 ([0, T ]; X ), x(t) ˙ w ∈ C([0, T ]; W) and x(t) ∈ V for all t ∈ (0, T ). This is obviously equivaw(t)

lent to Rx ∈ C 1 ([−T, 0]; X ), Rw ∈ C([−T, 0]; W) and ⎡ d ⎤ ⎤ ⎡ −x(t) ˙ dt (Rx)(−t) ⎣ (Rx)(−t) ⎦ = ⎣ x(t) ⎦ ∈ V ← , t ∈ [0, T ], w(t) (Rw)(−t) x i.e., to R [ w ] ∈ V← [−T, 0]. This computation remains valid if we replace + [0, T ] by R and [−T, 0] by R− . Properties (i) and (ii) in Definition 3.1 are the same as in Definition −1 0 0 1.2 and they are invariant under premultiplication of V by 0 1 0 . Assume 0 01 z0 x0 that these conditions are met by V and note that w ∈ V if and only if 0 −z 0 x0 ∈ V ← . Then by Deﬁnition 1.2, V is a s/s node if and only if for all w0 x(0) ˙ x such elements, there exists a trajectory [ w ] ∈ V[0, T ], such that x(0) = w(0) z0 x0 . By the computation we just made, that same trajectory then satisﬁes w 0

x R[w ] ∈ V← [−T, 0] and ⎤ ⎡ d ⎤ ⎡ ⎤ ⎡ −x(0) ˙ −z0 dt (Rx)(0) ⎣ (Rx)(0) ⎦ = ⎣ x(0) ⎦ = ⎣ x0 ⎦ . w0 w(0) (Rw)(0)

Therefore (V ; X , W) is a s/s node if and only if (V ← ; X , W) is a time-reﬂected s/s node. The equivalence (3.1) is trivial. Looking at Definition 2.4, we see that conditions (i), (iii) and (iv) are independent of the sign of A&B. Actually,

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condition (ii) is also independent of the sign of A&B, because α ∈ Res (A) if and only if −α ∈ Res (−A). Thus V has operator node representation A&B ; X , U, Y if and only if V ← has operator node representation Vop = C&D ← ; X , U, Y . Vop = −A&B C&D One can formulate properties of the trajectories generated by a timereﬂected s/s node similar to those listed in Lemma 2.1 using Definition 3.1 and Proposition 3.2. Preparing for the next definition, we recall that V [⊥] denotes the space of all vectors which are orthogonal to V in K; see (1.9). Definition 3.3. Let (V ; X , W) be a (time-reﬂected or ordinary) s/s node. The triple (V [⊥] ; X , W) is the s/s dual of (V ; X , W). For any subinterval I of R with positive length, we denote the space of classical trajectories generated by V [⊥] on I by Vd (I). By Wd (I) we denote the space of generalised trajectories generated by V [⊥] , i.e., Wd (I) is the closure of Vd (I) in

C(I;X ) L2loc (I;W)

.

The following example shows that the dual of a s/s node is in general neither a s/s node nor a time-reﬂected s/s node. Example 3.4. The space V := {0} ⊂ C3 is a s/s node but V [⊥] violates C condition (ii) of Definition 1.2, because {0} ⊂ V [⊥] . {0}

Note that the s/s node in Example 3.4 lacks operator node representations. We show that the s/s dual of a s/s node which has an i/s/o system node representation is a time-reﬂected s/s node in Theorem 3.6. Proposition 3.5. The dual of a s/s node is always closed. The double dual of z a s/s node is the s/s node itself. For all V ⊂ K, the property 0 ∈ V [⊥] =⇒ 0 z = 0 is equivalent to denseness of 0 1 0 V in X . Proof. The ﬁrst two claims follow from standard Kre˘ın-space theory, since z V [⊥] is always closed and (V [⊥] )[⊥] = V . For the last claim, note that 0 ∈ 0 z z z x , 0 V [⊥] if and only if = −(x, z )X = 0 for all x ∈ V , which is w w 0 ⊥K equivalent to z ∈ 0 1 0 V . This orthogonal complement contains only the vector 0 if and only if 0 1 0 V is dense in X . We identify the dual X of X with X itself, as is common for Hilbert spaces, and moreover, we identify the dual of W with W itself as well. The correctness of the following argument follows from [7, Sect. 2.3], where the reader can also ﬁnd more details. Note, however, that the dual of W is identiﬁed with −W in [7]. We explain this discrepancy after Definition 4.1. By [7, Lemma 2.3], if W = U Y then also W = Y [⊥] U [⊥] . This allows us to identify the duals U and Y of U and Y as U = Y [⊥]

and

Y = U [⊥]

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using the duality pairings u, u U ,Y [⊥] = [u, u ]W ,

y, y Y,U [⊥] = [y, y ]W ,

u ∈ U, u ∈ Y [⊥] y ∈ Y,

y ∈U

[⊥]

and .

We thus obtain the following duality pairings between [ X U ] and their respective duals: & ' x z , and ( ) = (x, z )X + u, u U ,Y [⊥]

u u X X [ U ], Y [⊥] & ' z x , ( ) = (z, x )X + y, y Y,U [⊥] . y y X , X Y

(3.2) X Y

and

(3.3)

U [⊥]

Adjoint operators computed with respect to these X duality pairings are denoted by †. For instance, if S : [ X U ] ⊃ Dom (S) → Y is densely deﬁned, then S † : UX[⊥] ⊃ Dom S † → YX[⊥] is the maximally deﬁned operator, such that for all [ ux ] ∈ Dom (S) and xy ∈ Dom S † : & ' & ' x x x † x S , ,S (3.4) ( ) = ( ) . u u y y X , X X ], X [ [⊥] [⊥] Y U Y U The operator A&B d appearing in (3.6) below is not obtained solely from A&B in (3.5). It should be interpreted as the tophalf of S d deﬁned in (3.6) A&B . in the same way as A&B is the top half of C&D Theorem 3.6. Let V ⊂ K and W = U Y. Assume that there exists a densely A&B : [ X ] ⊃ Dom (S) → X , such that V has the defined operator S = C&D Y U graph representation ⎡ ⎤ A&B ⎢ ⎥ 10 V =⎣ (3.5) ⎦ Dom (S) . C&D + 0 1 Let S † be the adjoint of S, as given in (3.4), and define A&B d −1 0 0 † 1 S d := := S on 0 1 0 −1 C&Dd † d X 1 0 Dom S ⊂ Dom S = . 0 −1 U [⊥] Then V [⊥] is given by V [⊥]

(3.6)

⎡

⎤ A&B d d ⎢ ⎥ 1 0 =⎣ ⎦ Dom S . C&Dd + 0 1

(3.7)

If S is an operator node, then so are S d and S † . In this case, the main operator of S d is −A∗ , where A∗ is the adjoint of A as an unbounded operator on the Hilbert space X .

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If S is an ordinary i/s/o system node, then so is S † and in this case S d is a time-reflected i/s/o system node; see Definition 2.4. z x ∈ V [⊥] with u ∈ Y [⊥] and y ∈ U [⊥] Proof. In the described setup, u +y

if and only if for all [ ux ] ∈ Dom (S) we have: ⎡⎡ ⎤ ⎤⎤ ⎡ A&B z ⎥ ⎢⎢ ⎥ x ⎣ 10 x ⎦⎦ 0 = ⎣⎣ , ⎦ u u + y C&D + 0 1 K & ' x −z = , ( ) u u X , [X U ] Y [⊥] & ' A&B x x − , ( ) . C&D u −y X , X Y

(3.8)

U [⊥]

x Due to the assumed denseness (S) in [ X U ], (3.8) holds for all [ u ] ∈ of Dom x x ∈ Dom S † and −z = S † −y Dom (S) if and only if −y . The latter u condition is easily seen to be equivalent to (3.7). According to [26, Lemma 6.2.14], the adjoint S † of an operator node S is an operator node with main operator A∗ , and S † is an i/s/o system node if and only if S is an i/s/o system node. By Definition 2.4, it is immediate 0 † 1 0 that S † is an i/s/o system node if and only if S d = −1 0 −1 is a 0 1 S time-reﬂected i/s/o system node. A&B in (3.4) is usually referred to as Remark 3.7. The operator S † =: C&D the causal dual of S. Looking at (3.6) and (3.7), we see that thes/s dual −A&B [⊥] (V ; X , W) corresponds to the so-called anti-causal dual C&D of S. d ) induced by −A∗ , the main Note that the Gelfand triple (X1d , X , X−1 d operator of the dual S , in general differs from (X1 , X , X−1 ) when A is unbounded on X . To be more precise, we identify the dual of X1d by X−1 and d by X1 , using X as pivot space. The Gelfand triple induced the dual of X−1 ∗ by A , the main operator of S † , is the same as that induced by −A∗ . We have the following important corollary to Theorem 3.6.

Corollary 3.8. An i/o pair (U, Y) is admissible for the (ordinary or timereflected) s/s node (V ; X , W) if and only if the “dual i/o pair” (U [⊥] , Y [⊥] ) is admissible for the dual s/s node (V [⊥] ; X , W). In Theorem 3.6 we characterised the dual s/s node in terms of the primal s/s node. We now characterise the classical trajectories of the dual in terms of the classical trajectories of the primal s/s node. d x ∈ Proposition 3.9. Let (V ; X , W) be a s/s node, let T < 0 and let w d 1 d d C ([T,0];X ) x x . Then w ∈ Vd [T, 0], i.e., w is a classical trajectory gend d C([T,0];W)

erated by V [⊥] on [T, 0], if and only if for all a and b, such that T ≤ a < b ≤ 0,

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x ] ∈ V[a, b], it holds that: and for all [ w

x(b), x (b) X − x(a), xd (a) X =

b

d

Moreover,

d

x wd

∈

for all a < b ≤ 0 and for

1

−

C (R ; X ) lies in C(R− ; W) x all [ w ] ∈ V[a, b].

w(s), wd (s) W ds.

(3.9)

a

Vd (R− ) if and only if (3.9) holds

Proof. First of all, (3.9) is equivalent to ⎡⎡ ⎤ ⎡ ⎤⎤ b x(s) ˙ x˙d (s) ⎣⎣ x(s) ⎦ , ⎣ xd (s) ⎦⎦ ds = 0, w(s) wd (s) a K

(3.10)

because d (x(s), xd (s))X = −(x(s), ˙ xd (s))X − (x(s), x˙d (s))X ds ⎡⎡ ⎤ ⎡ ⎤⎤ x(s) ˙ x˙d (s) = ⎣⎣ x(s) ⎦ , ⎣ xd (s) ⎦⎦ . 0 0 K d x ∈ Vd [T, 0] then x˙d , xd and wd are all continuous on (T, 0) and If w d −

x˙d (s) xd (s) wd (s)

∈ V [⊥] for all s ∈ [T, 0], according to Definition 3.3. This implies d x ∈ Vd (R− ), then for every a < 0 we (3.10) for all T ≤ a < b ≤ 0. If w d d x ∈ Vd [a, 0] according to Proposition 3.2 and Lemma 2.1(ii). have ρ[a,0] w d

Therefore (3.10) holds for every a < b ≤ 0. Conversely, ﬁx T < 0 and assume that (3.10) holds for all T ≤ a < b ≤ 0. Fix a ∈ [T, and b ∈ (a, 0] arbitrarily. By Definition 1.2 and Lemma z0) a x x ] ∈ V[a, b] such that 2.1, we can let wa ∈ V be arbitrary and ﬁnd a [ w a x(a) za ˙ x(a) = xa . Divide both sides of (3.10) by b − a > 0 and let b → a+ . By w w(a)

a

the assumed continuity from the right of x, ˙ x, w, x˙d , xd and wd at a we get that ⎡⎡ ⎤ ⎡ ⎡⎡ ⎤ ⎡ ⎤⎤ ⎤⎤ ˙d (s) ˙d (a) b x(s) ˙ x(a) ˙ x x 1 ⎣⎣ x(s) ⎦ , ⎣ xd (s) ⎦⎦ ds → ⎣⎣ x(a) ⎦ , ⎣ xd (a) ⎦⎦ . (3.11) 0= b−a w(s) w(a) wd (s) wd (a) a K K x(a) za ˙ xa ∈ V , which implies that Thus the limit in (3.11) is zero for all x(a) = w a w(a) x˙d (a) xd (a) wd (a)

∈ V [⊥] for all a ∈ [T, 0). By the closedness of V [⊥] and continuity

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from the left of xd , x˙d and wd at 0 we obtain that also ⎡ ⎡ ⎤ ⎤ x˙d (0) x˙d (t) ⎣ xd (0) ⎦ = lim ⎣ xd (t) ⎦ ∈ V [⊥] . t→0− wd (0) wd (t) d x We have now proved that w ∈ Vd [T, 0]. d d 1 − (R ; X ) x ∈ CC(R and (3.9) holds for We still need to show that if w − d ; W) d x all a < b ≤ 0, then w ∈ Vd (R− ). Combining Lemma 2.1(vii) and Propod d d x x ∈ Vd (R− ) if and only if ρ[T ,0] w ∈ Vd [T , 0] sition 3.2, we see that w d d d x ∈ for all T < 0. Now ﬁx T < 0 arbitrarily and note that ρ[T ,0] w d 1 C ([T ,0]; X ) and that (3.9) by assumption holds for all T ≤ a < b ≤ 0 and C([T ,0]; W) x for all[ w ]∈ V[a, b]. The ﬁnite-interval case of this yields that proposition ρ[T ,0]

xd wd

∈ Vd [T , 0], and we have proved that

xd wd

∈ Vd .

We are now ﬁnally ready to introduce passive s/s systems properly.

4. Passive and Conservative State/Signal Nodes In this section we add the concept of passivity to the s/s framework and study what additional structure passive and conservative s/s nodes have. We begin with the following definition, which was motivated in the introduction of the article. Definition 4.1. An ordinary s/s node (V ; X , W) is dissipative (in the forwardtime direction) if V ≥ 0, i.e., [v, v]K ≥ 0 for all v ∈ V . A time-reflected s/s node (V ← ; X , W) is dissipative (in the backward-time direction) if V ← ≤ 0. An ordinary or time-reﬂected s/s node (V ; X , W) is energy preserving if V is neutral: [v, v]K = 0 for all v ∈ V . An ordinary or time-reﬂected s/s node is passive or conservative if both (V ; X , W) and its dual (V [⊥] ; X , W) are dissipative or energy preserving, respectively. An ordinary or time-reﬂected s/s system is said to be dissipative, passive, energy preserving or conservative if one of its generating s/s nodes is of the corresponding type. In Definition 4.1 we deviate slightly from the terminology used by Arov and Staffans in [5]. In the setting of [5] all systems, also dual systems, evolve in forwards time, and this makes it natural for Arov and Staffans to identify the dual W of W by −W. Theorem 3.6 implies that the dual of a s/s node often is a time-reﬂected s/s node in our setting. Arov and Staffans call a dissipative s/s node “forward passive” and by a “backward passive” node they mean a s/s node whose dual s/s node is forward passive. While the terminology of Arov and Staffans is appropriate in their setting, it becomes confusing when some systems evolve in backward

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time. We also give the term “dissipative” a different meaning than Willems in [32,33]. More precisely, the class of s/s systems which are “dissipative” in Willems’ terminology are precisely those which we call passive. Recall that V ⊂ K is called maximally nonnegative if [v, v]K ≥ 0 for all v ∈ V and V has no proper extension that preserves this property. The subspace V is neutral if and only if V ⊂ V [⊥] and it is Lagrangian if V = V [⊥] ; see Lemma A.7. We have the following immediate corollary to Definition 4.1. Corollary 4.2. An ordinary s/s node is passive or conservative if and only if V is maximally nonnegative or Lagrangian, respectively. Every conservative s/s node is passive. Proof. According to Proposition A.8, a closed subspace V ≥ 0 is maximally nonnegative if and only if V [⊥] ≤ 0. Trivially both V and V [⊥] are neutral, i.e., V ⊂ V [⊥] and V [⊥] ⊂ V if and only if V = V [⊥] . In particular, V = V [⊥] by Proposition A.8 implies that V ≥ 0 and V [⊥] ≤ 0.

X {0} {0}

Let X = {0} and W = {0}. Note that V := is a Lagrangian z subspace of K, which does not satisfy 0 ∈ V =⇒ z = 0. Therefore the 0 triple (V ; X , W) needs not be a s/s node even if V is maximally nonnegative or Lagrangian. This is in contrast to the discrete-time case described in [5, Proposition 5.12]. We now characterise dissipative and energy-preserving s/s nodes in terms of their trajectories. Proposition 4.3. Let (V ; X , W) be an ordinary s/s node and let I = [a, b], with b > a, or I = [a, ∞). The s/s node (V ; X , W) is dissipative if and only if the inequality t ∀t ∈ I :

x(t)2X

−

x(a)2X

≤

[w(s), w(s)]W ds

(4.1)

a x ∈ V(I), or equivalently, for all [ w ] ∈ W(I). holds for all The s/s node is energy preserving if and only if (4.1) holds with equality x x ] ∈ V(I), or equivalently, for all [ w ] ∈ W(I). instead of inequality for all [ w x ] [w

Proof. The proof is divided into two parts for readability. Part 1: We begin by proving that (V ; X , W) is dissipative if and only if all x ] ∈ V(I) satisfy (4.1). Assume therefore ﬁrst that (V ; X , W) is dissipative, [w x ] ∈ V(I) and t ∈ I arbitrarily. Then we by (1.7) i.e., that V ≥ 0. Select [ w for all s ∈ [a, t] have ⎡⎡ ⎤ ⎡ ⎤⎤ x(s) ˙ x(s) ˙ ∂ x(s)2X 0 ≤ ⎣⎣ x(s) ⎦ , ⎣ x(s) ⎦⎦ = [w(s), w(s)]W − (4.2) ∂s w(s) w(s) K x and integrating this from a to t we get that (4.1) holds for all [ w ] ∈ V(I).

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za x xa Conversely assume that (4.1) holds for all [ w ] ∈ V(I). Let w ∈V a x(a) za ˙ x xa , cf. Lemma ] ∈ V(I) such that x(a) = w be arbitrary and choose [ w w(a)

a

2.1(iv). By (4.1) we for all h > 0, such that a + h ∈ I, have 1 h

a+h

[w(s), w(s)]W ds −

1 x(a + h)2X − x(a)2X ≥ 0. h

(4.3)

a

Letting h → 0+ , we get (4.2) with s = a and thus ⎡⎡ ⎤ ⎡ ⎤⎤ ⎤ ⎡ za za za ⎣⎣ xa ⎦ , ⎣ xa ⎦⎦ ≥ 0 for all ⎣ xa ⎦ ∈ V. wa wa wa K

(4.4)

x Part 2: If (4.1) holds for all [ w ] ∈ W(I) then (4.1) trivially holds for all x [ w ] ∈ V(I), because every classical trajectory is also generalised according to Definition 1.3. xn ] ∈ V(I), so that: Now conversely assume that (4.1) holds for all [ w n

t ∀t ∈ I :

xn (t)2X

−

xn (a)2X

−

[wn (s), wn (s)]W ds ≤ 0.

(4.5)

a xn x ] ∈W(I) be arbitrary and let the sequence [ w ] ∈ V(I), n ≥ 1, tend Let [ w n C(I; X ) x to [ w ] in L2 (I; W) . Then we for all t ∈ I have limn→∞ xn (t) = x(t) and loc

t lim

t [wn (s), wn (s)]W ds =

n→∞ a

[w(s), w(s)]W ds. a

x ] ∈ W(I) by letting n → ∞ in (4.5). We now obtain (4.1) for all [ w [⊥] The claim that V ⊂ V if and only if (4.1) holds with equality for all x x ] ∈ V(I), or equivalently for all [ w ] ∈ W(I), is proved by replacing the [w inequality signs in (4.1), (4.2), (4.3), (4.4) and (4.5) by equality signs.

Note that the conditions in Proposition 4.3 hold for some subinterval I ⊂ R of the type [a, b], b > a, or [a, ∞) if and only if they hold for all such subintervals, because the claim that the s/s node is dissipative (or energy preserving) does not depend on the choice of I. In order to characterise also passive and conservative s/s nodes, we need the following counterpart of Proposition 4.3 for time-reﬂected s/s nodes. Corollary 4.4. Let I = [a, b], where a < b, or I = (−∞, b]. A time-reflected x ] ∈ V(I), or s/s node (V ; X , W) is dissipative if and only if we for all [ w x equivalently, for all [ w ] ∈ W(I) have: b ∀t ∈ I :

x(b)2X

−

x(t)2X

≥

[w(s), w(s)]W ds.

(4.6)

t

The s/s node is energy preserving if and only if (4.6) holds with equality x x ] ∈ V(I), or equivalently, for all [ w ] ∈ W(I). instead of inequality for all [ w

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Proof. The argument is analogous to the proof of Proposition 4.3. First assume that V ≤ 0 and integrate (4.2) with a reversed inequality sign from t to b in order to obtain (4.6). Conversely, assume that (4.6) holds and let h > 0 be such that b−h > a. Letting t = b − h in (4.6) and dividing the inequality by h, we obtain V ≤ 0 by letting h → 0+ . The following theorem is of fundamental importance for the theory of passive s/s systems, because it establishes a.o. the very useful fact that every fundamental i/o pair is admissible for a passive s/s node. 4.5. Assume that V is a maximally nonnegative subspace of K, that Theorem z 0 ∈ V =⇒ z = 0 and that W = (W+ , W− ) is a fundamental i/o pair. 0 The following claims are valid: (i) (ii)

The triple (V ; X , W) is a passive s/s node for which (W+ , W− ) is admissible. A&B ; X , W+ , W− , the In the operator node representation Vop = C&D A&B X A&B operator C&D : W+ ⊃ Dom C&D → WX− is an i/s/o system node with a contraction semigroup A on X . The generator A of A satisfies C+ ⊂ Res (A).

Proof. We use [14, Proposition 6.7] to prove the claims we made. Deﬁne K± K by (A.2) and PK±∓ by (A.3) with α = 1, so that (K+ , K− ) is a fundamenK

tal decomposition of K and PK±∓ the corresponding fundamental projections, according to Proposition A.2. That proposition also yields that −1 1 0 X V = , (4.7) W W+ 0 0 P − W+

K

because PK+− V = K+ by the assumed maximal nonnegativity of V and Proposition A.8. The maximal nonnegativity also implies that V is closed; see Remark A.6. Thus conditions (a) and (d) of [14, Proposition 6.7] are fulﬁlled by V . Letting |W− | be as given in Definition A.1, we see that the nonnegativity property V ≥ 0 means that: ⎡ ⎤ z W 2 W 2 ⎣x⎦∈V − PW−+ w − 2Re (z, x)X ≥ 0. (4.8) =⇒ PW+− w W+ |W− | w 1 0 0 0 This implies that the space Vz := 0 1 V is closed, as we now W 0 0 PW − + X z zn show. Let therefore uxn ∈ Vz , n ∈ Z+ , tend to some x in X . The u W+ n zn inclusion uxn ∈ Vz means that there exists a sequence yn ∈ W− , such that n zn zm zn xn xn m ∈ V . Then − u x+y ∈ V for all m, n ∈ Z+ and (4.8) u +y u +y n

n

n

n

m

m

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yields that yn − ym 2|W− | ≤ un − um 2W+ − 2Re (zn − zm , xn − xm )X for all n, m ∈ Z+ . The right-hand side tends to zero as m, n → ∞, because zn , xn and un are all convergent, and thus Cauchy, sequences. Then also yn is a Cauchy sequence which tends z to some y in the complete space W− . By x ∈ V , and from the closedness of W+ the closedness of V , we have u+y z and W− we obtain that u ∈ W+ and y ∈ W− , i.e., that x ∈ Vz . We have u proved that Vz is closed. Another consequence of (4.8) is that V is given by ⎡

⎤ A&B A&B ⎢ ⎥ 10 V =⎣ ⎦ Dom C&D C&D + 0 1 for some operator Dom

Indeed, if

0 y

A&B C&D , which maps

A&B C&D

z

(4.9)

=

0 0

1 0

0

W PW+−

V ⊂

X W+

into

X . W−

∈ V and y ∈ W− then 0 ≤ −y2|W− | − 2Re (z, 0) = −y2|W− | , z

∈ V , which by assumption implies that z = 0. A&B Deﬁning the main operator A and the observation operator C of C&D by i.e., y = 0, and thus

0 0

A A&B x x := , C C&D 0

we obtain that [ yz ]

z x y

x∈

x0 A&B x0 ∈ Dom , 0 C&D

(4.10)

∈ V with y ∈ W− if and only if x ∈ Dom (A) and

A ] x. [C

= It still remains to prove that A generates a contraction semigroup on X and that C+ ⊂ Res (A). We use the Lumer–Phillips theorem [22, Theorem 1.4.6] for this purpose. We thus need to show that Dom (A) is dense in X , that 1 − A is surjective and that A is dissipative, i.e., that Re (Ax, x) ≤ 0 for all x ∈ Dom (A). According to [22, Theorem 1.4.6], denseness of Dom (A) is implied by the two latter properties, because X is a Hilbert space. It follows from (4.8), (4.9) and (4.10) that A is dissipative, because Ax x ∈ Dom (A) implies that x ∈ V and then Cx

02W+ − Cx2|W− | − 2Re (Ax, x)X ≥ 0 =⇒ 2Re (Ax, x)X ≤ 0.

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Moreover, z1− A is surjective, because by (4.7) and (4.9) there for all ξ ∈ X exists a x ∈ V , such that w ⎡ ⎤ z x 1 0 − A&B ξ −1 1 0 ⎣ ⎦ x = = , W W− 0 0 PW+− 01 0 PW+ w w which means that x ∈ Dom (A) and (1 − A)x = ξ. We have now proved that A generates a contraction semigroup on X and by [22, Corollary 1.3.6] we have C+ ⊂ Res (A). According to Proposition 2.7(i), (V ; X , W) is a s/s node with admissible i/o pair (W+ , W− ). This s/s node is passive by Corollary 4.2, because V was assumed to be maximally nonnegative. The next section deals with operator node representations that correspond to fundamental i/o pairs. In the rest of this section we present a few results which do not refer to speciﬁc i/o pairs. Remark 4.6. According to Definition 1.2, Corollary 4.2 and Theorem 4.5, the triple (V ; X , W) is a passive node if and only if V is a maximally z s/s nonnegative subspace of K and 0 ∈ V =⇒ z = 0. 0 z We now show that the condition 0 ∈ V =⇒ z = 0 is not crucial in the 0

context of passive s/s systems, because we can turn every maximally nonnegative subspace V ⊂ K into a s/s node by removing a certain “degenerate” part of V and shrinking the state space. Proposition 4.7. Let V be a maximally nonnegative subspace of K and define + * X1 X0 z X0 := z ∈ X 0 ∈ V , V0 := 0 and X1 := X X0 . Let K1 := X1 0

W

0

inherit the indefinite inner product from K and set V1 := V ∩ K1 . The following claims are true: [⊥]

(i) We have V = V0 V1 , where V0 [⊥]V1 and V0 is neutral: V0 ⊂ V0 . [⊥] [⊥] (ii) The orthogonal companion V1 1 of V1 in K1 is given by V1 1 = V [⊥] ∩ [⊥] K1 and, moreover, V [⊥] = V0 V1 1 . (iii) The triple (V1 ; X1 , W) is a passive s/s node, which is conservative if and only if V is Lagrangian: V = V [⊥] . (iv) The spaces V and V1 generate the same trajectories: V = V1 and W = W1 . The only trajectory generated by V0 is the zero trajectory: V0 = W0 = {[ 00 ]}. Proof. First ﬁx a fundamental decomposition W = (W+ , W− ) and let J := W W PW+− −PW−+ be the corresponding fundamental symmetry, cf. Definitions A.1 and A.3. Then it is readily veriﬁed that ⎛⎡ 1 ⎤ ⎡ 2 ⎤⎞ z z ⎝⎣ x1 ⎦ , ⎣ x2 ⎦⎠ := (z 1 , z 2 )X + (x1 , x2 )X + [w1 , Jw2 ]W (4.11) w1 w2 K

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is the admissible inner product on K corresponding to the fundamental decom position (A.2) of K with α = 1. Also note that V0 = V ∩

X {0} {0}

.

(i)

The maximal nonnegativity of the space V implies that it is closed, see Remark A.6, and it is then immediate that also V0 is closed. Deﬁne V%1 := V V0 , where the orthogonality is taken with respect to the Hilbert-space inner product (4.11). Then V = V0 ⊕ V%1 , by assumption [⊥] we have V ≥ 0, and it clearly holds that V0 ⊂ V0 . Lemma A.7 yields that V0 [⊥]V%1 and V%1 ≥ 0. We are done proving claim (i) once we have % established V1 = V1 . zthat Let x ∈ V%1 be arbitrary, and note that V%1 ⊂ V , V%1 [⊥]V0 and w V%1 ⊥ V0 imply that ⎡⎡ ⎤ ⎡ ⎤⎤ ⎛⎡ ⎤ ⎡ ⎤⎞ z z0 z z0 (x, z0 )X = − ⎣⎣ x ⎦ , ⎣ 0 ⎦⎦ = 0 = ⎝⎣ x ⎦ , ⎣ 0 ⎦⎠ = (z, z0 )X 0 0 w w K K z for all z0 ∈ X0 . This yields that z, x ∈ X0⊥ , i.e., that x ∈ V ∩K1 = V1 . w z Conversely, if x ∈ V ∩ K1 , then in particular z ∈ X1 , which implies w z that x ∈ V V0 = V%1 . We have proved that V%1 = V1 .

(ii)

Applying a slight modiﬁcation of the procedure in the proof of claim (i) to V [⊥] , which is maximally nonpositive by Proposition A.8, we X [⊥] [⊥] obtain that V = V$0 V$1 , where V$0 = V ∩ {0} is neutral and

w

{0}

V$1 = V [⊥] ∩ K1 is nonpositive. By Lemma A.7, we have V0 ⊂ V

[⊥]

and by definition V0 ⊂

X {0} {0}

,

and therefore V0 ⊂ V$0 . The same argument applied to V$0 yields that ⎡ ⎤ X [⊥] [⊥] V$0 ⊂ (V ) ∩ ⎣ {0} ⎦ = V0 {0} [⊥] and we thus have V$0 = V0 . Furthermore, V$1 ⊂ V1 1 , i.e. V$1 [⊥]V1 , because V [⊥] = V0 + V$1 is orthogonal to V = V0 + V1 . On the other [⊥] hand, assuming that v1 ∈ V1 1 , we obtain

[v0 + v1 , v1 ]K = [v0 , v1 ]K + [v1 , v1 ]K1 = 0,

vi ∈ Vi ,

[⊥] which implies that v1 ∈ V [⊥] ∩ K1= V$1 . Therefore V$1 = V1 1 . z z (iii) First note that if 0 ∈ V1 , then 0 ∈ V0 ∩ V1 = {0} and thus z = 0. 0 0 We showed that V1 ≥ 0 in the proof of claim (i), the space V1 = V ∩ K1 is closed since both V and K1 are closed, and the proof of claim (ii) [⊥] yields that V1 1 ≤ 0. According to Proposition A.8, we have that V1 is a maximally nonnegative subspace of K1 , and Theorem 4.5 yields that (V1 ; X1 , W) is a passive s/s node.

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If, moreover, V = V [⊥] , then V is also maximally nonpositive by Proposition A.8, and the above argument can be modiﬁed to yield [⊥] that V1 is a maximally nonpositive subspace of K1 . Then V1 = V1 1 by Proposition A.8, and thus (V1 ; X1 , W) is a s/s node and V1 is a Lagrangian subspace of K1 , i.e., (V1 ; X1 , W) is a conservative s/s node. Conversely, if (V1 ; X1 , W) is a conservative s/s node, then V1 = [⊥]1 V1 according to Corollary 4.2, and by claim (ii) we have V [⊥] = [⊥] V0 V 1 1 = V0 V 1 = V . (iv) Every classical trajectory generated by V1 is trivially a classical trajectory generated by V , because V1 ⊂ V . For the converse inclusion we x ] ∈ V be an arbitrary classical generated by V . Noting let [ w trajectory that V0 [⊥]V1 is equivalent to V ⊂

X X1 W

, we obtain from Definition 1.1

that x(t) ∈ X1 for all t > 0. Therefore

x(t + h) − x(t) ∈ X1 , t > 0, x(t) ˙ = lim h→0 h x(t) X1 ˙ and thus x(t) ∈ V1 = V ∩ X1 for all t > 0. W

w(t)

Every classical trajectory

x [w ]

generated by V0 ⊂

X {0} {0}

trivially

satisﬁes x(t) = 0 and w(t) = 0 for all t ≥ 0. According to Definition 1.3, V1 = V and V0 = {0} imply that W1 = W and W0 = {0}. The following corollary follows directly from Proposition 4.7(ii). z Corollary 4.8. Assume that V is maximally nonnegative. Then 0 ∈ V =⇒ 0 z [⊥] 0 z = 0 if and only if ∈V =⇒ z = 0. 0

We now demonstrate how Proposition 4.7 can be applied in practice by connecting two capacitors in parallel. Example 4.9. An ideal capacitor with capacitance Ci can be modelled by the equation √ 0 1/ Ci x˙ i (t) xi (t) = , (4.12) √ yi (t) ui (t) 1/ Ci 0 √ where xi is the charge in the capacitor divided by Ci , ui is the current entering the capacitor and yi is the voltage over the capacitor. This system has generating subspace ⎡ √ ⎤ 0 1/ Ci ⎢ 1 0 ⎥ ⎥ C2 , √ Vi = ⎢ (4.13) ⎣ 1/ Ci 0 ⎦ 0 1 C and Y = {0} , cf. (1.2). where U = {0} C

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Figure 1. Two initially disconnected capacitors C1 and C2 , which we interconnect by adding the dashed wire The appropriate external signal this case is W = C2 , and we in2 space y1 y , 2 equip W with the power product = y 1 u2 + u1 y 2 , because u1 u W electrical power equals voltage times current. Therefore the node space is Ki = C4 with the power product ⎡⎡ 1 ⎤ ⎡ 2 ⎤⎤ z z ⎢⎢ x1 ⎥ ⎢ x2 ⎥⎥ ⎢⎢ 1 ⎥ , ⎢ 2 ⎥⎥ = y 1 u2 + u1 y 2 − z 1 x2 − x1 z 2 . ⎣⎣ y ⎦ ⎣ y ⎦⎦ u1 u2 K Corollary A.9 yields that Vi is Lagrangian, because Vi is easily seen to be neutral and dim Vi = 2, which is precisely half of the dimension of C4 . This reﬂects the well-known fact that an ideal capacitor conserves energy. In Fig. 1 we have drawn two capacitors C1 and C2 , which are initially not interconnected. We consider these two capacitors as a single system, the so-called product of the two individual capacitors. This product system has generating subspace √ ⎡ ⎤ 0 0 1/ C1 0 √ ⎥ ⎢ ⎢ 0 0 0 1/ C2 ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 ⎥ ⎢ 1 ⎢ ⎥ ⎢ 0 1 0 0 ⎥ ⎢ ⎥ 4 V =⎢ √ ⎥C , ⎢ 1/ C1 0 0 0 ⎥ ⎢ ⎥ √ ⎢ ⎥ ⎢ 0 1/ C2 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ 0 1 0 ⎣ ⎦ 0 0 0 1 which is a Lagrangian subspace of C8 with the appropriate power product obtained as the sum of the power products on K1 and K2 . We use the horizontal lines to separate the two copies of the state space C2 from the external signal space C4 .

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We now connect the two capacitors in parallel. Making the dashed connections in Fig. 1 and applying Kirchhoﬀ’s laws√at the junction P, we get √ the additional constraint that y1 = y2 , i.e., x1 / C1 = x2 / C2 . Moreover, the total current ﬂowing into the parallel coupling is u1 + u2 . Let us therefore deﬁne y := (y1 + y2 )/2 and u := u1 + u2 , so that y is the voltage overthe parallel coupling and u is the current ﬂowing into it. Then the variables of the interconnected system live in the subspace ⎡ √ 0 0 1/ C1 ⎢ 0 0 0 ⎢ ⎢ ⎢ 1 0 0 V = ⎢ ⎢ 0 1 0 ⎢ ⎢ √ √ ⎣ 1/2 C1 1/2 C2 0 0 0 1 √ √ × N 1/ C1 −1/ C2 0

⎤ 0 √ 1/ C2 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦ 1 0 .

z x y u

(4.14)

The space V is a Lagrangian subspace of K := C6 equipped with the power product ⎡⎡ 1 ⎤ ⎡ 2 ⎤⎤ z z ⎢⎢ x1 ⎥ ⎢ x2 ⎥⎥ ⎢⎢ 1 ⎥ , ⎢ 2 ⎥⎥ = y 1 u2 + u1 y 2 − (z 1 , x2 )C2 − (x1 , z 2 )C2 . ⎣⎣ y ⎦ ⎣ y ⎦⎦ u1 u2 K because it does However, V is not the generating subspace of a s/ssystem, z not satisfy condition (ii) of Definition 1.2. Indeed, 00 ∈ V if and only if 0 √ z ∈ −√CC2 C =: X0 , and this space is nontrivial. One easily veriﬁes that 1

√ √ √ C X1 := X X0 = √ 1 C = N 1/ C1 −1/ C2 C2

and setting [ zz12 ] ∈ X1 in (4.14) yields that √ √ C u1 /√C1 z1 = = √ 1 a, a = (u1 + u2 )/(C1 + C2 ). z2 u2 / C2 C2 X1 Deﬁning V1 := V ∩ X21 , we thus obtain that ⎡

C

0 0

⎢ ⎢ ⎢ ⎢ 1 V1 = ⎢ ⎢ 0 ⎢ ⎢ √ ⎣ 1/2 C1 0

0 0 0 1 √ 1/2 C2 0

⎤ √ √C1 /(C1 + C2 ) ⎥ C2 /(C1 + C2 ) ⎥ ⎡ √ ⎤ ⎥ C ⎥⎢ √ 1 C⎥ 0 ⎥⎣ ⎦. C2 ⎥ 0 ⎥ C ⎥ ⎦ 0 1

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By Proposition 4.7(iii), the triple (V1 ; X1 , C2 ) is a conservative s/s node. Obviously this s/s node has operator node representation √ C1 {0} C √ C, , , (4.15) V1,op = S1 ; C {0} C2 √ √C1 C X C where the system node S1 is given by the restriction to U1 = 2 of

A1 B1 C1 D1

⎡

0 ⎢ 0 ⎢ =⎢ √ ⎣ 1/2 C1 0

⎤

{0} C

√ 0 √ C1 /(C1 + C2 ) 0 C2 /(C1 + C2 ) ⎥ ⎥ ⎥. √ ⎦ 1/2 C2 0 0 0 0 0 0 0

This example seemingly turns a simple task into very complicated one, but it is interesting that the same approach can be applied to quite general inﬁnite-dimensional systems, using the tools developed in Sects. 4 and 5. Some ﬁrst steps in the direction of interconnection of conservative systems, which are relevant for s/s systems, were taken in [15], but we will study this topic in more detail and generality elsewhere. The operator node representation (4.15) is a special case of a so-called impedance representation of C = U [⊥] and Y = {0} = Y [⊥] , a s/s node, due to the fact that U = {0} C so that W = U Y is a Lagrangian decomposition. We will also study more general impedance representations in a forthcoming article. The next step is to characterise conservative s/s nodes, but in order to do this we ﬁrst need to write down the following lemma. A&B is an operator node on (U, X , Y) and α ∈ Res (A), Lemma 4.10. If C&D [ α 0 ]−A&B A&B maps Dom C&D one-to-one onto [ X then the operator U ]. [ 0 1U ]

[ α 0 ]−A&B is injective, because Proof. Under the given assumptions, [ 0 1U ] α 0 − A&B x = 0 =⇒ x ∈ Dom (A) and (α − A)x = 0, u 0 1U and then x = 0 and u = 0. Moreover the operator is surjective, as we will now show. Let therefore z ∈ X and u ∈ U be arbitrary and let x be such that [ ux ] ∈ Dom (S). Since α ∈ Res (A), we can ﬁnd an x ∈ Dom (A) such that x (α − A)x = z − αx − A&B ∈ X. u = [ ux ] + x0 ∈ Dom (S) and Then x+x u x + x α 0 − A&B = z. u The proof is done.

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Recall from Corollary 3.8 that if (U, Y) is an admissible i/o pair for the s/s node (V ; X , W) then (U [⊥] , Y [⊥] ) is an admissible i/o pair for the s/s dual (V [⊥] ; X , W). In this case we denote the main operator of the operator node [⊥] representation Vop = (S d , X , U [⊥] , Y [⊥] ) by Ad , cf. Definitions 2.4 and 2.5. If (U [⊥] , Y [⊥] ) is admissible also for the primal s/s node (V ; X , W), then we denote the corresponding operator node by S × and its main operator by A× . z Theorem 4.11. Assume that V ⊂ K has the property that 0 ∈ V =⇒ z = 0. 0 Then the following claims are equivalent: The triple (V ; X , W) is a conservative s/s node. The space V is a Lagrangian subspace of K: V = V [⊥] . Both (V ; X , W) and (V [⊥] ; X , W) are ordinary s/s nodes and, for some subinterval [a, b] ⊂ R of finite positive length, the spaces of classical trajectories generated by V and V [⊥] on [a, b] coincide: V[a, b] = Vd [a, b]. (iv) Both (V ; X , W) and (V [⊥] ; X , W) are ordinary s/s nodes and, for every positive-length subinterval I ⊂ R, the spaces of classical trajectories generated by V and V [⊥] on I coincide: V(I) = Vd (I). (v) The triples (V ; X , W) and (V [⊥] ; X , W) are both passive ordinary s/s nodes, so that V and V [⊥] are maximally nonnegative. (vi) Both (V ; X , W) and (V [⊥] ; X , W) are passive time-reflected s/s nodes: V and V [⊥] are maximally nonpositive. (vii) The following two conditions hold for some decomposition W = U Y: (a) both (U, Y) and (U [⊥] , Y [⊥] ) are admissible i/o pairs for (V ; X , W), and (b) the operator node representations Vop = (S × ; X , U [⊥] , Y [⊥] ) and [⊥] Vop = (S d ; X , U [⊥] , Y [⊥] ) coincide: S × = S d . (viii) The following three conditions all hold: (c) the space V is neutral: V ⊂ V [⊥] , (d) there exists a decomposition W = U Y, such that (a) holds and (e) the main operators of the operator node representations of V and V [⊥] corresponding to the i/o pair (U [⊥] , Y [⊥] ), as given in (b), have non-disjoint resolvent sets: Res A× ∩ Res Ad = ∅. (i) (ii) (iii)

Assume that (d) holds and let A be the main operator of Vop = (S; X , U, Y). If either both A× and −A, or both −A× and A, generate C0 semigroups on X , then (e) also holds. Condition (a) of Theorem 4.11 needs some clariﬁcation. Let (V ; X , W) be a s/s node. By Definition 2.5, admissibility of the i/o pair (U, Y) means A&B with input space U and that V satisﬁes (2.4) for some operator node C&D output space Y. Thus there is no implication that A generates a C0 semigroup on X . As a consequence, there is no guarantee that A or A× generate semigroups even if (a) holds.

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Proof Theorem 4.11. We begin by proving the last claim. Assume therefore that A generates a C0 semigroup with growth bound ω; see [26, Deﬁnition 2.5.6]. By Theorem 3.6, −Ad = A∗ and, according to [26, Theorem 3.5.6], A∗ generates the C0 semigroup t → (At )∗ , which also has growth bound ω. If −A generates a C0 semigroup A , then (−A)∗ = Ad obviously generates the dual semigroup (A )∗ . Assume that A× and −A generate semigroups with growth bounds ω × and ω, respectively. By the argument we just made, Ad generates a semigroup with growth bound ω as well. Theorem 3.2.9(i) of [26] then yields that 0 0 11 λ ∈ C | Re λ > max ω × , ω ⊂ Res A× ∩ Res Ad , which obviously implies (e). Now drop the earlier assumptions on A and A× , and instead assume that both −A× and A generate C0 semigroups with growth bounds ω × and ω, respectively. Then (e) again holds, because 0 0 11 λ ∈ C | Re λ < − max ω × , ω ⊂ Res A× ∩ Res Ad . The following implications prove the equivalence of the claims (i–viii) listed in the theorem: (i)⇐⇒ (ii): if V = V [⊥] , then V is maximally nonnegative by Proposition A.8. In this case (V ; X , W) is a (passive) s/s node by Theorem 4.5. The rest was shown in Corollary 4.2. (ii) =⇒ (iv) and (iv) =⇒ (iii): these implications are both trivial once we know that (V ; X ,W) is a s/s node. za xa wa

∈ V . By Definition 1.2, V is closed and there exists x(a) za ˙ x xa . This by assumption a trajectory [ w ] ∈ V[a, b] such that x(a) = w a w(a) za x d [⊥] xa implies that [ w ] ∈ V [a, b], i.e., that w ∈ V . Thus V ⊂ V [⊥] . For (iii) =⇒ (ii): let

a

the converse inclusion, we apply the same argument to V [⊥] and obtain that V [⊥] ⊂ (V [⊥] )[⊥] = V . (ii) ⇐⇒ (v): claim (ii) implies claim (i), which says that (V ; X , W) is a conservative s/s node. In this case (V [⊥] ; X , W) = (V ; X , W) is a passive s/s node by Corollary 4.2. Conversely, Corollary 4.2 yields that (V ; X , W) and (V [⊥] ; X , W) are passive s/s nodes if and only if V and V [⊥] are both maximally nonnegative. According to Proposition A.8 this is equivalent to V being Lagrangian. (v) ⇐⇒ (vi): if V is maximally semidefinite, then V is closed by Remark A.6. According to Proposition A.8, V is then maximally nonnegative if and only if V [⊥] is maximally nonpositive. (vii) =⇒ (ii): the assumptions (a) and (b) immediately imply that [⊥] = (S d ; X , U [⊥] , Y [⊥] ) = Vop , Vop

i.e., that V = V [⊥] , cf. Definition 2.5. (ii) =⇒ (viii): assume that V = V [⊥] . Then trivially (c) holds and the triple (V ; X , W) is a passive s/s node by item (v). Thus every fundamental decomposition W = (W+ , W− ) is admissible for (V ; X , W) and C+ ⊂ Res (A)

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for the main operator of the corresponding operator node representation, according to Theorem 4.5. [⊥] [⊥] Theorem 3.6 yields that (W+ , W− ) is admissible for V [⊥] and that the corresponding operator node representation S d has main operator Ad = −A∗ . [⊥] [⊥] One easily shows that (W+ , W− ) = (W− , W+ ) and, by Corollary 3.8, this i/o pair is also admissible for V , because V = V [⊥] . Thus (d) holds and S d = S × is immediate from V = V [⊥] and the notation in (b), see Definition 2.5 again. In particular, Ad = A× and C− ⊂ Res (−A∗ ) = Res Ad = Res (A× ), because α ∈ Res (A) if and only if α ∈ Res (A∗ ), which is equivalent to −α ∈ Res (−A∗ ). (viii) =⇒ (vii): assume that V ⊂ V [⊥] . Then S × ⊂ S d , again bythenotation d in (b), and we are done if we can prove that Dom (S × ) = Dom S ×. maps Let now α ∈ Res (A× )∩Res Ad . By Lemma 4.10, [ α 0 []−A&B 0 1] X × × d Dom (S ) one-to one onto U [⊥] . The inclusion S ⊂ S implies that also [ α 0 ]−A&B d maps Dom (S × ) one-to one onto UX[⊥] . Since the latter oper[0 1] ator maps Dom S d one-to one onto UX[⊥] , again by Lemma 4.10, we con clude that Dom (S × ) = Dom S d . We made the following observation in the proof of Theorem 4.11. Let (V ; X , W) be a conservative s/s node. Then every fundamental i/o pair (U, Y) = (W+ , W− ) satisﬁes condition (a) of Theorem 4.11 and C− ⊂ Res A× = Res Ad , where A× and Ad are as given in (b) with U [⊥] = W− and Y [⊥] = W+ . Theorem 4.11(vi) has the following consequence, which should be compared to [19, Remark 4.3]. Let W = (U, Y) be an orthogonal i/o pair, i.e., let U[⊥]Y. Then U [⊥] = Y and if condition (a) holds for such an i/o pair, then V has the operator node representations Vop = (S; X , U, Y) and Vop = (S × ; X , Y, U). In this case S × has the interpretation of being the flow inverse of S, because (u, x, y) is a classical i/s/o trajectory generated by S if and only if (y, x, u) is a classical i/s/o trajectory generated by S × , cf. Definition 2.4. The condition S × = S d thus means that the ﬂow inverse equals the anticausal dual of S introduced in Remark 3.7. Flow inversion is described in more detail in [28] and Chapter 6 of [26]. I/s/o representations corresponding to general orthogonal i/o pairs are called transmission representations, and we will treat these in a forthcoming article. We conclude this section with an example of a conservative s/s node of boundary control type. Example 4.12. We continue ⎡ ∂ ⎤Example 2.9, using the notation we introduced − ∂z there. Deﬁning V := ⎣ 1 ⎦ Z, we obviously get a subspace of K with the δ0 property

z 0 0 2

∈ V =⇒ z = 0, and we may thus use Theorem 4.11 to show

that (V ; L (R ; C), C) is a conservative s/s node. +

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∂ x, with Combining the following short computation, where x = ∂z Lemma A.7(i), we obtain that V satisﬁes Theorem 4.11(c): ⎤ ⎡ ⎤⎤ ⎡⎡ ∞ ! −x −x " ⎣⎣ x ⎦ , ⎣ x ⎦⎦ = |x(0)|2 + x (z) x(z) + x(z) x (z) dz x(0) x(0) 0 K ∞ ∂ 2 2 |x(z)| = |x(0)| + dz ∂z 0

= |x(0)|2 + lim |x(z)|2 − |x(0)|2 = 0. z→+∞

From W = C we obtain C[⊥] = {0}. By Example 2.9, V has operator node representations Vop = (S; L2 (R+ ; C), C, {0}) ×

and

Vop = (S ; L (R ; C), {0}, C). 2

+

The i/o pair (C, {0}) thus proves that V satisﬁes Theorem 4.11(d) as well. ∂ × Note that A = − ∂z does not generate a C0 semigroup on X = Z

L2 (R+ ; C), but −A× does. By the last claim of Theorem 4.11, condition (e) of that theorem also holds, and we conclude that (V ; L2 (R+ ; C), C) is a conservative s/s node. We now proceed to study operator node representations corresponding to fundamental i/o pairs in more detail.

5. Scattering Representations and Passivity In this section we introduce the so-called scattering representations and use these to establish some additional properties of passive s/s systems. We also give a few characterisations of passive s/s systems in terms of scattering representations. Definition 5.1. Let V ⊂ K and assume that the fundamental decomposition W = (W+ , W− ) is an admissible i/o pair for V . Then we call the corresponding operator node representation a scatter A&B ; X , W+ , W− . ing representation of V and write Vsca = C&D If (V ; X , W) is a s/s node, then we call Vsca a scattering representation of both (V ; X , W) and the s/s system that the s/s node generates. According to Definition A.1, every fundamental decomposition of a Kre˘ın space is orthogonal. Thus a scattering representation is a special case of a transmission representation, as it was introduced in the previous section. The next theorem gives one characterisation of which dissipative s/s nodes are actually passive. Dissipativity is of course a necessary condition for passivity.

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Theorem 5.2. Let V be a nonnegative subspace of K and (W+ , W− ) a fundamental decomposition of W. Then the following claims are equivalent: (i) The triple (V ; X , W) is a passive s/s node. A&B ; X , W+ , W− . (ii) The subspace V has scattering representation C&D The corresponding main operator A defined in (2.3) generates a contraction semigroup on X and C+ ⊂ Res (A). (iii) The i/o pair (W+ , W− ) is admissible for V and the resolvent set of the associated main operator in (2.3) satisfies Res (A) ∩ C+ = ∅. z given (iv) The space V is closed, 0 =⇒ z = 0 and for some α ∈ C+ we have 0 −1 α 0 X . (5.1) W− V = W+ 0 0 PW+ z (v) The space V is closed, 0 =⇒ z = 0 and (5.1) holds for all α ∈ C+ . 0

Proof. We again divide the proof into a series of implications. (i) =⇒ (v): by Definition z 1.2, every s/s node (V ; X , W) has the properties that V is closed and 0 =⇒ z = 0. By assumption (i) and Corollary 4.2, 0 V is maximally nonnegative and, according to Propositions A.2 and A.8, we then have (5.1) for an arbitrary α ∈ C+ . (v) =⇒ (iv) and (ii) =⇒ (iii): these implications trivial. zare (iv) =⇒ (i), (ii): by assumption the implication 0 ∈ V =⇒ z = 0 holds. We 0 now show that V is maximally nonnegative, whereafter Theorem 4.5 yields that claims (i) and (ii) are true. Let α ∈ C+ satisfy (5.1) and let K = (K+ , K− ) be the fundamental K decomposition and PK+− the projection given in Proposition A.2, so that K

PK+− V = K+ . We assumed that V ≥ 0 and thus V is maximally nonnegative according to Proposition A.8. A&B , such that (iii) =⇒ (iv): by assumption there exists an operator node C&D (4.9) holds, and in particular the implication

z 0 0

∈ V =⇒ z = A&B [ 00 ] = 0

is valid. By Lemma 4.10, (5.1) holds for every α ∈ Res (A) ∩ C+ .

Conditions (ii–v) of Theorem 5.2 hold for some fundamental decomposition if and only if they hold for all fundamental decompositions, because condition (i) is independent of the fundamental decomposition. As the following example taken from [5, Example 5.5] shows, there exist energy-preserving s/s nodes, for which no fundamental i/o pair is admissible. If there on the contrary existed such an i/o pair, then the system would be passive by Theorem 5.2, because the condition Res (A) ∩ C+ = ∅ becomes trivial when X = {0}. Example 5.3. Let X = {0} and W = C3 with the power product y1 y2 1 1 = −y11 y12 + y21 y22 + u1 u2 . y21 , y22 Then V :=

1 0 1

u1

u2

W

C is neutral and (V ; {0}, W) is an energy-preserving s/s node.

414

M. Kurula Moreover,

0 1 0

∈ V [⊥] and

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0 0 1 , 1 0

0

W

= 1 ≥ 0, so that V [⊥] ≤ 0.

This implies that V is not maximally nonnegative, i.e. that the s/s node is not passive. In Theorem 5.2 we characterised passivity under the assumption of dissipativity. We now proceed to deﬁne a scattering-passive i/s/o system node in order to be able to prove that every scattering representation of a passive s/s node is of this type. The following definition uses classical i/s/o trajectories of an operator node, as these were introduced in Definition 2.4. A&B be an i/s/o node on (U, X , Y). Definition 5.4. Let C&D system 2 A&B The i/s/o system node C&D is L -well-posed if there exists aT > 0 A&B and a constant KT such that all classical i/s/o trajectories (u, x, y) of C&D satisfy ⎛ ⎞ t t ∀t ∈ [0, T ] : x(t)2X + y(s)2Y ds ≤ KT ⎝x(0)2X + u(s)2U ds⎠ . 0

0

(5.2) The i/s/o system node is scattering passive if it is L2 -well-posed with KT = 1, i.e., if all classical trajectories satisfy t ∀t ∈ [0, T ] :

x(t)2X

t y(s)2Y

+

ds ≤

x(0)2X

u(s)2U ds.

+

0

(5.3)

0

A&B

The i/s/o system node C&D is scattering energy preserving if (5.3) holds with equality of inequality. The i/s/o node is scattering conservative A&B ∗ A&B instead and C&D are scattering energy preserving. if both C&D Comparing Remark 4.6 to Definitions 2.4 and 5.4, we see that a passive s/s node is indeed much easier to describe than a passive i/s/o system node. This is only one example of how it can be more natural to study systems theory in the s/s framework than in the i/s/o counterpart. A&B on Remark 5.5. Theorem 4.7.13 in [26] says that a system node C&D 2 (U, X , Y) is L -well-posed if and only if there exists a T > 0, KT > 0 such A&B on [0, T ] satisfy the following for that all i/s/o trajectories (u, x, y) of C&D all t ∈ [0, T ]: 2 2 ⎛ ⎞ 3 t 3t 3 3 3 3 ⎜ ⎟ x(t)X + 4 y(s)2Y ds ≤ KT ⎝x(0)X + 4 u(s)2U ds⎠. (5.4) 0

0

Comparing this to (5.2), we see that the terms in (5.2) are the terms in (5.4) squared. This difference is non-essential, however, because (5.2) corresponds 7 a 2 2 to using the norm [ b ] = |a| + |b| on R2 and (5.4) corresponds to using the norm [ ab ] = |a| + |b|, and all norms in R2 are equivalent. If we change the powers, to which the terms in (5.2) are raised, then we may be forced to

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increase KT , but the claim that there exists some constant KT is either true in both cases or false in both cases. We use (5.2) as the definition for L2 -well-posedness, because it ﬁts passive systems better than (5.4). We now study passivity of s/s nodes which have a scattering representation. Proposition 5.6. Let the subspace V ⊂ K have the scattering representation A&B ; X , W + , W− . Then the following conditions are equivalent: C&D (i) The triple (V ; X , W) is a passive s/s node. (ii) The subspace V ⊂ K is nonnegative and Res (A) ∩ C+ = {0}. (iii) We have V ≥ 0 andC+ ⊂ Res (A). A&B is a scattering-passive i/s/o system node. (iv) The operator C&D A&B is a The triple (V ; X , W) is a conservative s/s node if and only if C&D scattering-conservative i/s/o system node. In this case conditions (i–iv) above hold. Proof. We ﬁrst note the following almost direct consequence of (4.9). It holds x ] ∈ V with u(t) ∈ W+ and y(t) ∈ W− for all t ≥ 0 if and only if that [ u+y A&B A&B . If is a i/s/o sys(u, x, y) is a classical i/s/o trajectory of C&D C&D tem node, then it is scattering passive if and only if V ≥ 0, because (5.3) is equivalent to (4.1) with I = [0, T ] when U = W+ and Y = W− . A&B has a C0 semigroup A. Theorem (iv) =⇒ (iii): by Definition 2.3, C&D 3.2.9(i) of [26] yields that any α ∈ R+ greater than the growth bound of A lies in Res (A) ∩ C+ , which is thus nonempty, cf. the beginning of the proof of Theorem 4.11. By the discussion at the beginning of this proof, V ≥ 0 and Theorem 5.2(ii) then yields that C+ ⊂ Res (A). (iii) =⇒ (ii): this implication is trivial. (ii) =⇒ (i): this follows from Theorem 5.2(iii). A&B gen(i) =⇒ (iv): Theorem 5.2 yields that the main operator A of C&D A&B is an i/s/o system node. erates a contraction semigroup, i.e., that C&D According to Corollary 4.2, V is a (maximally) nonnegative subspace, so that (4.1) holds I = [0, T ], T > 0, by Proposition 4.3. Therefore (5.3) also A&B for is scattering passive. holds and C&D The last claim follows from [9, Proposition 4.9] and Theorem 4.11. We now prove that all passive s/s nodes are L2 -well-posed. In order to do this, we ﬁrst need to recall the definition of an L2 -well-posed s/s node from [14]. Definition 5.7. The s/s node (V ; X , W) is L2 -well-posed if there exists a T > 0 and a direct sum decomposition W = U Y, such that V[0, T ] satisﬁes the following conditions: x 1 0 ] ∈ V[0, T ] is dense in X . (i) The space x(0) [ w (ii) The operator 0 PUY maps the space x(0) x =0 (5.5) V0 [0, T ] := ∈ V[0, T ] w(0) w

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densely into L2 ([0, T ]; U). x ] ∈ V[0, T ] satisfy for all (iii) There exists a KT > 0, such that all [ w t ∈ [0, T ]: ⎛ ⎞ t t x(t)2X + w(s)2W ds ≤ KT ⎝x(0)2X + PUY w(s)2U ds⎠ . 0

0

In this case we call (U, Y) an L2 -admissible i/o pair of (V ; X , W). The notion of an L2 -admissible i/o pair is related to the notion of admissibility given in Definition 2.5, but neither type of admissibility implies the other type. It follows from the following proposition that all of the theory developed in [14] is applicable to passive s/s systems. Proposition 5.8. Let T > 0 and let (V ; X , W) be a passive s/s node with generalised trajectories W[0, T ]. Then the following claims are true: (i) Every fundamental i/o pair is L2 -admissible for (V ; X , W). In particular, the space V generates an L2 -well-posed s/s system Σ. (ii) The space V is maximal in the sense that all subspaces that generate the same space W[0, T ] of generalised trajectories are contained in V . (iii) The space of classical trajectories generated by V on [0, T ] satisfies 1 C ([0, T ]; X ) . (5.6) V[0, T ] = W[0, T ] ∩ C([0, T ]; W) (iv)

The generating subspace V is uniquely determined by W[0, T ] through ⎧⎡ ⎫ ⎤ 1 ⎬ ˙ ⎨ x(0) x C ([0, T ]; X ) ∈ W[0, T ] ∩ V = ⎣ x(0) ⎦ . w C([0, T ]; W) ⎭ ⎩ w(0)

Any of the spaces V , V[0, T ], V, W[0, T ] and W determine the other four of these spaces uniquely. Proof. We begin by proving claims (i) and (ii) andwe therefore ﬁx a funda A&B ; X , W , W mental i/o pair W = (W+ , W− ). By Theorem 5.2, C&D + − is a A&B scattering representation of V , and by Proposition 5.6, C&D is a scatteringpassive system node. Thus (5.2) holds with KT = 1 and therefore every scattering representation of a passive s/s node is L2 -well-posed. Theorem 6.4 of [14] yields that (V ; X , W) is an L2 -well-posed s/s node, which is maximal in the sense that all other generating subspaces of the same s/s system are included in V . Claim (iii) was proved in the proof of [14, Theorem 6.4]. Combining (2.1) with (5.6), we see that claim (iv) holds. The generating subspace V determines V uniquely by Definition 1.1, and V in turn determines W uniquely by Definition 1.3. Moreover, W[0, T ] = ρ[0,T ] W by [14, Proposition 3.9], and W[0, T ] determines V[0, T ] as in claim (iii). Finally, V[0, T ] determines V through (2.1). One may ask if the converse of Proposition 5.8(ii) also is true, i.e., if assuming that V ≥ 0 contains all generating subspaces of the same s/s system is enough to imply that (V ; X , W) is passive. Example 5.3 shows that the

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answer is no, and the explanation is the following. According to [14, Theorem 6.4], the maximality of V as a generating subspace follows from the existence of an arbitrary L2 -well-posed i/o pair. However, for (V ; X , W) to be passive we need the i/o pair to be fundamental, which is a stronger condition.

Appendix A. Some Basics of Kre˘ın Spaces In this appendix we collect some standard terminology and results from the theory of Kre˘ın spaces. More background can be found e.g. in [5] and [11]. Definition A.1. The vector space (W; [·, ·]W ), where [·, ·]W is an indefinite sesquilinear product, is an anti-Hilbert space if −W := (W; − [·, ·]W ) is a Hilbert space. In this case we denote the Hilbert space −W by |W|. The space (W; [·, ·]W ) is a Kre˘ın space if it admits a direct-sum decomposition W = W+ W− , such that: (i) the spaces W+ and W− are [·, ·]W -orthogonal, i.e., [w+ , w− ]W = 0 for all w+ ∈ W+ and w− ∈ W− , and (ii) the space W+ is a Hilbert space and W− is an anti-Hilbert space. In this case we call the decomposition W = W+ W− a fundamental decomposition of W, and we always denote it by W = (W+ , W− ), so that the second space in the pair is the anti-Hilbert space. Let U and Y be subspaces of the Kre˘ın space W. By writing U[⊥]Y we mean that U and Y are orthogonal to each other with respect to [·, ·]W . The orthogonal companion of U is the space U [⊥] := {w ∈ W | ∀u ∈ U :

[u, w]W = 0}.

(A.1)

We now prove that the node space K in Definition 1.4 is a Kre˘ın space. Proposition A.2. Let α ∈ C+ and let W be a Kre˘ın space with fundamental decomposition W = (W+ , W− ). Then the node space K in Definition 1.4 is a Kre˘ın space with fundamental decomposition K = (K+ , K− ), where ⎤ ⎡ ⎡ ⎤ −α α X X ⎦ and K− = ⎣ 1 ⎦. K+ = ⎣ 1 (A.2) W+

W−

The projections of K onto K± along K∓ are given by ⎤ ⎡ −α 1 −1 α 0 K ⎦ and PK+− := ⎣ 2Re α 1 W− 0 PW+ ⎤ ⎡ α 1 1 α 0 K+ ⎦. PK− := ⎣ 2Re α 1 W+ 0 PW− K

For every V ⊂ K, the condition PK+− V = K+ is equivalent to −1 α 0 X V = . W W+ 0 0 PW+−

(A.3)

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Proof. The subspace K+ is a Hilbert space, because X and W+ are both Hilbert spaces by assumption and ⎡⎡ ⎤ ⎡ ⎤⎤ −α −α x ⎣⎣ 1 ⎦ , ⎣ 1 x ⎦⎦ = (w+ , w+ )W+ + (αx, x)X + (x, αx)X w+ w+ K = w+ 2W+ + (2Re α)x2X . Replacing −α by α and w+ by w− , we get that K− is an anti-Hilbert space. In particular K+ and K− are both closed. Moreover, K+ [⊥]K− , because for all x, z ∈ X and w± ∈ W± we have: ⎤ ⎡ ⎤⎤ ⎡⎡ α −α ⎣⎣ 1 x ⎦ , ⎣ 1 z ⎦⎦ = (αx, z)X − (x, αz)X + [w+ , w− ]W = 0. w+ w− K K

K

Checking that PK+− + PK−+ = 1K is trivial and it is also straightforward K

K

K

to verify that PK±∓ are projections onto K± , i.e., that (PK±∓ )2 = PK±∓ and " ! K Ran PK±∓ = K± . This implies that K = K+ + K− , because any k ∈ K can be K

written k = k+ +k− , where k± = PK±∓ k ∈ K± . The sum K = K+ +K− is direct, z because x ∈ K+ ∩K− implies that z = αx = −αx and w ∈ W+ ∩W− = {0}, w

and then, in particular, (2Re α)x = 0. As Re α > 0, we get z = αx = 0. The last claim is proved simply by noting that ⎤ ⎡ −α 1 0⎥ −1 α 0 ⎢ 2Re α 1 K− PK+ = ⎣ ⎦ W 0 PW++ 0 1 X and that the ﬁrst factor maps W+ one-to-one onto K+ .

Let W = (W+ , W− ) be a fundamental decomposition of the Kre˘ın space 1 1 2 2 + w− , w+ + w− ∈ W, W. Then it follows from Definition A.1 that all w+ 1 2 where w± , w± ∈ W± , satisfy 1 1 2 2 1 2 1 2 [w+ + w− , w+ + w− ]W = [w+ , w+ ]W+ + [w− , w− ]W− 1 2 1 2 = (w+ , w+ )W+ − (w− , w− )|W− | .

(A.4)

Therefore we can turn W into a Hilbert space by changing the sign on the restriction of [·, ·]W to W− , as described in the following definition. Definition A.3. We call the Hilbert-space inner products on W that arise from fundamental decompositions W = (W+ , W− ) through 1 1 2 2 1 2 1 2 (w+ + w− , w+ + w− )W = (w+ , w+ )W+ + (w− , w− )|W− |

(A.5)

admissible inner products. The inner product (A.5) can be compactly written (w1 , w2 )W = [w1 , Jw2 ]W , W

W

where the so-called fundamental symmetry J is given by J = PW+− − PW−+ . A norm induced by an admissible inner product is called an admissible norm.

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Only Hilbert and anti-Hilbert spaces have unique fundamental decompositions, but all admissible norms are equivalent. Every admissible inner product turns a closed subspace of a Kre˘ın space into a Hilbert space, and thus in particular, every closed subspace of a Kre˘ın space is a reﬂexive Banach space. In contrast to Hilbert spaces, not every closed subspace U of a Kre˘ın space W is itself a Kre˘ın Space. Indeed, in the state/signal theory we often encounter Lagrangian subspaces, which are closed non-Kre˘ın subspaces of Kre˘ın Spaces. More precisely, a closed subspace U is a Kre˘ın space if and only if it is ortho-complemented: U U [⊥] = W; see [11, Theorem V.3.4]. The orthogonal companion (A.1) of any subspace U of W is a closed subspace of W with respect to the admissible norms. Denoting the closure of a subspace U ⊂ W with respect to any admissible norm by U, we have that (U [⊥] )[⊥] = U. The following definition makes use of the continuous dual U of a Banach space U. Recall that this continuous dual is the space of all continuous linear functionals on U. Definition A.4. Let W = U Y be a direct-sum decomposition of a Kre˘ın space. According to [7, Sect. 2.3], we can identify the continuous duals of U and Y with Y [⊥] and U [⊥] , respectively, using the following restrictions of [·, ·]W as duality pairings: u, u U ,U = [u, u ]W ,

y, y Y,Y = [y, y ]W ,

u ∈ U, y ∈ Y,

u ∈ Y [⊥]

y ∈U

[⊥]

and

.

Let T map a dense subspace of U linearly into Y. By T † we denote the (possibly unbounded) adjoint of T computed with respect to these duality pairings, so that T † : Y → U is the maximally deﬁned operator that satisﬁes ∀u ∈ Dom (T ), y ∈ Dom T † : T u, y Y,Y = u, T † y U ,U . (A.6) Here Dom T † is the subspace consisting of those y ∈ Y , for which there exists some u ∈ U , such that T u, y Y,Y = u, u U ,U for all u ∈ Dom (T ). The condition (A.6) can also be written ∀u ∈ Dom (T ), y ∈ Dom T † : [T u, y ]W = [u, T † y ]W ,

(A.7)

but note that T is not densely deﬁned on W in general, and therefore (A.7) does not uniquely determine T † as an operator on W. However, if U = Y = W and this is a Hilbert space with inner product (·, ·)W , then the construction in Definition A.4 leads to an identiﬁcation W = W, using the standard Hilbert-space duality pairing w, w W,W = (w, w )W . In this case we denote the adjoint T † of T by T ∗ in order to emphasise that the adjoint is computed with respect to a Hilbert-space inner product. Definition A.5. A subspace V ⊂ W is nonnegative (nonpositive) if [v, v]W ≥ 0 ([v, v]W ≤ 0) for all v ∈ V . In both of these cases V is said to be semidefinite and V is maximally semidefinite if V has no proper extension to a semidefinite subspace of W.

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A vector v ∈ W is neutral if [v, v]W = 0. The space V is neutral if all v ∈ V are neutral and V is Lagrangian if V = V [⊥] . Remark A.6. The closure of a semidefinite subspace is semidefinite and, therefore, every maximally semidefinite subspace is closed. Obviously a subspace is neutral if and only if it is both nonnegative and nonpositive. Lemma A.7. Let W be a Kre˘ın Space, let V0 , V1 ⊂ W, and define V := V0 +V1 . Then the following claims are true: (i) The space V0 is neutral, i.e., [v, v]W = 0 for all v ∈ V , if and only if [⊥] V0 ⊂ V0 , i.e., [v, v ]W = 0 for all v, v ∈ V0 . (ii) Let V0 be neutral. Then V is nonnegative or nonpositive if and only if V1 is nonnegative or nonpositive, respectively, and V0 [⊥]V1 . (iii) If V0 is neutral and V0 [⊥]V1 then V0 ⊂ V [⊥] . Proof. We prove claim (ii) ﬁrst. Assume therefore that [v, v]W = 0 for all v ∈ V0 . Then we for all v0 + v1 ∈ V , vi ∈ Vi , have [v1 + v0 , v1 + v0 ]W = [v1 , v1 ] + 2Re [v1 , v0 ] + [v0 , v0 ] = [v1 , v1 ] + 2Re [v1 , v0 ]. (A.8) Thus, V1 ≥ 0 and V0 [⊥]V1 immediately imply that V ≥ 0. Now conversely assume that V ≥ 0 and that [v, v]W = 0 for all v ∈ V0 . Then trivially V1 ⊂ V is also nonnegative. Moreover, if there exists vi ∈ Vi such that [v1 , v0 ]W =: α = 0, then for all s ∈ R− we by (A.8) have v1 +sαv0 ∈ V and: [v1 + sαv0 , v1 + sαv0 ]W = [v1 , v1 ] + 2s|α|2 ∈ R. This expression tends to −∞ as s → −∞, which contradicts the assumption that V ≥ 0 and therefore necessarily V0 [⊥]V1 . We now give the proof of claim (i). If [v, v ]W = 0 for all v, v ∈ V0 then trivially [v, v]W = 0 for all v ∈ V0 . Conversely, if [v, v]W = 0 for all v ∈ V0 , then V0 = V0 + V0 is neutral and by item (ii) we have V0 [⊥]V0 , which [⊥] is equivalent to V0 ⊂ V0 . [⊥] Regarding claim (iii), note that if V0 [⊥]V1 and V0 ⊂ V0 , then by claim (i) we for all v0 + v1 ∈ V and v0 ∈ V0 have that: [v0 + v1 , v0 ] = [v0 , v0 ] + [v1 , v0 ] = 0. Thus v0 ∈ V [⊥] , i.e., V0 ⊂ V [⊥] .

The following characterisation of semidefinite subspaces of a Kre˘ın space is useful. For proof, see Theorems 11.7, 4.2 and 4.4, and Lemma 4.5 of [11]. Proposition A.8. Let K = (K+ , K− ) be a fundamental decomposition. Let K V ⊂ K and define V± := PK±∓ V . The space V is nonnegative if and only if there exists a Hilbert-space contraction A+ : K+ → |K− |, such that V = (1 + A+ )V+ . The space V is maximally nonnegative if and only if V+ = K+ .

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The subspace V is nonpositive if and only if there exists a contraction A− : |K− | → K+ , such that V = (1 + A− )V− . The space V is maximally nonpositive if and only if V− = K− . The subspace V is neutral if and only if it is nonnegative with an isometric A+ , which in turn is true if and only if V is nonpositive with an isometric A− . The subspace V is Lagrangian if and only if it is both maximally nonnegative and maximally nonpositive, in which case A+ and A− are both unitary. Let V be closed and nonnegative. Then V [⊥] is nonpositive if and only if V is maximally nonnegative. We can say even more, namely that V = (1 + A+ )K+ =⇒ V [⊥] = (1 + A∗+ )K− , where A∗+ is computed with respect to the inner product on |K− |, i.e., for all w− ∈ K− and w+ ∈ K+ : (A+ w+ , w− )|K− | = − [A+ w+ , w− ]K = w+ , A∗+ w− K . +

It is elementary to characterise Lagrangian subspaces V of ﬁnite-dimensional Kre˘ın spaces. Indeed, the following corollary to Proposition A.8 shows that it sufﬁces to check that V is neutral and has sufﬁciently large dimension. Corollary A.9. Assume that K is a Kre˘ın space with finite dimension n. If V ⊂ V [⊥] ⊂ K and dim V ≥ n/2 then V = V [⊥] and dim V = n/2. Proof. Assume that V ⊂ V [⊥] and let K = (K+ , K− ) be a fundamental decomposition. Then V = (1 + A+ )V+ for some isometric A+ : K+ → |K− |. Moreover, V+ ⊂ K+ and therefore necessarily n/2 ≤ dim V ≤ dim V+ ≤ dim K+ . Dually, V = (1 + A− )V− for some isometric A− : |K− | → K+ and V− ⊂ K− with n/2 ≤ dim V ≤ dim V− ≤ dim K− . From K = K+ K− we now get that dim K+ +dim K− ≤ n, which implies that dim K± = n/2. Then V± ⊂ K± with dim V± = dim K± = n/2, i.e., V± = K± . Thus dim V ≤ n/2 and Proposition A.8 yields that V = V [⊥] . We now end this paper by listing a few function spaces, which are frequently used throughout the article. Definition A.10. Let U be a closed subspace of a Kre˘ın space and let I = [a, b], where b > a, or I = [a, ∞). (i)

The space of continuous U-valued functions which are deﬁned on I is denoted by C(I; U). The space C([a, b]; U) is a Banach space with the supremum norm, whereas C([a, ∞); U) is a Fr´echet space with the following family of seminorms: f n :=

sup t∈[a,a+n]

f (t)U ,

n ∈ Z+ .

The space of U-valued functions with n ∈ Z+ continuous derivatives on I is denoted by C n (I; U).

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By L2 (I; U) we denote the space of all U-valued Lebesgue-measurable functions f deﬁned on I, such that ⎛ ⎞1/2 f L2 (I;U ) := ⎝ f (t)2U dt⎠ < ∞, I

(iii)

where · U denotes some arbitrary given admissible norm on U. The space L2loc ([a, ∞); U) consists of all functions f that lie locally in L2 : ρ[a,b] f ∈ L2 ([a, b]; W) for all b > a. This is a Fr´echet space when equipped with the following family of seminorms: f n := ρ[a,a+n] f L2 ([a,a+n];U ) ,

(iv)

n ∈ Z+ .

The space of functions f ∈ L2 (I; U) that have a distribution derivative g in L2 (I; U) is denoted by H 1 (I; U). By this we mean that f ∈ L2 (I; U) lies in H 1 (I; U) if and only if there exists a g ∈ L2 (I; U) such that t ∀t ≥ a :

g(s) ds.

f (t) =

(A.9)

a

If (A.9) holds for f, g ∈

L2loc (I; U),

1 then we write f ∈ Hloc (I; U).

If W is a Kre˘ın space and I is a subinterval R, then L2 (I; W) is a 8 of 1 2 1 Kre˘ın space with the inner product [w , w ] := I [w (t), w2 (t)]W dt, because every fundamental decomposition W = (W+ , W− ) induces the fundamental decomposition L2 (R+ ; W) = L2 (R+ ; W+ ), L2 (R+ ; W− ) . The space L2loc (I; U) is the same as L2 (I; U) for all finite intervals I ⊂ R.

References [1] Arov, D.Z.: A survey on passive networks and scattering systems which are ¨ lossless or have minimal losses. Archiv f¨ ur Elektronik Und Ubertragungstechnik 49, 252–265 (1995) [2] Arov, D.Z.: Passive linear systems and scattering theory. In: Dynamical Systems, Control Coding, Computer Vision. Progress in Systems and Control Theory, vol. 25, pp. 27–44, Birkh¨ auser, Basel (1999) [3] Arov, D.Z., Nudelman, M.A.: Passive linear stationary dynamical scattering systems with continuous time. Integr. Equ. Oper. Theory 24, 1–45 (1996) [4] Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory. Part I: discrete time systems. In: The State Space Method, Generalizations and Applications. Operator Theory: Advances and Applications, vol. 161, pp. 115–177. Birkh¨ auser, Bensel (2005) [5] Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory, passive discrete time systems. Int. J. Robust Nonlinear Control 17, 497–548 (2007)

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[6] Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory. Part III: Transmission and impedance representations of discrete time systems. In: Operator Theory, Structured Matrices, and Dilations. Tiberiu Constantinescu Memorial Volume, pp. 104–140, Bucharest, Romania, Theta Foundation (2007) [7] Arov, D.Z., Staffans, O.J.: State/signal linear time-invariant systems theory, part IV: aﬃne representations of discrete time systems. Comp. Anal. Oper. Theory 1, 457–521 (2007) [8] Arov, D.Z., Staffans, O.J.: Two canonical passive state/signal shift realizations of passive discrete time behaviors. J. Funct. Anal. 257, 2573–2634 (2009) [9] Ball, J.A., Staffans, O.J.: Conservative state-space realizations of dissipative system behaviors. Integr. Equ. Oper. Theory 54(2), 151–213 (2006) [10] Behrndt, J., Hassi, S., de Snoo, H.: Boundary relations, unitary colligations, and functional models. Comp. Anal. Oper. Theory 3(1), 57–98 (2009) [11] Bogn´ ar, J.: Indefinite inner product spaces. Springer, New York, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78 [12] Derkach, V., Hassi, S., Malamud, M., de Snoo, H.: Boundary relations and their Weyl families. Trans. Am. Math. Soc. 358(12), 5351–5400 (electronic) [13] Gorbachuk, V.I., Gorbachuk, M.L.: Boundary value problems for operator differential equations. Mathematics and its Applications (Soviet Series), vol. 48. Kluwer, Dordrecht, 1991. Translated and revised from the 1984 Russian original [14] Kurula, M., Staffans, O.J.: Well-posed state/signal systems in continuous time. 2009. Complex Analysis and Operator Theory (to appear), SpringerLink Online First version available with doi:10.1007/s11785-009-0021-5 [15] Kurula, M., Zwart, H., van der Schaft, A., Behrndt, J.: Dirac structures and their composition on Hilbert spaces (2009, submitted). http://users.abo.ﬁ/ mkurula/ [16] Malinen, J.: Conservativity of time-ﬂow invertible and boundary control systems. Technical Report A479, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 2004 [17] Malinen, J., Staffans, O.J.: Conservative boundary control systems. J. Differ. Equ. 231(1), 290–312 (2006) [18] Malinen, J., Staffans, O.J.: Impedance passive and conservative boundary control systems. Compl. Anal. Oper. Theory 1(2), 279–300 (2007) [19] Malinen, J., Staﬀans, O.J.,Weiss, G.: When is a linear system conservative? Q. Appl. Math. 64(1), 61–91 (2006) [20] Maschke, B., van der Schaft, A.: Compositional modelling of distributedparameter systems. In: Advanced Topics in Control Systems Theory. Lecture Notes in Control and Inform. Sci., vol. 311, pp. 115–154. Springer, London (2005) [21] Maschke, B.M., van der Schaft, A.: Hamiltonian representation of distributed parameter systems with boundary energy ﬂow. In: Nonlinear Control in the year 2000, vol. 2 (Paris). Lecture Notes in Control and Inform. Sci., vol. 259, pp. 137–142. Springer, London, 2001 [22] Pazy, A.: Semi-Groups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) [23] Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory: A Behavioral Approach. Springer, New York (1998)

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[24] Staffans, O.J.: Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Math. Control Signals Syst. 15, 291–315 (2002) [25] Staffans, O.J.: Passive and conservative inﬁnite-dimensional impedance and scattering systems (from a personal point of view). In: Mathematical Systems Theory in Biology, Communication, Computation, and Finance. IMA Volumes in Mathematics and its Applications, vol. 134, pp. 375–414, New York. Springer, Heidelberg (2002) [26] Staffans, O.J.: Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005) [27] Staffans, O.J.: Passive linear discrete time-invariant systems. In: International Congress of Mathematicians, vol. III, pp. 1367–1388. Eur. Math. Soc., Z¨ urich (2006) [28] Staffans, O.J., Weiss, G.: Transfer functions of regular linear systems. Part III: Inversions and duality. Integr. Equ. Oper. Theory 49, 517–558 (2004) [29] Tucsnak, M., Weiss, G.: How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability. SIAM J. Control Optim. 42, 907–935 (2003) [30] van der Schaft, A.J.: L2 -Gain and Passivity Techniques in Nonlinear Control, Springer Communications and Control Engineering series, vol. 218. Springer, London, 2nd revised and enlarged edition (2000) [31] Weiss, G., Staffans, O.J., Tucsnak, M.: Well-posed linear systems—a survey with emphasis on conservative systems. Int. J. Appl. Math. Comput. Sci. 11, 7–34 (2001) [32] Willems, J.C.: Dissipative dynamical systems Part I: General theory. Arch. Ration. Mech. Anal. 45, 321–351 (1972) [33] Willems, J.C.: Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Arch. Ration. Mech. Anal. 45, 352–393 (1972) Mikael Kurula Department of Mathematics ˚ Abo Akademi University F¨ anriksgatan 3B 20500 ˚ Abo, Finland e-mail: [email protected] Received: Ocotber 5, 2009. Revised: January 27, 2010.

Integr. Equ. Oper. Theory 67 (2010), 425–438 DOI 10.1007/s00020-010-1788-5 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc Trieu Le Abstract. Let ϑ be a measure on the polydisc Dn which is the product of n regular Borel probability measures so that ϑ([r, 1)n × Tn ) > 0 for all 0 < r < 1. The Bergman space A2ϑ consists of all holomorphic functions that are square integrable with respect to ϑ. In one dimension, it is well known that if f is continuous on the closed disc D, then the Hankel operator Hf is compact on A2ϑ . In this paper we show that for n n ≥ 2 and f a continuous function on D , Hf is compact on A2ϑ if and only if there is a decomposition f = h + g, where h belongs to A2ϑ and limz→∂Dn g(z) = 0. Mathematics Subject Classification (2010). Primary 47B35. Keywords. Bergman space, Hankel operator, compactness.

1. Introduction Fix a positive integer n ≥ 1. Let Dn be the open unit polydisc in Cn and let Tn be the n-torus, which is the Shilov boundary of Dn . The closure of Dn is n D , the product of n copies of the closed unit disc. For z = (z1 , . . . , zn ) ∈ Cn and ζ = (ζ1 , . . . , ζn ) ∈ Tn , we use z ·ζ and ζ ·z to denote the point (z1 ζ1 , . . . , zn ζn ). We write z = (z 1 , . . . , z n ), and for any m = (m1 , . . . , mn ) in Zn , we write z m = z1m1 · · · znmn whenever it is deﬁned. We use σ to denote the surface measure on Tn which is normalized so that σ(Tn ) = 1. Let μ be a regular Borel probability measure on [0, 1)n . Then there is a regular Borel probability measure ϑ on Dn so that ⎫ ⎧ ⎨ ⎬ f (z)dϑ(z) = f (r · ζ)dσ(ζ) dμ(r) (1.1) ⎭ ⎩ Dn

[0,1)n

Tn

for all continuous functions f with compact support on Dn . It then follows that the above identity also holds true for all functions f in L1 (Dn , ϑ). In this paper we are interested in those measures μ which satisfy the condition μ([r, 1)n ) > 0 for 0 < r < 1. This implies that ϑ({z ∈ Dn : |z1 | ≥

426

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r, . . . , |zn | ≥ r}) > 0 for 0 < r < 1. We write L2ϑ for L2 (Dn , ϑ) and · 2 for the norm on L2ϑ . The Bergman space A2ϑ is the closure in L2ϑ of the space of all holomorphic polynomials. The condition on μ will imply that all functions in A2ϑ are holomorphic on the polydisc. Let P denote the orthogonal projection from L2ϑ onto A2ϑ . For any function f in L2ϑ , the (big) Hankel operator Hf is a densely deﬁned operator from A2ϑ into L2ϑ A2ϑ by Hf (ϕ) = (1 − P )(f ϕ) for all holomorphic polynomials ϕ. The function f will be called a symbol of the operator Hf . It is clear that if f is bounded, then Hf is a bounded operator with Hf ≤ f ∞ . However, there are unbounded functions f such that Hf extends to a bounded operator on A2ϑ . In fact, if f belongs to A2ϑ , then since f ϕ belongs to A2ϑ for all holomorphic polynomials ϕ, we conclude that Hf = 0. Conversely, if Hf = 0, then since 0 = Hf (1) = (I − P )(f ), we see that f must belong to A2ϑ . Therefore, Hf = 0 if and only if f is in A2ϑ . This shows that a Hankel operator has many symbols and any two symbols of the same operator differ by a function in A2ϑ . It is well known that if a function g ∈ L2ϑ vanishes outside a compact subset of Dn , then the Hankel operator Hg is compact. Let ∂Dn be the topological boundary of Dn as a subset of Cn . If g ∈ L2ϑ such that limz→∂Dn g(z) = 0 (that is, for any > 0, there is a compact subset M of Dn so that |g(z)| < for ϑ-a.e. z in Dn \M ), then an approximation argument shows that Hg is also a compact operator. This together with the above fact about zero Hankel operators implies that if f = h + g, where h belongs to A2ϑ and limz→∂Dn g(z) = 0, then Hf is compact. In the one-dimensional case (n = 1), it is well known that if f is continuous on D, then Hf is compact. See [11, p. 226] for the case of weighted Bergman spaces. For generalized Bergman spaces, one can prove this by checking directly that Hz¯i zj is compact for all integers i, j ≥ 0. See Sect. 3 for more details. The case n ≥ 2 turns out to be completely different. Not all Hankel operators with continuous symbols are compact. More surprisingly, we will show, under the assumption that μ is the product of n measures on [0, 1), n that if f is continuous on D , then Hf is compact if and only if f admits a decomposition f = h + g, where h belongs to A2ϑ and limz→∂Dn g(z) = 0. If dμ(r1 , . . . , rn ) = 2n r1 · · · rn dr1 · · · drn , then A2ϑ is the usual Bergman space of the polydisc. Stroethoﬀ [8,9] and Zheng [10] gave necessary and sufﬁcient conditions on a bounded function f for which Hf is compact. However, their conditions, which involve the projection P and Mobius transformations, are difﬁcult to check. Indeed, even if a function f is assumed to be continn uous on D , it is not clear from their results what geometric conditions f needs to satisfy in order for Hf to be compact. Our approach (though works only for continuous functions) is different from theirs and our result is more transparent. To conclude the section, we would like to mention some results on the compactness of Hankel operators on the Hardy space H 2 = H 2 (Tn ). In the one-dimensional case, it is a classical theorem of Hartman (see [11, Chapter 10]) that Hf can be extended to a compact operator if and only if f = h + g, where h belongs to H 2 and g is continuous on the circle T. On the other

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hand, the case n ≥ 2 is much different. It was showed by Cotlar and Sadosky in [4] and Ahern et al. [1] with a different approach that if Hf is compact, then f must belong to H 2 . This means that there is no non-zero compact Hankel operator on H 2 . This result was also proved in the setting of Hardy– Sobolev spaces on the polydisc by Ahern et al., in the same paper with the same approach. Our analysis in the present paper was actually motivated by their results and techniques.

2. Preliminaries In this section we explain in more details some of the results that we mentioned in the Introduction. From Cauchy’s formula and the assumption that μ([r, 1)n ) > 0 for all 0 < r < 1, for any compact subset M of Dn , there is a positive constant CM so that |p(z)| ≤ CM p2 for all z ∈ M , and all holomorphic polynomials p. This implies that for f ∈ A2ϑ , f is holomorphic on Dn and we also have |f (z)| ≤ CM f 2 for all z ∈ M . In fact, it can be showed that A2ϑ is the space of all functions in L2ϑ that are holomorphic on Dn . Since |f (z)| ≤ CM f 2 , the valuation map z → f (z) is a continuous functional on A2ϑ for any z ∈ Dn . So there is a function Kz in A2ϑ such that f (z) = f, Kz for any f ∈ A2ϑ . The function Kz is called the reproducing kernel at z. For any compact subset M and for any z ∈ M , since 2 . Kz (z) ≤ CM Kz 2 = CM (Kz (z))1/2 , we have Kz (z) ≤ CM m n From (1.1), the monomials {z : m ∈ Z+ } are pairwise orthogonal. On the other hand, the linear span of these monomials is dense in A2ϑ . Therefore A2ϑ has the following orthonormal basis, usually referred to as the standard m orthonormal basis, {em (z) = √zcm : m ∈ Zn+ }, where cm =

z m z¯m dϑ(z) =

Dn

r12m1 · · · rn2mn dμ(r1 , . . . , rn ).

[0,1)n

Suppose f is a function in L2ϑ . Then Hf em 22 ≤ f em 22 m∈Zn +

m∈Zn +

=

|f (z)|2

=

⎩

|em (z)|2

m∈Zn +

Dn

⎧ ⎨

⎫ ⎬ ⎭

dϑ(z)

|f (z)|2 Kz (z)dϑ(z),

Dn

where the last equality follows from the well known formula Kz (z) = Kz 22 = | Kz , em |2 = |em (z)|2 . m∈Zn +

m∈Zn +

(2.1)

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If f vanishes outside a compact subset M of Dn , then (2.1) gives 2 Hf em 22 ≤ CM |f (z)|2 dϑ(z) < ∞, m∈Zn +

M

2 CM

since Kz (z) ≤ < ∞ for all z ∈ M . Thus, Hf is a Hilbert–Schmidt operator, hence it is compact. Suppose f belongs to L2ϑ so that limz→∂Dn f (z) = 0. Then for any > 0, there is a compact subset M ⊂ Dn so that |f (z)| < for ϑ-a.e. z ∈ Dn \M . This implies that f − f χM ∞ ≤ . And hence, Hf − Hf χM ≤ . As we have seen above, Hf χM is a compact operator for each . Therefore, Hf , being the limit of a net of compact operators, is also a compact operator. Thus we have showed the following well known result. Proposition 2.1. Suppose f = h + g, where h ∈ A2ϑ and g ∈ L2ϑ so that limz→∂Dn g(z) = 0. Then Hf is a compact operator on A2ϑ . In the rest of the section, we study a decomposition of L2ϑ into pairwise orthogonal subspaces. If a function belongs to one of these subspaces, the corresponding Hankel operator has a simple form which we can analyze easily. This is one of the key points in our study of the compactness of Hankel operators with continuous symbols. For any n-tuple l ∈ Zn , let Hl be the space of all functions f in L2ϑ such that for all ζ ∈ Tn , f (ζ · z) = ζ l f (z) for ϑ-a.e. z ∈ Dn . Following [7], we call each function in Hl quasi-homogeneous of multi-degree l. It is clear that Hl is a closed subspace of L2ϑ . Let Ql denote the orthogonal projection from L2ϑ onto Hl . The following lemma shows that these projections are pairwise orthogonal and they constitute a partition of the identity. Lemma 2.2. For s ∈ Zn and f ∈ L2ϑ , we have (Qs f )(z) = f (z · ζ)ζ¯s dσ(ζ),

(2.2)

Tn n

. Furthermore, H

for ϑ-a.e. z ∈ D

l ⊥Hs (which implies Ql Qs = 0) whenever l = s, and L2ϑ = s∈Zn Qs (L2ϑ ) = s∈Zn Hs . Proof. Since f belongs to L2ϑ , the integral on the right hand side of (2.2) is well-deﬁned for ϑ-a.e. z ∈ Dn . For such z, let fs (z) be the value of the integral. For other values of z, let fs (z) = 0. We will show Qs f = fs by proving that fs ∈ Hs and (f − fs )⊥Hs . For z and any γ ∈ Tn , if the integral in (2.2) is deﬁned, by the rotation invariance of σ, we have s ¯ f ((z · γ) · ζ)ζ dσ(ζ) = f (z · (γ · ζ))ζ¯s dσ(ζ) fs (z · γ) = Tn

=

Tn

f (z · ζ)γ s dσ(ζ) = γ s fs (z).

Tn

If the integral in (2.2) is not deﬁned, then fs (z · γ) = γ s fs (z) because they are both zero. Therefore, fs belongs to Hs .

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Now suppose g is a function in Hs . Then fs (z)¯ g (z)dϑ(z) = f (z · ζ)ζ¯s g¯(z)dσ(ζ)dϑ(z) Dn

D n Tn

=

f (z · ζ)ζ¯s g¯(z)dϑ(z)dσ(ζ)

Tn D n

f (z · ζ)¯ g (z · ζ)dϑ(z)dσ(ζ)

= Tn D n

(since g(z · ζ) = ζ s g(z) for ϑ-a.e. z) f (z)¯ g (z)dϑ(z)dσ(ζ) = Tn D n

f (z)¯ g (z)dϑ(z).

= Dn

This shows that f − fs , g = 0 for all g ∈ Hs . Since fs belongs to Hs , we conclude that fs = Qs f . Next, suppose l = k. Let f ∈ Hl and g ∈ Hk . For any ζ ∈ Tn , we have ζ l−k f (z)¯ g (z)dϑ(z) = f (z · ζ)¯ g (z · ζ)dϑ(z) Dn

Dn

f (z)¯ g (z)dϑ(z).

= Dn

Since l = k, we conclude that Dn f (z)¯ g (z)dϑ(z) = 0. Thus, Hl ⊥Hk .

2 To show Lϑ = l∈Zn Hl , it sufﬁces to show that for any f ∈ L2ϑ , the identity f 22 = l∈Zn Ql (f )22 holds true. Indeed, for f ∈ L2ϑ , f 22 = |f (z)|2 dϑ(z) Dn

=

|f (z · ζ)|2 dσ(ζ)dϑ(z)

D n Tn

=

2

f (z · ζ)ζ¯l dσ(ζ) dϑ(z)

n Dn l∈Z Tn

(since {ζ l : l ∈ Zn } is an orthonormal basis for L2 (Tn , σ))

2

=

f (z · ζ)ζ¯l dσ(ζ) dϑ(z) l∈ZnDn

=

l∈Zn

Tn

Ql (f )22 .

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It follows from the proof of Lemma 2.2 that for each s ∈ Zn , there is a function fs such that fs (z · γ) = γ s fs (z) for all z ∈ Dn and all ζ ∈ Tn and n Qs (f )(z) = fs (z) for ϑ-a.e. z. If f is continuous on the closed polydisc D , n then the integral in (2.2) is well-deﬁned for all z in D and fs is also con n tinuous on D . We have seen that the series s∈Zn fs converges to f in the n L2ϑ -norm. In general, for f in C(D ), the series does not converge uniformly to f . However, the Ces`aro means of the functions {fs : s ∈ Zn } do converge uniformly to f as we will see next. For any integer N ≥ 1, the Ces`aro mean ΛN (f ) is deﬁned by the formula ΛN (f )(z) = |s1 |,...,|sn |≤N

⎧ ⎨ = Tn

⎩

|s1 | |sn | 1− ··· 1 − fs1 ,...,sn (z) N +1 N +1

|s1 |,...,|sn |≤N

⎫ ⎬ |s1 | |sn | 1− ··· 1 − ζ1s1 · · · ζns1 f (z · ζ)dσ(ζ) ⎭ N +1 N +1

FN (ζ1 ) · · · FN (ζn )f (z · ζ)dσ(ζ),

= Tn

where FN is the N th Fej´er’s kernel. It now follows from a well known result in harmonic analysis (see, for example, Sections 2.2 and 9.2 in [5]) that ΛN (f ) → n n f uniformly on D as N → ∞ if f is continuous on D .

3. Hankel Operators with Quasi-Homogeneous Symbols Recall from Sect. 2 that A2ϑ has the standard orthonormal basis consisting of m monomials {em (z) = √zcm : m ∈ Zn+ }, where r12m1 · · · rn2mn dμ(r1 , . . . , rn ). cm = [0,1)n

We also recall that for l ∈ Zn , Ql is the orthogonal projection from L2ϑ onto the subspace Hl of quasi-homogeneous functions of multi-degree l. For two n-tuples of integers l = (l1 , . . . , ln ) and s = (s1 , . . . , sn ), we write l s if lj ≥ sj for all 1 ≤ j ≤ n and l s if otherwise. We will also use 0 to denote (0, . . . , 0). For m ∈ Zn+ and l ∈ Zn , Ql (em ) is either 0 (when l = m) or em (when l = m). Thus, Ql (A2ϑ ) = {0} if l 0 and Ql (A2ϑ ) = Cel if l 0. This shows that A2ϑ is an invariant subspace for Ql , hence it is also a reducing subspace since Ql is a projection. Let P be the orthogonal projection from L2 onto A2ϑ . Then we have P Ql = Ql P and this in turn shows that Hl is a reducing subspace for P . Lemma 3.1. Let s be in Zn . Suppose f is a bounded function on Dn so that we have f (r1 ζ1 , . . . , rn ζn ) = ζ s f (r1 , . . . , rn ) for all z = (r1 ζ1 , . . . , rn ζn ) in Dn .

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Then Hf∗ Hf is a diagonal operator with respect to the standard orthonormal basis. The eigenvalues of Hf∗ Hf are given by 1 1 n λm = |f (t1 , . . . , tn )|2 t2m · · · t2m dμ(t1 , . . . , tn ) n 1 cm [0,1)n

if m + s 0 and 1 1 n λm = |f (t1 , . . . , tn )|2 t2m · · · t2m dμ(t1 , . . . , tn ) n 1 cm [0,1)n

−

1 cm cm+s

2

1 +s1 n +sn f (t1 , . . . , tn )t2m · · · t2m dμ(t1 , . . . , tn )

n 1

[0,1)n

if m + s 0. Proof. For any m ∈ Zn+ , f em belongs to Hs+m , which is an invariant subspace for P . Therefore, P (f em ) and Hf em = f em − P (f em ) also belong to Hs+m . We have 0 if s + m 0, P (f em ), ek ek = P (f em ) = f em , es+m es+m if s + m 0. k∈Zn +

Now for k = m, Hf∗ Hf em , ek = Hf em , Hf ek = 0 since Hf em ∈ Hm+s , Hf ek ∈ Hk+s and Hm+s ⊥Hk+s by Lemma 2.2. Thus, Hf∗ Hf is a diagonal operator with respect to the standard orthonormal basis {em : m ∈ Zn+ }. The eigenvalues of Hf∗ Hf are given by λm = Hf∗ Hf em , em = Hf em 22 = f em 22 − P (f em )22 f em 22 = f em 22 − | f em , es+m |2 for m ∈ Zn+ . Since f em 22 =

1 cm

if s + m 0, if s + m 0,

1 n |f (t1 , . . . , tn )|2 t2m · · · t2m dμ(t1 , . . . , tn ), n 1

[0,1)n

and 1 f em , es+m = √ cm cs+m

1 +s1 n +sn f (t1 , . . . , tn )t2m · · · t2m dμ(t1 , . . . , tn ), n 1

[0,1)n

the conclusion of the lemma follows.

Remark 3.2. Let us consider the case n = 1 and f (z) = z u z¯v for integers u, v ≥ 0. We see that f belongs to Hs with s = u − v. From Lemma 3.1,

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Hf∗ Hf is a diagonal operator with eigenvalues λm for m ∈ Z+ . For all positive integers m ≥ v − u, we have

2

t2m+2(u+v) dμ(t) t2m+2u dμ(t) [0,1) [0,1) λm = − . t2m dμ(t) ( [0,1) t2m dμ(t))( [0,1) t2m+2(u−v) dμ(t)) [0,1) Since μ([r, 1)) > 0 for all 0 < r < 1, it can be showed that limm→∞ λm = 0 (see Lemma 3.3 below). Therefore, Hf∗ Hf is a compact operator, which implies that Hf is also a compact operator. Thus, for any polynomial n p = p(z, z¯), Hp is compact. Since any function g in C(D ) can be uniformly approximated by polynomials, we conclude that Hg is also a compact operator. Our characterization of compactness of Hankel operators when n ≥ 2 depends on the following lemma. For a sketch of its proof when δ = β = (0, . . . , 0), see Lemma 2.4 in [6]. The proof for arbitrary δ, β is similar. Lemma 3.3. Let μ1 , . . . , μN be positive measures on [0, 1) so that μj ([r, 1)) > 0 for all 0 < r < 1, all 1 ≤ j ≤ N . Suppose ϕ is a function on [0, 1)N so that lim(r1 ,...,rN )→(1,...,1) ϕ(r1 , . . . , rN ) = α. Then for any N -tuples of real numbers δ = (δ1 , . . . , δN ) and β = (β1 , . . . , βN ), we have mN +δN ϕ(r)r1m1 +δ1 · · · rN dμ(r) [0,1)N lim = α. m +β m +β (m1 ,...,mN )→(∞,...,∞) r 1 1 · · · rN N N dμ(r) [0,1)N 1 In the rest of the paper, we will consider only measures μ of the form dμ(r1 , . . . , rn ) = dμ1 (r1 ) · · · dμn (rn ), where μ1 , . . . , μn are regular Borel probability measures on the interval [0, 1) such that μj ([r, 1)) > 0 for all 0 < r < 1 and 1 ≤ j ≤ n. Recall that ϑ is the measure on Dn that is related to μ by Eq. (1.1). We now deﬁne a measure γ on the topological boundary ∂Dn associated with ϑ. It is clear that ∂Dn is the disjoint union of 2n − 1 parts of the form A1 × · · · × An , where Aj is either T or D and not all are D. Suppose W = A1 × · · · × An is a part of ∂Dn . Let 1 ≤ j1 < · · · < jp ≤ n be integers / {j1 , . . . , jp }. We deﬁne γ to so that Aj1 = · · · = Ajp = D and Aj = T if j ∈ be the unique regular Borel measure on W that satisﬁes f (w)dγ(w) W

⎧ ⎨ = [0,1)p

⎩

Tn

⎫ ⎬ f (ζ1 , . . . , rj1 ζj1 . . . , rjp ζjp , . . . , ζn )dσ(ζ) dμj1 (rj1 ) · · · dμjp (rjp ) ⎭

for all f ∈ Cc (W ). The following theorem characterizes compact Hankel operators with continuous quasi-homogeneous symbols when n ≥ 2. Theorem 3.4. Suppose n ≥ 2 and s ∈ Zn . Suppose f ∈ Hs is continuous on n D such that Hf is compact. If s 0, then f (z) = f (1, . . . , 1)z s for γ-a.e. z in ∂Dn . If s 0, then f (z) = 0 for γ-a.e. z in ∂Dn .

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Proof. By the remark after the proof of Lemma 2.2, we may assume that n f (ζ1 z1 , . . . , ζn zn ) = ζ s f (z1 , . . . , zn ) for all ζ ∈ Tn and all z ∈ D . In particun lar, for all z = (r1 ζ1 , . . . , rn ζn ) in D , we have f (r1 ζ1 , . . . , rn ζn ) = ζ s f (r1 , . . . , rn ).

(3.1)

Suppose 1 ≤ j < n. Put m1 = max{0, −s1 }, . . . , mj = max{0, −sj }. Since Hf is compact, Hf∗ Hf is also compact. By Lemma 3.1, Hf∗ Hf is diagonalizable and its eigenvalues are λm ’s for m = (m1 , . . . , mj , mj+1 , . . . , mn ). Therefore, lim(mj+1 ,...,mn )→(∞,...,∞) λm = 0. Using the formula for λm (when 0) in Lemma 3.1 together with Lemma 3.3 and the fact that cm = m + s 2m1 n t · · · t2m dμ1 (t1 ) · · · dμn (tn ), we conclude that n [0,1)n 1 2m 1 |f (t1 , . . . , tj , 1, . . . , 1)|2 t2m · · · tj j dμ1 (t1 ) · · · dμj (tj ) 1 [0,1)j

2

2mj +sj 1 +s1 · · · t dμ (t ) · · · dμ (t )

[0,1)j f (t1 , . . . , tj , 1, . . . , 1)t2m 1 1 j j 1 j = . 2m +2s t2m1 +2s1 · · · tj j j dμ1 (t1 ) · · · dμj (tj ) [0,1)j 1 m

m +s

j 1 1 +s1 and G(t) = tm · · · tj j j for Let F (t) = f (t1 , . . . , tj , 1, . . . , 1)tm 1 · · · tj 1 j t = (t1 , . . . , tj ) ∈ [0, 1) . Then the above identity shows that ⎫⎧ ⎫ ⎧ ⎪ ⎪ ⎪ ⎬⎪ ⎨ ⎬ ⎨ 2 2 |F (t)| dμ1 (t1 ) · · · dμj (tj ) G (t)dμ1 (t1 ) · · · dμj (tj ) ⎪ ⎪ ⎪ ⎭⎪ ⎩ n ⎭ ⎩ j

[0,1)

[0,1)

2

= (F G)(t)dμ1 (t1 ) · · · dμj (tj ) .

[0,1)j

This means that Holder’s inequality applied to F and G is in fact an equality. Therefore, we have F (t) = αG(t), or equivalently, mj

1 f (t1 , . . . , tj , 1, . . . , 1)tm 1 · · · tj

mj +sj

1 +s1 = αtm · · · tj 1

(3.2)

for μ1 × · · · × μj -a.e. t = (t1 , . . . , tj ) ∈ [0, 1)j , where α is a constant. Since 1 belongs to the support of all the measures μ1 , . . . , μj , we may let t1 = · · · = tj = 1 to obtain α = f (1, . . . , 1). Recall that in (3.2), m1 = max{0, −s1 }, . . . , mj = max{0, −sj }. Suppose ﬁrst s1 , . . . , sn ≥ 0. Since m1 = · · · = mj = 0, we obtain s from (3.2) that f (t1 , . . . , tj , 1, . . . , 1) = f (1, . . . , 1)ts11 · · · tj j for μ1 × · · · × μj j a.e. t = (t1 , . . . , tj ) ∈ [0, 1) . This together with (3.1) implies that f (z) = f (1, . . . , 1)z s for γ-a.e. z in W = Dj × Tn−j , which is a part of the boundary ∂Dn . Now suppose sp < 0 for some 1 ≤ p ≤ n. We will show that in this case f (1, . . . , 1) = 0. Without loss of generality, we may assume p = 1. For all large positive integers m2 , . . . , mn , let m = (0, m2 , . . . , mn ) (the assumption

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that n ≥ 2 is needed here). Since m + s 0, 2 n |f (t1 , . . . , tn )|2 t2m · · · t2m dμ1 (t1 ) · · · dμn (tn ) n 2 [0,1)n . λm = 2m2 2mn t · · · tn dμ1 (t1 ), · · · , dμn (tn ) [0,1)n 2 Letting (m2 , . . . , mn ) → (∞, . . . , ∞) and using Lemma 3.3 together with the fact that λm → 0, we conclude that |f (t1 , 1, . . . , 1)|2 dμ1 (t1 ). 0= [0,1)

This implies f (t1 , 1, . . . , 1) = 0 for μ1 -a.e. t1 on [0, 1). Since 1 is in the support of μ1 and f is continuous at the point (1, . . . , 1), it follows that f (1, . . . , 1) = 0. mj 1 = 0 for μ1 × · · · × μj Now (3.2) gives f (t1 , . . . , tj , 1, . . . , 1)tm 1 . . . tj mj 1 · · · t = 0, then we have a.e. t = (t1 , . . . , tj ) in [0, 1)j . For such t, if tm 1 j f (t1 , . . . , tj , 1 . . . , 1) = 0. Otherwise, there exists 1 ≤ i ≤ j so that ti = 0 and mi > 0. But mj = max{0, −si }, so si < 0. Since ti = 0 = ti ζi for any |ζi | = 1, we have f (t1 , . . . , ti , . . . , tj , 1, . . . , 1) = f (t1 , . . . , ti ζi , . . . , tj , 1, . . . , 1) = ζisi f (t1 , . . . , ti , . . . , tj , 1, . . . , 1). This implies f (t1 , . . . , tj , 1, . . . , 1) = 0 because ζi can be chosen so that ζisi = 1. Therefore, f (t1 , . . . , tj , 1, . . . , 1) = 0 for μ1 × · · · × μj -a.e. t = (t1 , . . . , tj ) in [0, 1)j , which implies f (z) = 0 for γ-a.e. z ∈ W = Dj × Tn−j . The same argument applies to other parts of ∂Dn which are different from Tn . On Tn , (3.1) gives f (ζ) = ζ s f (1, . . . , 1). If sp < 0 for some 1 ≤ p ≤ n then since f (1, . . . , 1) = 0 as showed above, we conclude that f (ζ) = 0 for ζ ∈ Tn . So the conclusions of the proposition also hold for z in Tn ⊂ ∂Dn . The proof of the proposition is now completed.

4. Compact Hankel Operators with More General Symbols

We have seen that any f in L2ϑ admits the decomposition f = l∈Zn fl , where fl = Ql (f ) is the orthogonal projection of f on the space of quasihomogeneous functions of multi-degree l. The next proposition shows that the compactness of Hf implies the compactness of each Hfl . We are then able to apply the results in the previous section. The Hardy space version of the proposition was proved in [1]. Our proof here, which also works for the Hardy space, is more direct. Proposition 4.1. Suppose f ∈ L2ϑ so that the operator Hf is compact on A2ϑ . Then for each s ∈ Zn , the operator Hfs is compact. Proof. For any m ∈ Zn+ , we have

Hf em = (I − P )(f em ) = (I − P ) =

l∈Zn

(I − P )(fl em ) =

l∈Zn

l∈Zn

Hfl em .

fl

em

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Since Qs+m (Hfl em ) = 0 if l = s and Qs+m (Hfs em ) = Hfs em , we obtain Qs+m (Hf em ) = Hfs em . From Lemma 3.1, Hf∗s Hfs is a diagonal operator with eigenvalues λm given by λm = Hfs em 22 = Qs+m (Hf em )22 ≤ Hf em 22 . Since Hf is compact, we have lim|m|→∞ Hf em 2 = 0 (here |m| = m1 + · · · + mn ). This implies lim|m|→∞ λm = 0 and hence, Hf∗s Hfs is compact. Thus, Hfs is a compact operator. Suppose g belongs to A2ϑ . It was showed by Axler [2] that when n = 1 and ϑ is the Lebesgue measure on the disk D, Hg is compact if and only if g is in the little Bloch space, that is, lim|z|↑1 (1−|z|2 )g (z) = 0. For n ≥ 2 and ϑ is the Lebesgue measure on the polydisc Dn , a special case of [3, Theorem D] gives that Hg is compact if and only if g is a constant function and in this case, Hg = 0. The following corollary to Proposition 4.1 shows that this holds true for general measures ϑ. Corollary 4.2. Suppose n ≥ 2. Let g be a function in A2ϑ so that Hg is compact. Then g is a constant function and hence Hg¯ = 0. Proof. Write g = m∈Zn cm em . For m ∈ Zn+ , since Q−m (g) = c¯m e¯m and + Hg is compact, Proposition 4.1 implies that Hc¯m e¯m is compact. Theorem 3.4 then shows that c¯m = 0 for all m = 0. Thus, g is a constant function. We are now ready for our main theorem in this paper. n

Theorem 4.3. Suppose n ≥ 2. Let f be continuous on D so that Hf is a n compact operator. Then there is a function h which is continuous on D and n holomorphic on D , and a bounded function g satisfying limz→∂Dn g(z) = 0 so that f (z) = h(z) + g(z) for ϑ-a.e. z in Dn . Proof. For any s ∈ Zn , Proposition 4.1 shows that Hfs is compact. Since f is continuous, each fs is also continuous. By Theorem 3.4, there is a holomorphic monomial hs so that (fs − hs )(w) = 0 for γ-a.e. w ∈ ∂Dn . (In fact, hs = 0 if s 0 and hs (w) = fs (1, . . . , 1)ws if s 0). For each integer N ≥ 1, deﬁne |s1 | |sn | n pN (z) = 1− ··· 1 − hs1 ,...,sn (z) for z ∈ D . N +1 N +1 |s1 |,...,|sn |≤N

Then pN is a holomorphic polynomial and pN (w) = ΛN (f )(w) for γ-a.e. w ∈ ∂Dn , where ΛN (f ) is the N th Ces`aro mean of f . Since γ restricted on n Tn ⊂ ∂D is the surface measure and pN − ΛN (f ) is continuous, we actually have pN (w) = ΛN (f )(w) for all w ∈ Tn . By the remark at the end of Sect. 2, n ΛN (f ) converges to f uniformly on D . In particular, pN |Tn = ΛN (f )|Tn converges to f |Tn uniformly. This implies that there is a function h which is n continuous on D and holomorphic on Dn so that pN converges uniformly to h n on D . Since pN (w) = ΛN (f )(w) for γ-a.e. w ∈ ∂Dn for each N , we conclude that h(w) = f (w) for γ-a.e. w ∈ ∂Dn . Let g˜ = f − h. Then g˜ is continuous n on D and g˜(w) = 0 for γ-a.e. w on ∂Dn . By Lemma 2.1 in [6], there is a

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function g such that g˜(z) = g(z) for ϑ-a.e. z in Dn and limz→w g(z) = 0 for all w ∈ ∂Dn . From the compactness of ∂Dn , it follows that limz→∂Dn g(z) = 0. We then have f (z) = h(z) + g˜(z) = h(z) + g(z) for ϑ-a.e. z ∈ Dn and h, g satisfy the requirements in the conclusion of the theorem. n

The continuity of f on D in Theorem 4.3 cannot be dropped. Indeed, there are bounded functions f which are continuous on the open polydisc Dn such that Hf is compact and no decomposition f = h + g with h ∈ A2ϑ and limz→∂Dn g(z) = 0 is possible. In the rest of the section, we will give a construction of such a function. Let 0 < r1 < r2 < · · · be an increasing sequence of positive real numn bers that converges to 1. Set j1 = 1. Since (rj1 , 1)n = ∪∞ j=j1 +1 (rj1 , rj ) and n n μ((rj1 , 1) ) > 0, there is an integer j2 ≥ j1 + 1 so that μ((rj1 , rj2 ) ) > 0. n n Since (rj2 , 1)n = ∪∞ j=j2 +1 (rj2 , rj ) and μ((rj2 +1 , 1) ) > 0, there is an integer n j3 ≥ j2 + 1 so that μ((rj2 , rj3 ) ) > 0. Continuing this process, we ﬁnd a n sequence of integers {jk }∞ k=1 such that jk+1 ≥ jk + 1 and μ((rjk , rjk+1 ) ) > 0 n for k = 1, 2, . . . . For each such k, let Rk = (rjk , rjk+1 ) . Choose an open subset Vk of Tn so that 0 < σ(Vk ) < 1/k and Ek Kz (z)dϑ(z) < 1/k 2 , where Ek = {r · ζ : r ∈ Rk , ζ ∈ Vk }. The existence of Vk follows from the fact that the function z → Kz (z) is bounded on compact sets. Since Ek is open in Dn and ϑ(Ek ) > 0, using the regularity of ϑ, we can choose a continuous function 0 ≤ fk ≤ 1 so that fk is supported in Ek and ϑ({z : f (z) = 1}) > 0. ∞ Put f = k=1 fk . Since the sets Ek ’s are pairwise disjoint, the function f is continuous on Dn and 0 ≤ f (z) ≤ 1 for all z ∈ Dn . We now show that Hf is a Hilbert–Schmidt operator, hence it is compact. Indeed, from (2.1), Hf em 22 ≤ |f (z)|2 Kz (z)dϑ(z) m∈Zn +

≤

Dn ∞ k=1E k

∞ 1 |f (z)| Kz (z)dϑ(z) < < ∞. k2 2

k=1

For each s ∈ Zn , we will show that limz→∂Dn Qs (f )(z) = 0. Since f ≥ 0, it follows from formula (2.2) in Lemma 2.2 that |Qs (f )(z)| ≤ |Q0 (f )(z)|. So it sufﬁces to prove limz→∂Dn Q0 (f )(z) = 0. For z = (z1 , . . . , zn ) ∈ Dn , we have Q0 (f )(z) = f (z · ζ)dσ(ζ) = f (|z1 |ζ1 , . . . , |zn |ζn )dσ(ζ) ≤

Tn ∞

Tn

χEk (|z1 |ζ1 , . . . , |zn |ζn )dσ(ζ).

k=1Tn

By the definition of Ek , χEk (|z1 |ζ1 , . . . , |zn |ζn ) = 1 if and only if the n-tuple (|z1 |, . . . , |zn |) belongs to Rk = (rjk , rjk+1 )n and (ζ1 , . . . , ζn ) belongs to Vk . Let k0 ≥ 2 be a positive integer. Suppose z = (z1 , . . . , zn ) so that |zi | > rjk0 for some 1 ≤ i ≤ n. If (|z1 |, . . . , |zn |) does not belong to any Rk , k ≥ 1, then

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Q0 (f )(z) = 0. Otherwise, there is exactly one k so that (|z1 |, . . . , |zn |) ∈ Rk , which is (rjk , rjk+1 )n . Since |zi | > rjk0 , we conclude that rjk+1 > rk0 , which implies k ≥ k0 . Therefore, 1 1 |Q0 (f )(z)| ≤ χEk (|z1 |ζ1 , . . . , |zn |ζn )dσ(ζ) = σ(Vk ) ≤ ≤ . k k0 Tn

Since this holds true for any z that does not belong to the compact set [0, rjk0 ]n × Tn , we conclude that limz→∂Dn Q0 (f )(z) = 0. Suppose there were a decomposition f = h + g, where h ∈ A2ϑ and g(z) → 0 as z → ∂Dn . We will show that there would be a contraction. For s 0, from formula (2.2) in Lemma 2.2 we see that Qs (g)(z) → 0 as z → ∂Dn . This implies Qs (h)(z) = Qs (f )(z) − Qs (g)(z) → 0 as z → ∂Dn . But Qs (h) is a multiple of z s , as explained at the beginning of Sect. 3, so we have Qs (h) = 0 for all s 0. It follows that h = 0 and hence, f = g. This is a contradiction because g(z) → 0 as z → ∂Dn but by the construction of f , for any compact subset M ⊂ Dn , the set {z ∈ Dn \M : f (z) = 1} has positive ϑ-measure.

References [1] Ahern, P., Youssﬁ, E.H., Zhu, K.: Compactness of Hankel operators on HardySobolev spaces of the polydisk. J. Oper. Theory 61(2), 301–312 (2009) [2] Axler, S.: The Bergman space, the Bloch space, and commutators of multiplication operators. Duke Math. J. 53(2), 315–332 (1986). MR 850538 (87m:47064) [3] B´ekoll´e, D., Berger, C.A., Coburn, L.A., Zhu, K.H.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal. 93(2), 310–350 (1990). MR 1073289 (91j:32034) [4] Cotlar, M., Sadosky, C.: Abstract, weighted, and multidimensional AdamjanArov-Kre˘ın theorems, and the singular numbers of Sarason commutants. Integr. Equ. Oper. Theory 17(2), 169–201 (1993). MR 1233667 (94k:47040) [5] Katznelson, Y.: An introduction to harmonic analysis, 3rd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004). MR 2039503 (2005d:43001) [6] Le, T.: On Toeplitz operators on Bergman spaces of the unit polydisk. Proc. Am. Math. Soc. 138, 275–285 (2010) [7] Louhichi, I.: Elizabeth Strouse, Lova Zakariasy: Products of Toeplitz operators on the Bergman space. Integr. Equ. Oper. Theory 54(4), 525–539 (2006). MR 2222982 (2007a:47033) [8] Stroethoﬀ, K.: Compact Hankel operators on the Bergman space. Ill. J. Math. 34(1), 159–174 (1990). MR 1031892 (91a:47030) [9] Stroethoﬀ, K.: Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in Cn . J. Oper. Theory 23(1), 153–170 (1990). MR 1054822 (91i:47040) [10] Zheng, D.C.: Toeplitz operators and Hankel operators. Integr. Equ. Oper. Theory 12(2), 280–299 (1989). MR 986598 (90b:47044)

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[11] Zhu, K.: Operator theory in function spaces, 2nd edn. In: Mathematical Surveys and Monographs, vol. 138. American Mathematical Society, Providence (2007). MR 2311536 Trieu Le Department of Mathematics University of Toledo Mail Stop 942 Toledo, OH 43606 USA e-mail: [email protected] Received: October 11, 2009. Revised: February 15, 2010.

Integr. Equ. Oper. Theory 67 (2010), 439–448 DOI 10.1007/s00020-010-1791-x Published online March 17, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On the Salas Theorem and Hypercyclicity of f (T ) Vladimir M¨ uller Abstract. We study hypercyclicity properties of functions of Banach space operators. Generalizations of the results of Herzog–Schmoeger and Bermudez–Miller are obtained. As a corollary we also show that each non-trivial operator commuting with a generalized backward shift is supercyclic. This gives a positive answer to a conjecture of Godefroy and Shapiro. Furthermore, we show that the norm-closures of the set of all hypercyclic (mixing, chaotic, frequently hypercyclic, respectively) operators on a Hilbert space coincide. This implies that the set of all hypercyclic operators that do not satisfy the hypercyclicity criterion is rather small—of first category (in the norm-closure of hypercyclic operators). Mathematics Subject Classification (2000). Primary 47A16. Keywords. Hypercyclic operators, hypercyclicity criterion, Salas’ theorem, generalized backward shift.

1. Introduction In [19], Salas proved that for any backward weighted shift T on p (1 ≤ p < ∞) or c0 , the operator I + T is hypercyclic. Surprisingly, the proof was direct and did not use the hypercyclicity criterion (or any of its variants), which is the usual tool for showing the hypercyclicity. However, in [15] it was shown that I + T does satisfy the hypercyclicity criterion. In [9] it was shown that I + T is even mixing for all backward weighted shifts. Moreover, the Salas theorem is true not only for weighted backward shifts but for a much larger class of operators, see e.g. [1,2,4]. In Sect. 2 we study the hypercyclicity properties of functions f (T ) of an operator T with large generalized kernel. We generalize the results of [14] and [5]. As a corollary we also show that each non-trivial operator commuting with a generalized backward shift is supercyclic. This gives a positive answer to a conjecture of Godefroy and Shapiro [11], Remark 4.10 (c). ˇ and by IRP AV OZ The research was supported by grant No. 201/09/0473 of GA CR 10190503.

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The results become stronger if we consider operators with closed ranges. In Sect. 3 we show that if T is Kato with dense generalized kernel and |f (0)| = 1 then f (T ) is not only mixing, but also chaotic and frequently hypercyclic. In Sect. 4 we consider approximation in the algebra of all operators on a separable Hilbert space. Recall that the norm-closure of hypercyclic operators was characterized by Herrero [13]. Analyzing the proof of [13] we show that norm-closures of hypercyclic, mixing, chaotic and frequently hypercyclic operators coincide. It was shown recently by Bayart and Matheron [3] (modifying an earlier result of Read and De La Rosa [7]) that there are hypercyclic Hilbert space operators which do not satisfy the hypercyclicity criterion. We show that the set of such operators is relatively small, of first category in the norm-closure of hypercyclic operators. The author wishes to thank to the referee for useful comments, especially for improving Theorem 4.

2. Hypercyclicity of f (T ) Let X be a complex Banach space. Denote by B(X) the algebra of all bounded linear operators acting on X. For T ∈ B(X) denote by N (T ) and R(T ) ∞ ∞ k (T ) = the kerneland range of T , respectively. Write N k=1 N (T ) and ∞ ∞ k R (T ) = k=1 R(T ). We denote by the closed linear span. An operator T ∈ B(X) is called hypercyclic if there exists a vector x ∈ X which is hypercyclic for T , i.e., such that the set {T n x : n = 0, 1, . . . } is dense in X. It is well known [11] that T is hypercyclic if and only if for all non-empty open sets U, V ⊂ X there exists n ∈ N such that T n U ∩ V = ∅. An operator T ∈ B(X) is called supercyclic if there exists a vector x ∈ X supercyclic for T , i.e., such that the set {λT n x : λ ∈ C, n = 0, 1, . . . } is dense in X. A usual tool for showing hypercyclicity of an operator is the hypercyclicity criterion. There are many equivalent formulations of this condition; the simplest one is that T ∈ B(X) satisfies the hypercyclicity criterion if and only if T ⊕ T is hypercyclic. These operators are sometimes called weakly mixing. Furthermore, T ∈ B(X) is called mixing if for all non-empty open subsets U, V ⊂ X we have T n U ∩ V = ∅ for all n sufficiently large. The following implications are true: T mixing =⇒ T ⊕ T hypercyclic =⇒ T hypercyclic =⇒ T supercyclic; the reverse implications are not true. The following generalization of the Salas theorem follows from [4], Theorem 2.2 (the result is attributed there to Grivaux and Shkarin). Theorem 1. Let X be a separable Banach space, T ∈ B(X). Let (N ∞ (T − z) ∩ R∞ (T − z)) = X. z∈C,|z|=1

Then T is mixing.

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We shall need the following lemma. Lemma 2. Let X be a separable Banach space, T ∈ B(X), w ∈ σ(T ), let f be a function analytic on a neighbourhood of σ(T ) which is non-constant on a neighbourhood of w. Then N ∞ (T − w) ∩ R∞ (T − w) ⊂ N ∞ (f (T ) − f (w)) ∩ R∞ (f (T ) − f (w)) . ∞ Proof. Let f (z) = i=0 αi (z − w)i be the Taylor expansion of f in a neighbourhood of w. Since f is non-constant in a neighbourhood of w, there exists n ≥ 1 such that αn = 0 and αi = 0(1 ≤ i ≤ n − 1). Write f (z) = α0 + αn (z − w)n + (z − w)n+1 g(z) where g is a function analytic on a neighbourhood of σ(T ). Let M = N ∞ (T − w) ∩ R∞ (T − w) and let S = f (T ) − f (w) = αn (T − w)n + (T − w)n+1 g(T ). Clearly SN (T − w)k ⊂ N (T − w)k−1 for all k ∈ N. So S k N (T − w)k = {0} and M ⊂ N ∞ (S). We show that M ⊂ R∞ (S). To this end we must showthat M ⊂ R(S m ) for each m. Fix m ∈ N. By induction on k we show that N (T − w)k ∩ M ⊂ R(S m ). This is clear for k = 0. Let k ≥ 0 and suppose that N (T − w)k ∩ u ∈ X such M ⊂ R(S m ). Let x ∈ N (T − w)k+1 ∩ M . Then there exists that x = αnm (T − w)mn u. Clearly u ∈ N (T − w)mn+k+1 . We have S m = αnm (T − w)mn + h(T )(T − w)mn+1 for some function h analytic on a neighbourhood of σ(T ). So S m u = x + h(T )(T − w)mn+1 u = x + αn−m h(T )(T − w)x ∈ x + N ((T − w)k ) ∩ M . By the induction assumption, x ∈ R(S m ) and N ((T − w)k+1 ) ∩ M ⊂ R(S m ). Hence M ⊂ R(S m ). Since m was arbitrary, we have M ⊂ R∞ (S). Hence N ∞ (S) ∩ R∞ (S) ⊃ M . Theorem 1 and Lemma 2 imply the following generalization of [5]. Theorem 3. Let X be a separable Banach space, T ∈ B(X), let f be a function analytic on a neighbourhood of σ(T ) which is non-constant on each component of its domain of definition. Let {N ∞ (T − w) ∩ R∞ (T − w) : w ∈ σ(T ), |f (w)| = 1} = X. Then f (T ) is mixing. Proof. By Lemma 2, {N ∞ (f (T ) − z) ∩ R∞ (f (T ) − z) : z ∈ C, |z| = 1} ⊃ {R∞ (f (T ) − f (w)) ∩ N ∞ (f (T ) − f (w)) : w ∈ σ(T ), |f (w)| = 1} ⊃ {R∞ (T − w) ∩ N ∞ (T − w) : w ∈ σ(T ), |f (w)| = 1} = X. Hence f (T ) is mixing by Theorem 1.

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Corollary 4 (cf. [14]). Let X be a separable Banach space, T ∈ B(X), let N ∞ (T ) ∩ R∞ (T ) be dense in X. Let f be a function analytic on a neighbourhood of σ(T ) which is non-constant on a neighbourhood of 0. Suppose that |f (0)| = 1. Then: (i) f (T ) is supercyclic; (ii) if |f (0)| = 1 then f (T ) is mixing. Proof. Note that 0 ∈ σ(T ), and so f (0) is defined. Note also that σ(T ) is connected. Indeed, suppose that σ(T ) = K1 ∪ K2 where K1 and K2 are disjoint compact sets. We may assume that 0 ∈ K1 . Let X1 , X2 be the spectral subspaces corresponding to K1 , K2 . Then T |X2 is invertible, and so N ∞ (T ) ⊂ X1 . Hence X2 = {0} and K2 = ∅. (ii) follows from Theorem 3. −1 (i) If f (0) = 0 then f (0) f (T ) is mixing by (ii), and so f (T ) = f (0) · −1 f (0) f (T ) is supercyclic. So it is sufficient to consider the case f (0) = 0. By Lemma 2 for w = 0, we have N ∞ (f (T )) ∩ R∞ (f (T )) ⊃ N ∞ (T ) ∩ ∞ R (T ). So N ∞ (f (T )) and R(f (T )) are dense. Hence f (T ) is supercyclic by [[4], Exercise 1.3]. Conditions of Corollary 4 are clearly satisfied by any backward weighted shift on p (1 ≤ p < ∞) or c0 (we always assume that the shift is non-break degenerated, with all weights non-zero). Another example of an operator satisfying the conditions of Theorem 4 is any Hilbert space operator which has in some orthonormal basis the form T = (aij ) with aij = 0(i ≥ j) and ai,i+1 = 0 (the entries aij with j ≥ i + 2 are arbitrary). Recall that an operator T acting on an infinite-dimensional Banach space X is called a generalized backward shift if N ∞ (T ) is dense and ⊂X dim N (T ) = 1. By [11], Proposition 3.3 there exists a sequence (xk )∞ k=0 ∞ k−1 such that T x0 = 0, T xk = xk−1 (k ≥ 1), k=0 xk = X and N (T k ) = i=0 xi for all k ≥ 1. Clearly the conditions of Theorem 4 are satisfied by any generalized backward shift or, more generally, by any finite direct sum of generalized backward shifts. Moreover, for generalized backward shifts it is possible to formulate Theorem 4 not only for analytic functions f (T ) but also for operators in the commutant of T . The next result gives a positive answer to the conjecture [11], Remark 4.10 (c). Theorem 5. Let T ∈ B(X) be a generalized backward shift and let A ∈ B(X) be an operator commuting with T which is not a scalar multiple of the identity. Then A is supercyclic. Proof. Let x 0 , x1 , · · · ∈ X be the vectors satisfying T x0 = 0, T xk = xk−1 ∞ (k ≥ 1) and k=0 xk = X. Let A ∈ B(X), AT = T A. Since AN (T k ) ⊂ N (T k ) and N (T k ) = k−1 k = i=0 xi for all k, we can express Axk i=0 βk,i xi for some complex k β coefficients βk,i (0 ≤ i ≤ k). We have i=0 k,i xi = Axk = AT xk+1 = k+1 T Axk+1 = i=1 βk+1,i xi−1 . Since the vectors xi are linearly independent,

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we have βk,i = βk+1,i+1 for all 0 ≤ i ≤ k. Hence there exist complex numbers α0 , α1 , . . . such that Axk =

k

αi xk−i

(k = 0, 1, . . . ).

i=0

Since A is not a scalar multiple of the identity, there exists n ≥ 1 such that αn = 0 and αi = 0(1 ≤ i ≤ n − 1). Suppose first that α0 = 1. We show that A is even mixing. Set S = A−I, k k−1 so Sxk = i=1 αi xk−i ∈ i=0 xi for all k. Thus S k+1 xk = 0. Let M be the set of all finite linear combinations of the vectors x0 , x1 , . . .. Then M ⊂ N ∞ (S). Clearly SM ⊂ M . We show by induction on k that xk ∈ SM for all k. We have S(αn−1 )xn = x0 , so x0 ∈ SM . Let k ≥ 1 and suppose that x0 , . . . , xk−1 ∈ SM . We have S(αn−1 xn+k ) ∈ k−1 xk + i=0 xi , so by the induction assumption xk ∈ SM . Hence SM = M and M ⊂ N ∞ (S) ∩ R∞ (S). So A = S + I is mixing by Theorem 1. If α0 = 0 then the operator α0−1 A is mixing and so A is supercyclic. If α0 = 0 then Ax0 = 0 and N (A) ⊃ N (T ). By [11], Proposition 3.6, A is supercyclic. We finish this section by the following observation which is an analogy to the result of [16]. Denote by T the unit circle in the complex plane, T = {z ∈ C : |z| = 1}. Proposition 6. Let T ∈ B(X) be mixing, λ ∈ C, |λ| = 1. Then λT is mixing. Proof. Let U, V ⊂ X be non-empty open subsets of X. Choose y ∈ V, y = 0. Let ε > 0 satisfy B(y, ε) ⊂ V , where B(y, ε) denotes the open ball centered at y with radius ε. ε Let μ1 , . . . , μk ∈ T be a finite 2y - net in T. Since T is mixing, for each j = 1, . . . , k there exists nj ∈ N such that T n U ∩ B(μj y, ε/2) = ∅ for all n ≥ nj . ε Let n ≥ max{nj : j = 1, . . . , k}. Find j such that |λ−n − μj | ≤ 2y . n Find u ∈ U such that T u − μj y < ε/2. Then (λT )n u − y = T n u − λ−n y ≤ T n u − μj y + μj y − λ−n y < ε/2 +y · |μj − λ−n | ≤ ε. Hence (λT )n U ∩ V = ∅ and λT is mixing.

3. Operators with Closed Range The results from the previous section can be improved if we consider operators with closed ranges. Operators T ∈ B(X) with closed range and satisfying N ∞ (T ) ⊂ R(T ) are called Kato (sometimes also semiregular) and have been studied intensely

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by a number of authors. For T ∈ B(X) denote by σK (T ) = {z ∈ C : T − zis not Kato} the Kato spectrum of T . Recall that σK (T ) is always a non-empty compact subset of C, ∂σ(T ) ⊂ σK (T ) ⊂ σ(T ). The functions z → N ∞ (T − z) and z → R∞ (T − z) are constant on each connected subset of C\σK (T ). For a survey of results concerning Kato operators and the corresponding Kato spectrum see e.g. [17], pp. 117–130. We need the following Lemma, cf. [12], 7, Chap. 3. Lemma 7. Let X be a Banach space, let T ∈ B(X) be a Kato operator, let G be a connected open set such that 0 ∈ G and ∞T − z is Kato for each z ∈ G. Let αk ∈ G\{0}, αk → 0. Then N ∞ (T ) = k=1 N (T − αk ). ∞ Proof. By [17], p. 124, N ∞ (T ) = N ∞ (T − αk ) for all k, so N ∞ (T ) ⊃ k=1 N (T − αk ). Conversely, let n ∈ N and x ∈ N (T n ). Set x1 = T n−1 x, x2 = T n−2 x, . . . , xn−1 = T x, xn = x. Let γ(T ) = inf{T x : dist {x, N (T )} = 1} denote the reduced minimum modulus. Since R(T ) is closed, we have γ(T ) > 0. Since N ∞ (T ) ⊂ R(T ), we can find inductively vectors xn+1 , xn+2 , · · · ∈ So xj ≤ X such that T xj = xj−1 and xj ≤ 2γ(T )−1 xj−1 (j ≥ n + 1). ∞ j−1 (2γ(T )−1 )j−n x for all j ≥ n, and so the series f (z) := j=1 xj z is convergent in a certain neighbourhood U of 0. It is easy to check that (T − z)f (z) = 0 for all z ∈ U . Thus f (αk ) ∈ N (T − αk ) for all k large enough. We prove by induction on j that xj ∈ k N (T − αk ). We have x1 = f (0) = limk→0 f (αk ) ∈ k N (T − αk ). Suppose that j ≥ 1 and xi ∈ k N (T − αk ) for i = 1, . . . , j. Then xj+1 = lim

k→∞

f (αk ) − (x1 + x2 αk + · · · + xj αkj−1 ) αkj

by the induction assumption. In particular, x = xn ∈ N ∞ (T ) = k N (T − αk ).

∈

N (T − αk )

k

k

N (T − αk ). Hence

If we replace the condition of Theorem 3 by the condition that ∞ {N (T − w) : w ∈ / σK (T ), |f (w)| = 1} is dense, then the operator f (T ) is not only mixing but also chaotic and frequently hypercyclic. Recall that an operator T ∈ B(X) is called chaotic if it is hypercyclic and the set of its periodic vectors (i.e., the vectors x ∈ X satisfying T n x = x for some n ≥ 1) is dense. T ∈ B(X) is called frequently hypercyclic if there exists x ∈ X such that for each non-empty open subset U ⊂ X the set {n ∈ N : T n x ∈ U } has a positive lower density. Theorem 8. Let X be a separable Banach space, T ∈ B(X), let f be a function analytic on a neighbourhood of σ(T ) which is non-constant on each component of its domain of definition. Suppose that {N ∞ (T − w) : w ∈ σ(T )\σK (T ), |f (w)| = 1} = X.

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Then f (T ) is mixing and chaotic. If X is a Hilbert space then f (T ) is also frequently hypercyclic. Proof. f (T ) is mixing by Theorem 3. Let w ∈ σ(T )\σK (T ) and |f (w)| = 1. Since f is non-constant in a neighbourhood of w, there exists n ≥ 1 and a function g analytic on a neighbourhood of σ(T ) such that f (z) = f (w) + (z − w)n g(z) and g(w) = 0. Let U be an open disc centered at w such that U ∩ σK (T ) = ∅ and |g| is bounded below on U . Then V := f (U ) is an open neighbourhood of f (w) ∈ T. Find a sequence (αk ) ⊂ (V \{f (w)}) ∩ T such that αk → f (w) and each αk = e2πitk for some rational tk . Choose wk ∈ U such that f (wk ) = αk . This implies that wk → w. By Lemma 7, we have k N (T − wk ) = N ∞ (T − w). Moreover, if x ∈ N (T − wk ) then x ∈ N (f (T ) − f (wk )) = N (f (T ) − αk ), and so x is periodic for f (T ). Clearly the set of all periodic vectors for f (T ) is a linear manifold. Its closure contains N (T − wk ) for each k, and so also N ∞ (T − w) whenever w∈ / σK (T ), |f (w)| = 1. Hence the set of all vectors periodic for f (T ) is dense and f (T ) is chaotic. Let X be a Hilbert space. Since ∂σ(T ) ⊂ σK (T ) ⊂ σ(T ), the set σ(T )\σK (T ) is open. So A := f (σ(T )\σK (T )) ∩ T is open in T. Let μ be the normalized Lebesgue measure on A, i.e., μ(A) = 1. Let B ⊂ A be a measurable subset with μ(B) = 1. We show that z∈B N (f (T ) − z) = X. Let w ∈ σ(T )\σK (T ), |f (w)| = 1. Let U be an open disc centered at w chosen as above. Let A0 = f (U )∩T and B0 = B ∩A0 . Then μ(B0 ) = μ(A0 ) > 0, and so there exists a sequence (zk ) ⊂ B converging to a point z ∈ A0 such that zk = z (k ∈ N). Let w , wk ∈ U , f (w ) = z , f (wk ) = zk for all k ∈ N. As N ∞ (T − w). Moreabove, wk → w , and so k N (T − wk ) = N ∞ (T − w ) = over, for each k we have N (T − wk ) ⊂ N (f (T ) − zk ) ⊂ z∈B N (f (T ) − z). So z∈B N (f (T ) − z) ⊃ N ∞ (T − w). Hence N (f (T ) − z) ⊃ {N ∞ (T − w) : w ∈ σ(T )\σK (T ), |f (w)| = 1} = X, z∈B

and so f (T ) is frequently hypercyclic by [4], Corollary 6.24.

Corollary 9. Let T ∈ B(X) be a Kato operator such that N ∞ (T ) is dense, let f be a function analytic on a neighbourhood of σ(T ) which is nonconstant on each component of its domain of definition. Suppose that |f (0)| = 1. Then f (T ) is mixing and chaotic. If X is a Hilbert space then f (T ) is also frequently hypercyclic. Remark 10. By a recent result of Grivaux [10], Corollary 6.24 of [4] remains true also for Banach space operators. So the operator f (T ) in Theorem 8 and Corollary 9 is frequently hypercyclic even in the Banach space case. Remark 11. The conditions of Theorem 3 do not imply that f (T ) is chaotic or frequently hypercyclic. The simplest example (for f ≡ 1) is the operator

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I + B on a Hilbert space, where B is a quasinilpotent backward shift, see [20], Theorem 1.1.

4. Norm-closures of Hilbert Space Operators Let H be an infinite-dimensional separable complex Hilbert space. Denote by hc(H) the set of all hypercyclic operators on H. The norm-closure hc(H) was characterized in [13] as the set of all operators T ∈ B(H) satisfying: σ0 (T ) = ∅, where σ0 (T ) is the set of all λ ∈ C such that λ is an isolated point of the spectrum σ(T ) and the corresponding Riesz spectral space is finitedimensional; (b) ind (T − λ) ≥ 0 for all complex numbers λ such that T − λ is semiFredholm; (c) T ∪ σW (T ) is connected, where σW (T ) = {σ(T + K) : K ∈ B(H) is compact} is the Weyl spectrum. (a)

The proof that conditions (a–c) characterize the norm-closure hc(H) consists of two parts: A.

B.

The set of all operators satisfying (a–c) is norm-closed and every hypercyclic operator satisfies these conditions. This part is simple and is true for any Banach space. The second inclusion that every operator satisfying (a–c) can be approximated by hypercyclic operators is much more difficult and uses deep approximating techniques that are available only for Hilbert space operators. Using several approximations it is shown that for each T ∈ B(H) satisfying (a–c) and every ε > 0 there exists a hypercyclic operator S such that S − T < ε.

In fact, it is proved more in [13]. It is proved that for each operator T ∈ B(H) satisfying (a–c) and every ε > 0 there exists

n S ∈ B(H) satisfying S − T < ε and S is a finite direct sum S = j=1 Sj , where for each Sj ∈ B(Hj ) there exists αj ∈ T such that Sj − αj is a semi-Fredholm operator, and Sj − αj is upper-triangular (in a certain orthonormal basis) with zero main diagonal. Clearly by an arbitrarily small perturbation we can achieve that all the entries in the first diagonal over the main diagonal are nonzero. Thus we may assume that N ∞ (Sj − αj ) is dense in Hj and Sj − αj is surjective. By Corollary 9, Sj is mixing, chaotic and frequently hypercyclic. Moreover, S is mixing by Theorem 1. Furthermore, S has a dense set of periodic vectors, so S is chaotic. For j = 1, . . . , n choose a non-empty open connected subset Aj ⊂ T\σK (Sj ). Without loss of generality we may assume that the sets Aj are mutually disjoint. Let μ be the normalized Lebesgue measure on the set A : n A . Let B ⊂ A be a measurable subset satisfying μ(B) = μ(A) = 1. For j j=1 each j we have μ(B ∩ Aj ) = μ(Aj ) > 0. As above, z∈B∩Aj N (Sj − z) = Hj ,

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and so z∈B N (S − z) = H. Hence S is frequently hypercyclic by [4], Corollary 6.24. Hence for each T ∈ B(H) satisfying (a–c) and each ε > 0 there exists a mixing, chaotic and frequently hypercyclic operator S ∈ B(H) with S−T < ε. So we have proved the following theorem. Theorem 12. Let H be a separable infinite-dimensional Hilbert space. Then the set of all operators satisfying (a–c) is equal to the norm-closure of any of the following sets: (i) The set of all hypercyclic operators; (ii) The set of all operators satisfying the hypercyclicity criterion; (iii) The set of all mixing operators; (iv) The set of all chaotic operators; (v) The set of all frequently hypercyclic operators; (vi) The set of all mixing, chaotic and frequently hypercyclic operators. Corollary 13. The set of all operators satisfying the hypercyclicity criterion is residual in hc(H). Proof. Denote by L the set of all operators on H satisfying the hypercyclicity criterion, i.e., operators S such that S ⊕ S is hypercyclic. By Theorem 12, L is dense in hc(H). Moreover, L is a Gδ set. Indeed, let (Uj ) be a countable base of open subsets of H. It is easy to see that L= {T ∈ B(H) : T n Uj1 ∩ Uj3 = ∅, T n Uj2 ∩ Uj4 = ∅}, j1 ,j2 ,j3 ,j4 ∈N n∈N

which is obviously a Gδ set. Hence L is residual in hc(H).

Remark 14. By [6] and [20], there are Banach spaces without chaotic operators and frequently hypercyclic operators. On the other hand, each Banach space admits hypercyclic operators, [1], [2]. So Theorem 12 cannot be true for general Banach spaces. Remark 15. By [18], the set of all operators in B(H) satisfying (a–c) is also equal to the norm-closure of the set of all weakly hypercyclic operators.

References [1] Ansari, S.I.: Existence of hypercyclic operators on topological vector spaces. J. Funct. Anal. 148, 384–390 (1997) [2] Bernal-Gonz´ alez, L.: On hypercyclic operators on Banach spaces. Proc. Am. Math. Soc. 127, 1003–1010 (1999) [3] Bayart, F., Matheron, E.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250, 426–441 (2007) [4] Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics 179. Cambridge University Press, Cambridge (2009) [5] Bermudez, T., Miller, V.: On operators T such that f (T ) is hypercyclic. Integr. Equ. Oper. Theory 37, 332–340 (2000)

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[6] Bonet, J., Martinez-Gimenez, F., Peris, A.: A Banach space which admits no chaotic operators. Bull. Lond. Math. Soc. 33, 196–198 (2001) [7] De La Rosa, M., Read, C.: Hypercyclic operator whose direct sum is not hypercyclic. J. Oper. Theory 61, 369–380 (2009) [8] Grivaux, S.: Sums of hypercyclic operators. J. Oper. Theory 202, 486– 503 (2003) [9] Grivaux, S.: Hypercyclic operators, mixing operators, and the bounded step problem. J. Oper. Theory 54, 147–168 (2005) [10] Grivaux, S.: New class of frequently hypercyclic operators, preprint [11] Godefroy, G., Shapiro, J.H.: Operators with dense, invariant cyclic vector manifolds. J. Funct. Anal. 48, 229–269 (1991) [12] Herrero, D.A.: Approximation of Hilbert space operators, vol. I. 2nd edn. Pitman Research Notes in Math. Series, vol. 224, Longman Scientific and Technical, Harlow, Essex (1989) [13] Herrero, D.A.: Limits of hypercyclic and supercyclic operators. J. Funct. Anal. 99, 179–190 (1991) [14] Herzog, G., Schmoeger, C.: On operators T such that f (T ) is hypercyclic. Studia Math. 108, 209–216 (1996) [15] Leon-Saavedra, F., Montes-Rodriguez, A.: Linear subspaces of hypercyclic vectors. J. Funct. Anal. 148, 524–545 (1997) [16] Leon-Saavedra, F., M¨ uller, V.: Rotations of hypercyclic operators. Integr. Equ. Oper. Theory 50, 385–391 (2004) [17] M¨ uller, V.: Spectral Theory of Linear Operators, Operator Theory Advances and Applications, vol. 139, 2nd edn. Birkh¨ auser, Basel (2007) [18] Prajitura, G.: Limits of weakly hypercyclic and supercyclic operators. Glasgow Math. J. 47, 255–260 (2005) [19] Salas, H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347, 993– 1004 (1995) [20] Shkarin, S.: On the spectrum of frequently hypercyclic operators. Proc. Am. Math. Soc. 136, 123–134 (2009) Vladimir M¨ uller Mathematical Institute Czech Academy of Sciences Zitna 25 11567 Prague 1 Czech Republic e-mail: [email protected] Received: November 12, 2009. Revised: January 29, 2010.

Integr. Equ. Oper. Theory 67 (2010), 449 DOI 10.1007/s00020-010-1794-7 Published online June 3, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

ERRATUM

Erratum to: On Passive and Conservative State/Signal Systems in Continuous Time Mikael Kurula

Erratum to: Integr. Equ. Oper. Theory DOI 10.1007/s00020-010-1787-6 Due to a processing error, the title of this paper and Fig. 1 were incorrect. The corrected title and Fig. 1 is given below: Title: On Passive and Conservative State/Signal Systems in Continuous Time

Figure 1.

Mikael Kurula Department of Mathematics ˚ Abo Akademi University F¨ anriksgatan 3B 20500 ˚ Abo, Finland e-mail: [email protected] The online version of the original article can be found under doi:10.1007/s00020-010-1787-6.

Integr. Equ. Oper. Theory 67 (2010), 451–466 DOI 10.1007/s00020-010-1812-9 Published online July 14, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Topology and Smooth Structure for Pseudoframes Esteban Andruchow, Jorge Antezana and Gustavo Corach In memoriam Roberto Fontanarrosa Abstract. Given a closed subspace S of a Hilbert space H, we study the sets FS of pseudo-frames, CFS of commutative pseudo-frames and XS of dual frames for S, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({fn }n∈N , {hn }n∈N ), cn fn , F : 2 → H, F ({cn }n∈N ) = n

and H : 2 → H, H = ({cn }n∈N ) =

cn hn .

n

We prove that, with this identiﬁcation, the sets FS , CFS and XS are complemented submanifolds of B(2 , H)×B(2 , H). We examine in more detail XS , which carries a locally transitive action from the general linear group GL(2 ). For instance, we characterize the homotopy theory of XS and we prove that XS is a strong deformation retract both of FS and CFS ; therefore these sets share many of their topological properties. Mathematics Subject Classification (2010). Primary 42C15, 57N20. Keywords. Pseudoframes, dual frames.

1. Introduction Let H be a Hilbert space, B(2 , H) the Banach space of bounded (linear) operators acting from 2 to H, and S ⊂ H a closed subspace of H. In this paper we study the set FS = {(F, H) ∈ B(2 , H) × B(2 , H) : F H ∗ |S = idS }.

(1)

The first author partially supported by CONICET (PIP 5690); the second author partially supported by PICT 808 (ANPCYT), MTM-2008-05561-C02-02, 2009-SGR-1303; and the third author partially supported by UBACYT I030, CONICET (PIP 2083/00).

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If Q is an idempotent with R(Q) = S, then FS = {(F, H) ∈ B(2 , H) × B(2 , H) : F H ∗ Q = Q}. This set parametrizes the set of what S. Li and H. Ogawa call pseudoframes for the subspace S [21]. Let us recall their definition: A pair of sequences {fn }n∈N , {hn }n∈N of vectors in H is a pseudoframe for the subspace S if 1. {fn }n∈N and {hn }n∈N are Bessel sequences of H (see definition in Sect. 2). 2. For any f ∈ S, f=

∞

f, hn fn .

n=1

As these authors remark, pseudoframes for S are in one to one correspondence with pairs of bounded operators F, H, deﬁned by H ∗ : H → 2 , H ∗ x = ( x, hn )n , and F : 2 → H, F ({cn }n∈N ) =

cn fn

n

such that F H ∗ |S = idS . In fact, Li and Ogawa require less, namely that {fn }n∈N be a Bessel sequence with respect to S, which means that the condition n | f, fn |2 ≤ cf 2 holds only for f ∈ S. This weaker condition gives naturally representations using unbounded operators. We shall consider this setting elsewhere. Therefore we may call the present version bounded pseudoframes for S. We shall use this one-to-one correspondence between bounded pseudoframes for the subspace S and the set FS ⊂ B(2 , H) × B(2 , H) to endow the former with a topological structure, namely the one given by the operator norm in the space B(2 , H) × B(2 , H). This point of view is not new. There are many papers dealing with frames by examining their analysis and synthesis operators. Among them, let us mention [10,12,15,20,21]. What is gained by this abstract presentation of frames, is that different special sets of frames, as the ones considered below, are regarded as spaces in a common ambient Banach space. Questions about the local structure of these spaces can be examined, for instance, if any given pair of frames in one of these spaces can be joined by continuous or a smooth path inside the space. As it turns out, FS is a smooth complemented submanifold of the Banach space B(2 , H) × B(2 , H). We study its properties, as well as those of the following subsets: 1. The set of CFS ⊂ FS of commutative pseudoframes [21], which consists of pairs (F, H) in FS which also verify that P F H ∗ = P (where P is the orthogonal projection onto S). This set parametrizes the set of pairs of sequences ({fn }n∈N , {hn }n∈N ) such that f=

∞ n=1

f, hn fn =

∞ n=1

f, fn hn ,

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for all f ∈ S. This set is shown to be a complemented submanifold of FS . 2. The set XS ⊂ FS of dual frames for S [7,14,21]. Recall that dual frames for S are pairs of sequences ({fn }n∈N , {hn }n∈N ) which verify that Pf =

∞

f, hn fn ,

n=1

for all f ∈ H. As operators, they are parametrized by XS = {(F, H) ∈ FS : R(F ) = R(H) = S}. The set XS is also a complemented submanifold of FS , and a strong deformation retract both of FS and CFS , a fact which allows to examine many of the topological properties of FS and CFS in XS . This space is easier to handle than Fs because it carries a locally transitive action of the general linear group GL(2 ). We characterize the orbits (=connected components of) XS under this action, and analyze their homotopy groups. Apparently, XS CFS FS . The contents of the paper are the following. Section 2 contains preliminaries and notations. In Sect. 3 we prove the existence of a smooth structure for FS (Proposition 3.4). The same result for CFS is proven in Sect. 4, by means of a trivial ﬁbre bundle. In Sect. 5 we introduce XS and state its basic properties: the smooth structure for XS (Proposition 5.2) and the existence of strong deformation retractions from FS and CFS onto XS (Proposition 5.3). Section 6 examines the ﬁner topological features of XS by means of the action of GL(2 ): existence of continuous local cross sections for the action (Proposition 6.1), characterization of the connected components of XS (Proposition 6.3), description of the homotopy structure of this space (Corollary 6.8 and Theorem 6.9).

2. Preliminaries Given two separable Hilbert spaces H1 and H2 , the space of bounded linear operators from H1 to H2 is denoted by B(H1 , H2 ). For an operator A ∈ B(H1 , H2 ), R(A) denotes the range or image of A, N (A) the null space of A, A∗ the adjoint of A, A the usual operator norm of A and, if R(A) is closed, A† denotes the Moore–Penrose pseudoinverse of A. The space B(H2 , H2 ) is denoted B(H2 ), GL(H2 ) denotes the group of invertible operators on H2 , and B(H2 )+ denotes the cone of positive (semi-definite) operators of B(H2 ). If H = W ⊕ M⊥ then the projection Q onto W deﬁned by this decomposi∗ tion is denoted by PW||M⊥ . Observe that PW||M ⊥ = PM||W ⊥ . In the case of orthogonal projections, i.e. W = M, we write PW instead of PW||W ⊥ . Frames We introduce some basic facts about frames in Hilbert spaces. For complete descriptions of frame theory and applications, the reader is referred to the

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paper by Dufﬁn and Schaeffer [13], the review by Heil and Walnut [17] or the books by Young [27], Daubechies [12] and Christensen [6]. A sequence {fn }n∈N of elements of a Hilbert space H is called a Bessel sequence if there exists a positive number B such that | f, fn |2 ≤ Bf 2 ∀ f ∈ H. n∈N

If the inequality holds for all f in a certain subspace S, then {fn }n∈N is called a Bessel sequence for S. A frame for a closed subspace W of H is a sequence {fn }n∈N such that each fn belongs to W and there exist numbers A, B > 0 such that, for every f ∈ W, Af 2 ≤ | f, fn |2 ≤ Bf 2 . (2) n∈N

The optimal constants A, B in (2) are called the frame bounds for F. F is a Parseval frame if A = B = 1. Note that, as each fn ∈ W, for every f ∈ W ⊥ it holds that f, fn = 0. This shows that every frame for W is in particular a Bessel sequence in H. Any Bessel sequence F = {fn }n∈N deﬁnes a bounded linear operator T : 2 → H by T en = fn , where {en }n∈N denotes the “canonical” basis of 2 . This operator is called the synthesis operator of F, T ∗ ∈ B(H, 2 ) is called ∗ the analysis operator of F, and S = T T is called the frame operator of F. ∗ It is easy to see that T f = f, fn en and therefore n∈N

Sf =

f, fn fn

∀ f ∈ H.

(3)

n∈N

Observe that, in the case of a frame for a (closed) subspace W, from (2) we obtain the operator inequality A · PW ≤ S ≤ B · PW . Hence, S|W is invertible in B(W) and R(T ) = W. The dimension of N (T ) is called sometimes the excess of F. A Riesz basis for a closed subspace W is a frame for this subspace with excess equal to zero. (see Balan et al. [3])

3. Differentiable Structure of FS Our goal in this section is to prove that FS is a smooth submanifold of B(2 , H) × B(2 , H). To do this we shall use an auxiliary set AFS which contains FS . Namely, AFS = {(F, H) ∈ B(2 , H) × B(2 , H) : P F H ∗ P = P } Throughout P = PS denotes the orthogonal projection from H onto S. Clearly FS ⊂ AFS . In this section we shall use the sesquilinear map ΠB A, for A, B ∈ B(2 , H) ﬁxed: 2 2 B ∗ ∗ ΠB A : B( , H) × B( , H) → B(H), ΠA (X, Y ) = AX Y B .

Let OS = {(X, Y ) ∈ B(2 , H)×B(2 , H) : P XY ∗ P is invertible in S}. Clearly OS is open in B(2 , H) × B(2 , H).

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Lemma 3.1. The set AFS is a C ∞ submanifold of B(2 , H) × B(2 , H). Proof. Consider the map ΠP P |OS : OS → GL(S), where we identify P B(H)P with B(S). We claim that ΠP P |OS is a submersion. In that case, we have that P −1 AFS = ΠP |OS ({P }) is a submanifold of OS , which is open in B(2 , H) × B(2 , H), and the proof follows. Apparently, ΠP P |OS is onto. On the other hand, the differential of ΠP at (F, H) is P ∗ ∗ 2 d(ΠP P )(F,H) (X, Y ) = P (XH + F Y )P, X, Y ∈ B( , H).

Let us exhibit a (bounded linear) cross section for d(ΠP P )(F,H) . For Z ∈ (T Gl(S))P F H ∗ P = B(S), let S = (P F H ∗ P )† and put 1 1 ZSF, Z ∗ S ∗ F . σ(F,H) (Z) = 2 2 Note that 1 P (ZSF H ∗ + F H ∗ SZ)P = P ZP = Z. 2 It follows that ΠP P is a submersion. d(ΠP P )(F,H) (σ(F,H) (Z)) =

Remark 3.2. From the proof of the above lemma it can be deduced that the tangent space of AFS at a pair (F, H) is given by (TAFS )(F,H) = ker d(ΠP P )(F,H) = {(X, Y ) ∈ B(2 , H) × B(2 , H) : P (XH ∗ + F Y ∗ )P = 0}. (4) Remark 3.3. The set AFS can be interpreted as a parametrization of the pairs of Bessel sequences ({fn }n∈N , {hn }n∈N ) such that f, g =

∞

f, hn fn , g ,

n=1

for every f, g ∈ S. Proposition 3.4. The set FS is a C ∞ submanifold of B(2 , H) × B(2 , H). Proof. Let RP = {X ∈ B(H) : P X = P and (I − P )X(I − P ) = 0}. Note that RP is an aﬃne and complemented submanifold of B(H). Indeed, RP = P + {Y ∈ B(H) : P Y = 0, (I − P )Y (I − P ) = 0}, and this linear subset of B(H) is complemented in B(H). Note that if X ∈ RP , X = P + (I − P )XP and in particular XP = X. Let ΠP : AFS → RP , ΠP (F, H) = F H ∗ P. Clearly it is well deﬁned and onto. Since FS ⊂ AFS , it follows that FS = Π−1 P ({P }). Let us show that ΠP is also a submersion. Its differential at a point (F, H) ∈ AFS is given by d(ΠP )(F,H) (X, Y ) = (XH ∗ + F Y ∗ )P. Consider the linear map τ(F,H) (Z) = (ZP F, 0).

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First note that Z is tangent to RP if P Z = 0 and (I − P )Z(I − P ) = 0. This implies that ZP = Z and also that the pair (ZP F, 0) is tangent to AFS at (F, H). Indeed, this follows by Eq. (4) and the fact that P (ZP F H ∗ + 0)P = P ZP = 0. Then, d(ΠP )(F,H) (τ(F,H) (Z)) = (ZP F H ∗ + 0)P = ZP = Z. Therefore ΠP is a submersion, which concludes the proof.

Corollary 3.5. FS is a submanifold of AFS and the tangent space of FS at (F, H) is given by: (T FS )(F,H) = ker d(ΠP )(F,H) = {(X, Y ) ∈ B(2 , H) × B(2 , H) : (XH ∗ + F Y ∗ )P = 0}.

(5)

We emphasize the fact that both AFS and FS are complemented submanifolds of B(2 , H) × B(2 , H). In other words, the linear subspaces (T FS )(F,H) are complemented in B(2 , H) × B(2 , H). Let us exhibit natural supplements for them, or more precisely, for each (F, H) ∈ FS , a linear idempotent Q(F,H) acting on B(2 , H) × B(2 , H) whose range is (T FS )(F,H) . This is done by composing the linear maps τ and σ of the preceding results with d(ΠP )(F,H) . This gives, after tedious but straightforward computations, the linear map 1 P − I (XH ∗ + F Y ∗ )P F, Y Q(F,H) (X, Y ) = X + 2 1 ∗ ∗ (6) − P (XH + F Y )P H 2 for (F, H) ∈ FS . Proposition 3.6. The map Q(F,H) is a linear idempotent with range equal to (T FS )(F,H) . Proof. First note that Q(F,H) acts as the identity map on (T FS )(F,H) : if (X, Y ) ∈ (T FS )(F,H) then (XH ∗ +F Y ∗ )P = 0, and therefore Q(F,H) (X, Y ) = (X, Y ). Next note that R(Q(F,H) ) ⊂ (T FS )(F,H) . Recall the basic relation ∗ Y ) = Q(F,H) (X, Y ), F H P = P , and abbreviate Z = (XH ∗ + F Y ∗ )P . If (X, then ∗ + F Y ∗ )P (XH 1 = XH ∗ P + − I (XH ∗ + F Y ∗ )P F H ∗ P + F Y ∗ P 2 1 − F H ∗ P (XH ∗ + F Y ∗ )P 2 1 1 − I ZP + P ZP = 0. = ZP + 2 2

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Y ) = Q(F,H) (X, Y ); Finally, we show that Q2(F,H) = Q(F,H) . Write again (X, ∗ + F Y ∗ )P = 0, and therefore then by the ﬁrst fact above, (XH Y ) = (X, Y ), Q(F,H) (X,

which proves the result.

4. Commutative Pseudoframes A pseudoframe ({fn }n∈N , {hn }n∈N ) for S is called commutative [21] if the roles of {fn }n∈N and {hn }n∈N are interchangeable (see [21]). That is, if f ∈ S, one has two reconstruction formulas: f, hn fn and f = f, fn hn . (7) f= n

n

In operator theoretic terms, a pseudoframe (F, H) is commutative if F H ∗P = P F H ∗ = P Let us denote by CFS the set of such pairs: CFS = {(F, H) ∈ FS : P F H ∗ = P }. More generally, given an oblique projection Q such that R(Q) = S we can consider the set CFQ = {(F, H) ∈ FS : P F H ∗ = Q}. Equivalently, a pair (F, H) of FS belongs to CFQ if HF ∗ P = Q∗ . This means that instead of the second reconstruction formula in (7), one has f, fn hn = Q∗ f. n ∗

Note that Q |S is an isomorphism between S and Q∗ (S) (because N (Q∗ ) = S ⊥ ). In other words, in principle, no information is lost, and one recovers equivalent data. Let us also consider the set QS , which consists of the oblique projections whose range is S. This set is an aﬃne subspace of B(H). Lemma 4.1. Given an oblique projection Q ∈ QS , the set CFQ is diffeomorphic to CFS . Proof. Let GQ = Q + (I − P ). Straightforward but simple computations show that the set {GQ }Q∈QS is a multiplicative subgroup of GL(H) and the action (F, H) → (F, G∗ H) deﬁnes a diffeomorphism between CFS and CFQ . This proves that the sets CFQ are diffeomorphic. Now, let us consider the map Φ : FS → QS deﬁned by Φ(F, H) = P F H ∗ . Note that CFQ = Φ−1 (Q). As we will see in the next proposition, this map induces in FS a structure of trivial ﬁber bundle with base space Qs and ﬁber CFS . Recall that given two manifolds M and N and a submersion π : M → N , we say that M is a trivial ﬁber bundle with base space N if there exist another

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manifold F , called ﬁber, and a diffeomorphism φ : M → N × F such that the following diagram commutes: φ

/ N ×F zz zz z z zz π zpr z zz 1 zz z z |z N M

Proposition 4.2. The map Φ : FS → QS deﬁned by Φ(F, H) = P F H ∗ is a smooth submersion that induces a structure of trivial ﬁber bundle on FS where the ﬁber is CFS . Moreover, the diffeomorphism Ψ : FS → QS × CFS which makes the following diagram commutative Ψ

/ QS × CFS (8) w w w ww ww w Φ ww ww pr1 w w ww w {w QS

∗ is given by Ψ(F, H) = P F H ∗ , F, G∗(P−1 ∗ H) , where G(P F H ∗ ) = P F H + FH FS

(I − P ). Proof. As we observe in the proof of Lemma 4.1, given (F, H) ∈ FS , then (F, G∗(P−1 F H ∗ ) H) ∈ CFS . Hence, the map Ψ is well deﬁned. Clearly it is smooth, and its inverse is given by Ψ−1 (Q, (F , H )) = F , (Q∗ + I − P )−1 H , which is apparently smooth. The commutativity of the diagram in Eq. (8) is straightforward to verify. Remark 4.3. Essentially, the last result shows that all information about pseudoframes for S is contained in the Cartesian product of the set of all oblique projections onto S and the set of all commutative pseudoframes for S. In particular, the topology and the geometry of FS are determined by those of QS and CFS .

5. Dual Frames Inside FS there is another distinguished subset: XS := {(F, H) ∈ B(2 , H) × B(2 , H) : F H ∗ = P, R(F ) = R(H) = S}. Note that XS is non empty: if V : 2 → H is a partial isometry with ﬁnal space S (for instance, take V = P U with U a unitary operator), then (V, V ) ∈ XS .

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This set is the operator version of the set of all dual frames ({fn }n∈N , {hn }n∈N ) for the subspace S, i.e., pairs of frames such that for every f ∈ H ∞ Pf = f, hn fn . n=1

In [2], the following alternative characterizations of the set XS were proved: Proposition 5.1. If F, H ∈ B(2 , H), then the following statements are equivalent: 1. (F, H) ∈ XS ; 2. F H ∗ F = F , H ∗ F H ∗ = H ∗ and F H ∗ = P ; 3. R(F ) = S, H = (QF † P )∗ where Q is an oblique projection such that N (Q) = N (F ); 4. F H ∗ = P , P F = F and P H = H. From the geometrical point of view, we have the next result: Proposition 5.2. The set XS is a smooth submanifold of FS . Since XS is a submanifold of B(2 , H) × B(2 , H) (see [2,11] for more details), the proof of this proposition is a consequence of the following result, which will be used in the sequel to study the topological properties of the set FS . Proposition 5.3. There is a smooth strong deformation retraction from FS onto XS . Moreover, when restricted to CFS , it is a strong deformation retraction from this space. Proof. Deﬁne Γ : [0, 1] × FS → FS by Γ (t, (F, H)) = ( P F + (1 − t)(1 − P )F, P H + (1 − t)(1 − P )H ) .

(9)

Straightforward computations show that Γ (t, (F, H)) ∈ FS for every t ∈ [0, 1], and, if (F, H) ∈ XS , then Γ(t, (F, H)) = (F, H) for every t ∈ [0, 1]. So, it only remains to prove that for every (F, H) ∈ FS it holds that Γ(1, (F, H)) ∈ XS . Let (F, H) ∈ FS and deﬁne (F1 , H1 ) = Γ(1, (F, H)) = (P F, P H). Then ∗

∗

F1 H1 = P F H P = P. ∗

∗

On the other hand, since P is selfadjoint, we get F1 H1 = H1 F1 = P . This implies that S ⊆ R(F1 ) and S ⊆ R(H1 ). The other inclusions are trivial because F1 = P F and H1 = P H, and then (F1 , H1 ) ∈ XS . If the pair (F, H) belongs to CFS , then P Γ(t, (F, H)) = P F (H ∗ P + (1 − t)H ∗ (1 − P ) = P F H ∗ P + (1 − t)P F H ∗ (1 − P ) = P, because in CFS , equation P F H ∗ = P holds. Then Γ(t, (F, H)) ∈ CFS

In particular, any element (F, H) in FS (resp. CFS ) can be joined in FS (resp. CFS ) to (P F, P H) ∈ XS with the path Γ(t, (F, H)). Therefore the connected components of FS and CFS are parametrized by the components of XS . We characterize these in the next section.

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6. Topological Structure of XS In this section we shall study some topological properties of the set XS . For all topological notions needed here, we refer the reader to [22,26]. By Proposition 5.3, many of these properties will hold for FS . With this aim, we consider the action ρ : GL(2 ) × XS → XS deﬁned by ρ(G, (F, H)) = G · (F, H) = (F G, H (G−1 )∗ ). This action of GL(2 ) on XS induces a partition of XS in orbits: the orbit of (F, H) is O(F, H) = {G · (F, H) : G ∈ GL(2 )}. Denote by C(F, H) the connected component of (F, H) in XS . Since GL(2 ) is connected, O(F, H) is automatically contained in C(F, H) . The following result establishes the existence of continuous local cross sections for the action and, as a consequence, the equality O(F, H) = C(F, H) . Proposition 6.1. For every (F, H) ∈ XS there is an open neighborhood U of (F, H) in XS and a continuous map σ : U → GL(2 ) such that

˜ (F, H) = (F˜ , H) ˜ ρ σ(F˜ , H), ˜ ∈ U. for all (F˜ , H) Proof. Fix (F, H) ∈ XS , and let σ0 : B(2 , H) × B(2 , H) → B(2 ) be deﬁned by

˜ = H ∗ F˜ + (1 − H ∗ F ) 1 − H ˜ ∗ F˜ . σ0 (F˜ , H) Since H ∗ F is a projection, σ0 ((F, H)) = I. Thus the fact that σ0 is continuous implies that there is an open neighborhood U0 of (F, H) in B(2 , H)×B(2 , H) such that σ0 |U0 takes values in GL(2 ). Let U = U0 ∩ XS and let σ be the restriction of σ0 to U. Note that

˜ ˜ ∗ F˜ F σ(F˜ , H) = F H ∗ F˜ + (1 − H ∗ F ) 1 − H

˜ ∗ F˜ = QF˜ + (F − F H ∗ F ) 1 − H = F˜ . Analogously,

∗

˜ σ(F˜ , H) ˜ ˜ (1 − F ∗ H) ˜ F˜ ∗ H + 1 − F˜ ∗ H H =H

˜ (1 − F ∗ H) ˜ −H ˜ F˜ ∗ H = Q∗ H + H

= H.

˜ (F, H) = (F˜ , H) ˜ for all (F˜ , H) ˜ ∈ U. So, ρ σ(F˜ , H),

Corollary 6.2. The orbits are open, and therefore also closed, subsets of XS . In particular, for every (F, H) ∈ XS it holds that O(F, H) = C(F, H) . Given (F, H) ∈ XS , note that dim N (F ) = dim N (H) and this quantity is invariant in O(F, H) .

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Our next goals are to characterize C(F, H) in terms of dim N (F ) and to characterize the homotopy groups of each connected component. Proposition 6.3. Given (F, H) ∈ XS , then C(F, H) = {(S, T ) ∈ XS : dim N (S) = dim N (F )} To prove this result, we need the following technical lemmas: Lemma 6.4. Let V and W be two partial isometries of B(2 , H) such that V V ∗ = W W ∗ and dim N (V ) = dim N (W ). Then, there is a unitary operator U ∈ B(2 ) such that V U = W . Proof. Let U0 be a partial isometry with initial space N (W ) and ﬁnal space N (V ). Then, U = V ∗ W + U0 is a unitary operator that satisﬁes the identity V U = W. Lemma 6.5. Let (F, H) ∈ XS and deﬁne A = F ∗ F + (1 − F ∗ H)(1 − H ∗ F ). Then, A is a positive invertible operator such that V = HA1/2 = F A−1/2 is a partial isometry with the following properties: VV∗ =P

and

dim N (V ) = dim N (F ),

(10)

where P denotes the orthogonal projection from H onto S. Proof. Straightforward computations show that A−1 = H ∗ H + PN (F ) . On the other hand, by Proposition 5.1 and using that F H ∗ = HF ∗ = P , we get H ∗ − H ∗ F H ∗ = H − HF ∗ H = 0 and, therefore HA = HF ∗ F + (H − HF ∗ H)(H − HH ∗ F ) = P F = F,

(11)

and V V ∗ = (HA1/2 )(HA1/2 )∗ = HAH ∗ = HF ∗ F H ∗ − (H − HF ∗ H)(H ∗ − H ∗ F H ∗ ) = P.

(12) 1/2

−1/2

= FA , whereas Eq. (12) implies that Equation (11) implies that HA V = HA1/2 = F A−1/2 is a partial isometry. Finally, by construction, V satisﬁes that VV∗ =P

and

dim N (V ) = dim N (F ).

Proof of Proposition 6.3. Since C(F, H) = O(F, H) and the action preserves the dimension of the kernels of F and H, it is sufﬁcient to prove that, given a pair (S, T ) with dim N (S) = dim N (F ), then (S, T ) ∈ O(F, H) . Take (S, T ) ∈ XS with dim N (S) = dim N (F ). By Lemma 6.5, there are positive invertible operators A, B ∈ B(2 ) such that V = F A−1/2 = HA1/2 and W = SB −1/2 = T B 1/2 are partial isometries which satisfy V V ∗ = WW∗ = P and dim N (V ) = dim N (F ) = dim N (S) = dim N (W ).

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Then, by Lemma 6.4, there is a unitary operator U such that V U = W . Therefore, the operator C = A−1/2 U B 1/2 satisﬁes that C · (F, H) = (S, T ), which concludes the proof. The characterization of the homotopy groups of each connected component is based in the following proposition. For the topological definitions see [22]. Proposition 6.6. Let Cn be the connected component of XS consisting of all the pairs (F, H) with nullity (of F and H) equal to n ∈ N ∪ {∞}. Let Jn be the subset of Cn of pairs that have the form (V, V ), where V is a partial isometry with nullity equal to n. Then there is a smooth strong deformation retraction from Cn onto Jn . Proof. By Lemma 6.5, given (F, H) ∈ Cn , A(F, H) = F ∗ F + (1 − F ∗ H)(1 − H ∗ F ) is a positive invertible operator of B(2 ) such that −1/2

−1/2

1/2

A(F, H) · (F, H) = (F A(F, H) , HA(F, H) ) ∈ Jn . Also note that, given (V, V ) ∈ Jn , then A(V, V ) = I. Deﬁne Γ : [0, 1]×Cn → Cn by −t/2

Γ (t, (F, H)) = A(F, H) · (F, H) = (F ∗ F + (1 − F ∗ H)(1 − H ∗ F )) Γ is a strong deformation retraction from Cn onto Jn .

−t/2

· (F, H).

In particular, the above result implies that the homotopy type of Cn is the same as the homotopy type of the set Jn of partial isometries with ﬁxed ﬁnal space P and nullspace of dimension n. The sets Jn were studied in [1]. More precisely, the space studied there is the set of partial isometries with ﬁxed initial space. Apparently, one can pass from one set to the other by taking adjoints, an operation which preserves topological and differentiable properties. Among these, let us cite a few: 1.

The action of the unitary group U(H) given by right multiplication is transitive in Jn , and deﬁnes, for a ﬁxed V ∈ Jn , a locally trivial ﬁbration map πV : U(H) → Jn , πV (U ) = V U. The ﬁbre of this bundle is the subgroup JV = {U ∈ U(H) : V U = V }, which consists of unitaries whose matrices in terms of the orthogonal projection PV = V ∗ V are of the form I 0 . 0 U22

2.

It is apparent that it is homeomorphic to the unitary group U(PV (H)⊥ ). The sets Jn are C ∞ submanifolds of B(H).

We may use this ﬁbration map to compute the homotopy groups of Jn (and therefore of Cn ):

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Proposition 6.7. If n = ∞, then Jn is contractible. If n < ∞, then for all k ≥ 1, πk (Jn ) πk−1 (U(n)), where U(n) denotes the unitary group on Cn . Proof. The ﬁbration map πV induces the homotopy exact sequence · · · → πk (JV ) → πk (U(H)) → πk (Jn ) → πk−1 (JV ) → πk−1 (U(H)) → · · · By Kuiper’s theorem (see [18]), U(H) is contractible, as well as JV in the n = ∞ case. Thus in this case πk (Jn ) = 0 for all k, and Jn is therefore contractible, being a differentiable manifold, by Palais’ theorem (see [26]). If n < ∞, the result follows by observing that JV U(n). Corollary 6.8. If n = ∞, then Cn is contractible. If n < ∞, then for every k ≥ 1, πk (Cn ) πk−1 (U(n)). The next theorem summarizes the previous results about XS . Theorem 6.9. The following facts hold: 1. Given (F, H) ∈ XS then: C(F, H) = {(S, T ) ∈ XS : dim N (S) = dim N (F )}. 2. 3.

There is a smooth strong deformation retraction from Cn onto Jn . If n ∈ N, then πk (Cn ) = πk−1 (U(n)), and C∞ is contractible. In particular, the different connected components of XS are not homeomorphic.

Following Li and Ogawa [20,21] and Christensen and Eldar [7], we shall consider now the ﬁbers of the map (F, H) → F . We start, as usual, with a decomposition H = S ⊕S ⊥ . By definition of the set XS , for every (F, H) ∈ XS the range of the operator F ∈ B(2 , H) is S. Denote by CRS the set F ∈ B(2 , H) : R(F ) = S . This set has an interesting geometrical structure as a discrete union of homogeneous spaces of GL(2 ) × GL(H) (see [9]). If S = H we write E instead of CRH . Theorem 6.10. The map pr1 : XS → CRS deﬁned by pr1 ((F, H)) = F is onto, it has aﬃne ﬁbers, and it admits a global continuous cross section. Proof. The surjectivity of pr1 follows directly from Proposition 5.1. Moreover, this result parametrizes the ﬁber pr1−1 (F ) = {(F, H) ∈ XS } for every F ∈ CRS as pr1−1 (F ) = (F, QF † P ) : Q ∈ B(2 ), N (Q) = N (F ), Q2 = Q . Observe that, since the set of all projections with a ﬁxed kernel is an aﬃne manifold, the same holds for pr1−1 (F ). In particular, pr1−1 (F ) is contractible. To end the proof it sufﬁces to show that the Moore–Penrose map F → F † is continuous when it is restricted to CRS . This result easily follows from the next lemma. Lemma 6.11. If F ∈ CRS then F F ∗ + PS ⊥ ∈ GL(H) and F † = F ∗ (F F ∗ ) + −1 PS ⊥ .

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Proof. One ﬁrst checks that the inverse of F F ∗ + PS ⊥ is (F F ∗ )† + PS ⊥ . Then, a simple computation, using elementary properties of † , shows that −1 F ∗ ((F F ∗ ) + PS ⊥ ) = F † . Corollary 6.12. XS is homotopy equivalent to CRS . Proof. The ﬁber bundle pr1−1 (F ) −−→ XS −−→ CRS has a global cross section and contractible ﬁbers. Therefore, as in the proof of Proposition 6.7, we obtain that XS is homotopy equivalent to CRS . Remark 6.13. It should be mentioned that the results of this section generalize those about the connected components of the set E = {T ∈ B(H) : R(T ) = H} studied in [10]. Recall that E is an open subset of B(H) with a natural action GL(H) × E −−→ E deﬁned by (G, T ) → T G−1 . The orbits by this action are exactly the connected components of E. Moreover, the components are determined by the nullity of T : T, T ∈ E belong to the same component if and only if dim N (T ) = dim N (T ). Fix an orthonormal basis {en }n∈N of H and deﬁne the unilateral shift S ∈ B(H) by Sen = en+1 (n ∈ N). Then S ∗ ∈ E and all “ﬁnite” connected components of E have the form GL(H) · S ∗n . The map E → E deﬁned by T → (T T ∗ )−1/2 T is a retraction from E onto the subset E0 = {T ∈ B(H) : T T ∗ = I}. It is well known that E corresponds naturally to the set of all frames on H and, under this correspondence, E0 is mapped onto the set of all Parseval frames. Remark 6.14. As remarked above, any operator whose range equals S, is the ﬁrst coordinate of a pair in XS . Note that any F : 2 → H such that S ⊂ R(F ) is the ﬁrst coordinate of a pair in FS . Indeed, consider F |F −1 (S) : F −1 (S) → S ⊂ H, and let F0 = F |F −1 (S) ⊕ 0F −1 (S)⊥ whose range equals S, then (F, F0† ∗ ) belongs to FS .

References [1] Andruchow, E., Corach, G., Stojanoﬀ, D.: Geometry of the sphere of a Hilbert module. Math. Proc. Camb. Phil. Soc. 127, 295–315 (1999) [2] Andruchow, E., Antezana, J., Corach, G.: Sampling formulae and optimal factorizations of projections. Sampl. Theory Signal Image Process. 7, 291–309 (2008) [3] Balan, R., Casazza, P.G., Heil, C., Landau, Z.: Excesses of Gabor frames. Appl. Comput. Harmon. Anal. 14, 87–106 (2003)

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[4] Ben-Israel, A., Greville, T.N.E.: Generalized inverses, theory and applications, 2nd edn. In: CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, vol. 15. Springer-Verlag, New York (2003) [5] Campbell, S.L., Meyer, C.D. Jr.: Generalized Inverses of Linear Transformations. Corrected reprint of the 1979 original. Dover, New York (1991) [6] Christensen, O.: An Introduction to Frames and Riesz Bases. Birkh¨ auser, Boston (2003) [7] Christensen, O., Eldar, Y.: Oblique dual frames and shift-invariant spaces. Appl. Comput. Harmon. Anal. 17, 48–68 (2004) [8] Christensen, O., Eldar, Y.C.: Generalized shift-invariant systems and frames for subspaces. J. Fourier Anal. Appl. 11, 299–313 (2005) [9] Corach, G., Maestripieri, A., Mbekhta, M.: Metric and homogeneous structure of closed range operators. J. Operator Theory 61, 171–190 (2009) [10] Corach, G., Pacheco, M., Stojanoﬀ, D.: Geometry of epimorphisms and frames. Proc. Am. Math. Soc. 132, 2039–2049 (2004) [11] Corach, G., Porta, H., Recht, L.: Differential geometry of spaces of relatively regular operators. Integr. Equ. Oper. Theory 13, 771–794 (1990) [12] Daubechies, I.: Ten lectures on wavelets. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, xx+357 pp (1992). ISBN: 0-89871274-2 [13] Dufﬁn, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952) [14] Eldar, Y.C.: Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors. J. Fourier Anal. Appl. 9, 77–96 (2003) [15] Eldar, Y.C., Werther, T.: General framework for consistent sampling in Hilbert spaces. Int. J. Wavelets Multiresolution Inf. Process. 3, 347–359 (2005) [16] Gramsch, B.: Relative Inversions in der St¨ orungstheorie von Operatoren und ψ-algebren. Math. Ann. 269, 27–71 (1984) [17] Heil, C.E., Walnut, D.F.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989) [18] Kuiper, N.H.: The homotopy type of the unitary group of Hilbert space. Topology 3, 19–30 (1965) [19] Labrousse, J.P., Mbekhta, M.: Les op´erateurs points de continuit´e pour la conorme et l’inverse de Moore-Penrose. Houston J. Math. 18, 7–23 (1992) [20] Li, S., Ogawa, H.: Pseudo-duals of frames with applications. Appl. Comput. Harmon. Anal. 11, 289–304 (2001) [21] Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10, 409–431 (2004) [22] Munkres, J.R.: Topology: A First Course. Prentice Hall, New Jersey (1975) [23] Nashed, M.Z.: Generalized Inverses and Applications. Academic Press, London (1976) [24] Ogawa, H., Berrached, N.E.: A theory of extended pseudo-biorthogonal bases and its application to generalized sampling theorem. Contemp. Math. 190, 305– 321 (1995) [25] Ogawa, H., Berrached, N.E.: EPBOBs (extended pseudo-biorthogonal bases) for signal recovery. IEICE Trans. Inf. Syst. E83-D, 223–232 (2000)

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[26] Palais, R.S.: Homotopy theory of inﬁnite dimensional manifolds. Topology 5, 1–16 (1966) [27] Young, R.M.: An Introduction to Nonharmonic Fourier Series, revised 1st edn. Academic Press, San Diego (2001) Esteban Andruchow (B) Instituto de Ciencias, UN de Gral. Sarmiento and IAM-CONICET Buenos Aires, Argentina e-mail: [email protected] Jorge Antezana Departament de Matem` atiques, Universitat Aut` onoma de Barcelona Barcelona, Spain e-mail: [email protected] Gustavo Corach Depto. de Matem´ atica, FI-UBA and IAM-CONICET Buenos Aires, Argentina e-mail: [email protected] Received: June 26, 2009. Revised: May 11, 2010.

Integr. Equ. Oper. Theory 67 (2010), 467–479 DOI 10.1007/s00020-010-1795-6 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Boundedness, Compactness and Schatten-class Membership of Weighted Composition Operators E. A. Gallardo-Guti´errez, R. Kumar and J. R. Partington Abstract. The boundedness and compactness of weighted composition operators on the Hardy space H2 of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces A2α (D) are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation. Mathematics Subject Classification (2010). Primary 47B38. Keywords. Weighted composition operators, Carleson embeddings, reproducing kernels, angular derivative.

1. Introduction Let D denote the open unit disk of the complex plane. For 1 ≤ p < ∞, recall that the classical Hardy space Hp consists of holomorphic functions f on D for which the norm ⎞1/p ⎛ 2π dθ ⎠ f p = ⎝ sup |f (reiθ )|p 2π 0≤r<1 0

is finite. If p = ∞, H∞ is the space of holomorphic functions f on D such that f ∞ = sup |f (z)| < ∞. D

E. A. Gallardo-Guti´ errez and J. R. Partington are partially supported by Plan Nacional I + D grant no. MTM2007-61446 and Gobierno de Arag´ on research group An´ alisis Matem´ atico y Aplicaciones, ref. DGA E-64. R. Kumar is partially supported by the Royal Society (UK) and the Department of Science and Technology (India).

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Fatou’s Theorem asserts that any Hardy function f has radial limit at eiθ ∈ ∂D except on a set Lebesgue measure zero (see [7], for instance). Throughout this work, f (eiθ ) will denote the radial limit of f at eiθ , i. e., f (eiθ ) = limr→1− f (reiθ ). If ϕ is an analytic function on D which takes D into itself, the Littlewood Subordination Principle [18] ensures that the composition operator induced by ϕ, Cϕ f = f ◦ ϕ,

(f ∈ Hp ),

is bounded on Hp , 1 ≤ p ≤ ∞. On the other hand, given h ∈ Hp it is possible to define for those functions f ∈ Hp for which it makes sense the weighted composition operator Wh,ϕ : Wh,ϕ f (z) = h(z)f (ϕ(z)). Note that if f belongs to the domain of the operator, which will denoted by D(Wh,ϕ ), the weighted composition operator may be expressed by Wh,ϕ = Th Cϕ , where Th denotes the Toeplitz operator with symbol h. Observe that the condition h ∈ H∞ is always a sufficient condition for boundedness of Wh,ϕ . By considering the image of the constant functions, it is clear that h ∈ Hp is a necessary condition. Nevertheless, the most interesting feature in this sense is that there exists symbols h (unbounded) and ϕ ∈ H∞ with ϕ(D) ⊆ D such that Wh,ϕ is a bounded operator in Hp . Similar considerations, with the obvious modifications, apply to other spaces of analytic functions such as the Bergman space A2 (D). For some general properties of these operators on Hardy and Bergman spaces we refer to the survey [17], where it is remarked that characterising the boundedness of weighted composition operators on spaces of functions on the disc enables one to characterise the boundedness of such operators on Hardy and Bergman spaces of other simply-connected domains, such as the half-plane. Though weighted composition operators have attracted recently the attention of operator theorists (see for instance the recent papers [3,4,11,12]), we would like to point out that its study traces back to the sixties. Indeed, de Leeuw showed that the isometries in the Hardy space H1 are weighted composition operators and Forelli obtained the same result for the Hardy spaces Hp when 1 < p < ∞, p = 2 (see [8,14]). The aim of this work is taking further the study of weighted composition on spaces of analytic functions of D. In particular, we deal with the boundedness of weighted composition operators on Hardy and Bergman spaces of the unit disc. In this sense, we are primarily interested in the behaviour of the weights h. Let us point out in this direction that an applicable characterization of those weights h so that Wh,ϕ is bounded on the Bergman space would solve the famous Brennan conjecture (see Shimorin’s work [23] and the related work by Smith [24]). In Sect. 2 we use recent results of Harper and Smith [11,12] (of which the main tool is the reproducing kernel of the Hilbert space H2 ) to derive

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new results on the boundedness, compactness and Schatten-class membership of Wh,ϕ . A case of particular interest is when ϕ is inner, and in this case the properties of the angular derivative are seen to play an important role. In particular, we give extensions of recent results due to Jury [15] and Gunatillake [10]. Most of our results apply principally to operators on the Hilbert space H2 , but some results and proofs make sense also for all Hp with 1 ≤ p < ∞. In Sect. 3 we extend some results due to Harper and Smith [11,12], concerning the link between weighted composition operators and Carleson embeddings, to the case of weighted Bergman spaces.

2. Boundedness and Compactness on Hardy Spaces 2.1. Conditions for Boundedness Contreras and Hern´ andez-D´ıaz [3] gave a necessary and sufficient condition for boundedness of Wh,ϕ , namely that μh,ϕ is a Carleson measure on D, where μh,ϕ (E) = |h|2 dm, (2.1) ϕ−1 (E)∩T

for measurable subsets E ⊆ D. Harper [11, Thm. 3.3] exploited the theory of admissibility for semigroups to show (as part of a family of more general results) that a more explicit necessary and sufficient condition for boundedness of Wh,ϕ is the following: (1 − |w|2 )1/2 h < ∞. sup (2.2) 1 − wϕ |w|<1 2 Moreover, Wh,ϕ is compact if and only if (1 − |w|2 )1/2 h →0 1 − wϕ 2

as |w| → 1.

(2.3)

Some independent generalizations to mappings between Hp and Hq spaces were given by Cu˘ckovi´c and Zhao [6]. In fact (2.2) follows directly from (2.1) on using the standard reproducing kernel test for Carleson embeddings (see [9, p. 231] or [21, p. 105]), namely that an embedding J : H2 → L2 (μ) is bounded (i.e., μ is a Carleson measure on D) if and only if J is uniformly bounded on the set of normalized reproducing kernels { kw : w ∈ D} on H2 , where (1 − |w|2 )1/2 , kw (z) = 1 − wz For

D

| kw |2 dμh,ϕ =

T

for z ∈ D.

|h|2 | kw ◦ ϕ|2 dm,

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from which (2.2) follows easily. Similar arguments involving vanishing Carleson measures establish (2.3). On the other hand, Jury [15] has obtained the estimate Wh,ϕ ≤ hH(ϕ) , but only for h ∈ H(ϕ), where H(ϕ) is the de Branges–Rovnyak space, that is, the reproducing kernel Hilbert space on D with kernel 1 − ϕ(w)ϕ(z) . 1 − wz In general this is not a necessary condition for boundedness. If ϕ is inner, then H(ϕ) = Kϕ := H2 ϕH2 , and PKϕ h = ϕP− (ϕh), where P− is the orthogonal projection onto L2 H2 . In this case, it is easy to check that, if h ∈ Kϕ , ∞ 2 ∞ an ϕn h = h22 |an |2 , ϕ (z) = kw

n=0

n=0

2

since

ϕn h, ϕm h =

0

for n = m,

h22

for n = m,

and so Wh,ϕ f 2 = h2 f 2 . Thus the condition is correct for h ∈ Kϕ , although not in general. Indeed, if ϕ(z) = z, then Wh,ϕ = Th = h∞ . Suppose that ϕ is inner, and let α be an automorphism of the disc such that α(ϕ(0)) = 0. Then α ◦ ϕ is also inner, and Wh,ϕ Cα = Wh,α◦ϕ . For questions of boundedness and compactness, we may therefore assume without loss of generality that ϕ(0) = 0, and hence {1, ϕ, ϕ2 , . . .} is an orthonormal set and Cϕ is an isometry. A further observation is that Wh,ϕ is bounded as soon as ϕ is inner (w.l.o.g. ϕ(0) = 0 again) and h ∈ H2 satisfies h, hϕn = 0 for all n ≥ 1 (which is a weaker condition than h ∈ Kϕ ); for 2 ∞ ∞ h 2 k k 2 = hw ϕ = h |w|2k , 1 − wϕ 2 k=0

2

k=0

since hϕm , hϕn = 0 for m = n. Then by (2.2) it follows that Wh,ϕ is bounded. In order to make a more systematic analysis of the weights that induce a bounded weighted composition operator, we make the following definition. Definition 2.1. For ϕ : D → D analytic, the multiplier space of ϕ is defined by Mϕ = {h ∈ H2 : Wh,ϕ := Th Cϕ is bounded}. Evidently, H∞ ⊆ Mϕ ⊆ H2 for all analytic self-maps ϕ of the unit disc. It is easily verified that Mϕ is a Banach space with the norm hMϕ = Wh,ϕ . Contreras and Hern´ andez-D´ıaz [4] and Matache [20] showed that Mϕ = H∞ if and only if ϕ is a finite Blaschke product. In Corollary 2.5 we shall present a stronger result.

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The other extreme situation is the following. Theorem 2.2. Mϕ = H2 if and only if ϕ∞ < 1. Proof. If ϕ∞ < 1, and f 2 = 1, then ∞ n Cϕ f ∞ = fˆ(n)ϕ n=0 ∞ ∞ 1/2 ∞ 1/2 ≤ |fˆ(n)|2 ϕ2n , ∞

n=0

n=0

and we conclude easily that Cϕ maps H2 boundedly into H∞ , whence Mϕ = H2 . Conversely, if Mϕ = H2 , then clearly Cϕ maps H2 into H∞ , and the mapping is bounded by the closed graph theorem. In particular the functions (1 − |a|2 )1/2 , 1 − aϕ(z)

(z ∈ D),

are uniformly bounded for a ∈ D. However, if |ϕ(z0 )| = 1 − δ for some z0 ∈ D and 0 < δ < 1, then for a = ϕ(z0 ) we obtain (1 − |a|2 )1/2 (1 − (1 − δ)2 )1/2 = (2δ − δ 2 )−1/2 , ≥ 1 − aϕ 1 − (1 − δ)2 ∞

which tends to ∞ as δ → 0. Hence if Mϕ = H2 we must have ϕ∞ < 1.

There is a close relation between the angular derivative of ϕ and the boundedness of Wh,ϕ , as we explain in the next section. 2.2. Relation with the Angular Derivative Once more, let kw be the reproducing kernel at w in H2 , that is, 1 , (z ∈ D). kw (z) = 1−wz ∗ If Wh,ϕ denotes the adjoint of Wh,ϕ , it is not difficult to see that ∗ kw = h(w) kϕ(w) , Wh,ϕ

(w ∈ D).

Hence, if Wh,ϕ is bounded, it follows that

1/2 1 − |ϕ(w)|2 |h(w)| ≤ Wh,ϕ 1 − |w|2

(2.4)

for any w ∈ D. Note that from Eq. (2.4) one gets another proof of the fact that if ϕ is a finite Blaschke product then Mϕ = H∞ ; for then we have 1 − |ϕ(w)|2 < ∞, 2 w∈D 1 − |w| sup

(as noted in [20]), and so h ∈ H∞ . The next result generalizes this observation giving a necessary condition for boundedness and compactness of weighted composition operators.

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Proposition 2.3. Let ϕ be an analytic self-map of D. Let Eϕ = {ζ ∈ T : ϕ has finite angular derivative at ζ}. Then: i) If Wh,ϕ is bounded on Hp for some 1 ≤ p < ∞, then h is pointwise bounded on every Stolz domain whose vertex is a point of Eϕ . ii) If Wh,ϕ is compact on Hp for some 1 ≤ p < ∞, then h tends to zero as |z| → 1 in any Stolz domain whose vertex is a point of Eϕ . Proof. Note that if ϕ has a finite angular derivative at ζ ∈ T, then by the Julia–Carath´eodory Theorem, the nontangential limit satisfies ∠ lim

w→ζ

1 − |ϕ(w)| <∞ 1 − |w|

(see [5, pp. 51] or [22]). Indeed, it holds that |ϕ (ζ)| = lim |ϕ (r ζ)| = ∠ lim r→1

w→ζ

1 − |ϕ(w)| 1 − |ϕ(w)| = lim inf . w→ζ 1 − |w| 1 − |w|

This along with (2.4) and a standard argument on reproducing kernels yield the statement of the proposition in H2 . The result in Hp is analogous. At this point, one may ask whether Proposition 2.3 is also sufficient for boundedness of weighted composition operators. The next example shows that this is no longer true. √ Example. Let ϕ be the map ϕ(z) = 1 − 1 − z. Note that ϕ sends D onto a subdomain of D shaped roughly like a (two dimensional) ice cream cone with vertex at 1. It is easy to see that this map has finite angular derivative nowhere on the unit circle. So for any h ∈ H2 the boundedness requirement on h given by Proposition 2.3(i) holds trivially. Let us consider the H2 function h(z) = (1 − z)−3/8 and the weighted composition operator operator Wh,ϕ . Observe that if f (z) = (1−z)−1/4 , then (Wh,ϕ f )(z) = (1 − z)−1/2 , so Wh,ϕ does not take H2 into itself. Thus Wh,ϕ satisfies the conclusion of Proposition 2.3(i), but not the hypothesis. If ϕ is an inner function, there is a well known characterization of the set where ϕ has finite angular derivative, essentially due to Carath´eodory. Let ⎛ ⎞ |an | an − z ζ + z ϕ(z) = eiα z N exp ⎝− dμ(ζ)⎠ , (2.5) an 1 − an z ζ −z n≥1

T

where a ∈ R and N ∈ N ∪ {0}, be its canonical representation in terms of the zeroes {an }n and a singular measure μ. We define the Ahern–Clark set Eϕ [1] by: ⎧ ⎫ ⎨ ⎬ 1 − |an |2 dμ(t) + 2 < +∞ . Eϕ := ζ ∈ T : ⎩ ⎭ |ζ − an |2 |t − ζ|2 n≥1

T

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Proposition 2.4 (See, for example, [2]). Let ϕ be an inner function and ζ0 ∈ T. Then the following assertions are equivalent: (i) ϕ has an angular derivative in the sense of Carath´eodory at ζ0 . (iii) ζ0 ∈ Eϕ . It is known that Eϕ ⊇ T\σ(ϕ), where σ(ϕ) denotes the spectrum of ϕ, namely, σ(ϕ) = clos{an }n ∪ supp μ. In general we do not have equality. The following result may be seen as a generalization of the result that Mϕ = H∞ whenever ϕ is a finite Blaschke product (in which case Eϕ = T and σ(ϕ) = ∅). Corollary 2.5. Let ϕ be an inner function. Then any function h ∈ Mϕ is essentially bounded on all relatively compact subsets of T\σ(ϕ). Proof. On sufficiently small neighbourhoods of any relatively compact subset K of T\σ(ϕ), one may extend ϕ analytically across T; we have |ϕ(ζ)| = 1 and |ϕ (ζ)| is uniformly bounded for ζ ∈ K. It is now easily verified that (1 − |ϕ(w)|2 )/(1 − |w|2 ) is uniformly bounded as w approaches K. The result follows from (2.4). Remark 2.6. If ϕ is an inner function whose zero set, if infinite, forms a finite union of subsequences tending nontangentially to points of T, and whose singular part is given by a measure of finite support, then in fact Eϕ = T \ σ(ϕ), as is shown in [2, Thm. 5.3]. Example. The following example shows that the condition of Corollary 2.5 is not sufficient for membership of Mϕ , even for h ∈ H2 . Let

1+z ϕ(z) = exp − , (z ∈ D), 1−z which is a singular inner function with σ(ϕ) = {1}. For w = reiα ∈ D, it is straightforward to verify that

1 − |ϕ(w)|2 1 1 − r2 = 1 − exp −2 , 1 − |w|2 1 − r2 |1 − reiα |2 which, as r → 1, tends to 2/|α|2 . Hence by (2.4), for each h ∈ Mϕ we have that |h(eiα )| = O(|α|−1 ) as α → 0. It is easy to construct examples of functions h ∈ H2 , bounded on any closed subarc of T disjoint from {1}, that nonetheless do not satisfy this condition, and hence induce unbounded operators Wh,ϕ . 2.3. Schatten Class Operators In this section we provide stronger versions of the two main theorems of [10]. To begin with, in [10, Thm. 2], it was shown that if ϕ∞ < 1, then for every h ∈ H2 the weighted composition operator T = Wh,ϕ is compact. Indeed, much more is true, as the next result shows. Theorem 2.7. Suppose that ϕ∞ < 1. Then for every h ∈ H2 the weighted composition operator T = Wh,ϕ is trace-class (i.e., lies in S1 ).

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Proof. A result of Harper and Smith [12, Thm. 3.1] asserts that for 1 ≤ p < 2, a sufficient condition for a weighted composition operator T to lie in Sp (H2 ) is dA(w) T k˙ w p < ∞, (2.6) (1 − |w|2 )2 D

where k˙ w = k˙ w /k˙ w and k˙ w (z) = Note that for w ∈ D

z (1−wz)2 ,

T k˙ w 2 ≈ (1 − |w|2 )3/2

so that k˙ w ≈ (1 − |w|2 )−3/2 .

hϕ 2 (1 − wϕ) 2

and since |1 − wϕ| ≥ 1 − ϕ∞ , this is bounded by a constant multiple of (1 − |w|2 )3/2 h2 . We see now that the integral in (2.6) converges, even for p = 1, and the result follows. The other main result of [10], its Theorem 1, asserts that if Wh,ϕ is bounded on H2 , if ϕ lies in the disc algebra, and if h = 0 on {ζ ∈ T : |ϕ(ζ)| = 1} and h is continuous at all such points ζ, then Wh,ϕ is compact. Here again, we may give a significantly stronger result. Theorem 2.8. Suppose that ϕ : D → D is holomorphic and that for some δ > 0 and cδ > 0 the function h ∈ H2 satisfies |h(z)| ≤ cδ a.e. on the set Aδ := {z ∈ T : |ϕ(z)| ≥ 1 − δ}. Then Wh,ϕ is a bounded operator. If, in addition, cδ → 0 as δ → 0, then Wh,ϕ is a compact operator. Proof. We estimate the H2 norm of (1 − |w|2 )1/2 h/(1 − wϕ) for w ∈ D. Now (1 − |w|2 )|h|2 ≤ c2δ (1 − |w|2 )Cϕ kw 22 ≤ c2δ Cϕ 2 , |1 − wϕ|2 Aδ

whereas

T\Aδ

Thus

(1 − |w|2 )|h|2 (1 − |w|2 )h22 h22 (1 − |w|2 ) ≤ ≤ . 2 2 |1 − wϕ| 1 − |w|(1 − δ) δ2

(1 − |w|2 )1/2 h 2 ≤ c2δ Cϕ 2 + h22 (1 − |w|2 )/δ 2 . 1 − wϕ 2

This is uniformly bounded, independently of w, and so Wh,ϕ is a bounded operator; moreover, if cδ → 0 as δ → 0, then given ε > 0, we may choose δ > 0 sufficiently small such that 2c2δ Cϕ 2 < ε, and then find η > 0 such that, if |w| > 1 − η, one has h22 (1 − |w|2 )/δ 2 < c2δ Cϕ 2 . Hence condition (2.3) is satisfied, and Wh,ϕ is compact. Remark 2.9. There is a related estimate due to Kriete and Moorhouse [16, Thm. 3.1], which implies that if h ∈ H∞ and |h| is continuous on Eϕ , then Wh,ϕ is compact if and only if h vanishes on Eϕ . Theorem 2.8 shows that this condition is sufficient for compactness even without the assumptions on h.

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3. Weighted Bergman Spaces For α > −1 the weighted Bergman space A2α = A2α (D) consists of all analytic functions in D for which the norm, given by ⎛ ⎞1/2 1 |f (z)|2 (1 − |z|2 )α dA(z)⎠ f A2α = ⎝ π D

is finite. It is a reproducing kernel Hilbert space with kernel function given by hw (z) =

α+1 (1 − wz)α+2

(cf. [5, p. 27] and [25, p. 135]). We shall also require the normalized functions hw = hw /hw A2α , given by (1 − |w|2 )(α+2)/2 hw (z) = . (1 − wz)α+2 A necessary and sufficient condition for boundedness of Wh,ϕ on the Bergman space A2 (D) = A20 (D) is given in [17, Thm. 3.1], namely that the measure νh,ϕ defined by νh,ϕ (E) = |h(z)|2 dA(z), ϕ−1 (E)

should be a Hastings–Carleson measure, i.e., the embedding J : A2 (D) → L2 (D, dνh,ϕ ) should be bounded. Such measures were characterised in [13,19]. These can also be tested on normalized reproducing kernels hw , for w ∈ D as in [25, Thm. 7.5]. Zen Harper [11, Thm. 3.3] shows (as part of a more general theory, which we describe in Theorem 3.2, below) that Wh,ϕ is bounded on A2 (D) if and only if (1 − |w|2 )h < ∞. (3.1) sup (1 − wϕ)2 A2 (D) |w|<1 This result generalizes to A2α (D) as follows. Proposition 3.1. Let ϕ be a holomorphic self-map of the disc and h ∈ A2α (D). Then the following conditions are equivalent: (i) The weighted composition operator Wh,ϕ is bounded on A2α (D). (ii) The measure νh,ϕ defined by νh,ϕ (E) = |h(z)|2 (1 − |z|2 )α dA(z) ϕ−1 (E)

is an A2α -Carleson measure, in the sense that the canonical embedding J : A2α (D) → L2 (D, dνh,ϕ ) is bounded.

476 (iii)

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(1 − |w|2 )1+α/2 h < ∞. sup (1 − wϕ)α+2 A2 |w|<1

(3.2)

α

Moreover, Wh,ϕ is compact if and only if νh,ϕ is a vanishing Carleson measure, or, equivalently, if (1 − |w|2 )1+α/2 h (3.3) (1 − wϕ)α+2 2 → 0 as |w| → 1. Aα

Proof. The equivalence of (i) and (ii) follows from the formula |h(z)|2 |f (ϕ(z))|2 (1 − |z|2 )α dA(z) = |f (z)|2 dνh,ϕ (z), D

D 2

adapting the calculation for A (D), given in [17], without significant alteration. Next, observing that | hw |2 dμh,ϕ = |h(z)|2 | hw ◦ ϕ(z)|2 (1 − |z|2 )α dA(z), D

D

and using the fact that A2α -Carleson measures can be tested on normalized reproducing kernels (which may be found in [25, Sec. 7.2]), we see that (iii) implies (ii). Trivially, (i) implies (iii), this being the property that the weighted composition operator is uniformly bounded on the normalized kernels. The proof of the characterization of compactness is similar, and is omitted. To conclude, we mention briefly how one may also derive results that provide characterizations of Schatten class weighted composition operators on A2α (D). The key is the following result of Harper [11, Thm. 3.1, Cor. 3.2]. Theorem 3.2. Let T be a subnormal operator on a Hilbert space H, with spectral radius r(T ) ≤ 1. Let h ∈ H. Then there exists a finite positive Borel measure μ on D such that 2 f (T )h = |f (z)|2 dμ(z) D

for all functions f analytic in a neighbourhood of D. The application to weighted composition operators is the following. Let T = Tϕ denote the operator of multiplication by ϕ on A2α (D), which is subnormal, since A2α (D) is a closed subspace of L2 (D, (1 − |z|2 )α dA(z)). Then f (T )h = h(f ◦ ϕ) = Wh,ϕ f . We thus have the following corollary. Corollary 3.3. For ϕ : D → D holomorphic and h ∈ A2α (D), there exists a finite positive Borel measure μ on D such that Wh,ϕ f 2A2α = |f (z)|2 dμ(z) D

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for all functions f analytic in a neighbourhood of D. Thus Wh,ϕ f = Iμ f , where Iμ : A2α (D) → L2 (D, μ) is the canonical Carleson embedding. It follows that Wh,ϕ is respectively bounded, compact or of Schatten class Sp if and only if Iμ has the same property. We shall omit the detailed calculations, which are entirely analogous to those of [12, Prop. 2.4], but it may now be verified that the Schatten class membership of Iμ , and hence Wh,ϕ , can be tested on reproducing kernels. Namely Iμ ∈ Sp (A2α (D), L2 (μ)) for 1 < p < ∞ if and only if dA(w) Iμ < ∞, hw p (1 − |w|2 )2 D

and Iμ ∈ S1 (A2α (D), L2 (μ)) for 1 < p < ∞ if and only if dA(w) Iμ h˙ w p < ∞, (1 − |w|2 )2 D

where we write h˙ w = stants, we have h˙ w (z) ≈

∂ ∂w hw

and h˙ w = h˙ w /h˙ w . To within irrelevant con-

z (1 − wz)α+3

(1 − |w|2 )(α+4)/2 z and h˙ w (z) ≈ . (1 − wz)α+3

References [1] Ahern, P., Clark, D.: Radial limits and invariant subspaces. Am. J. Math. 2, 332–342 (1970) [2] Chalendar, I., Fricain, E., Partington, J.R.: Overcompleteness of sequences of reproducing kernels in model spaces. Integral Equ. Oper. Theory 56(1), 45–56 (2006) [3] Contreras, M.D., Hern´ andez-D´ıaz, A.G.: Weighted composition operators on Hardy spaces. J. Math. Anal. Appl. 263(1), 224–233 (2001) [4] Contreras, M.D., Hern´ andez-D´ıaz, A.G.: Weighted composition operators between different Hardy spaces. Integral Equ. Oper. Theory 46, 165–188 (2003) [5] Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995) ˘ ckovi´c, Z., Zhao, R.: Weighted composition operators between different [6] Cu˘ weighted Bergman spaces and different Hardy spaces. Ill. J. Math. 51(2), 479– 498 (2007) [7] Duren, P.L.: Theory of Hp Spaces. Academic Press, New York (1970) [8] Forelli, F.: The isometries of H p . Can. J. Math. 16, 721–728 (1964) [9] Garnett, J.B.: Bounded Analytic Functions. Revised ﬁrst edition. Springer, New York (2007) [10] Gunatillake, G.: Compact weighted composition operators on the Hardy space. Proc. Am. Math. Soc. 136(8), 2895–2899 (2008) [11] Harper, Z.: Applications of the discrete Weiss conjecture in operator theory. Integral Equ. Oper. Theory 54(1), 69–88 (2006)

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[12] Harper, Z., Smith, M.P.: Testing Schatten class Hankel operators, Carleson embeddings and weighted composition operators on reproducing kernels. J. Oper. Theory 55(2), 349–371 (2006) [13] Hastings, W.W.: A Carleson measure theorem for Bergman spaces. Proc. Am. Math. Soc. 52, 237–241 (1975) [14] Hoﬀman, K.: Banach Spaces of Analytic Functions. Dover Publications, Inc., NY (1988) [15] Jury, M.: Reproducing kernels, de Branges–Rovnyak spaces, and norms of weighted composition operators. Proc. Am. Math. Soc. 135(11), 3669– 3675 (2007) [16] Kriete, T., Moorhouse, J.: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359(6), 2915–2944 (2007) [17] Kumar R., Partington J.R.: Weighted composition operators on Hardy and Bergman spaces. Recent advances in operator theory, operator algebras, and their applications. Oper. Theory Adv. Appl., vol. 153, pp. 157–167. Birkh¨ auser, Basel (2005) [18] Littlewood, J.E.: On inequalities in the theory of functions. Proc. Lond. Math. Soc. 23(2), 481–519 (1925) [19] Luecking, D.H.: Inequalities on Bergman spaces. Ill. J. Math. 25(1), 1–11 (1981) [20] Matache, V.: Weighted composition operators on H 2 and applications. Complex Anal. Oper. Theory 2(1), 169–197 (2008) [21] Nikolski N.K.: Operators, functions, and systems: an easy reading. Hardy, Hankel, and Toeplitz, vol. 1. Translated from the French by Andreas Hartmann. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence, RI (2002) [22] Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993) [23] Shimorin, S.: Weighted composition operators associated with conformal mappings. Quadrature domains and their applications. Oper. Theory Adv. Appl., vol. 156, pp. 217–237. Birkh¨ auser, Basel (2005) [24] Smith, W.: Brennan’s conjecture for weighted composition operators. Recent advances in operator-related function theory. Contemp. Math., vol. 393, pp. 209–214. Amer. Math. Soc., Providence, RI (2006) [25] Zhu, K.: Operator theory in function spaces. Second edition. Mathematical Surveys and Monographs, vol. 138. American Mathematical Society, Providence, RI (2007)

E. A. Gallardo-Guti´errez Departamento de Matem´ aticas, Universidad de Zaragoza e IUMA Plaza San Francisco s/n, 50009 Zaragoza, Spain e-mail: [email protected] R. Kumar Department of Mathematics, University of Jammu Jammu 180 001, India e-mail: romesh [email protected]

Vol. 67 (2010)

Weighted Composition Operators

J. R. Partington (B) School of Mathematics, University of Leeds Leeds LS2 9JT, UK e-mail: [email protected] Received: July 14, 2009. Revised: April 11, 2010.

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Integr. Equ. Oper. Theory 67 (2010), 481–497 DOI 10.1007/s00020-010-1796-5 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Weyl’s Theorem for Functions of Operators and Approximation Chun Guang Li, Sen Zhu and You Ling Feng Abstract. Let H be a complex separable infinite dimensional Hilbert space. In this paper, we characterize those operators T on H satisfying that Weyl’s theorem holds for f (T ) for each function f analytic on some neighborhood of σ(T ). Also, it is proved that, given an operator T on H and ε > 0, there exists a compact operator K with K < ε such that Weyl’s theorem holds for T + K. Mathematics Subject Classification (2010). Primary 47A10, 47A53; Secondary 47A60, 47A58. Keywords. Weyl’s theorem, function of operators, small-compact closure.

1. Introduction Throughout this paper, C and N denote the set of complex numbers and the set of natural numbers, respectively. H will always denote a complex separable inﬁnite dimensional Hilbert space. We let B(H) denote the algebra of all bounded linear operators on H, and let K(H) denote the ideal of compact operators in B(H). Let T ∈ B(H). We denote by σ(T ) and σp (T ) the spectrum of T and the point spectrum of T , respectively. Denote by N (T ) and R(T ) the kernel of T and the range of T , respectively. T is called a semi-Fredholm operator, if R(T ) is closed and either nul T or nul T ∗ is ﬁnite, where nul T := dim N (T ) and nul T ∗ := dim N (T ∗ ); in this case, ind T := nul T − nul T ∗ is called the index of T . In particular, if −∞ < ind T < ∞, then T is called a Fredholm operator. T is called a Weyl operator if it is Fredholm of index 0. The Wolf spectrum σlre (T ), the essential spectrum σe (T ) and the Weyl spectrum σw (T ) of T are deﬁned by σlre (T ) := {λ ∈ C : T − λ is not semi-Fredholm}, σe (T ) := {λ ∈ C : T − λ is not Fredholm} Supported by NSF of China (10971079) and the Research Foundation for Young Teachers of Department of Mathematics at Jilin University.

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and σw (T ) := {λ ∈ C : T − λ is not Weyl}, respectively. ρs−F (T ) := C\σlre (T ) is called the semi-Fredholm domain of T and ρF (T ) := C\σe (T ) is called the Fredholm domain of T . We shall denote ρ0s−F (T ) := {λ ∈ C : T − λ is Weyl}. Given a subset σ of C, we denote by iso σ the set of all isolated points of σ. If T ∈ B(H), then we denote π00 (T ) := {λ ∈ iso σ(T ) : 0 < nul(λ − T ) < ∞}. Following Coburn [6], we say that Weyl’s theorem holds for T ∈ B(H), denoted by T ∈ (W), if there is equality σ(T )\σw (T ) = π00 (T ). We let T ∈ / (W) denote that Weyl’s theorem does not hold for T . The study of Weyl’s theorem for bounded linear operators has a long history. In 1909, Weyl [21] proved that Weyl’s theorem holds for self-adjoint operators. In 1966, Coburn [6] studied Weyl’s theorem for several classes of non-normal operators including hyponormal operators. Thereafter Weyl’s theorem was extended to several classes of operators including seminormal operators by Berberian [3]. Today, Weyl’s theorem has been extended to various operators acting on both Hilbert spaces and Banach spaces, and a lot of work has been done (see [1,4,5,8–10,12,13,18]). People also investigate Weyl’s theorem for functions of operators. If A ∈ B(H), then, given f ∈ Hol(σ(A)), we let f (A) denote the Riesz-Dunford functional calculus of A with respect to f . In this paper, we denote by Hol(σ(A)) the set of all functions f which are analytic on some neighborhood of σ(A) (the neighborhood depends on f ). The reader is referred to [11, Chapter VII] for more results on Riesz-Dunford functional calculus. Lee and Lee [18] proved that if T is a hyponormal operator on H, then f (T ) ∈ (W) for all f ∈ Hol(σ(T )). Subsequently, the above result was extended to p-hyponormal operators [10], algebraically hyponormal operators [13], algebraically paranormal operators [9], w-hyponormal [12], algebraically M-hyponormal operators [5], analytically Class A operators [4] and algebraically quasi-class A operators [1]. In particular, C. Schmoeger [20] obtained the following result. Theorem 1.1 ([20], Theorem 1). Let T be a Banach space operator and suppose that iso σ(T ) ⊆ σp (T ). If T ∈ (W), then the following assertions are equivalent: (a) ind(λ − T ) · ind(μ − T ) ≥ 0 for all λ, μ ∈ / σe (T ). (b) f (T ) ∈ (W) for all f ∈ Hol(σ(T )). (c) p(T ) ∈ (W) for each polynomial p. It follows immediately from the above result that if T ∈ B(H) and iso σ(T ) ⊆ σp (T ), then f (T ) ∈ (W) for all f ∈ Hol(σ(T )) if and only if T ∈ (W) and ind(λ − T ) · ind(μ − T ) ≥ 0 for all λ, μ ∈ / σe (T ). In this paper, we investigate Weyl’s theorem for functions of Hilbert space operators and extend Theorem 1.1 to general operators. A necessary

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and sufﬁcient condition is given for a Hilbert space operator T to satisfy that f (T ) ∈ (W) for all f ∈ Hol(σ(T )). In order to state our main theorem we ﬁrst recall some notations and terminologies ([16, page 5]). Let T ∈ B(H). If σ is a clopen subset of σ(T ), then there exists an analytic Cauchy domain Ω such that σ ⊆ Ω and [σ(T )\σ] ∩ Ω = ∅. We let E(σ; T ) denote the Riesz idempotent of T corresponding to σ, i.e. 1 (λ − T )−1 dλ, E(σ; T ) = 2πi Γ

where Γ = ∂Ω is positively oriented with respect to Ω in the sense of complex variable theory. In this case, we denote H(σ; T ) = R(E(σ; T )). If λ ∈ iso σ(T ), then {λ} is a clopen subset of σ(T ) and we simply write H(λ; T ) instead of H({λ}; T ); if, in addition, dim H(λ; T ) < ∞, then λ is called a normal eigenvalue of T . The set of all normal eigenvalues of T will be denoted by σ0 (T ). The main result of this paper is the following theorem which extends Theorem 1.1. Theorem 1.2 (Main theorem). Let T ∈ B(H). Then f (T ) ∈ (W) for all f ∈ Hol(σ(T )) if and only if the following conditions hold: (i) T ∈ (W); (ii) ind(λ − T ) · ind(μ − T ) ≥ 0 for all λ, μ ∈ / σe (T ); (iii) If σ0 (T ) = ∅, then iso σ(T ) ⊆ σp (T ). Also, we investigate related approximation problems of operators. Given sc a subset E of B(H), we denote by E the set of all operators T ∈ B(H) satisfying that for any ε > 0, there exists K ∈ K(H) with K < ε such sc that T + K ∈ E. We call E the small-compact closure of E. Obviously sc E ⊆ E ⊆ E. We obtain the following result which implies that both the closure and the small-compact closure of {X ∈ B(H) : X ∈ (W)} coincide with B(H). Theorem 1.3. Given T ∈ B(H) and ε > 0, there exists K ∈ K(H) with K < ε such that T + K ∈ (W). On the other hand, we characterize the small-compact closure of the set {X ∈ B(H) : X ∈ / (W)} and obtain the following result. Theorem 1.4. Let T ∈ B(H). Then T is in the small-compact closure of {X ∈ B(H) : X ∈ / (W)} if and only if at least one of the following conditions holds: (i) T ∈ / (W); (ii) iso[σ(T )\σ0 (T )] = ∅; (iii) C\σw (T ) consists of infinitely many connected components. It follows immediately from Theorem 1.4 that S is not in the small-compact closure of {X ∈ B(H) : X ∈ / (W)}, where S denotes the unilateral shift on H. In this paper it is also proved that {X ∈ B(H) : X ∈ / (W)} is a dense subset of B(H)(Theorem 4.6). Therefore we can conclude that {X ∈ B(H) : X ∈ / (W)}

sc

= {X ∈ B(H) : X ∈ / (W)} = B(H).

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The rest of this paper is organized as follows. In Sect. 2, we shall make some preparations for the proofs of main results. Sect. 3 is devoted to the proof of Theorem 1.2. In Sect. 4, we shall give the proofs of Theorem 1.3 and Theorem 1.4.

2. Preparations We ﬁrst give some useful lemmas. Lemma 2.1 ([19], Theorem 2.10). Let T ∈ B(H) and suppose that σ(T ) = σ1 ∪ σ2 , where σi (i = 1, 2) are clopen subsets of σ(T ) and σ1 ∩ σ2 = ∅. Then H(σ1 ; T ) + H(σ2 ; T ) = H, H(σ1 ; T ) ∩ H(σ2 ; T ) = {0} and T admits the following matrix representation T1 0 H(σ1 ; T ) T = , 0 T2 H(σ2 ; T ) where σ(Ti ) = σi (i = 1, 2). Using [16, Corollary 3.22] and the above lemma, we can obtain the following lemma whose proof is left to the reader. Corollary 2.2. Let T ∈ B(H) and suppose that σ is a clopen subset of σ(T ). Then A ∗ H(σ; T ) A 0 H(σ; T ) T = ∼ , 0 B H(σ; T )⊥ 0 B H(σ; T )⊥ where σ(A) = σ and σ(B) = σ(T )\σ. In this paper, if S, T ∈ B(H), then we let S ∼ T denote that S, T are similar. Lemma 2.3 ([7], Proposition 6.9). Let T ∈ B(H) and λ0 ∈ iso σ(T ). Then the following statements are equivalent. (i) λ0 ∈ σ0 (T ). (ii) λ0 ∈ ρ0s−F (T ). (iii) λ0 ∈ ρs−F (T ). By Lemma 2.1 and Lemma 2.3, it is easy to see that σ0 (T ) ⊆ π00 (T ). Let T ∈ B(H). For λ ∈ ρs−F (T ), the minimal index(see [2]) of λ − T is deﬁned by min ind(λ − T ) := min{nul(λ − T ), nul(λ − T )∗ }. Lemma 2.4 ([16], Corollary 1.14). Let T ∈ B(H). Then the function λ → min · ind(λ − T ) is constant on every component of ρs−F (T ) except for an at most denumerable subset ρss−F (T ) of ρs−F (T ) without limit points in ρs−F (T ). Furthermore, if μ ∈ ρss−F (T ) and λ is a point of ρs−F (T ) in the same component as μ but λ ∈ / ρss−F (T ), then min · ind(λ − T ) < min · ind(μ − T ). Corollary 2.5. Let T ∈ B(H). Then σ0 (T ) ⊆ ρss−F (T ) ∩ ρ0s−F (T ).

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In this paper we denote by card σ the cardinality of a subset σ of C. Lemma 2.6. Let T ∈ B(H) and f ∈ Hol(σ(T )). If 0 ∈ σ(f (T )) and nul f (T ) < ∞, then card{λ ∈ σ(T ) : f (λ) = 0} < ∞; if, in addition, 0 ∈ iso σ(f (T )), then {λ ∈ σ(T ) : f (λ) = 0} ⊆ iso σ(T ). Proof. For a proof by contradiction we assume that {λ ∈ σ(T ) : f (λ) = 0} is an inﬁnite subset of σ(T ). Then we can choose a limit point λ0 of {λ ∈ σ(T ) : f (λ) = 0}. Since f ∈ Hol(σ(T )), there exists a neighborhood Ω of σ(T ) such that f is analytic on Ω. Then there is a component Ω1 of Ω such that λ0 ∈ Ω1 and f ≡ 0 on Ω1 . Set σ1 = σ(T ) ∩ Ω1 and σ2 = σ(T )\σ1 . Then σi (i = 1, 2) are clopen subsets of σ(T ) and σ1 = ∅. By Lemma 2.1, T can be written as T 0 H(σ1 ; T ) T = 1 , 0 T2 H(σ2 ; T ) where σ(Ti ) = σi (i = 1, 2). Then we have f (T1 ) H(σ1 ; T ) 0 0 f (T ) = = 0 0 f (T2 ) H(σ2 ; T )

0 H(σ1 ; T ) . f (T2 ) H(σ2 ; T )

Since λ0 is a limit point of {λ ∈ σ(T ) : f (λ) = 0}, λ0 ∈ σ1 and σ1 is a clopen subset of σ(T ), it is easy to see that card{λ ∈ σ1 : f (λ) = 0} = ∞ and hence dim H(σ1 ; T ) = ∞. Therefore we obtain nul f (T ) = ∞, a contradiction. Thus we have proved that card{λ ∈ σ(T ) : f (λ) = 0} < ∞. If, in addition, 0 is an isolated point of σ(f (T )), then {λ ∈ σ(T ) : f (λ) = 0} = f −1 (0) is a clopen subset of σ(T ) since f is continuous. Then it follows from card{λ ∈ σ(T ) : f (λ) = 0} < ∞ that {λ ∈ σ(T ) : f (λ) = 0} ⊆ iso σ(T ). Lemma 2.7. Let T ∈ B(H) and f ∈ Hol(σ(T )). If 0 ∈ σ(f (T )) and nul f (T ) < ∞, then there exists g ∈ Hol(σ(T )) such that f (T ) = g(T ) and g(λ) = (λ − λ1 )k1 (λ − λ2 )k2 · · · (λ − λn )kn g0 (λ), where λi ∈ σ(T )(1 ≤ i ≤ n), g0 ∈ Hol(σ(T )) and g0 (λ) = 0 for all λ ∈ σ(T ). Proof. Obviously, we need only prove that there exists g ∈ Hol(σ(T )) such that g(T ) = f (T ) and each zero of g on σ(T ) is of ﬁnite multiplicity. We assume that f is analytic on a neighborhood Ω of σ(T ). By Lemma 2.6, 1 ≤ card{λ ∈ σ(T ) : f (λ) = 0} < ∞. Then we may assume that {λi }ni=1 is an enumeration of {λ ∈ σ(T ) : f (λ) = 0}. For each i, the multiplicity of λi is either ﬁnite or ∞. Without loss of generality, we assume that λi (1 ≤ i ≤ m) are all zeroes of multiplicity ∞ of f on σ(T ), where 1 ≤ m ≤ n. For 1 ≤ i ≤ m, assume that Ωi is the component of Ω containing λi (some of these Ωi ’ may coincide). Let 1 ≤ i ≤ m be ﬁxed. Since f is analytic on Ωi and has a zero λi of multiplicity ∞ on Ωi , we deduce that f ≡ 0 on Ωi . Set σi = σ(T ) ∩ Ωi and σi = σ(T )\σi . Then σi and σi are clopen subsets of

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σ(T ), σi ∪ σi = σ(T ) and σi ∩ σi = ∅. By Lemma 2.1, T can be represented as Ti 0 H(σi ; T ) T = , 0 Ti H(σi ; T ) where σ(Ti ) = σi and σ(Ti ) = σi . Then f (Ti ) 0 0 f (T ) = = 0 0 f (Ti )

0 H(σi ; T ) . f (Ti ) H(σi ; T )

Since nul f (T ) < ∞, we have dim H(σi ; T ) < ∞ and hence σi is a ﬁnite set. Note that λi ∈ σi , we obtain λi ∈ iso σ(T ), dim H(λi ; T ) < ∞ and λi ∈ σ0 (T ). Likewise, λj ∈ σ0 (T ) for all 1 ≤ j ≤ m. Set σ = σ(T )\{λi : 1 ≤ i ≤ m}. Then σ is a clopen subset of σ(T ) and, by Lemma 2.1, T can be represented as ⎤ ⎡ H(λ1 ; T ) A1 ⎥ H(λ2 ; T ) ⎢ A 2 ⎥ ⎢ ⎥ ⎢ .. .. T =⎢ , ⎥ . . ⎥ ⎢ ⎦ H(λm ; T ) ⎣ Am A0 H(σ; T ) where σ(Ai ) = {λi }(1 ≤ i ≤ m) and σ(A0 ) = σ. Obviously, we can choose δj > 0(1 ≤ i ≤ m) and an open neighborhood D of σ such that 1. D ∪ [∪m j=1 Bδj (λj )] ⊆ Ω, and 2. Bδ1 (λ1 ), . . . , Bδm (λm ), D are pairwise disjoint. Then D ∪ [∪m j=1 Bδj (λj )] is a neighborhood of σ(T ). For each 1 ≤ i ≤ m, set ki = dim H(λi ; T ). λi ∈ σ0 (T ) implies that ki < ∞ for all i. Since Ai is an operator on H(λi ; T ) and σ(Ai ) = {λi }, it is easy to see that (Ai − λi )ki = 0(1 ≤ i ≤ m). We deﬁne an analytic function g on D ∪ [∪m j=1 Bδj (λj )] as follows (λ − λj )kj , if λ ∈ Bδj (λj ) for some 1 ≤ j ≤ m, g(λ) = f (λ), if λ ∈ D. It is evident that g ∈ Hol(σ(T )) and g(Ai ) = 0 = f (Ai ) for all 1 ≤ i ≤ m. Then ⎤ ⎡ g(A1 ) ⎥ ⎢ g(A2 ) ⎥ ⎢ ⎥ ⎢ . .. g(T ) = ⎢ ⎥ ⎥ ⎢ ⎦ ⎣ g(Am ) f (A0 ) ⎡ ⎤ 0 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ . .. =⎢ ⎥ = f (T ). ⎢ ⎥ ⎣ ⎦ 0 f (A0 )

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Note that λ1 , λ2 , . . . , λn are all zeroes of g on σ(T ) and each zero is of ﬁnite multiplicity. Thus we complete the proof. Given an operator T on H, the essential minimum modulus me (T ) of T is deﬁned by me (T ) = min{λ ∈ σe ((T ∗ T )1/2 )}. Lemma 2.8 ([16], Proposition 6.10). Let T ∈ B(H). Then (i) me (T ) > 0 if and only if R(T ) is closed and nul T < ∞; (ii) me (λ − T ) is continuous for λ ∈ C; moreover, |me (λ1 − T ) − me (λ2 − T )| ≤ |λ1 − λ2 |, ∀λ1 , λ2 ∈ C.

3. Proof of Theorem 1.2 We ﬁrst give a useful lemma. Lemma 3.1. Let T ∈ B(H). Then T ∈ (W) if and only if the following conditions hold: (i) σ(T )\σw (T ) ⊆ σ0 (T ); (ii) π00 (T ) ⊆ σ0 (T ). Proof. “⇐=”. By Lemma 2.3, it is obvious that σ0 (T ) ⊆ σ(T )\σw (T ) and σ0 (T ) ⊆ π00 (T ). Then it follows from (i) and (ii) that σ(T )\σw (T ) = σ0 (T ) = π00 (T ). “=⇒”. By Lemma 2.3, it follows from σ(T )\σw (T ) = π00 (T ) that σ(T )\σw (T ) ⊆ ρ0s−F (T ) ∩ iso σ(T ) = σ0 (T ) and hence π00 (T ) ⊆ σ0 (T ).

Let T ∈ B(H). Then, by Lemma 2.3, it is obvious that σ0 (T ) ⊆ σ(T )\ σw (T ) and σ0 (T ) ⊆ π00 (T ). Hence the following result is obvious. Corollary 3.2. Let T ∈ B(H). Then T ∈ (W) if and only if σ(T ) = σw (T ) ∪ σ0 (T ) and σ0 (T ) = π00 (T ). In this paper, if λ ∈ C and δ > 0, then we denote Bδ (λ) = {z ∈ C : |z − λ| < δ}. Now, we are going to give the proof of Theorem 1.2. Proof of Theorem 1.2. “=⇒”. Assume that f (T ) ∈ (W) for all f ∈ Hol(σ(T )). (i) Set f1 (λ) = λ(λ ∈ C). Then f1 ∈ Hol(σ(T )) and obviously T = f1 (T ) ∈ (W). / σe (T ) such that (ii) If (ii) does not hold, then we can choose λ1 , λ2 ∈ ind(T − λ1 ) = m, ind(T − λ2 ) = n and mn < 0. Without loss of generality, we may assume that m > 0 and n < 0. Deﬁne f2 (z) = (z − λ1 )|n| (z − λ2 )|m| . Then f2 (T ) is a Fredholm operator and ind f2 (T ) = |n|m + |m|n = 0. Hence 0 ∈ σ(f2 (T ))\σw (f2 (T )) = π00 (f2 (T )).

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Since Fredholm index function ind(·) is a continuous function, there exists δ > 0 such that ind(T − μ) = m for all μ ∈ Bδ (λ1 ). Then Bδ (λ1 ) ⊆ σ(T ) and 0 ∈ f2 (Bδ (λ1 )) ⊆ f2 (σ(T )) = σ(f2 (T )). Since f is an open mapping, it follows that f2 (Bδ (λ1 )) is an open set and 0 is an interior point of f2 (Bδ (λ1 )). Furthermore, 0 is an interior point of σ(f2 (T )), contradicting that 0 ∈ π00 (f2 (T )) ⊆ iso σ(f2 (T )). (iii) If (iii) does not hold, then we can choose λ1 ∈ σ0 (T ) and λ2 ∈ iso σ(T ) / σp (T ). By Lemma 2.3, it is obvious that λ2 ∈ / σ0 (T ) and with λ2 ∈ hence λ2 ∈ σlre (T ). By Lemma 2.1, T can be written as ⎡ ⎤ T1 0 0 H(λ1 ; T ) 0 ⎦ H(λ2 ; T ), T2 T =⎣0 0 0 T3 M where M = H(σ; T ) and σ = σ(T )\{λ1 , λ2 }. Then σ(Ti ) = {λi }(i = / σ(T3 ). Since λ2 ∈ / σ0 (T ), we 1, 2), 0 < dim H(λ1 ; T ) < ∞ and λ1 , λ2 ∈ have dim H(λ2 ; T ) = ∞. Deﬁne f3 (z) = (z − λ1 )(z − λ2 ). Then ⎤ ⎡ H(λ1 ; T ) 0 0 f3 (T1 ) 0 ⎦ H(λ2 ; T ). f3 (T2 ) f3 (T ) = ⎣ 0 0 0 f3 (T3 ) M It is not difﬁcult to see that (1) (2) (3)

/ f3 (σ(T3 )) = σ(f3 (T1 )) = f3 (σ(T1 )) = {0} = f3 (σ(T2 )) = σ(f3 (T2 )) and 0 ∈ σ(f3 (T3 )), 0 ∈ iso σ(f3 (T )), H(0; f3 (T )) = H(λ1 ; T ) + H(λ2 ; T ), and 0 < nul f3 (T1 ) < ∞, nul f3 (T2 ) = 0 = nul f3 (T3 ).

Hence nul f3 (T ) = nul f3 (T1 ) + nul f3 (T2 ) + nul f3 (T3 ) = nul f3 (T1 ) and 0 ∈ / σ0 (f3 (T )) since dim H(0; f3 (T )) = ∞. Then, by π00 (f3 (T )). However, 0 ∈ / (W), a contradiction. Lemma 3.1, we can conclude that f3 (T ) ∈ “⇐=”. Arbitrarily choose an f ∈ Hol(σ(T )). It sufﬁces to prove that f (T ) ∈ (W). By Lemma 3.1, it sufﬁces to verify that σ(f (T ))\σw (f (T )) ⊆ σ0 (f (T )) and π00 (f (T )) ⊆ σ0 (f (T )). Step 1. σ(f (T ))\σw (f (T )) ⊆ σ0 (f (T )). Arbitrarily choose a λ0 ∈ σ(f (T )) such that ind(λ0 − f (T )) = 0. Now we shall prove that λ0 ∈ σ0 (f (T )). Obviously, 0 < nul(λ0 −f (T )) < ∞, then, by Lemma 2.6 and Lemma 2.7, we may assume that {λi }ni=1 is an enumeration of {z ∈ σ(T ) : f (z) − λ0 = 0} and f (z) − λ0 = (z − λ1 )k1 · · · (z − λn )kn g(z), where g(z) = 0 for all z ∈ σ(T ). Then f (T ) − λ0 = (T − λ1 )k1 · · · (T − λn )kn g(T ), where g(T ) is invertible. n / σw (f (T )), we have λi ∈ / σe (T ) and i=1 ki · ind(T − λi ) = 0. Since λ0 ∈ It follows from (ii) that ind(T − λi ) = 0. Then λi ∈ σ(T )\σw (T )(1 ≤ i ≤ n).

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Since T ∈ (W), using Lemma 3.1, we obtain λi ∈ σ0 (T )(1 ≤ i ≤ n). Hence T can be represented as ⎤ ⎡ T1 0 ··· 0 0 H(λ1 ; T ) ⎢0 0 0⎥ T2 · · · ⎥ H(λ2 ; T ) ⎢ ⎢ .. . . . .. . .. .. .. .. ⎥ , T =⎢ . ⎥ . ⎥ ⎢ ⎦ ⎣0 0 H(λn ; T ) 0 · · · Tn 0 0 ··· 0 T0 H(σ; T ) where σ = σ(T )\{λi : 1 ≤ i ≤ n}. Then σ(T0 ) = σ, σ(Ti ) = {λi } and dim H(λi ; T ) < ∞(1 ≤ i ≤ n). Hence we get ⎤ ⎡ H(λ1 ; T ) f (T1 ) 0 ··· 0 0 ⎥ ⎢ 0 ) · · · 0 0 f (T H(λ 2 2; T ) ⎥ ⎢ ⎥ ⎢ .. . . . . . .. .. .. .. ⎥ .. . f (T ) = ⎢ . ⎥ ⎢ ⎦ ⎣ 0 0 0 · · · f (Tn ) H(λn ; T ) 0 0 ··· 0 f (T0 ) H(σ; T ) By the spectral mapping theorem, it is not difﬁcult to see that / σ(f (T0 )), σ(f (Ti )) = f (σ(Ti )) = {λ0 }(1 ≤ i ≤ n) and λ0 ∈ λ0 ∈ iso σ(f (T )), and n (5) H(λ0 ; f (T )) = i=1 H(λi ; T ). (4)

Note that dim H(λ0 ; f (T )) < ∞, then we can conclude that λ0 ∈ σ0 (f (T )). Step 2. π00 (f (T )) ⊆ σ0 (f (T )). Arbitrarily choose λ0 ∈ π00 (f (T )), it sufﬁces to prove that λ0 ∈ σ0 (f (T )). Obviously, 0 < nul(f (T ) − λ0 ) < ∞, then, by Lemma 2.6 and Lemma 2.7, we may assume that {λi }ni=1 is an enumeration of {λ ∈ σ(T ) : f (λ) = 0} and f (z) − λ0 = (z − λ1 )k1 . . . (z − λn )kn g(z), where g(z) has no zero on σ(T ). Then f (T ) − λ0 = (T − λ1 )k1 · · · (T − λn )kn g(T ), where g(T ) is invertible. Since λ0 ∈ π00 (f (T )), it follows from Lemma 2.6 that λi ∈ iso σ(T ) and nul(T − λi ) < ∞(i = 1, 2, . . . , n). Obviously, there exists 1 ≤ i0 ≤ n such that 0 < nul(T − λi0 ) < ∞. Then, by Lemma 3.1, it follows immediately from T ∈ (W) that λi0 ∈ σ0 (T ). Hence σ0 (T ) = ∅ and, by (iii), λi ∈ σp (T )(1 ≤ i ≤ n). Then we have 0 < nul(T − λi ) < ∞ and λi ∈ π00 (T )(i = 1, 2, . . . , n). Thus it follows from T ∈ (W) that λi ∈ σ0 (T )(i = 1, 2, . . . , n). Hence f (T ) − λ0 is Fredholm of index 0. Since λ0 ∈ π00 (f (T )) ⊆ iso σ(f (T )), by Lemma 2.3, we can conclude that λ0 ∈ σ0 (f (T )). If one checks the proof of Theorem 1.2, then one can easily obtain the following result. Corollary 3.3. Let T ∈ B(H). Then f (T ) ∈ (W) for all f ∈ Hol(σ(T )) if and only if p(T ) ∈ (W) for all polynomial p(λ).

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Example. Let V be the classical Volterra operator deﬁned on L2 [0, 1] given by 1 (V f )(t) = f (x)dx, f ∈ L2 [0, 1]. t

It is well known that σ(V ) = {0} and σp (V ) = ∅. Hence iso σ(V ) is not a subset of σp (V ). However, using Theorem 1.2, we have f (V ) ∈ (W) for all f ∈ Hol(σ(V )). Remark 3.4. One can easily check that the result of Theorem 1.2 can be extended to Banach space operators.

4. Proofs of Theorem 1.3/1.4 Lemma 4.1 ([16], Theorem 3.48). Let T ∈ B(H). Then, given ε > 0, there exists K ∈ K(H) such that K < ε + max{dist[λ, ∂ρs−F (T )] : λ ∈ σ0 (T )} and min ind(T + K − λ) = 0 for all λ ∈ ρs−F (T ). Corollary 4.2. Let T ∈ B(H) and suppose that σ0 (T ) = ∅. Then, given ε > 0, there exists K ∈ K(H) with K < ε such that σ(T + K) = σw (T + K). Lemma 4.3 ([17], Lemma 3.2.6). Let T ∈ B(H) and suppose that ∅ = Γ ⊆ σlre (T ). Then, given ε > 0, there exists a compact operator K with K < ε such that N ∗ H1 , T +K = 0 A H2 where N is a diagonal normal operator of uniformly infinite multiplicity, σ(N ) = σlre (N ) = Γ, σ(T ) = σ(A), σlre (T ) = σlre (A) and ind(T − λ) = ind(A − λ) for all λ ∈ ρs−F (T ). Proof of Theorem 1.3. For given ε > 0, set

ε . σ1 = λ ∈ σ0 (T ) : dist[λ, ∂ρs−F (T )] ≥ 2 Then σ1 is a ﬁnite, clopen subset of σ(T ). Set σ2 = σ(T )\σ1 . By Corollary 2.2, T admits the following representation T1 ∗ H(σ1 ; T ) , T = 0 T2 H(σ1 ; T )⊥ where σ(Ti ) = σi (i = 1, 2). Then one can verify that ε . 2 Then, by Lemma 4.1, there exists a compact operator K1 on H(σ1 ; T )⊥ such that K1 < 2ε and min ind(T2 + K1 − λ) = 0 for all λ ∈ ρs−F (T2 ). Then σ(T2 + K1 ) = σw (T2 + K1 ). Set 0 0 H(σ1 ; T ) . K1 = 0 K1 H(σ1 ; T )⊥ max{dist[λ, ∂ρs−F (T2 )] : λ ∈ σ0 (T2 )} <

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Then K1 ∈ K(H), K1 <

ε 2

and T T + K1 = 1 0

491

∗ H(σ1 ; T ) . T2 + K1 H(σ1 ; T )⊥

One can easily check that (1) T1 is an operator on H(σ1 ; T )(a ﬁnite dimensional Hilbert space) and hence σ(T1 ) = σ0 (T1 ) = σ1 ; (2) σ(T2 + K1 ) = σw (T2 + K1 ) = σw (T2 ) ⊆ σ2 and hence σ(T2 + K1 ) ∩ σ(T1 ) = ∅; (3) σ(T + K1 ) = σw (T + K1 ) ∪ σ0 (T + K1 ), σw (T + K1 ) = σ(T2 + K1 ) and σ0 (T + K1 ) = σ1 ; (4) iso σ(T + K1 ) = σ1 ∪ iso σ(T2 + K1 ); (5) iso σ(T2 + K1 ) ⊆ σlre (T2 + K1 ) ⊆ σlre (T + K1 ). Denote Γ = iso σ(T2 + K1 ). Then Γ is an at most countable set. Without loss of generality, we assume that Γ = ∅. By Lemma 4.3, there exists a compact operator K2 on H(σ1 ; T )⊥ with K2 < 2ε such that N ∗ , T 2 + K1 + K2 = 0 A where N is a diagonal normal operator of uniformly inﬁnite multiplicity, σ(N ) = Γ, σ(A) = σ(T2 + K1 ), σlre (A) = σlre (T2 + K1 ) and ind(A − λ) = ind(T2 + K1 − λ) for all λ ∈ / σlre (A). It is easy to see that σw (T2 + K1 + K2 ) = σ(T2 + K1 + K2 ) = σ(T2 + K1 ) ⊆ σ2 and hence σ(T2 + K1 + K2 ) ∩ σ(T1 ) = ∅. Note that Γ = iso σ(T2 + K1 ), then Γ ⊆ iso Γ = iso σ(N ) ⊆ σe (N ). Since N is normal, we have nul(T2 + K1 + K2 − λ) ≥ nul(N − λ) = ∞ for all λ ∈ Γ. Set 0 0 H(σ1 ; T ) K2 = . 0 K2 H(σ1 ; T )⊥ Then K2 ∈ K(H), K2 <

ε 2

and

⎡ T1 T + K1 + K2 = ⎣ 0 0

∗ N 0

⎤ ∗ ∗⎦ . A

Set K = K1 + K2 . Then K ∈ K(H) and K < ε. Also, one can easily check that (6) σ(T + K) = σw (T + K) ∪ σ0 (T + K) and σ0 (T + K) = σ1 ; (7) iso σ(T + K) = σ1 ∪ Γ; (8) π00 (T + K) = σ1 (since nul(T + K1 + K2 − λ) = ∞ for all λ ∈ Γ). Therefore we have proved that T + K ∈ (W). Definition 4.4. We say that a certain property (P) (of operators on a Hilbert spaces H) is a bad property if the following three conditions are fulﬁlled: 1. if A has property (P), then α + βA has the property (P) for all α ∈ C and β = 0, 2. if A has the property (P) and T is similar to A, then T has the property (P), and

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if A has the property (P) and σ(A) ∩ σ(B) = ∅, then A ⊕ B has the property (P), where A ⊕ B denotes the orthogonal direct sum of A and B.

Lemma 4.5 ([16], Theorem 3.51). If (P) is a bad property and there exists some operator A with the property (P), then the set {T ∈ B(H) : T satisﬁes (P)} is dense in B(H). Using the above lemma we can obtain the following result. Theorem 4.6. The set {X ∈ B(H) : X ∈ / (W)} is dense in B(H). Proof. By Corollary 3.2, one can prove that “X ∈ / (W)” is a bad property of Hilbert space operators. Then, by Lemma 4.5, the result is obvious. The details are omitted. Let T ∈ B(H). T is said to be quasitriangular if ind(λ − T ) ≥ 0 for all λ ∈ ρs−F (T ). If min ind(λ−T ) ≡ 0 on ρs−F (T ), then we say that T is smooth (see [2]). Obviously, if σ(T ) is a singleton, then T is a smooth quasitriangular operator. Lemma 4.7 ([14], Theorem 2.3). Let T be a smooth quasitriangular operator on H, and let Γ = {λj }∞ j=1 be a sequence of complex numbers such that (i) (ii)

λj ∈ σ(T ) for all j = 1, 2 . . ., every nonempty clopen subset σ of σ(T ) satisfies card{j : λj ∈ σ} = ℵ0 .

Then, given ε > 0, there exists Kε ∈ K(H) with Kε < ε such that ⎡ ⎤ λ1 ∗ ∗ ∗ · · · e1 ⎢0 ∗ ∗ · · ·⎥ λ2 ⎢ ⎥ e2 ⎢0 ⎥ e3 ∗ · · · 0 λ 3 T + Kε = ⎢ ⎥ , ⎢0 ⎥ e4 · · · 0 0 λ 4 ⎣ ⎦ . .. .. .. .. .. . .. . . . . where {en }∞ n=1 is an orthonormal basis of H. Lemma 4.8. Let T ∈ B(H) and suppose that σ(T ) = {0}. Then, given ε > 0, there exists K ∈ K(H) with K < ε such that σ(T + K) = {0} and T + K ∈ / (W). Proof. Obviously, T is a smooth, quasitriangular operator. Then, by Lemma 4.7, for given ε > 0 there exists K1 ∈ K(H) with K1 < 2ε such that T + K1 can be written as ⎤ ⎡ 0 ∗ ∗ ∗ · · · e1 ⎢ 0 0 ∗ ∗ · · · ⎥ e2 ⎥ ⎢ e1 0 ∗ ⎢ 0 0 0 ∗ · · · ⎥ e3 T + K1 = ⎢ = , ⎥ 0 A {e1 }⊥ ⎢ 0 0 0 0 · · · ⎥ e4 ⎦ ⎣ .. .. .. .. . . .. . . . . . .

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where {ei }∞ i=1 is an orthonormal basis of H. It is obvious that σ(A) = {0} = σ(A∗ ) and hence A∗ is also a smooth, quasitriangular operator (see [16, Theorem 3.40]). Using Lemma 4.7 again, we can choose a compact operator K2 with K2 < 4ε such that ⎤ ⎡ 0 0 0 0 · · · f1 ⎢∗ 0 0 0 · · ·⎥ f2 ⎥ ⎢ ⎥ ⎢ A + K2 = ⎢∗ ∗ 0 0 · · ·⎥ f3 , ⎢∗ ∗ ∗ 0 · · ·⎥ f4 ⎦ ⎣ .. .. .. .. . . . . .. . . . . ⊥ where {fi }∞ i=1 is an orthonormal basis of {e1 } . We can choose a compact operator K3 with ⎡ 0 0 0 ⎢λ1 0 0 ⎢ ⎢ 0 λ2 A = A + K2 + K3 = ⎢ ∗ ⎢∗ ∗ λ 3 ⎣ .. .. .. . . .

K3 < 0 0 0 0 .. .

ε 4

⎤

··· · · ·⎥ ⎥ · · ·⎥ ⎥ · · ·⎥ ⎦ .. .

such that f1 f2 f3 , f4 .. .

where λi = 0 for all i ≥ 1. Then it is easy to see that σ(A) = {0} and nul A = 0. Set 0 0 e1 . K = K1 + 0 K2 + K3 {e1 }⊥ Then K ∈ K(H), K < ε ⎡ 0 ∗ ⎢0 0 ⎢ ⎢0 λ1 ⎢ T + K = ⎢0 ∗ ⎢ ⎢0 ∗ ⎣ .. .. . .

and ∗ 0 0 λ2 ∗ .. .

∗ 0 0 0 λ3 .. .

∗ 0 0 0 0 .. .

⎤ · · · e1 · · ·⎥ ⎥ f1 · · ·⎥ 0 ⎥ f2 = · · ·⎥ f 3 0 ⎥ · · ·⎥ f 4 ⎦ . .. . ..

∗ e1 . A {e1 }⊥

One can easily check that σ(T + K) = σw (T + K) = {0}, nul(T + K) = 1 / (W) and we conclude the and hence π00 (T + K) = {0}. Therefore T + K ∈ proof. Corollary 4.9. Let T ∈ B(H) and suppose that σ(T ) = {λ}. Then, given ε > 0, there exists K ∈ K(H) with K < ε such that σ(T + K) = {λ} and T +K ∈ / (W). Proposition 4.10. Let T ∈ B(H). If σ(T ) = σw (T ) ∪ σ0 (T ) and C\σw (T ) consists of at most finite connected components, then there exists δ > 0 such that σ(T + K) = σw (T + K) ∪ σ0 (T + K) for all K ∈ K(H) with K < δ. Proof. Without loss of generality, we may assume that {Ωk }nk=1 is an enumeration of all connected components of C\σw (T ), where n ∈ N. It is obvious that ∪nk=1 Ωk = ρ0s−F (T ). For each 1 ≤ k ≤ n, we can choose λk ∈ Ωk such that T − λk is invertible since σ(T ) = σw (T ) ∪ σ0 (T ). Note that the group of

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invertible operators is an open subset of B(H). Then it follows immediately that there exists δ > 0 such that T + X − λk is invertible for all 1 ≤ k ≤ n and for all X ∈ B(H) with X < δ. Arbitrarily choose K ∈ K(H) with K < δ. It sufﬁces to prove that σ(T + K) = σw (T + K) ∪ σ0 (T + K). Obviously, there exists ε > 0 such that T + K − λ is invertible for all λ ∈ ∪nk=1 Bε (λk ). Hence min ind(T + K − λ) = 0 for all λ ∈ ∪nk=1 Bε (λk ). It follows from Lemma 2.4 that min ind(T +K−λ) = 0 on ∪nk=1 Ωk except for an at most countable set E without limit points in ρ0s−F (T + K). Since ∪nk=1 Ωk = ρ0s−F (T + K), we have σ(T + K)∩(∪nk=1 Ωk ) = E = σ0 (T + K). Note that σ(T + K) = σw (T + K) ∪ (σ(T + K) ∩ ρ0s−F (T + K)). Thus we have proved σ(T + K) = σw (T + K) ∪ σ0 (T + K). Given a subset σ of C, we denote σ c = C\σ. Lemma 4.11 ([15], Theorem 3.1). Let T ∈ B(H) and Φ be the union of a collection of bounded component of (σ(T )\σ0 (T ))c . Then, given ε > 0, there exists K ∈ K(H) such that K < max{me (λ − T ) : λ ∈ Φ} + ε and σ(T + K) = σ(T ) ∪ Φ. Proof of Theorem 1.4. “=⇒”. For a proof by contradiction we assume that T ∈ (W), iso[σ(T )\σ0 (T )] = ∅ and C\σw (T ) consists of ﬁnite connected components. Then, by Corollary 3.2, σ(T ) = σw (T ) ∪ σ0 (T ) and iso σw (T ) = ∅. By Proposition 4.10, there exists δ > 0 satisfying that σ(T + K) = σw (T + K) ∪ σ0 (T + K) for all K ∈ K(H) with K < δ. To complete the proof for the necessity, we need only prove that T +K ∈ (W) for all K ∈ K(H) with K < δ. Fix a K ∈ K(H) with K < δ. Then σ(T + K)\σw (T + K) = σ0 (T + K) ⊆ π00 (T + K). By Corollary 3.2, it remains to prove that π00 (T + K) ⊆ σ0 (T + K). If λ0 ∈ π00 (T + K)\σ0 (T + K), then λ0 ∈ iso σ(T + K) and λ0 ∈ σlre (T + K). Hence we obtain λ0 ∈ iso σw (T + K) = iso σw (T ) = ∅, a contradiction. Thus π00 (T +K) ⊆ σ0 (T +K). So we have proved that T +K ∈ (W). “⇐=”. Arbitrarily choose an ε > 0. Then it sufﬁces to ﬁnd a compact / (W). operator Kε on H with Kε < ε such that T + Kε ∈ Case 1. If T ∈ / (W), then we set Kε = 0. It is obvious that Kε satisﬁes all requirements. Case 2. If iso[σ(T )\σ0 (T )] = ∅ and λ0 ∈ iso[σ(T )\σ0 (T )], then λ0 ∈ σlre (T ) and, for given ε > 0, there exists 0 < δ < 2ε such that σ1 := Bδ (λ0 ) ∩ σ(T ) is a clopen subset of σ(T ) and σ1 consists of λ0 and at most countable normal eigenvalues of T . Moreover, if card σ1 ∩ σ0 (T ) = ∞, then λ0 is the unique limit point of σ1 ∩ σ0 (T ). By Corollary 2.2, T can be represented as A ∗ H(σ1 ; T ) T = , 0 B H(σ1 ; T )⊥

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where σ(A) = σ1 and σ(B) = σ(T )\σ1 . It is not difﬁcult to verify that σ(A) = σlre (A) ∪ σ0 (A), σlre (A) = {λ0 }, σ0 (A) ⊆ σ0 (T ) and max{dist(λ, ∂ρs−F (A)) : λ ∈ σ0 (A)} < 2ε . Then, by Lemma 4.1, there exists a compact K1 acting on H(σ1 ; T ) with K1 < 2ε such that σ(A + K1 ) = σlre (A + K1 ) = {λ0 }. Using Corollary 4.9, there exists a compact K2 acting on H(σ1 ; T ) with K2 < 2ε such that / (W). Note that σ(A0 ) ∩ σ(B) = ∅. σ(A0 ) = {λ0 } and A0 := A + K1 + K2 ∈ Set K1 + K2 0 H(σ1 ; T ) Kε = . 0 0 H(σ1 ; T )⊥ Then Kε ∈ K(H), Kε < ε and, using [16, Corollary 3.22], we have A0 A0 ∗ H(σ1 ; T ) 0 H(σ1 ; T ) T + Kε = ∼ . 0 B H(σ1 ; T )⊥ 0 B H(σ1 ; T )⊥ Since “X ∈ / (W)” is a bad property of Hilbert space operators and invariant under similarity, it follows from σ(A0 ) ∩ σ(B) = ∅ that T + Kε ∈ / (W). Case 3. If C\σw (T ) consists of inﬁnitely many components, then we may assume that {Ωi }∞ i=1 is an enumeration of the bounded connected components of C\σw (T ). Obviously, Σ∞ i=1 m(Ωi ) ≤ m(σ(T )) < ∞, where m(·) denotes the ) is the polynomially convex hull of σ(T ). planar Lebesgue measure and σ(T 2 Then, for given ε > 0, there exists n such that m(Ωn ) < πε4 . Obviously we may directly assume that T ∈ (W). Then, by Corollary 3.2, σ(T ) = σw (T ) ∪ σ0 (T ) and Ωn is a component of [σ(T )\σ0 (T )]c . By Lemma 4.11, there exists Kε ∈ K(H) with Kε < max{me (λ − T ) : λ ∈ Ωn } + 2ε such that σ(T + Kε ) = σ(T ) ∪ Ωn . Obviously, Ωn ⊆ σ(T + Kε )\σw (T + Kε ) and hence σ(T + Kε )\σw (T + Kε ) is not a subset of σ0 (T + / (W). Now it remains to prove Kε ). Therefore we can conclude that T + Kε ∈ that Kε < ε. Since ∂Ωn ⊆ σlre (T ), it follows from Lemma 2.8 that me (λ − T ) = |me (λ − T ) − me (μ − T )| ≤ |λ − μ|, ∀λ ∈ Ωn , μ ∈ ∂Ωn . Then for each λ ∈ Ωn , we have me (λ − T ) ≤ dist(λ, ∂Ωn ). For each λ ∈ Ωn , note that {z ∈ C : |z − λ| < dist(λ, ∂Ωn )} ⊆ Ωn , then π · dist(λ, ∂Ωn )2 ≤

m(Ωn ) and dist(λ, ∂Ωn ) ≤

m(Ωn ) π

< 2ε . Hence we have

Kε < max{me (λ − T ) : λ ∈ Ωn } + This completes the proof.

ε ε ε < + = ε. 2 2 2

Acknowledgements The authors wish to thank Professor You Qing Ji for valuable suggestions concerning this paper.

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References [1] An, I.J., Han, Y.M.: Weyl’s theorem for algebraically quasi-class A operators. Integr. Equ. Oper. Theory 62(1), 1–10 (2008) [2] Apostol, C.: The correction by compact perturbation of the singular behavior of operators. Rev. Roumaine Math. Pures Appl. 21(2), 155–175 (1976) [3] Berberian, S.K.: An extension of Weyl’s theorem to a class of not necessarily normal operators. Michigan Math. J. 16, 273–279 (1969) [4] Cao, X.H.: Analytically Class A operators and Weyl’s theorem. J. Math. Anal. Appl. 320(2), 795–803 (2006) [5] Ch¯ o, M., Han, Y.M.: Riesz idempotent and algebraically M -hyponormal operators. Integ. Equ. Oper. Theory 53(3), 311–320 (2005) [6] Coburn, L.A.: Weyl’s theorem for nonnormal operators. Michigan Math. J. 13, 285–288 (1966) [7] Conway, J.B.: A course in functional analysis, 2nd edn. Graduate Texts in Mathematics, vol. 96. Springer, New York (1990) [8] Curto, R.E., Han, Y.M.: Weyl’s theorem, a-Weyl’s theorem, and local spectral theory. J. Lond. Math. Soc. (2) 67(2), 499–509 (2003) [9] Curto, R.E., Han, Y.M.: Weyl’s theorem for algebraically paranormal operators. Integ. Equ. Oper. Theory 47(3), 307–314 (2003) [10] Duggal, B.P.: The Weyl spectrum of p-hyponormal operators. Integ. Equ. Oper. Theory 29(2), 197–201 (1997) [11] Dunford, N., Schwartz, J.T.: Linear operators. Part I, Wiley Classics Library, General theory, With the assistance of Bade, W.G., Bartle, R.G., Reprint of the 1958 original. Wiley, New York (1988) [12] Han, Y.M., Lee, J.I., Wang, D.M.: Riesz idempotent and Weyl’s theorem for w-hyponormal operators. Integ. Equ. Oper. Theory 53(1), 51–60 (2005) [13] Han, Y.M., Lee, W.Y.: Weyl’s theorem holds for algebraically hyponormal operators. Proc. Am. Math. Soc. 128(8), 2291–2296 (Electronic) (2000) [14] Herrero, D.A.: The diagonal entries in the formula “quasitriangular— compact = triangular” and restrictions of quasitriangularity. Trans. Am. Math. Soc. 298(1), 1–42 (1986) [15] Herrero, D.A.: Economical compact perturbations. II. Filling in the holes. J. Oper. Theory 19(1), 25–42 (1988) [16] Herrero, D.A.: Approximation of Hilbert space operators, vol. 1, 2nd edn. Pitman Research Notes in Mathematics Series, vol. 224. Longman Scientific & Technical, Harlow (1989) [17] Jiang, C.L., Wang, Z.Y.: Structure of Hilbert Space Operators. World Scientific Publishing, Hackensack (2006) [18] Lee, W.Y., Lee, S.H.: A spectral mapping theorem for the Weyl spectrum. Glasgow Math. J. 38(1), 61–64 (1996) [19] Radjavi, H., Rosenthal, P.: Invariant Subspaces, 2nd edn. Dover Publications, Mineola (2003) [20] Schmoeger, C.: On operators T such that Weyl’s theorem holds for f (T ). Extracta Math. 13(1), 27–33 (1998) [21] Weyl, H.: Uber beschrankte quadratische formen, deren differenz, vollsteig ist. Rend. Circ. Mat. Palermo 27, 373–392 (1909)

Vol. 67 (2010)

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Chun Guang Li, You Ling Feng Institute of Mathematics Jilin University Changchun 130012 People’s Republic of China e-mail: [email protected] Sen Zhu (B) Department of Mathematics Jilin University Changchun 130012 People’s Republic of China e-mail: [email protected] You Ling Feng Department of Applied Mathematics Changchun Taxation College Changchun 130117 People’s Republic of China e-mail: youling− [email protected] Received: August 2, 2009. Revised: March 10, 2010.

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Integr. Equ. Oper. Theory 67 (2010), 499–557 DOI 10.1007/s00020-010-1797-4 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Differential and Analytical Properties of Semigroups of Operators Marko Kosti´c Abstract. We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the uniﬁcation of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23:413– 429, 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded ‘commuting’ perturbations. Mathematics Subject Classification (2000). Primary 47D03; Secondary 47D06, 47D60, 47D99. Keywords. Differentiability, analyticity, integrated C-semigroup, convoluted C-semigroup, C-distribution semigroup, ultradistribution semigroup.

1. Introduction and Preliminaries In his celebrated paper of 1987, Arendt [1] generalized Widder’s representation theorem for Laplace transforms and initiated the study of integrated semigroups. The non-existence of an appropriate reference which treats the theoretical aspects of differentiability of integrated semigroups served as a powerful starting point for this research. In the present paper, we deal with the notion of differentiability and analyticity of (local) convoluted C-semigroups [6,19,20,50,63,64,66], C-distribution semigroups and (ultra-)distribution semigroups (cf. [15,16,18, 34,51,56,62,65,72–75,78,84,86,107,109] and references therein) and continue the researches of regularity properties of strongly continuous, integrated and This research was supported by grant 144016 of Ministry of Science and Technological Development, Republic of Serbia.

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C-regularized semigroups [4,5,8–10,22,23,28,33,36,44–46,66,93,94,101,104, 112–116]. Smoothness of (a, k)-regularized C-resolvent families has been recently considered in [59]. For various topics related to differentiability and analyticity of nonlinear semigroups, we refer the reader to [40,42,49,57,58, 89,98,100]. The paper is organized as follows. In Sect. 2, we completely transfer results proved by Barbu [7] to non-dense distribution semigroups. In Theorems 2.5 and 2.6, we precisely describe differentiable (C-)distribution semigroups in terms of spectral properties of their generators and differentiable integrated (C-)semigroups. Following the approach of Pazy [93,94], we establish in Theorem 2.8 the sufﬁcient conditions for the generation of differentiable convoluted C-semigroups. The central theme of the second section is Theorem 2.10 which almost completely characterizes differentiable fractionally integrated semigroups. For further information, we refer to the formulation of Theorem 2.10 and the list of open problems stated at the end of the paper. We discuss in Theorem 2.12 differential properties of automatic extensions of local convoluted C-semigroups and prove that the study of differentiability of local integrated semigroups can be always reduced to the global case. In Theorem 2.19, Corollary 2.20 and Theorem 2.22, we analyze differential properties of perturbed fractionally integrated C-semigroups and (C-)distribution semigroups, and continue the researches of Da Prato and Mosco [26,27] and Fujiwara [38] concerning analytic distribution semigroups. We revisit the backwards heat equation and answer aﬃrmatively to the problem proposed in [63]. Section 3 is devoted to the study of corresponding properties of ultradistribution semigroups. Since the generator of an ultradistribution semigroup need not be stationary dense [72], it is quite questionable how one can remove density assumptions from the formulation of our results given in the third section. Contrary to the case of distribution semigroups, it is not clear whether a differentiable ultradistribution semigroup must be inﬁnitely differentiable (exponential). In the present situation, the best we can do (cf. Theorem 3.5) is to show that every dense differentiable ultradistribution semigroup is always exponential and that the generator of such a semigroup possesses the ultra-polynomially bounded resolvent existing on the regions introduced by Pazy [94]. Regrettably, the spectral characterizations established in Theorem 3.5 are not sufﬁcient for the generation of differentiable ultradistribution semigroups and the problem of ﬁnding a satisfactory Hille–Yosida’s type theorem for differentiable ultradistribution semigroups is quite non-trivial. An additional difﬁculty is that the use of structural theorems for operator-valued ultradistributions [37,55] does not take any eﬀect. The main purpose of Theorem 3.6 is to proﬁle some speciﬁc classes of differentiable ultradistribution semigroups. Distribution semigroups of class CL unify the notion of C0 -semigroups of Gevrey’s class, introduced independently by Taylor [106] and Chen and Triggiani [17], and the proof of generation theorem for this class of distribution semigroups essentially follows from the technique developed in the proof of Theorem 3.6. We consider (pseudo-)differential operators as generators of differentiable integrated semigroups of class CL (C L ). Although far from being satisfactory, Theorem 3.10 clariﬁes

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the sufﬁcient conditions for the generation of (real) analytic ultradistribution semigroups. Finally, the assertions of Theorem 2.2, Theorem 3.2 and Proposition 3.12 may be of some independent interest. Now we will collect the material needed later on. By E and L(E) are denoted a non-trivial complex Banach space and the Banach algebra of bounded linear operators on E. For a closed linear operator A acting on E, D(A), N (A), R(A) and ρ(A) denote its domain, kernel, range and resolvent set, respectively, while [D(A)] stands for the Banach space D(A) equipped ∞ with the graph norm. Put D∞ (A) := n=0 D(An ) and recall that A is stationary dense [72] if n(A) := inf{k ∈ N0 : D(Am ) ⊆ D(Am+1 ) for all m ≥ k} < ∞. If Y is a closed subspace of E, denote by AY the part of A in Y, i.e., AY = {(x, y) ∈ A : x ∈ Y, y ∈ Y }. Henceforth, we assume τ ∈ (0, ∞], K is a complex-valued locally integrat ble function in [0, τ ) and K is not identical to zero. Put Θ(t) := 0 K(s)ds and t k−1 Θ(−k) (t) := 0 (t−s) (k−1)! Θ(s)ds, t ∈ [0, τ ); then it is clear that Θ is an absolutely continuous function in [0, τ ) and that Θ (t) = K(t) for a.e. t ∈ [0, τ ). We employ occasionally the following condition for the function K : (P1) K is Laplace transformable, i.e., K ∈ L1loc ([0, ∞)) and there exists b ∞ ˜ β ∈ R so that K(λ) := L(K)(λ) := limb→∞ 0 e−λt K(t)dt := 0 e−λt K(t)dt exists for all λ ∈ C with Reλ > β. Put abs(K) :=inf{Reλ : ˜ K(λ) exists}. Let us remind that a function K ∈ L1loc ([0, τ )) is called a kernel, if for every t φ ∈ C([0, τ )), the assumption 0 K(t−s)φ(s)ds = 0, t ∈ [0, τ ), implies φ ≡ 0; due to the famous Titchmarsh’s theorem [2], the condition 0 ∈ suppK implies that K is a kernel. Schwartz spaces of test functions on the real line R are denoted by D = C0∞ , E = C ∞ and S. Their strong duals are D , E and S , respectively. By D0 we denote the subspace of D which consists of the elements supported by [0, ∞). Further on, D (L(E)) = L(D, L(E)), E (L(E)) = L(E, L(E)) and S (L(E)) = L(S, L(E)) denote the spaces consisted of all continuous linear mappings D → L(E), E → L(E) and S → L(E), respectively, equipped with the topology of uniform convergence on bounded subsets of D, E and S, respectively; D0 (L(E)), E0 (L(E)) and S0 (L(E)) denote the subspaces of D (L(E)), E (L(E)) and S (L(E)), respectively, containing the elements supported by [0, ∞). Given a non-empty subset K of R, set DK := {ϕ ∈ D : suppϕ ⊆ K}. In this paper, the convolution of operator-valued distributions is taken in the sense of [73, Proposition 1.1]. Let us recall the basic structural properties of operator-valued ultradifferentiable functions and ultradistributions. From now on, we will always assume that (Mp ) is a sequence of positive real numbers such that M0 = 1 and that the following conditions hold: (M.1) Mp2 ≤ Mp+1 Mp−1 , p ∈ N, (M.2) Mn ≤ AH n minp,q∈N,p+q=n Mp Mq , n ∈ N, for some A > 1 and H > 1, and ∞ Mp−1 (M.3)’ p=1 Mp < ∞.

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Every employment of the condition: ∞ Mp+1 Mq−1 < ∞ will be explicitly accented. (M.3) supp∈N q=p+1 pM p Mq The Gevrey sequences (p!s ), (pps ) and (Γ(1 + ps)) satisfy the above conditions, where Γ(·) denotes the Gamma function and s > 1. Put mp := Mp Mp−1 , p ∈ N; then (M.1) implies that (mp ) is increasing and (M.3)’ sim∞ ply means that p=1 m1p < ∞. The associated function of (Mp ) is deﬁned by

M (λ) := supp∈N0 ln |λ| Mp , λ ∈ C\{0}, M (0) := 0. The function t → M (t), t ≥ 0 is increasing, absolutely continuous, and satisﬁes limt→∞ M (t) = +∞ and limt→∞ Mt(t) = 0 (cf. [54] for further information). Since no confusion seems likely, we also denote by M a general positive constant appearing in our estimations. Let l ≥ 1, α > 0, and β ∈ R. By Λα,β,l we denote the ultra-logarithmic region of type l, introduced by Chazarain [15] as follows Λα,β,l := {λ ∈ C : Reλ ≥ αM (l|Imλ|) + β}. We assume that the boundary of Λα,β,l , denoted by Γl , is upwards oriented. The exponential region E(a, b) (a > 0, b > 0) was introduced by Arendt et al. [3]: E(a, b) =: {λ ∈ C : Reλ ≥ b, |Imλ| ≤ eaReλ }. Set also Σθ := {λ ∈ C : λ = 0, | arg λ| < θ}, θ ∈ (0, π]. Recall, the spaces of Beurling, respectively, Roumieu ultradifferentiable (M ) functions are deﬁned by setting D(Mp ) := D(Mp ) (R) := ind limK⊆⊆R DK p , {M } (M ) respectively, D{Mp } := D{Mp } (R) :=ind limK⊆⊆R DK p , where DK p :=proj M ,h {M } M ,h limh→∞ DK p , respectively, DK p :=ind limh→0 DK p , p

M ,h

DK p

:= {ϕ ∈ C ∞ (R) : suppϕ ⊆ K, ||ϕ||Mp ,h,K < ∞} and

||ϕ||Mp ,h,K := sup{

hp |ϕ(p) (t)| : t ∈ K, p ∈ N0 }. Mp

We refer the reader to [54, p. 44] for the notion of the spaces E (Mp ) = E (Mp ) (R) and E {Mp } = E {Mp } (R). With the notation ∗ for both cases of brackets, we denote by D∗ (E) := L(D∗ , E) the space which consists of all continuous linear mappings from D∗ into E; D0∗ , respectively E0∗ denotes the space of elements of D∗ , respectively E ∗ , whose supports are compact subsets of [0, ∞), and E0∗ denotes the space of scalar-valued ultradistributions with the above property. The convolution of Banach space valued ultradistributions (cf. [55,75]) is taken in the sense of the following proposition. Proposition 1.1. Suppose X, Y and Z are Banach spaces and b : X × Y → Z is bilinear and continuous. Then there is a unique bilinear, separately continuous mapping ∗b : D0∗ (X) × D0∗ (Y ) → D0∗ (Z) such that (S ⊗ x) ∗b (T ⊗ y) = S ∗ T ⊗ b(x, y) for all S, T ∈ D0∗ and x ∈ X, y ∈ Y. Moreover, this mapping is hypo-continuous with respect to bounded sets. The spaces of tempered ultradistributions of the Beurling, respectively, Roumieu type, were introduced in [99] as duals of the next test function

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spaces: S (Mp ) := S (Mp ) (R) := proj limh→∞ S Mp ,h (R), respectively, S {Mp } := S {Mp } (R) := ind limh→0 S Mp ,h (R), where, for every h > 0, S Mp ,h (R) := {ϕ ∈ C ∞ (R) : ||ϕ||Mp ,h < ∞} and ||ϕ||Mp ,h := sup{

hα+β (1 + t2 )β/2 |ϕ(α) (t)| : t ∈ R, α, β ∈ N0 }. Mα Mβ

The space S ∗ (E) consists of all linear continuous mappings from S ∗ into E; hereon we assume that the space S ∗ (E) is equipped with the topology of uniform convergence on bounded subsets of S ∗ . Some references on (operatorvalued) ultradistributions and ultradistribution semigroups are [12,13,15– 18,21,34,41–48,54–56,64,65,70,71,75,86,107]. It could be of importance to stress that D(Mp ) (R) → S (Mp ) (R) → S(R) and D{Mp } (R) → S {Mp } (R) → S(R), where → means the continuous and dense embedding. By a regularizing sequence in D, respectively, D∗ , we mean a sequence ∗ (ρ n ) of the form ρn (·) := nρ(n·), where ρ ∈ D, respectively ρ ∈ D , ρ ≥ 0 and ∞ ρ(t)dt = 1. Given a real number t, deﬁne a scalar-valued distribution −∞ (ultradistribution) δt by δt (ϕ) := ϕ(t), ϕ ∈ D(ϕ ∈ D∗ ). Assume (Rk ) is a sequence of positive real numbers such that R0 = 1 and that (M.1) and (M.2) hold for (Rk ). For consistency of terminology with [7], ∞ k 1/k we also employ the sequence (Lk := Rk ) and set ωL (t) := k=0 Lt k , t ≥ 0. k ∞ The non-quasianalyticity of (Lk ) is equivalent to say that k=0 L1k < ∞, or that (Rk ) satisﬁes (M.3)’. The notion of dense distribution semigroups of class C L , respectively, A , ≥ 1(Aγ , ≥ 1, γ ≥ 0) is understood in the sense of [7]. In the continuation of the paper, the use of symbol A is clear from the context. Henceforth we assume C ∈ L(E), C is injective and CA ⊆ AC. Let us recall that the C resolvent set of A, denoted by ρC (A), is the set which consists of all complex numbers λ such that the operator λ − A is injective and that R(C) ⊆ R(λ − A). Definition 1.2. [19,64] Suppose τ ∈ (0, ∞], A is a closed operator and K ∈ L1loc ([0, τ )). If there exists a strongly continuous family (SK (t))t∈[0,τ ) in L(E) such that: (i) SK (t)A ⊆ ASK (t), t ∈ [0, τ ), (ii) SK (t)C = CSK (t), t ∈ [0, τ ), t (iii) 0 SK (s)xds ∈ D(A), x ∈ E, t ∈ [0, τ ) and t SK (s)xds = SK (t)x − Θ(t)Cx, x ∈ E, t ∈ [0, τ ),

A

(1.1)

0

then it is said that A is a subgenerator of a (local) K-convoluted C-semigroup (SK (t))t∈[0,τ ) . It is said that (SK (t))t≥0 is an exponentially bounded, K-convoluted C-semigroup with a subgenerator A if, in addition, there exist M > 0 and ω ≥ 0 such that ||SK (t)|| ≤ M eωt , t ≥ 0.

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Definition 1.3. [63,64] Let α ∈ (0, π2 ] and let (SK (t))t≥0 be a K-convoluted C-semigroup. Then we say that (SK (t))t≥0 is an analytic K-convoluted Csemigroup of angle α, if there exists an analytic function SK : Σα → L(E) which satisﬁes: (i) SK (t) = SK (t), t > 0 and (ii) limz→0,z∈Σγ SK (z)x = 0 for all γ ∈ (0, α) and x ∈ E. It is said that (SK (t))t≥0 is an exponentially bounded, analytic K-convoluted C-semigroup of angle α, if additionally, for every γ ∈ (0, α), there exist Mγ > 0 and ωγ ≥ 0 such that ||SK (z)|| ≤ Mγ eωγ Rez , z ∈ Σγ . Since there is no risk for confusion, we also write SK for SK . The integral generator Aˆ of (SK (t))t∈[0,τ ) is deﬁned by t Aˆ := {(x, y) ∈ E × E : SK (t)x − Θ(t)Cx =

SK (s)yds

for all t ∈ [0, τ )},

0

and it is a closed linear operator which is an extension of any subgenerator of (SK (t))t∈[0,τ ) . Let A be a subgenerator of (SK (t))t∈[0,τ ) . Then we know that ˆ = C −1 AC. Recall that the set consisting of all subgenerators of Aˆ = C −1 AC a (local) convoluted C-semigroup (SK (t))t∈[0,τ ) need not be ﬁnite [64]. The notion of (local) convoluted C-cosine functions is understood in the sense of [64]. r−1 Plugging K(t) = tΓ(r) , where r > 0, in Definitions 1.2 and 1.3, we obtain the well known classes of (analytic) r-times integrated C-semigroups; 0-times integrated C-semigroup is said to be the usual C-regularized semigroup. We need the next proﬁling of analytic convoluted C-semigroups; let us only mention that the assertions (i) and (iv) can be formulated in the case of a general convoluted C-semigroup. Lemma 1.4. [66] Suppose α ∈ (0, π2 ] and K satisﬁes (P1). ˜ can be analytically continued to (i) Let ω ≥ max(0, abs(K)) and let K a function g : ω + Σ π2 +α → C. Suppose, further, that A is the generator of an analytic K-convoluted semigroup (SK (t))t≥0 of angle α and that supz∈Σγ ||e−ωz SK (z)|| < ∞ for all γ ∈ (0, α). Set N = {λ ∈ ω + Σ π2 +α : g(λ) = 0}. Then N ⊆ ρ(A), sup λ∈N ∩(ω+Σ π +γ1 )

||(λ − ω)g(λ)R(λ : A)|| < ∞ for all γ1 ∈ (0, α) and

2

lim

˜ λ→+∞,K(λ)

=0

(ii)

˜ λK(λ)R(λ : A)x = 0,

x ∈ E.

Assume α ∈ (0, π2 ], K satisﬁes (P1), ω ≥ max(0, abs(K)), A is a closed ˜ linear operator with {λ ∈ C : Reλ > ω, K(λ)

= 0} ⊆ ρC (A) and there exists an analytic function q˜ : ω + Σ π2 +α → L(E) satisfying q˜(λ) = ˜ ˜ K(λ)(λ − A)−1 C, λ ∈ C, Reλ > ω, K(λ)

= 0, supλ∈ω+Σ π +γ ||(λ − 2

ω)˜ q (λ)|| < ∞ for all γ ∈ (0, α), and limλ→+∞ λ˜ q (λ)x = 0, x ∈ E, if D(A) = E. Then A is a subgenerator of an exponentially bounded, analytic K-convoluted C-semigroup of angle α.

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Suppose r ≥ 0 and α ∈ (0, π2 ]. Then A is a subgenerator of an exponentially bounded, analytic r-times integrated C-semigroup (Sr (t))t≥0 of angle α iﬀ for every γ ∈ (0, α), there exist Mγ > 0 and ωγ ≥ 0 such that: ωγ + Σ π2 +γ ⊆ ρC (A), −1

||(λ − A)

C|| ≤ Mγ (1 + |λ|)

r−1

,

λ ∈ ωγ + Σ π2 +γ ,

(1.2) (1.3)

the mapping λ → (λ − A)−1 C, λ ∈ ωγ + Σ π2 +γ is analytic (continuous) and

(1.4)

−1

lim

λ→+∞

(λ − A) Cx = χ{0} (r)Cx, x ∈ E, if D(A) = E. λr−1

(1.5)

(Kato’s analyticity criterion for integrated C-semigroups) Suppose r ≥ 0, α ∈ (0, π2 ] and ω ∈ [0, ∞) if r > 0, respectively, ω ∈ R if r = 0. Then A is a subgenerator of an analytic r-times integrated C-semigroup (Sr (t))t≥0 of angle α satisfying supλ∈Σβ ||e−ωz Sr (z)|| < ∞ for all β ∈ (0, α) iﬀ the following conditions hold: (iv.1) For every θ ∈ (−α, α), eiθ A is a subgenerator of an r-times integrated C-semigroup (Sθ (t))t≥0 , and (iv.2) for every β ∈ (0, α), there exists Mβ > 0 such that ||Sθ (t)|| ≤ Mβ eωt cos θ , t ≥ 0, θ ∈ (−β, β). (v) Let θ ∈ (−α, α) and let A be a subgenerator of an exponentially bounded, analytic r-times integrated C-semigroup (Sr (t))t≥0 of angle α. Then eiθ A is a subgenerator of an exponentially bounded, analytic r-times integrated C-semigroup (e−iθr Sr (teiθ ))t≥0 of angle z zr α − |θ|, Sr (z)A ⊆ ASr (z) and A 0 Sr (s)xds = Sr (z)x − Γ(r+1) Cx, z ∈ Σα , x ∈ E.

(iv)

Lemma 1.5. [63,64] Suppose K satisﬁes (P1) and A is a closed linear operator. (i) Suppose M > 0, ω ≥ 0, A is a subgenerator of an exponentially bounded, K-convoluted C-semigroup (SK (t))t≥0 satisfying SK (t) ≤ M eωt , t ≥ 0 and ω1 = max(ω, abs(K)). Then ˜ {λ ∈ C : Reλ > ω1 , K(λ)

= 0} ⊆ ρC (A) and (1.6) ∞ 1 ˜ e−λt SK (t)xdt, Reλ > ω1 , K(λ)

0, x ∈ E. = (λ − A)−1 x = ˜ K(λ) 0

(1.7) (ii)

(iii)

Suppose M > 0, ω ≥ 0, (SK (t))t≥0 is a strongly continuous family in L(E), ||SK (t)|| ≤ M eωt , t ≥ 0 and ω1 = max(ω, abs(K)). If (1.6) and (1.7) are fulﬁlled, then (SK (t))t≥0 is an exponentially bounded, K-convoluted C-semigroup having A as a subgenerator. Suppose ω > max(0, abs(K)), A is a closed linear operator with {λ ∈ ˜ C : Reλ > ω, K(λ)

= 0} ⊆ ρC (A) and there exists an analytic function −1 ˜ C, Reλ > ω, F : {z ∈ C : Rez > ω} → L(E) so that F (λ) = K(λ)(λ−A)

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˜ K(λ)

= 0, and that ||F (λ)|| ≤ M0 |λ|r , Reλ > ω, for some r ≥ −1. Then, for every α > 1, there exist a continuous function S : [0, ∞) → L(E) with S(0) = 0 and M1 > 0 such that ||S(t)|| ≤ M1 eωt , t ≥ 0 and that ∞ −1 α+r ˜ K(λ)(λ − A) Cx = λ e−λt S(t)xdt, Reλ > ω, 0

˜ K(λ)

= 0,

x ∈ E.

Moreover, subgenerator of the norm continuous, exponentially A is aα+r−1 t bounded K ∗0 Γ(α+r) -convoluted C-semigroup (S(t))t≥0 . We need the following family of kernels [20]. Since (Mp ) satisﬁes (M.1) and (M.3)’, one can deﬁne, for every l > 0, the following entire function ∞ lλ ωl (λ) := p=1 (1 + mp ), λ ∈ C. Then it is clear that: |ωl (λ)| ≥ k k (l|λ|)k lλ supk∈N p=1 |1 + m | ≥ supk∈N p=1 l|λ| mp ≥ supk∈N Mp , λ ∈ C, Reλ ≥ 0. p

Hence, |ωl (λ)| ≥ eM (l|λ|) , λ ∈ C, Reλ ≥ 0. It is also worth noting that, for every α ∈ (0, π ), p ∈ N and λ ∈ Σ π , we have |1 + lλ | ≥ l|Imλ| ≥ 0

2

l(1+tan α)−1 |λ| . mp

2

+α

mp

mp

Thereby,

|ωl (λ)| ≥ eM (l(1+tan α)

−1

|λ|)

,

π , l > 0, λ ∈ Σ π2 +α . α ∈ 0, 2

(1.8)

Put now Kl (t) := L−1 ( ωl1(λ) )(t), t ≥ 0, l > 0, where L−1 denotes the inverse Laplace transform. Then, for every l > 0, 0 ∈ suppKl and Kl is inﬁnitely differentiable in t ≥ 0. The proof of the next auxiliary lemma follows from that of [60, Lemma 1.10].

0 I Lemma 1.6. Suppose A is a closed linear operator, λ ∈ C, A ≡ and A 0

C 0 C≡ . Then λ ∈ ρC (A) ⇔ λ2 ∈ ρC (A). In this case, 0 C

λ(λ2 − A)−1 C (λ2 − A)−1 C −1 , (λ − A) C = A(λ2 − A)−1 C λ(λ2 − A)−1 C ||(λ − A)−1 C|| ≤ (1 + |λ|) 1 + |λ|2 ||(λ2 − A)−1 C|| and ||(λ2 − A)−1 C|| ≤ ||(λ − A)−1 C||. Lemma 1.7. [68] Assume α > 0, a > 0, b > 0, M > 0, E(a, b) ⊆ ρC (A),

||(λ − A)−1 C|| ≤ M (1 + |λ|)α ,

λ ∈ E(a, b),

and the mapping λ → (λ − A)−1 C, λ ∈ E(a, b) is continuous. Then, for every β ∈ (α + 1, ∞), the operator A is a subgenerator of a local β-times integrated C-semigroup (Sβ (t))t∈[0,a(β−α−1)) .

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The following extension type theorem for local convoluted C-semigroups essentially follows from the analysis obtained by Cior˘ anescu and Lumer [19]. Here we give the precise formulation. Lemma 1.8. Assume A is a subgenerator of a local K-convoluted C-semigroup (SK (t))t∈[0,τ ) , τ0 ∈ ( τ2 , τ ) and K = K1|[0,τ ) for an appropriate complext (−1) valued function K1 ∈ L1loc ([0, 2τ )). (Put Θ1 (t) = 0 K1 (s)ds and Θ1 (t) = t Θ1 (s)ds, t ∈ [0, 2τ ); since it makes no misunderstanding, we will also write 0 K, Θ and Θ(−1) for K1 , Θ1 and Θ−1 1 , respectively, and denote by K ∗0 K the restriction of this function to any subinterval of [0, 2τ ).) Then A is a subgenerator of a local (K ∗0 K)-convoluted C 2 -semigroup (SK∗0 K (t))t∈[0,2τ0 ) , t which is given by SK∗0 K (t)x := 0 K(t − s)SK (s)Cxds, t ∈ [0, τ0 ], and SK∗0 K (t)x := SK (τ0 )SK (t − τ0 )x ⎞ ⎛ t−τ 0 τ0 +⎝ + ⎠ K(t − r)SK (r)Cxdr, t ∈ (τ0 , 2τ0 ), 0

0

for any x ∈ E. Furthermore, the condition 0 ∈ suppK implies that A is a subgenerator of a local (K ∗0 K)-convoluted C 2 -semigroup on [0, 2τ ). In Theorem 2.12(i) and Remark 2.16(ii) we will assume that the function K has the property stated in the formulation of Lemma 1.8.

2. Differentiability and Analyticity of Convoluted C-Semigroups and C-Distribution Semigroups First of all, we recall the following definition of a C-distribution semigroup. Definition 2.1. [62] Suppose G ∈ D0 (L(E)). If the following condition holds: (qd1) : G(ϕ ∗0 ψ)C = G(ϕ)G(ψ), ϕ, ψ ∈ D, t where ϕ ∗0 ψ(t) := 0 ϕ(t − u)ψ(u)du, t ∈ R, then G is called a pre-(C-DSG) and if, additionally, (qd2) : N (G) := N (G(ϕ)) = {0}, ϕ∈D0

then G is called a C-distribution semigroup. We use the shorthand (C-DSG), respectively (DSG), to denote a C-distribution semigroup, respectively I-distribution semigroup. A pre-(C-DSG) is called dense iﬀ the set R(G) := ϕ∈D0 R(G(ϕ)) is dense in E. Let G be a (C-DSG) and let T ∈ E0 . We deﬁne the operators G(T ) and GC (T ) by setting: G(T ) := {(x, y) ∈ E × E : G(T ∗ ϕ)x = G(ϕ)y for all ϕ ∈ D0 } and GC (T ) := {(x, y) ∈ E × E : G(T ∗ ϕ)Cx = G(ϕ)y for all ϕ ∈ D0 }.

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Owing to (qd2), G(T ) is a function, and it can be simply proved that G(T ) is a closed linear operator and that G(T )C = GC (T ), T ∈ E0 . The inﬁnitesimal generator of a (C-DSG) G is deﬁned by A := G(−δ ). A (C-DSG) G is said to be an exponential C-distribution semigroup, in short (C-EDSG), iﬀ there exists ω ∈ R such that e−ωt G ∈ S (L(E)). Arguing as in the proofs of [62, Theorems 4.2, 4.4], we have that a closed linear operator A generates a (C-DSG) iﬀ for every τ > 0 there exists nτ ∈ N such that A is the integral generator of a local nτ -times integrated C-semigroup on [0, τ ), and that every (C-DSG) is uniquely determined by its generator. Theorem 2.2. Suppose a > 0, b > 0, α > 0, M > 0, E(a, b) ⊆ ρC (A), the mapping λ → (λ − A)−1 C, λ ∈ E(a, b) is continuous and ||(λ − A)−1 C|| ≤ ∞ M (1 + |λ|)α , λ ∈ E(a, b). Set ϕ(λ) ˜ := −∞ eλt ϕ(t)dt, ϕ ∈ D and G(ϕ)x := 1 ˜ − A)−1 Cxdλ, x ∈ E, ϕ ∈ D, where Γ is the upwards oriented 2πi Γ ϕ(λ)(λ boundary of E(a, b). Then G is a (C-DSG) generated by C −1 AC. Proof. By [29, Remark 2.7], we may assume that the mapping λ → (λ − A)−1 C is analytic on some open neighborhood of the region E(a, b). Using the Paley–Wiener theorem, the Cauchy formula, the inverse Fourier transform as well as the simple equalities A(λ − A)−1 Cx = λ(λ − A)−1 Cx − Cx, λ ∈ ρC(A), x ∈ E and ϕ (λ) = −λϕ(λ), ˜ λ ∈ C, it follows that: suppG ⊆ 1 ϕ(λ)dλ ˜ = ϕ(0), ϕ ∈ D and [0, ∞), 2πi Γ AG(ϕ)x = G(−ϕ )x − ϕ(0)Cx, ϕ ∈ D, x ∈ E.

(2.1)

Let ψ ∈ D and x ∈ E be ﬁxed. Put P := δ ⊗ I − δ ⊗ A, U := G(·)G(ψ)x, V := G(· ∗0 ψ)Cx and consider G as an element of the space D0 (L(E, [D(A)])); clearly, P ∈ D0 (L([D(A)], E)), U ∈ D0 ([D(A)]) and V ∈ D0 ([D(A)]). Since G(ϕ)A ⊆ AG(ϕ), ϕ ∈ D and (2.1) holds, we have G ∗ P = δ ⊗ C[D(A)] ∈ D0 (L([D(A)])), P ∗G = δ ⊗C ∈ D0 (L(E)) and P ∗U = P ∗V = δ ⊗G(ψ)Cx ∈ D0 (E). The associativity of convolution implies (G ∗ P ) ∗ U = (G ∗ P ) ∗ V, i.e., (δ ⊗ C[D(A)] ) ∗ U = (δ ⊗ C[D(A)] ) ∗ V ∈ D0 ([D(A)]) and CG(ϕ)G(ψ)x = CG(ϕ ∗0 ψ)Cx, ϕ ∈ D. The injectiveness of C implies (qd1). The proof of (qd2) can be obtained by means of the arguments employed in [73,84]. In fact, the supposition G(ϕ)x = 0, ϕ ∈ D0 implies that suppG(·)x ⊆ {0} and k that there exist k ∈ N and y0 , . . . , yk ∈ D(A) such that G(·)x = j=0 δ (j) ⊗yj k k and that j=0 δ (j+1) ⊗ yj − j=0 δ (j) ⊗ Ayj = δ ⊗ Cx. This implies yk = · · · = y0 = Cx = x = 0 and (qd2). Hence, G is a (C-DSG) whose gener˜ satisﬁes C −1 AC ⊆ A. ˜ By [62, Theorem 4.2(iii)], there ator, denoted by A, ˜ exist n1 ∈ N and τ1 ∈ (0, ∞) such that A is the integral generator of a local n1 -times integrated C-semigroup (Sn1 (t))t∈[0,τ1 ) . Furthermore, Lemma 1.7 implies that there exist n2 ∈ N and τ2 ∈ (0, ∞) such that C −1 AC is the integral generator of a local n2 -times integrated C-semigroup (Sn2 (t))t∈[0,τ2 ) . Without loss of generality, we may assume n = n1 = n2 and τ = τ1 = τ2 . t Denote S(t)x = Sn (t)x − Sn (t)x, t ∈ [0, τ ), x ∈ E. Then A˜ 0 S(s)xds = S(t)x, t ∈ [0, τ ), x ∈ E and arguing as in the proofs of [87, Proposition 2.6] and [77, Lemma 1.2], one gets S(t) = 0, t ∈ [0, τ ). This implies A˜ = C −1 AC and completes the proof of theorem.

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We introduce differentiable and analytic (C-DSG)’s following the approach of Barbu [7] (cf. [26,27,38]). Definition 2.3. Suppose G is a (C-DSG), j ∈ N and α ∈ (0, π2 ]. Then it is said that G is an (inﬁnitely, j-times) differentiable C-distribution semigroup, respectively, an analytic C-distribution semigroup of angle α, iﬀ the mapping t → GC (δt ) ∈ L(E), t > 0 is (inﬁnitely, j-times) differentiable, respectively iﬀ the mapping t → GC (δt ) ∈ L(E), t > 0 can be analytically extended to the sector Σα . Hereon we assume that the space L(E) is equipped with the strong topology and, in the case when G is an analytic (C-DSG) of angle α, we designate the analytic extension of the mapping t → GC (δt ), t > 0 by the same symbol. In the case C = I, there is no risk for confusion and we use the shorthand G, respectively (EDSG), to denote a distribution semigroup, respectively an exponential distribution semigroup; we also write G(δt ) for GI (δt ), t > 0. The next characterization of differentiable (DSG)’s with densely deﬁned generators was proved by Barbu. Theorem 2.4. [7]. Suppose that A is a closed, densely deﬁned linear operator. Then A generates a differentiable (DSG) iﬀ there exist n ∈ N and ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that Υσ,ω := {λ ∈ C : Reλ ≤ ω, Reλ ≥ −σ ln |Imλ| + Cσ } ∪ {λ ∈ C : Reλ ≥ ω} ⊆ ρ(A)

(2.2)

and ||R(λ : A)|| ≤ Mσ (1 + |λ|)n , λ ∈ Υσ,ω .

(2.3)

Now we state the following important extension of Theorem 2.4. Theorem 2.5. Suppose A is a closed, linear operator. Then (i) ⇔ (ii) ⇔ (iii) ⇔ (iv), where: (i) There exists n ∈ N such that A generates an exponentially bounded n-times integrated semigroup (Sn (t))t≥0 such that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable. (ii) There exists n ∈ N such that A generates an exponentially bounded n-times integrated semigroup (Sn (t))t≥0 such that the mapping t → Sn (t), t > 0 is (n + 1)-times differentiable. (iii) The operator A generates a differentiable (DSG). (iv) There exist n ∈ N and ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that (2.2) and (2.3) hold. Proof. The implication (i) ⇒ (ii) is trivial. Suppose that (ii) holds and put ∞ n ϕ(n) (t)Sn (t)xdt, x ∈ E, ϕ ∈ D. G(ϕ)x := (−1) 0

Then G is an (EDSG) [62] and, by the closed graph theorem, the differendn tiability of G follows immediately if one shows that dt n Sn (t) ⊆ G(δt ), t > 0.

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Since G(δt ) = {(x, y) ∈ E × E : G(ϕ(· − t))x = G(ϕ)y for all ϕ ∈ D0 }, t > 0, one has to prove that, for every x ∈ E, t > 0 and ϕ ∈ D(0,∞) : ∞ ϕ

(n)

∞ (s − t)Sn (s)xds =

0

ϕ(n) (s)Sn (s)Sn(n) (t)xds

0

∞ =

ϕ(n) (s)

0

dn Sn (s)Sn (t)xds. dtn

(2.4)

To prove (2.4), notice that, for every ϕ ∈ D(0,∞) , t > 0 and x ∈ E, ∞ 0

dn Sn (s)Sn (t)xds dtn ⎡⎛ t+s t ⎞ ⎤ ∞ n n−1 (t + s − r) d − ⎠ = ϕ(n) (s) n ⎣⎝ Sn (r)xdr⎦ ds dt (n − 1)! 0 s 0 ⎡⎛ t+s t ⎞ ⎤ ∞ n−2 n−1 (t + s − r) dn−1 ⎣⎝ s (n) − ⎠ = ϕ (s) n−1 Sn (r)xdr − Sn (t)x⎦ ds dt (n − 2)! (n − 1)! 0 s 0 ⎡⎛ t+s t ⎞ ⎤ ∞ n−2 (t + s − r) dn−1 ⎣⎝ (n) − ⎠ = ϕ (s) n−1 Sn (r)xdr⎦ ds. dt (n − 2)! ϕ(n) (s)

0

s

0

Repeating this procedure sufﬁciently many times leads us to the following: ∞ ϕ

(n)

(s)Sn (s)Sn(n) (t)xds =

0

∞ ϕ

(n)

∞ (s)Sn (t + s)xds =

0

ϕ(n) (s − t)Sn (s)xds.

0

Thus, (2.4) holds and G is differentiable, as claimed. Suppose now that A generates a differentiable (DSG) G and that t > 0. By [72, Theorem 3.5], the operator A is stationary dense. Put F := D(An(A) ) and G|F (ϕ)x := G(ϕ)x, ϕ ∈ D, x ∈ F. Then one can simply verify that G|F is a dense (DSG) in the Banach space F with the generator AF . This implies that there exists n ∈ N such that (cf. [62,73,109,110]), for every k ∈ N, AF generates a F (s))s∈[0,kt) in F which additionally local (kn)-times integrated semigroup (Skn satisﬁes: ∞ G|F (ϕ)x = (−1)kn

F ϕ(kn) (s)Skn (s)xds,

x ∈ F, ϕ ∈ D(−∞,kt) .

0

Let x ∈ D∞ (A). We will prove that G(δt )x ∈ F and that G|F (δt )x = d2n F dt2n S2n (t)x. Suppose ϕ ∈ D[0,(k−1)t) for some k ∈ N with k ≥ 3. Since

Vol. 67 (2010) F Skn (s)x =

s 0

Differential and Analytical Properties

511

(s−r)(k−2)n−1 F ((k−2)n−1)! S2n (r)xdr, s

F (s) Skn

∈ [0, 2t), we have

kn

d2n F d F F S (r)x = Skn (s) S (r)x dr2n 2n drkn kn r=t r=t kn

d F = S F (s)Skn (r)x , drkn kn r=t

for all s ∈ [0, (k − 1)t). Then the proof of implication (ii) ⇒ (iii) yields: ∞

F ϕ(kn) (s − t)Skn (s)xds =

0

∞ 0

ϕ(kn) (s)(

dkn F F S (s)Skn (r)x)r=t ds. drkn kn

2n

d F Hence, G(δt )x = G|F (δt )x = dt 2n S2n (t)x, x ∈ D∞ (A). Exploitation of [72, Proposition 2.1(iv)] gives that D∞ (A) = F, and the continuity of the mapping s → G(δs ), s > 0 implies G(δt )x = G|F (δt )x ∈ F. Thereby, G|F is a dense differentiable (DSG) in F generated by AF . The assertion (iv) is a consequence of Theorem 2.2 and [72, Corollary 2.2]. We will prove the implication (iv) ⇒ (i) by the next insignificant modiﬁcation of the proof of [94, Theorem 4.7, p. 54]. Suppose ω0 > ω and put Γ1 := {λ ∈ C : Reλ = 2Cσ − σ ln(−Imλ), −∞ < 2Cσ 2Cσ 2Cσ Imλ ≤ −e σ }, Γ2 := {λ ∈ C : Reλ = ω0 , −e σ ≤ Imλ ≤ e σ }, Γ3 := 2Cσ {λ ∈ C : Reλ = 2Cσ − σ ln(Imλ), e σ ≤ Imλ < +∞}, Γ := Γ1 ∪ Γ2 ∪ Γ3 and Γk := {λ ∈ Γ : |λ| ≤ k}, k ∈ N. The curves Γ and Γk are oriented so that Imλ increases along Γ and Γk , k ∈ N. Set, for a sufﬁciently large 1 eλt R(λ:A) k0 ∈ N, S k (t) := 2πi n+2 dλ, t ≥ 0, k ≥ k0 . It is straightforwardly Γk λ λt j−n−2 j d 1 k checked that dtj S (t) = 2πi Γk e λ R(λ : A)dλ, t ≥ 0, k ≥ k0 , j ∈ N. Suppose k0 < k < l and ς > 0. Then there exists c > 0 such that dj 1 dj l k λt j−n−2 = e λ R(λ : A)dλ j S (t) − j S (t) 2π dt dt Γl ∩{λ∈C:k≤|λ|≤l} Mσ 2Cσ t ≤c e |Imλ|−σt (1 + |Cσ − ln |Imλ| + iImλ|)j−2 |dλ|, 2π Γl ∩{λ∈C:k≤|λ|≤l}

for all j ∈ N0 . Since |Imλ|−σt (1 + |Cσ − ln |Imλ| + iImλ|)j−2 ∼ |Imλ|j−2−σt , |λ| → ∞, λ ∈ Γ, j

d k one gets that, for every j ∈ N0 and t > max( j−1 σ , 0), the sequence ( dtj S (t))k is convergent in L(E) and that the convergence is uniform on every compact dj k subset of [max( j−1 σ , 0) + ς, ∞). Put Sj (t) := limk→∞ dtj S (t), j ∈ N0 , t > j d max( j−1 σ , 0). It is obvious that dt Sj (t) = Sj+1 (t), j ∈ N0 , t > σ + ς. This j implies that the mapping t → S0 (t), t > σ +ς is j-times differentiable and that ω0 +i∞ λt R(λ:A) j dj 1 dtj S0 (t) = Sj (t), t > σ + ς. Set also S(t) := 2πi ω0 −i∞ e λn+2 dλ, t ≥ 0. Then the proof of [2, Theorem 2.5.1] combined with Lemma 1.5(iii) implies that (S(t))t≥0 is an exponentially bounded (n + 2)-times integrated semigroup, and thanks to the residue theorem, we have that S0 (t) = S(t), t > σ1 .

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Consequently, the mapping t → S(t), t > σj + ς is j-times differentiable. The arbitrariness of σ and ς shows that the mapping t → S(t), t > 0 is inﬁnitely differentiable, as required. The proof of theorem is completed. Theorem 2.6. Let A be a closed, linear operator and let j ∈ N. (i) Consider the following assertions: (i.1) There exist n ∈ N and ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that: Υσ,ω ⊆ ρC (A), −1

||(λ − A)

(2.5)

C|| ≤ Mσ (1 + |λ|) , λ ∈ Υσ,ω and n

−1

the mapping λ → (λ − A)

C, λ ∈ Υσ,ω is continuous.

(2.6) (2.7)

There exists n ∈ N such that A is a subgenerator of an exponentially bounded n-times integrated C-semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable. (i.3) There exists n ∈ N such that A is a subgenerator of an exponentially bounded n-times integrated C-semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is (n + j)-times differentiable. (i.4) The operator C −1 AC generates an inﬁnitely differentiable (C-EDSG). (i.5) The operator C −1 AC generates a j-times differentiable (C-EDSG). Then (i.1) ⇒ (i.2) ⇒ (i.4) and (i.2) ⇒ (i.3) ⇒ (i.5). Furthermore, if A generates a dense (inﬁnitely, j-times) differentiable (C-EDSG), then there exists n ∈ N such that A is the integral generator of an exponentially bounded n-times integrated C-semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is (inﬁnitely, (n + j)-times) (n + 1)-times differentiable. (ii) We have (ii.1) ⇔ (ii.2) and (ii.3) ⇒ (ii.1), where: (ii.1) For every τ > 0 there exists nτ ∈ N such that A is the integral generator of an nτ -times integrated C-semigroup (Snτ (t))t∈[0,τ ) satisfying that the mapping t → Snτ (t), t ∈ (0, τ ) is (inﬁnitely) (nτ + j)-times differentiable. (ii.2) There exists n ∈ N such that A is the integral generator of a global n-times integrated C-semigroup (Sn (t))t≥0 such that the mapping t → Sn (t), t > 0 is (inﬁnitely) (n + j)-times differentiable for some j ∈ N. (ii.3) The operator A is the generator of a dense (inﬁnitely) j-times differentiable (C-DSG) G. (i.2)

Proof. The implication (i.1) ⇒ (i.2) follows from the proof of Theorem 2.5 and [29, Remark 2.7] while the implications (i.2) ⇒ (i.4) and (i.3) ⇒ (i.5) follow from the proofs of Theorem 2.5 and [62, Theorem 4.4, Proposition 4.6(ii)]. The implication (ii.2) ⇒ (ii.1) is trivial. Let us prove that (ii.1) implies (ii.2). Suppose to the contrary that (ii.2) does not hold. Choose τ > 0 arbitrarily and assume that ε ∈ (0, τ ) and that A is the integral generator

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of a local n-times integrated C-semigroup (Sn (t))t∈[0,τ ) such that the mapping t → Sn (t), t ∈ (0, τ ) is (inﬁnitely) (n + j)-times differentiable. By the hypothesis, there exists m ∈ N such that m > n + j + 1 and that A is the integral generator of a local m-times integrated C-semigroup (Sm (t))t∈[0,2τ ) such that the mapping t → Sm (t), t ∈ (0, τ ) is (inﬁnitely) (m + j)-times difdm−n ferentiable. Deﬁne S˜n (t) := Sn (t), t ∈ [0, 1 − ε] and S˜n (t) := dt m−n Sm (t), t ∈ t (t−s)m−n−1 ˜ (1 − ε, 2τ ). Since Sn (t)x = Sn (s)xds, t ∈ [0, τ ), x ∈ E, it fol0

(m−n−1)!

lows that (S˜n (t))t∈[0,2τ ) is a strongly continuous operator family and that the mapping t → S˜n (t), t ∈ (0, 2τ ) is (inﬁnitely) (n + j)-times differentiable. Furthermore, it can be easily seen that (S˜n (t))t∈[0,2τ ) is a local n-times integrated C-semigroup having A as a subgenerator. Continuing this procedure yields the validity of (ii.2). To show that (ii.3) implies (ii.1), we argue as follows. By the proof of [62, Theorem 4.2], we have that for every τ > 0 there exists nτ ∈ N such that A is the integral generator of a local, norm continuous nτ -times integrated C-semigroup (Snτ (t))t∈[0,τ ) which satisﬁes G(ϕ) = ∞ (−1)nτ 0 ϕ(nτ ) (t)Snτ (t)dt, ϕ ∈ D(−∞,τ ) . Since G is dense, one obtains that ∞ G(ϕ) = 0 ϕ(t)GC (δt )dt, ϕ ∈ D(0,∞) . Let τ > 0 be ﬁxed and let a ∈ (0, τ ). Using integration by parts, we get the existence of operators Bi ∈ L(E), i = 0, . . . , nτ − 1 such that t tnτ −1 t1 n τ −1 · · · GC (δt0 )dt0 · · · dtnτ −1 = ti Bi , Snτ (t) − a

a

t ∈ (a, τ ).

i=0

a

Hence, the mapping t → Snτ (t), t ∈ (0, τ ) is nτ -times differentiable, dnτ dtnτ Snτ (t) = GC (δt ), t ∈ (0, τ ) and (ii.1) holds. The remnant of the proof of (i) can be deduced along the same lines. Remark 2.7. (i) Suppose A generates a differentiable (DSG) G, (ρk ) is a regularizing sequence in D and t > 0. Then the proof of Theorem 2.5 yields that the mapping t → G(δt ), t > 0 is inﬁnitely differentiable and ∞ that G(ϕ)x = 0 ϕ(t)G(δt )xdt, x ∈ E, ϕ ∈ D(0,∞) . Let us also notice that G is an (EDSG), G|F is a dense differentiable (DSG) in the Banach space F generated by AF and that G(δt )x = G|F (δt )x, t > 0, x ∈ F. By Theorem 2.5, there exists n ∈ N such that A generates an exponentially bounded n-times integrated semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable. Hence, ∞ G(δt ∗ ρk )x = (−1)

(n)

ρk (s − t)Sn (s)xds

n 0

∞ ρk (s − t)

= 0

dn Sn (s)xds dsn

and limk→∞ G(δt ∗ ρk )x = G(δt )x, x ∈ E. Furthermore, the dominated convergence theorem implies

514

M. Kosti´c d G(δt ∗ ρk )x = (−1)n+1 dt

∞

ρk (s − t) 0

(s − t)Sn (s)xds

0

∞ =

(n+1)

ρk

IEOT

dn+1 Sn (s)xds, dsn+1 n+1

d d d and limk→∞ dt G(δt ∗ ρk )x = dt G(δt )x = dt n+1 Sn (t)x, x ∈ E. Induc ∞ l ∞ (n+l) d n+l ρk (s − t)Sn (s)xds = 0 ρk tively, dtl G(δt ∗ ρk )x = (−1) 0 n+l

d (s − t) ds n+l Sn (s)xds and

dl dl dn+l G(δt ∗ ρk )x = l G(δt )x = n+l Sn (t)x, l ∈ N0 , x ∈ E. l k→∞ dt dt dt lim

(2.8)

Having in mind [72, Proposition 2.1(iii)], we get that the spectral characterizations clariﬁed in [7, Theorem 4, Corollaries p. 423, 427, Theorem 5] are necessary for the generation of non-dense (DSG)’s of class C L , respectively A , ≥ 1(Aγ , ≥ 1, γ ≥ 0). The sufﬁciency of such spectral characterizations follows from (2.8) and the proofs of cited results. In particular, [7, Corollary, p. 423] completely describes the spectral properties of generators of (non-dense) real analytic (DSG)’s. Scalar type operators which generate C0 -semigroups of a Gevrey class s > 1 (cf. [85, Theorem 5.1]) as well as generators of Crandall–Pazy class of C0 -semigroups (cf. [9, Theorem 2.6], [33] and [46, Deﬁnition 5.4, Theorem 5.5]) can be used for the construction of (DSG)’s of class As . Finally, one can introduce and study inﬁnitely differentiable (C-DSG)’s of class C L , respectively A (Aγ ). Using Theorem 2.2, [62, Proposition 5.5] and the proofs of quoted results from [7], we are in a position to clarify the sufﬁcient conditions for the generation of dense (C-DSG)’s of class C L , respectively A (Aγ ). (ii) Suppose that A generates a global n-times integrated semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is n-times continuously differentiable. By [61, Corollary 3.3], we have that dn ( dt n Sn (t))t≥0 is a semigroup [44] and this implies that (Sn (t))t≥0 must be exponentially bounded. (iii) In the case C = I, a dense inﬁnitely differentiable (C-DSG) need not be exponential and the estimates (2.5)–(2.7) are not necessary for the generation of (dense) inﬁnitely differentiable (C-EDSG)’s. To justify the ﬁrst statement, consider the operator A := −Δ with the Dirichlet boundary conditions on E := L2 [0, π]. It is well known [28] that −Δ generates a dense non-exponential (C-DSG) G and that the mapping t → GC (δt ), t > 0 can be extended to an entire function taking values in L(E). To justify the second statement, put E := l1 , C(t)xn := e−n cos(nt)xn , t ≥ 0, xn ∈ l1 , C := C(0), D(A) := {xn : −n2 xn ∈ l1 }, Axn := −n2 xn , xn ∈ D(A) and, for every

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515

t ≥ 0, xn ∈ l1 and yn ∈ l1 : t

t xn xn C(s)ds 0 (t − s)C(s)ds 0 t S(t) := . yn yn C(t) − C C(s)ds 0 Then the operator A is the integral generator of a bounded C-regularized cosine function (C(t))t≥0 in l1 (cf. [102] for the notion) and the operator A is the integral generator of a once integrated C-semigroup (S(t))t≥0 in l1 × l1 satisfying that ||S(t)|| = O(1 + t), t ≥ 0 and that the mapping t → S(t), t ≥ 0 is inﬁnitely differentiable. Notice / ρC (A) and that Lemma 1.6 implies that {λ : Reλ > 0} ⊆ ρC (A), −n2 ∈ ±ni ∈ / ρC (A), n ∈ N. Thereby, the estimate (2.5) is not necessary for the generation of differentiable exponentially (polynomially) bounded integrated C-semigroups which satisfy the condition (ii) quoted in the formulation of Theorem 2.6. For further information related to differentiability of hermitian and positive integrated C-semigroups, see [80,81]. Theorem 2.8. (i) Suppose A is a closed linear operator, K satisﬁes (P1), r ≥ 1 and there exists ω ≥ 0 such that, for every σ > 0, there exists ˜ can be analytically extended to an open Cσ > 0 such that the function K neighborhood Ωσ,ω of the region Υσ,ω . Denote, for every σ > 0, by gσ (·) ˜ to the region Ωσ,ω and suppose the analytic extension of the function K that, for every σ > 0 : (i.1) Λσ,ω := {λ ∈ Υσ,ω : gσ (λ) = 0} ⊆ ρC (A), (i.2) there exists an analytic function hσ : Ωσ,ω → L(E) such that hσ (λ) = gσ (λ)(λ − A)−1 C, λ ∈ Λσ,ω and (i.3) there exists Mσ > 0 such that ||hσ (λ)|| ≤ Mσ |λ|r , λ ∈ Υσ,ω . Then, for every ζ > 1, A is a subgenerator of a norm continuous, expotζ+r−1 )-convoluted C-semigroup (S(t))t≥0 satnentially bounded (K ∗0 Γ(ζ+r) isfying that the mapping t → S(t), t > 0 is inﬁnitely differentiable. (ii) Suppose α > 0, A is a closed linear operator, r ≥ 1 and there exists ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 which satisfy: (ii.1) Υσ,ω ⊆ ρC (A), (ii.2) there exist an open neighborhood Ωσ,ω of Υσ,ω and an analytic function hσ : Ωσ,ω → L(E) such that hσ (λ) = λ1α (λ − A)−1 C, λ ∈ Υσ,ω and (ii.3) ||hσ (λ)|| ≤ Mσ |λ|r , λ ∈ Υσ,ω . Then, for every ζ > 1, A is a subgenerator of a norm continuous, exponentially bounded (α + ζ + r)-times integrated C-semigroup (S(t))t≥0 satisfying that the mapping t → S(t), t > 0 is inﬁnitely differentiable. (iii) Suppose K satisﬁes (P1) and A is a subgenerator of a K-convoluted C-semigroup (SK (t))t≥0 satisfying ||SK (t)|| ≤ M eω t , t ≥ 0 for appro priate constants M > 0 and ω ≥ max(0, abs(K)). If there exists ω > ω such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 so that: (iii.1) there exist an open neighborhood Ωσ,ω of the region Υσ,ω and an ˜ λ∈ analytic mapping gσ : Ωσ,ω → C such that gσ (λ) = K(λ), C, Reλ ≥ ω,

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˜

= 0} ⊆ ρC (A), Υσ,ω \{λ ∈ C : Re(λ) > ω, K(λ) there exists an analytic mapping hσ : Ωσ,ω → L(E) such that ˜ hσ (λ) = gσ (λ)(λ − A)−1 C, λ ∈ Υσ,ω \{λ ∈ C : Re(λ) > ω, K(λ)

= 0}, (iii.4) ||hσ (λ)|| ≤ Mσ |Imλ|, λ ∈ Υσ,ω , Reλ ≤ ω and ||gσ (λ)|| ≤ Mσ , λ ∈ Υσ,ω , then the mapping t → SK (t)x, t > 0 is inﬁnitely differentiable for every ﬁxed x ∈ D(A2 ). Furthermore, if D(A2 ) is dense in E, then the mapping t → SK (t), t > 0 is inﬁnitely differentiable in the uniform operator topology. (iv) Suppose α > 0 and A is a subgenerator of an α-times integrated C-semigroup (Sα (t))t≥0 satisfying ||Sα (t)|| ≤ M eω t , t ≥ 0 for appropriate constants M > 0 and ω > 0. If there exists ω > ω such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 so that the following conditions hold: (iv.1) Υσ,ω ⊆ ρC (A), (iv.2) there exist an open neighborhood Ωσ,ω of the region Υσ,ω and an analytic mapping fσ : Ωσ,ω → L(E) such that fσ (λ) = λ1α (λ − A)−1 C, λ ∈ Υσ,ω , and (iv.3) ||fσ (λ)|| ≤ Mσ |Imλ|, λ ∈ Υσ,ω , Reλ ≤ ω, then the mapping t → Sα (t)x, t > 0 is inﬁnitely differentiable for every ﬁxed x ∈ D(A2 ). Furthermore, if D(A2 ) is dense in E, then the mapping t → Sα (t), t > 0 is inﬁnitely differentiable in the uniform operator topology. (iii.2) (iii.3)

Proof. To prove (i), let ζ > 1 and σ > 0. Put, with the same notation as in the (λ) 1 proof of Theorem 2.5: S k (t) =: 2πi eλt hλσr+ζ dλ, t ≥ 0, k ≥ k0 . Let ς > 0. Γk Then the proof of Theorem 2.5 implies that, for every j ∈ N0 , the sequence j+1−ζ dj k ( dt ) =: aj,σ,ζ and that the j S (t))k is convergent in L(E) for t > max(0, σ convergence is uniform on every compact subset of [aj,σ,ζ +ς, ∞). Put Sj (t) := dj k limk→∞ dt j S (t), j ∈ N0 , t > aj,σ,ζ . As before, one gets that the mapping dj t → S0 (t), t > aj+1,σ,ζ + ς is j-times differentiable as well as that dt j S0 (t) = ω0 +i∞ λt hσ (λ) 1 Sj (t), t > aj+1,σ,ζ +ς and that S0 (t) = S(t) := 2πi ω0 −i∞ e λr+ζ dλ, t ≥ σ1 , where ω0 > ω. The arbitrariness of σ > 0, combined with the proof of [2, Theorem 2.5.1] and Lemma 1.5(iii), implies that (S(t))t≥0 is a norm contintζ+r−1 uous, exponentially bounded (K ∗0 Γ(ζ+r) )-convoluted C-semigroup having A as a subgenerator and that the mapping t → S(t), t > 0 is inﬁnitely differentiable, ﬁnishing the proof of (i). The proof of (ii) follows instantly from α−1 that of (i) with K(t) = tΓ(α) , t > 0. To prove (iii), one can slightly modify the argumentation given in the proof of [94, Theorem 4.7]. Let ω0 > ω . Notice that (λ−A)−1 Cx =

Cx CAx (λ − A)−1 CA2 x + 2 + , λ λ λ2

λ ∈ ρC (A),

x ∈ D(A2 ), (2.9)

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1 and that, for given σ > 0, the integral SΘ,0 (t) := 2πi eλt hσλ(λ) dλ (deﬁned Γσ as in the proof of Theorem 2.5 and (i), with ω0 replaced by ω in the definition of the contour Γσ ) converges for every t > σ1 . Furthermore, the mapping t → SΘ,0 (t), t > j+1 σ is j-times differentiable for all j ∈ N and σ > 0. By (2.9), (iii.4), the residue theorem and [2, Theorem 2.5.1], we have that, for every x ∈ D(A2 ) and t > σ1 , (−1)

SΘ,0 (t)x = Θ

(−2)

(t)Cx + Θ

1 (t)CAx + 2πi

ω+i∞

hσ (λ)CA2 x dλ. (2.10) λ3

ω−i∞

Denote by (SΘ (t))t≥0 the exponentially bounded Θ-convoluted C-semigroup ˜ with a subgenerator A. Owing to Lemma 1.5(i), Θ(λ)(λ − A)−1 Cx = ∞ −λt ˜ e SΘ (t)xdt, Reλ > ω , K(λ) = 0, x ∈ E. Taken together with (2.9), 0 ˜ Θ(λ) ˜ the previous equality implies hσ (λ) = Θ(λ)Cx+ Cx+ hσ (λ) CA2 x, Reλ ≥ 2 λ

λ

ω, x ∈ D(A2 ). Then one gets, for every t ≥ 0 and x ∈ D(A2 ) :

hσ (λ)x −1 SΘ (t)x = L (t) λ

Cx CAx hσ (λ)CA2 x ˜ + 2 + (t) = L−1 Θ(λ) λ λ λ3 ω+i∞ hσ (λ)CA2 x 1 (−1) (−2) (t)Cx + Θ (t)CAx + eλt dλ. =Θ 2πi λ3 ω−i∞

With (2.10) in view, we obtain that SΘ,0 (t)x = SΘ (t)x, x ∈ D(A2 ), t > σ1 . This implies that the required properties hold for (SΘ (t))t≥0 and the proof α−1 of (iii) is completed; (iv) follows from (iii) with K(t) = tΓ(α) . Remark 2.9. The assertion of preceding theorem can be simply formulated in the case of regularized semigroups and the items (i.2), (ii.2), (iii.3) and (iv.2) can be slightly rearranged by using [29, Proposition 2.6, Remark 2.7, Corollary 2.8]. Theorem 2.10. (i) Suppose n ∈ N0 , ω ≥ 0, M > 0 and A is the densely deﬁned generator of an n-times integrated semigroup (Sn (t))t≥0 which satisﬁes ||Sn (t)|| ≤ M eω t , t ≥ 0. Then the following assertions are equivalent: (i.1) The mapping t → Sn (t), t > 0 is ((n + 1)-times) inﬁnitely differentiable. (i.2) There exists ω ≥ ω such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0, such that Υσ,ω ⊆ ρ(A) and that ||R(λ : A)|| ≤ Mσ |λ|n |Imλ|, λ ∈ Υσ,ω , Reλ ≤ ω. (ii)

(2.11)

Suppose α ≥ 0, ω ≥ 0, M > 0 and A is the densely deﬁned generator of an α-times integrated semigroup (Sα (t))t≥0 which satisﬁes ||Sα (t)|| ≤ M eω t , t ≥ 0. Then the following assertions are equivalent:

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The mapping t → Sα (t), t > 0 is ((α + 1)-times) inﬁnitely differentiable. There exists ω ≥ ω such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that Υσ,ω ⊆ ρ(A) and that

(ii.1) (ii.2)

||R(λ : A)|| ≤ Mσ |λ| α +1 |Imλ|, λ ∈ Υσ,ω , Reλ ≤ ω. (iii)

Suppose α ≥ 0 and A generates an α-times integrated semigroup (Sα (t))t≥0 . Then the following assertions are equivalent: (iii.1) The mapping t → Sα (t), t > 0 is ((α + 1)-times) inﬁnitely differentiable. (iii.2) There exist m ∈ N and ω > 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 satisfying Υσ,ω ⊆ ρ(A) and ||R(λ : A)|| ≤ Mσ (1 + |λ|)m , λ ∈ Υσ,ω , Reλ ≤ ω.

Proof. We will prove (i) provided n ∈ N. The implication (i.2) ⇒ (i.1) is a consequence of Theorem 2.8(iv). To prove the implication (i.1) ⇒ (i.2), set ∞ n G(ϕ)(t)x = (−1) ϕ(n) (t)Sn (t)xdt, ϕ ∈ D, x ∈ E. 0

By Theorem 2.5, G is a dense differentiable (DSG) generated by A. Let ε > 0, ψ ∈ D, ψ(t) = 1, 0 ≤ t ≤ 12 , ψ(t) = 0, t ≥ 1 and ψ(t) = 0, t ≤ −1. Set ψε (t) := ψ( εt ), t ∈ R. Then (λ − A)G(e−λt ψε (t)) = I + G(e−λt ψε (t)), λ ∈ C.

(2.12)

Since the set R(G) is dense in E, we have ε −λt G(e ψε (t))x = e−λt ψε (t)G(δt )xdt, x ∈ E, λ ∈ C. ε 2

Using integration by parts, one gets ε d 1 −λt G(e ψε (t))x = e−λt (ψε (t)G(δt )x)dt, x ∈ E, λ ∈ C\{0}. λ dt ε 2

This implies ||G(e−λt ψε (t))|| ≤

1 1 ε sup e−tReλ Mε , |λ| 2 1 ε≤t≤ε

(2.13)

2

d where Mε = sup || dt (ψε (t)G(δt ))||. Deﬁne 1 2 ε≤t≤ε

Φε := {λ ∈ C : λ = 0, Reλ ≤ 0, ln |λ| ≥ −εReλ + ln(εMε )} ε ∪{λ ∈ C : λ = 0, Reλ ≥ 0, ln |λ| ≥ − Reλ + ln(εMε )}. (2.14) 2 Then ||G(e−λt ψε (t))|| ≤ 12 , λ ∈ Φε , Φε ⊆ ρ(A) and (cf. also [7, (2.5), p. 417] and the proof of Theorem 3.5 given below): ||R(λ : A)|| ≤ 2||G(e−λt ψε (t))||, λ ∈ Φε .

(2.15)

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On the other hand, ε

G(e−λt ψε (t))x = λn

2

e−λt Sn (t)xdt

0

ε n

+ (−1)

(e−λ· ψε (·))(n) (t)Sn (t)xdt, x ∈ E, (2.16)

ε 2

and the proof of (2.11) follows from a simple computation with σ = 1ε . This completes the proof of (i). The proof of (ii) is clear in the case α ∈ N0 . Assume α ∈ (0, ∞)\N. Let (ii.1) holds. Put n := α and Sn (t)x := t (t−s)n−α−1 Γ(n−α) Sα (s)xds, t ≥ 0, x ∈ E. Then (Sn (t))t≥0 is an exponentially 0 bounded n-times integrated semigroup generated by A. To see that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable, we argue as follows. Let t > 0 and j ∈ N be ﬁxed. Choose an arbitrary a ∈ (0, t) and notice that the partial integration gives a t (t − s)n−α−1 (t − s)n−α−1 Sα (s)xds + Sα (s)xds Sn (t)x = Γ(n − α) Γ(n − α) 0

a = 0

a

j (t − s) (t − a)n−α+i Sα (s)xds + Sα(i) (a)x Γ(n − α) Γ(n − α + i + 1) i=0 n−α−1

t + a

(t − s)n−α+j dj Sα (s)xds, x ∈ E. Γ(n − α + j + 1) dsj

Hence, the mapping t → Sn (t), t > 0 is inﬁnitely differentiable and dj Sn (t)x = dtj

a 0

j j (t − s)n−α−1−j (t − a)n−α+i−j (n − α − i)Sα (s)xds + Γ(n − α) Γ(n − α + i + 1) i=1 i=0 j−1

×

k=0

(n − α + i − k)Sα(i) (a)x +

t a

(t − s)n−α dj Sα (s)xds,(2.17) Γ(n − α + 1) dsj

for every t > 0 and x ∈ E. The validity of (ii.2) follows from (i). To prove that (ii.2) implies (ii.1), notice only that Theorem 2.8(iv) implies that A generates an exponentially bounded (α + 2)-times integrated semigroup (Sα+2 (t))t≥0 satisfying that the mapping t → Sα+2 (t), t > 0 is inﬁnitely differentiable. With (2.17) in view, the assertion (iii) is an immediate consequence of Theorem 2.5 and Theorem 2.8(ii). Example. For every n ≥ 1 there exists a closed densely deﬁned operator B acting on a Banach (Hilbert) space such that B generates a global exponentially bounded n-times integrated semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable, and that B does not generate a local (n − 1)-times semigroup. To this end, suppose that A generates a contractive, immediately differentiable C0 -semigroup (T (t))t≥0 in a

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Banach (Hilbert) space E which additionally satisﬁes that (T (t))t≥0 cannot be analytically extended in a sector around the nonnegative real axis. For concrete examples, we refer to [35, p. 24–33, p. 409]. Let E n+1 be equipped with the sup-norm and let D(B) := D(A)n+1 and B(x1 , . . . , xn+1 ) := (Ax1 + Ax2 , Ax2 + Ax3 , . . . , Axn + Axn+1 , Axn+1 ), (x1 , . . . , xn+1 ) ∈ D(B). Arguing as in [90, Proposition 2.4] (cf. also [2, Theorem 3.2.13, p. 133] and Theorem 2.12), we have that B generates a global exponentially bounded n-times integrated semigroup (Sn (t))t≥0 and that B does not generate a local (n−1)times semigroup. It remains to be proved that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable. Let σ > 0. Then there exist ω ≥ 0, Cσ > 0 and Mσ > 0 such that Υσ,ω ⊆ ρ(A) and that ||R(λ : A)|| ≤ Mσ |Imλ|. The claimed assertion follows from Theorem 2.10(iii) and the computation given in the proof of [90, Proposition 2.4]: ||R(λ : B)|| ≤

n

||Ak R(λ : A)k+1 || ≤ Mσ |Imλ|

k=0

n

||Ak R(λ : A)k ||

k=0

≤ Mσ |Imλ|

n

||λR(λ : A)−I||k ≤ Mσ |Imλ|

k=0

n

(1+Mσ |λ||Imλ|)k

k=0

≤ (n + 1)Mσ |Imλ|(1 + Mσ |λ|2 )n+1 ≤ (n + 1)Mσ |λ|2n+2 |Imλ|, λ ∈ Υσ,ω , for some Mσ > 0. The above example can be set in the context of integrated C-semigroups [83]. The proof of Theorem 2.10 implies the following. Corollary 2.11. Suppose α ≥ 0, β > α, j ∈ N and (Sα (t))t∈[0,τ ) is a (local) α-times integrated C-semigroup with a subgenerator A. Set, for every t ∈ t β−α−1 [0, τ ) and x ∈ E, Sβ (t)x := 0 (t−s) Γ(β−α) Sα (s)xds. Then (Sβ (t))t∈[0,τ ) is a β-times integrated C-semigroup with a subgenerator A. Furthermore, if the mapping t → Sα (t), t ∈ (0, τ ) is j-times differentiable, then the mapping t → Sβ (t), t ∈ (0, τ ) is likewise j-times differentiable. If this is the case, then we have, for every t ∈ [0, τ ), a ∈ (0, t) and x ∈ E : dj Sβ (t)x = dtj

a 0

j j (t − s)β−α−1−j (t − a)β−α+i−j (β − α − i)Sα (s)xds + Γ(β − α) Γ(β − α + i + 1) i=1 i=0 j−1

×

k=0

(β − α + i −

k)Sα(i) (a)x

t + a

(t − s)β−α dj Sα (s)xds. Γ(β − α + 1) dsj (2.18)

Theorem 2.12. (i) Suppose j ∈ N, τ > 0, K ∈ L1loc ([0, τ )), 0 ∈ suppK, K ∈ C j ((0, τ ))(K ∈ C ∞ ((0, τ ))) and A is a subgenerator of a local K-convoluted C-semigroup (SK (t))t∈[0,τ ) satisfying that the mapping t → SK (t), t ∈ (0, τ ) is j-times (inﬁnitely) differentiable. Then A is

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(ii)

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a subgenerator of a local (K ∗0 K)-convoluted C 2 -semigroup (SK∗0 K (t))t∈[0,2τ ) satisfying that the mapping t → SK∗0 K (t), t ∈ (0, 2τ ) is j-times (inﬁnitely) differentiable. Suppose α ≥ 0, j ∈ N and A is a subgenerator of a local α-times integrated C-semigroup (Sα (t))t∈[0,τ ) . Then A is a subgenerator of a local (2α)-times integrated C 2 -semigroup (S2α (t))t∈[0,2τ ) and the following holds: if the mapping t → Sα (t), t ∈ (0, τ ) is j-times (inﬁnitely) differentiable, then the mapping t → S2α (t), t ∈ (0, 2τ ) is j-times (inﬁnitely) differentiable. Let α ≥ 0 and A generate a local α-times integrated semigroup (Sα (t))t∈[0,τ ) satisfying that the mapping t → Sα (t), t ∈ (0, τ ) is (α + 1)-times differentiable. Then A generates a global α-times integrated semigroup (S˜α (t))t≥0 satisfying that the mapping t → S˜α (t), t > 0 is inﬁnitely differentiable.

Proof. By Lemma 1.8, we have that A is a subgenerator of a local (K ∗0 K)convoluted C 2 -semigroup (SK∗0 K (t))t∈[0,2τ ) , which satisﬁes, for every τ0 ∈ t ( τ2 , τ ) and x ∈ E : SK∗0 K (t)x = 0 K(t − s)SK (s)Cxds, provided t ∈ [0, τ0 ] and ⎞ ⎛ t−τ 0 τ0 SK∗0 K (t)x = SK (τ0 )SK (t − τ0 )x + ⎝ + ⎠ K(t − r)SK (r)Cxdr, 0

0

provided t ∈ (τ0 , 2τ0 ). Let t ∈ (0, τ ) and x ∈ E be ﬁxed. Choose afterwards a ∈ (0, t) and notice that the condition 0 ∈ suppK implies, for every t ∈ [τ0 , τ ), ⎞ ⎛ t−τ 0 τ0 + ⎠ K(t − r)SK (r)Cxdr SK (τ0 )SK (t − τ0 )x + ⎝ 0

0

t K(t − s)SK (s)Cxds.

= 0

Proceeding as in the proof of Theorem 2.10, one yields a

t K(t − s)SK (s)Cxds +

SK∗0 K (t)x = 0

a K(t − s)SK (s)Cxds +

=

K(t − s)SK (s)Cxds a j

Θ(−i) (t − a)(

i=0

0

t − a

Θ(−j) (t − s)

dj SK (s)Cxds. dsj

di SK (s)Cx)s=a dsi

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Hence, dj SK∗0 K (t)x = dtj

a 0

+

dj K(t − s)SK (s)Cxds dtj j

Θ(j−i) (t − a)(

i=0

t −

Θ(t − s)

di SK (s)Cx)s=a dsi

dj SK (s)Cxds dsj

a

and the mapping t → SK∗0 K (t), t ∈ (0, τ ) is j-times (inﬁnitely) differentiable. Suppose now τ0 ∈ ( τ2 , τ ) and ε ∈ (0, τ − τ0 ). Then the mappings −ε t → SK∗0 K (t), t ∈ (0, τ − ε) and t → SK (τ0 )SK (t − τ0 ), t ∈ ( τ0 +τ , 2τ 2 0τ ) are j-times (inﬁnitely) differentiable. Furthermore, the mapping t → 0 0 −ε K(t − r)SK (r)Cxdr, t ∈ ( τ0 +τ , 2τ0 ) is j-times (inﬁnitely) differentiable 2 with τ0 dj K(t − r)SK (r)Cxdr dtj 0

τ0 = 0

dj K(t − r)SK (r)Cxdr, dtj

t∈

τ0 + τ − ε , 2τ0 2

the dominated convergence theorem implies that the mapping t → and t−τ0 −ε K(t − r)SK (r)Cxdr, t ∈ ( τ0 +τ , 2τ0 ) is also j-times (inﬁnitely) dif2 0 ferentiable with t−τ 0 dj K(t − r)SK (r)Cxdr dtj 0

=

j−1 i=0

K (i) (τ0 )

dj−1−i SK (t − τ0 )Cx + dtj−1−i

−ε ( τ0 +τ , 2τ0 ) 2

t−τ 0

K (j) (t − r)SK (r)Cxdr,

0

for any t ∈ and x ∈ E. This completes the proof of (i) while the proof of (ii) in the case α > 0 follows from Lemma 1.8 and the assertion α−1 (i) with K(t) = tΓ(α) . The proof of (ii) in the case α = 0 is much easier. To prove (iii), set n := α and apply [72, Corollary 1.8] in order to see that A is stationary dense. Let F = D(An(A) ) and let SnF (t)x = Sn (t)x, t ∈ [0, τ ), x ∈ F. Then one can simply prove that the operator AF is the generator of a local n-times integrated semigroup (SnF (t))t∈[0,τ ) in the Banach space F, and that the mapping t → SnF (t), t ∈ (0, τ ) is (n + 1)-times differentiable. This implies that AF generates a dense (DSG) GF which satisﬁes n ∞ (n) GF (ϕ)x = (−1) 0 ϕ (t)SnF (t)xdt, x ∈ F, ϕ ∈ D(−∞,τ ) . Let ε ∈ (0, τ ) and let the functions ψ(·) and ψε (·) possess the same meaning as in the proof of Theorem 2.10. Repeating literally the argumentation given in the proof of

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cited theorem, one gets that there exist ω ≥ 0 and m ∈ N such that, for every σ > τ1 , there exist Cσ > 0 and Mσ > 0 such that Υσ,ω ⊆ ρ(AF ) and that ||R(λ : AF )|| ≤ Mσ (1 + |λ|)m , λ ∈ Υσ,ω . Using [72, Corollary 2.2] we infer that there exists ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that Υσ,ω ⊆ ρ(A) and that ||R(λ : A)|| ≤ Mσ (1 + |λ|)m , λ ∈ Υσ,ω . By Theorem 2.5, A generates a differentiable (DSG) G and that there exists k ∈ N such that A generates a global k-times integrated semigroup (Sk (t))t≥0 satisfying that the mapping t → Sk (t), t > 0 is inﬁnitely differentiable. Hence, the mapping t → Sα (t), t ∈ (0, τ ) is inﬁnitely differentiable. By (ii) and Lemma 2.11, we have that A generates a local (α + n)-times integrated semigroup (Sα+n (t))t∈[0,2τ0 ) which satisﬁes that the mapping t → Sα+n (t), t ∈ (0, 2τ ) is inﬁnitely differentiable. Let ε ∈ (0, τ ) be ﬁxed. Deﬁne dn S˜α (t) := Sα (t), if t ∈ [0, τ − ε) and S˜α (t) := dt n Sα+n (t), if t ∈ [τ − ε, 2τ ). Then it is checked at once that (S˜α (t))t∈[0,2τ ) is a local α-times integrated semigroup generated by A and that the mapping t → S˜α (t), t ∈ (0, 2τ ) is inﬁnitely differentiable. Repeating successively this procedure, one gets that A generates a global α-times integrated semigroup (S˜α (t))t≥0 satisfying that the mapping t → S˜α (t), t > 0 is inﬁnitely differentiable. Remark 2.13. (i) In the assertions (i) and (ii) of Theorem 2.12, one can consider differentiability of (SK (t))t∈[0,τ ) , respectively (Sα (t))t∈[0,τ ) , for t ∈ [0, τ ). The same comment could be applied to Theorem 2.6. In order to avoid unnecessary discussion, let us only point out that the inﬁnite differentiability of (SK (t))t∈[0,τ ) , respectively (Sα (t))t∈[0,τ ) , for t ∈ [0, τ ), implies the inﬁnite differentiability of (SK∗0 K (t))t∈[0,2τ ) , respectively (S2α (t))t∈[0,2τ ) , for t ∈ [0, 2τ ). (ii) Suppose that α ≥ 0 and that A generates a (local) α-times integrated cosine function (Cα (t))t∈[0,τ ) satisfying that the mapping t → Cα (t), t > 0 is (α + 2)-times differentiable. Then [64, Proposition 3] implies that the operator A generates a local (α + 1)-times integrated semigroup (S(t))t∈[0,τ ) satisfying that the mapping t → S(t), t ∈ (0, τ ) is (α + 2)-times differentiable. By Theorem 2.12(iii) and Lemma 1.6, we obtain that A must be bounded. As [64, Example 3] shows, the situation is quite different in the case of convoluted cosine functions; it would take too long to go into a more detailed analysis of differential properties of convoluted C-cosine functions. (iii) Let α ∈ (0, ∞)\N and let A be the generator of a global α-times integrated semigroup (Sα (t))t≥0 satisfying that the mapping t → Sα (t), t > 0 is inﬁnitely differentiable. It is not clear whether (Sα (t))t≥0 must be exponentially bounded. Example. Let α > 0 and let A be the generator of a (local) α-times integrated semigroup (Sα (t))t∈[0,τ ) satisfying that the mapping t → Sα (t), t ∈ (0, τ ) is (α + 1)-times differentiable. This implies that the mapping t → Sα (t), t ∈ (0, τ ) is inﬁnitely differentiable. The above assertion does not hold in the case of general C. To illustrate this, we will slightly modify [82, Example 3.4] given by Li and Shaw. Let E := l1 , D(A) := {xn :

524

M. Kosti´c

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−n2 xn ∈ l1 }, Axn := −n2 xn , xn ∈ D(A), Cxn := n13 xn , xn ∈ l1 , and S(t)xn := n14 sin(nt)xn , xn ∈ l1 , t ≥ 0. Then it can be easily veriﬁed that (S(t))t≥0 is a bounded once integrated C-cosine function with the integral generator A. Furthermore, the mapping t → S(t), t ≥ 0 is 3-times d3 1 differentiable, with dt 3 S(t)xn = − n cos(nt)xn , t ≥ 0, xn ∈ l1 , and the mapping t → S(t), t > 0 is not 4-times differentiable. Put now, for every t ≥ 0, xn ∈ l1 and yn ∈ l1 , t t

S(s)ds (t − s)S(s)ds xn xn 0 0 S(t) := . t yn yn S(t) − tC S(s)ds 0 By [64, Proposition 3], we have that the operator A is the integral generator of a twice integrated C-semigroup (S(t))t≥0 in l1 × l1 satisfying that ||S(t)|| = O(1 + t2 ), t ≥ 0 and that the mapping t → S(t), t ≥ 0 is 3-times differentiable. Clearly, the mapping t → S(t), t > 0 is not 4-times differentiable. Put

∞ xn xn G := ϕ (t)S(t) dt, ϕ ∈ D, xn , yn ∈ l1 . yn yn 0

Then G is a differentiable (C-EDSG) in l1 × l1 and the mapping t → GC (δt ), t > 0 is not two times differentiable. Finally, one can simply construct an example of a local C-regularized semigroup (S(t))t∈[0,1) which cannot be extended beyond the interval [0, 1) and satisﬁes that the mapping t → S(t), t ∈ [0, 1) is inﬁnitely differentiable. Definition 2.14. Let (SK (t))t∈[0,τ ) be a (local) K-convoluted C-semigroup (C-regularized semigroup) having A as a subgenerator and let the mapping t → SK (t), t ∈ (0, τ ) be inﬁnitely differentiable. Then it is said that (SK (t))t∈(0,τ ) is of class C L , respectively, of class CL , iﬀ for every compact set K ⊆ (0, τ ) there exists hK > 0, respectively, iﬀ for every compact set K ⊆ (0, τ ) and for every h > 0: k

hk d k SK (t) || < ∞, respectively, sup || K dt k Lk t∈K, k∈N0

k

hk d k SK (t) sup || dt k || < ∞; Lk t∈K,k∈N0 (2.19)

(SK (t))t∈[0,τ ) is said to be ρ-hypoanalytic, 1 ≤ ρ < ∞, if (SK (t))t∈[0,τ ) is of class C L with Lk = k!ρ/k . An inﬁnitely differentiable (C-DSG) G is of class CL iﬀ for every compact set K ⊆ (0, ∞) and for every h > 0 : supt∈K,k∈N0 ||

hk

dk dtk

GC (δt )

Lk k

|| < ∞.

By the proof of scalar-valued version of the Pringsheim theorem (cf. for example [53, Theorem 2.1, p. 34]), it follows that the mapping t → SK (t), t ∈ (0, τ ) is real analytic iﬀ (SK (t))t∈[0,τ ) is ρ-hypoanalytic with ρ = 1. Furthermore, it can be simply proved that a closed linear operator A generates a (C-DSG) of class CL with Lk = k!1/k iﬀ A is the integral generator of an entire C-regularized group; in the particular case C = I, the above is equivalent with the boundedness of A. Of concern is to emphasize that

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the important examples of Gevrey’s C0 -semigroups given by Belinskiy and Lasiecka [11], Chen and Triggiani [17], Favini and Triggiani [39], Markin [85] and Shubov [103] (cf. also Xiao and Liang [113, Chapter 6]) can be used for the construction of dense (DSG)’s of class CL with Lj = j!s/j , s > 1, j ∈ N0 . Notice also that Zhang and Zheng (cf. [117, Theorem 3.2, Theorem 3.3]) considered Lp -realizations of pseudodifferential operators as generators of global ρ-hypoanalytic regularized semigroups with ρ ≥ 1; we also refer the reader to [118] and [119, Sect. 3] for more examples of operators generating ρ-hypoanalytic (differentiable) regularized semigroups. The following lemma can be reformulated and proved in the case of a general (local) convoluted C-semigroup. Lemma 2.15. Suppose α ≥ 0, β > α, j ∈ N and (Sα (t))t∈[0,τ ) is a (local) α-times integrated C-semigroup with a subgenerator A. Set, for every t ∈ t β−α−1 [0, τ ) and x ∈ E, Sβ (t)x := 0 (t−s) Γ(β−α) Sα (s)xds. Then (Sβ (t))t∈[0,τ ) is a β-times integrated C-semigroup with a subgenerator A and: (i)

(ii)

Let (Lk ) be non-quasianalytic. If (Sα (t))t∈[0,τ ) is of class C L , respectively, of class CL , then (Sβ (t))t∈[0,τ ) is likewise of class C L , respectively, of class CL . If (Sα (t))t∈[0,τ ) is ρ-hypoanalytic, 1 ≤ ρ < ∞, then (Sβ (t))t∈[0,τ ) is likewise ρ-hypoanalytic.

Proof. We will prove only (i). Assume x ∈ E, j ∈ N, K ⊆ (0, τ ) is a compact set, a ∈ (0, inf K) and Ka = [a, sup K]. Denote, formally, for every j ξ j || d j Sα (t)|| ξ dt := sup . Clearly, L := supn∈N | β−α ξ > 0, IK t∈Ka , j∈N0 n | < ∞ a Lj j

and (2.18) implies that there exists CK > 0 such that, for every h > 0, ξ > 0 and t ∈ K : j

d hj || dt j Sβ (t)||

Ljj

≤ CK +

( sup hK−a )j j! Ljj

j j−1 (sup K − a)β−α+i−j i=0 sup K

+ a

Γ(β − α + i + 1)

k=0

Lii hj ξ IK Ljj ξ i a

j (sup K − s)β−α ξ h dsIK . a j Γ(β − α) ξ

Denote, for every i ∈ {0, . . . , j}, Si,j := Li j k) Lji hξi . j

(β − α + i − k)

(sup K−a)β−α+i−j Γ(β−α+i+1)

j−1

k=0 (β

−α+i−

Then the functional equality of the Gamma function and the loga-

rithmic convexity

Lii Ljj

≤

1 Lj−i j−i

imply:

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M. Kosti´c

Si,j = ≤

IEOT

Lii hj (sup K − a)β−α+i−j (β − α)(β − α − 1) · · · (β − α + i − (j − 1)) ξi Γ(β − α + 1) Ljj hj (sup K − a)β−α+i−j L(j − i)! j−i ξi Γ(β − α + 1) Lj−i

≤L

j−i h (sup K − a)β−α hi ( sup K−a ) (j − i)! . Γ(β − α + 1) ξ i Lj−i j−i

Hence, j

|Si,j | ≤ L

i=0

∞ i ( sup hK−a )p p! (sup K − a)β−α h sup <∞ Γ(β − α + 1) i=0 ξ p∈N0 Lpp

provided h < ξ.

This implies that there exists CK,α > 0 such that: p h ∞ i dj p! hj || dt S (t)|| sup K−a h j β ≤ C sup , sup K,α ξ p∈N0 Lpp Ljj t∈K,j∈N0 i=0

h < ξ. (2.20)

The non-quasianaliticity of (Lpp ) (cf., for example, [54, p. 74]) enables one to p see that, for every ζ > 0, sup ζLppp! < ∞, and the proof follows from (2.20). p∈N0

Remark 2.16. The careful inspection of the proof of Theorem 2.12 shows the following: (i)

(ii)

Let α ≥ 0 and let (Sα (t))t∈[0,τ ) be an α-times integrated semigroup which is of class C L , respectively CL . Then there exists a global α-times integrated semigroup (S˜α (t))t≥0 that is of class C L , respectively CL , and satisﬁes S˜α (t) = Sα (t), t ∈ [0, τ ). Let (Lk ) be non-quasianalytic. If 0 ∈ suppK and K is of class C L , respectively, CL (the notion is clear) and (SK (t))t∈[0,τ ) is a K-convoluted C-semigroup which is of class C L , respectively, CL , then (SK∗0 K (t))t∈[0,2τ ) is a (K ∗0 K)-convoluted C 2 -semigroup which is also of class C L , respectively, CL . In particular, the above holds in the case α−1 K(t) = tΓ(α) , α > 0.

Theorem 2.17. (i) Suppose (Lk ) is non-quasianalytic, α ≥ 0 and A generates a (local) α-times integrated semigroup (Sα (t))t∈[0,τ ) . Then (Sα (t))t∈[0,τ ) is of class C L iﬀ there exist ω ≥ 0 and m ∈ N such that, for every ε ∈ (0, 1), there exist Cε > 0 and Mε > 0 satisfying: Fε,ω =: {λ ∈ C : Reλ ≥ − ln ωL (Cε |Imλ|) + ω} ⊆ ρ(A),

(2.21)

, λ ∈ Fε,ω , Reλ ≤ ω and ||R(λ : A)|| ≤ Mε (1 + |λ|) e m ||R(λ : A)|| ≤ Mε (1 + |λ|) , λ ∈ C, Reλ ≥ ω.

(2.22) (2.23)

m ε|Reλ|

(ii)

Suppose ρ ≥ 1, α ≥ 0 and A generates a local α-times integrated semigroup (Sα (t))t∈[0,τ ) . Then (Sα (t))t∈[0,τ ) is ρ-hypoanalytic iﬀ there exist

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527

ω ≥ 0 and m ∈ N such that, for every ε ∈ (0, 1), there exist Cε > 0 and Mε > 0 satisfying: Gε,ω,ρ := {λ ∈ C : Reλ ≥ −Cε |Imλ|1/ρ + ω} ⊆ ρ(A), ||R(λ : A)|| ≤ Mε (1 + |λ|) e

m ε|Reλ|

, λ ∈ Gε,ω,ρ , Reλ ≤ ω and

||R(λ : A)|| ≤ Mε (1 + |λ|) , λ ∈ C, Reλ ≥ ω. m

(2.24) (2.25) (2.26)

Proof. We will prove only (i). Suppose ﬁrst that (Sα (t))t∈[0,τ ) is of class C L . t n−α−1 Set n := α and Sn (t)x := 0 (t−s) Γ(n−α) Sα (s)xds, t ∈ [0, τ ), x ∈ E. By Lemma 2.15, (Sn (t))t∈[0,τ ) is of class C L and Remark 2.16(i) implies that there exists a global n-times integrated semigroup (S˜n (t))t≥0 that is of class C L , respectively CL , and satisﬁes S˜n (t) = Sn (t), t ∈ [0, τ ). Denote, for every ∞ x ∈ E and ϕ ∈ D, G(ϕ)x = (−1)n 0 ϕ(n) (t)Sn (t)xdt. Clearly, G is a differentiable (DSG) and, for every compact set K ⊆ (0, ∞), there exists hK > 0 n+j+1 n+j dj Ln+j , j ∈ N0 . Since (Lkk ) satisﬁes (M.2), it such that || dt j G(δt )|| ≤ hK follows that G is a (DSG) of class C L and the necessity is a consequence of [7, Theorem 4, p. 421] and Remark 2.7(i). To prove the sufﬁciency, notice that (2.21)–(2.23) implies that G is of class C L and that A generates an expo nentially bounded (m + 2)-times integrated semigroup (S m+2 (t))t≥0 that is L of class C . An employment of Lemma 2.15 gives that there exists k ∈ N so that (Sα+k (t))t∈[0,τ ) is of class C L . The following theorem establishes a sufﬁcient criterion for the generation of K-convoluted C-semigroups of class C L ; the proof is analogous to those of Theorem 2.8 and [7, Theorems 3, 4]. Theorem 2.18. Suppose K satisﬁes (P1), A is a subgenerator of a (local) K-convoluted C-semigroup (SK (t))t∈[0,τ ) , ω > max(0, abs(K)), m ∈ N and, for every ε ∈ (0, 1), there exist Cε > 0 and Mε > 0 such that: ˜ (i) Gε,ω := {λ ∈ C : Reλ > ω, K(λ)

= 0} ∪ {λ ∈ Fε,ω : Reλ ≤ ω} ⊆ ρC (A), (ii) there exist an open neighborhood of the region Fε,ω , denoted by Oε,ω , and analytic mappings gε : Oε,ω → C and hε : Oε,ω → L(E) such that ˜ λ ∈ C, Reλ > ω and that: gε (λ) = K(λ), (iii) hε (λ) = gε (λ)(λ − A)−1 C, λ ∈ Gε,ω , (iv) ||hε (λ)|| ≤ Mε (1 + |λ|)m eε|Reλ| , λ ∈ Fε,ω , Reλ ≤ ω and ||hε (λ)|| ≤ Mε (1 + |λ|)m , λ ∈ C, Reλ ≥ ω. Then (SK (t))t∈[0,τ ) is of class C L . By Theorem 2.5 and the assertions (i) and (ii) of the next theorem, the classes of (differentiable) (DSG)’s and (EDSG)’s are stable under bounded commuting perturbations. Concerning the regularity properties of perturbed, strongly continuous semigroups, the references [8–10,14,32,46,88,101] are crucially important. Theorem 2.19. Suppose α ≥ 0, A is a subgenerator, respectively the integral generator, of a (local, global exponentially bounded) α-times integrated C-semigroup (Sα (t))t∈[0,τ ) , B ∈ L(E), BA ⊆ AB and BC = CB.

528 (i)

M. Kosti´c

[69, Theorem 2.5] Then A+B is a subgenerator, respectively the integral generator, of a (local, global exponentially bounded) α-times integrated C-semigroup (SαB (t))t∈[0,τ ) , which satisﬁes, for every t ≥ 0 and x ∈ E, SαB (t)x

tB

= e Sα (t)x +

∞

α i=1

(ii) (iii) (iv)

IEOT

i

t i

(−B)

0

(t − s)i−1 Bs e S(s)xds. (2.27) (i − 1)!

Let j ∈ N. If the mapping t → Sα (t), t ∈ (0, τ ) is j-times differentiable, then the mapping t → SαB (t), t ∈ (0, τ ) is also j-times differentiable. If (Sα (t))t∈[0,τ ) is of class C L , respectively CL , then (SαB (t))t∈[0,τ ) is also of class C L , respectively CL . If (Sα (t))t∈[0,τ ) is ρ-hypoanalytic, 1 ≤ ρ < ∞, then (SαB (t))t∈[0,τ ) is also ρ-hypoanalytic.

Proof. We will consider the case α > 0. The proof of (ii) follows immediately from (2.27), with the following ﬁnal formula, which holds for every t ∈ (0, τ ) and x ∈ E : j

j di dj B S (t)x = B j−i etB i Sα (t)x α j i dt dt i=0

α + (−B)j etB Sα (t)x j t ∞

α (t − s)i−1−j sB i e Sα (s)xds (−B) + i (i − 1 − j)! i=j+1 j−1

+

i=1

0

α (−B)i i

j−i k=0

j−i dk B j−i−k etB k Sα (t)x k dt

:= I(t) + II(t) + III(t) + IV (t). In order to prove (iii), we must estimate the contribution of I(t), II(t), III(t) and IV (t). Given a compact set K ⊆ (0, τ ) and a number σ > 0, denote σ j ||

dj

Sα (t)||

dtj . Let ||Sα (t)|| ≤ M, t ∈ [0, sup K], L := ||Sα ||K,σ = supt∈K, j∈N0 Ljj α supn∈N0 | n |, h > 0 and ξ > 2h. Then the proof of (iii) follows from the following computations which show that there exists CK > 0 such that, for every t ∈ K and j ≥ 2 :

hj ||I(t)|| Ljj

≤

j

j i=0

≤e

t||B||

i

||B||j−i et||B||

||Sα ||K,ξ

j

Lii ||Sα ||K,ξ hj ξi Ljj

2j−i 2i ||B||j−i hj−i

i=0

≤ et||B|| ||Sα ||K,ξ

j (2h||B||)j−i 2h i ( ) ξ Lj−i j−i i=0

hi ξ i Lj−i j−i

Vol. 67 (2010)

Differential and Analytical Properties ≤ et||B|| ||Sα ||K,ξ ωL (2||B||h)

hj ||III(t)|| Ljj

≤ ML

∞

||B||i et||B||

i=j+1

≤ M Let||B|| ≤ h ||IV (t)|| j

Ljj

529

∞ 2h ( )i ; ξ i=0

ti−j hj (i − j)! Ljj

∞ (||B||t)i (h||B||)j

i!

i=1 2t||B|| M Le ωL (h||B||),

≤ Let||B|| ||Sα ||K,ξ

Ljj

j−1 j−i j−i k

i=1 k=0

≤ Let||B|| ||Sα ||K,ξ

j−i j−1

||B||j−k

2j−k 2k−i ||B||j−k

i=1 k=0

≤ Let||B|| ||Sα ||K,ξ

j−1 i=1

2−i

hj 1 ξ k Lj−k j−k hj−k hk ξk Lj−k j−k

j−i 2h ( )k ωL (2||B||h) ξ

k=0

∞ 2h ≤ Let||B|| ||Sα ||K,ξ ( )k ωL (2||B||h) ξ k=0

and

hj ||II(t)|| Ljj

≤ CK ωL (h||B||). The assertion (iv) is a consequence of (iii).

By Theorems 2.19 and 2.6(ii), we have that the class of (exponential) (C-DSG)’s (dense (inﬁnitely) j-times differentiable (C-DSG)’s) persists under bounded commuting perturbations. Furthermore, it is clear (cf. the proof of Theorem 2.17) that a closed linear operator A generates a (DSG) of class CL , respectively of class CL , iﬀ there exists n ∈ N such that A generates an exponentially bounded, n-times integrated semigroup of class C L , respectively of class CL ; (DSG)’s of class A (Aγ ) can be also described in terms of a corresponding class of integrated semigroups. With Theorem 2.19 in view, we obtain the following corollary which can be reformulated in the case of dense inﬁnitely differentiable (C-DSG)’s. Corollary 2.20. Suppose A is the generator of a (DSG) G, B ∈ L(E) and BA ⊆ AB. Then the following holds: (i) If G is of class C L , respectively of class CL , then the operator A + B generates a (DSG) GB that is also of class C L , respectively of class CL . (ii) If G is ρ-hypoanalytic, 1 ≤ ρ < ∞, then the operator A + B generates a (DSG) GB that is also ρ-hypoanalytic. The next definition of an (analytic) semigroup of growth order r > 0 is a modiﬁcation of the definitions given by Periago and Straub [97] and Tanaka [105]. Definition 2.21. An operator family (T (t))t>0 in L(E) is a semigroup of growth order r > 0 if the following conditions hold: (i) T (t + s) = T (t)T (s), t, s > 0,

530

M. Kosti´c

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(ii) the mapping t → T (t)x, t > 0 is continuous for every ﬁxed x ∈ E, (iii) ||tr T (t)|| = O(1), t → 0+ and (iv) T (t)x = 0 for all t > 0 implies x = 0. Suppose that α ∈ (0, π2 ] and that (T (t))t>0 has an analytic extension to the sector Σα , denoted by the same symbol. If, additionally, for every γ ∈ (0, α), there exist Mγ > 0 and ωγ ≥ 0 with ||z r T (z)|| ≤ Mγ eωγ Rez , z ∈ Σγ , then (T (z))z∈Σα is called an analytic semigroup of growth order r. Since we do not require that the set E0 := t>0 T (t)E is dense in E, the introduced notion generalizes the former one given by Da Prato [25]. The integral generator of (T (t))t>0 , respectively (T (z))z∈Σα , is deﬁned in [76] by ⎧ ⎫ t ⎨ ⎬ Aˆ := (x, y) ∈ E ×E : T (t)x − T (s)x = T (r)ydr for all t, s > 0 with t ≥ s ⎩ ⎭ s

and it is a closed linear operator which contains the complete inﬁnitesimal generator, c.i.g. in short, of (T (t))t>0 , respectively (T (z))z∈Σα . Recall that the c.i.g. of (T (t))t>0 , respectively, (T (z))z∈Σα , is deﬁned as the closure of the inﬁnitesimal generator A of (T (t))t>0 , respectively (T (z))z∈Σα , given by ( ' T (t)x − x =y . A := (x, y) ∈ E × E : lim t→0+ t Suppose that (T (z))z∈Σα is an analytic semigroup of growth order r. With the help of regularized semigroups, one can prove that, for every γ ∈ (−α, α), the operator eiγ A is the integral generator of a semigroup (T (teiγ ))t>0 of growth order r. The following result is closely related to [27, Theorem 1.1] and [38, Theorem 4], and clariﬁes the basic structural properties of (non-dense) analytic (C-DSG)’s. Theorem 2.22. Let α ∈ (0, π2 ] and let A be a closed linear operator. (i)

(ii)

(iii)

Consider the next assertions: (i.1) The operator C −1 AC generates an analytic (C-DSG) G of angle α. (i.2) There exist n ∈ N, M > 0, ω ≥ 0 and an analytic function Sn : Σα → L(E) so that A is a subgenerator of an n-times integrated C-semigroup (Sn (t))t≥0 satisfying Sn (t) = Sn (t), t > 0 and ||Sn (t)|| ≤ M eωt , t ≥ 0. (i.3) There are an n ∈ N and an analytic function Sn : Σα → L(E) so that A is a subgenerator of an n-times integrated C-semigroup (Sn (t))t≥0 which satisﬁes Sn (t) = Sn (t), t > 0. Then (i.2) ⇒ (i.3) ⇒ (i.1) and, in the case C = I, (i.1) ⇔ (i.2) ⇔ (i.3). Suppose that, for every γ ∈ (0, α), there exist Mγ > 0, ωγ ≥ 0 and rγ > 0 such that (1.2)–(1.4) hold with rγ instead of r. Then the operator C −1 AC is the generator of an analytic (C-DSG) of angle α. Let r ∈ (0, 1). Then the following assertions are equivalent:

Vol. 67 (2010) (iii.1)

(iii.2) (iii.3)

Differential and Analytical Properties

531

The operator A generates an analytic (DSG) G of angle α satisfying that (G(δz ))z∈Σα is an analytic semigroup of growth order r. The operator A generates an exponentially bounded, analytic r-times integrated semigroup of angle α. For every γ ∈ (0, α), there exist Mγ > 0 and ωγ ≥ 0 such that (1.2), (1.3) and (1.5) hold.

Proof. We will prove (i) provided C = I. Notice that the implication (i.2) ⇒ (i.3) is trivial and that the proof of implication (ii) ⇒ (iii) of Theorem 2.5 shows that the analyticity of G follows automatically from the supposition (i.3). Let A generate an analytic (DSG) G of angle α. By Theorem 2.5, we know that there exists n ∈ N such that A generates an exponentially bounded n-times integrated semigroup (Sn (t))t≥0 satisfying that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable. One can use again the proof of the dn implication (ii) ⇒ (iii) of Theorem 2.5 to deduce that G(δt ) = dt n Sn (t), t > 0. dn S (t), t > 0 can be analytically extended to the Hence, the mapping t → dt n n sector Σα . Now it is checked at once that the mapping t → Sn (t), t > 0 can be analytically extended to the sector Σα . This yields (i.2) and completes the proof of (i). The hypothesis quoted in the formulation of (ii), combined with Lemma 1.4(iii), implies that, for every γ ∈ (0, α), A is a subgenerator of an exponentially bounded, analytic (rγ + 1)-times integrated C-semigroup of angle γ. By (i), the operator C −1 AC generates an analytic (DSG) of angle α. Now we will prove the implication (iii.2) ⇒ (iii.1). Suppose that A is the generator of an exponentially bounded, analytic r-times integrated semigroup (Sr (t))t≥0 of angle α. Then it is clear that A generates an exponentially bounded, analytic once integrated semigroup (S1 (t))t≥0 of angle α, where t −r S1 (t)x = 0 (t−s) Γ(1−r) Sr (s)xds, t ≥ 0, x ∈ E. Let γ ∈ (−α, α) be ﬁxed. Then Lemma 1.4(v) yields that, for every r ≥ r, the operator eiγ A generates an exponentially bounded r -times integrated semigroup (Sr ,γ (t))t≥0 and that, for every t ≥ 0 and x ∈ E, e−iγ S1 (teiγ )x = S1,γ (t)x t t (t − s)−r (t − s)−r Sr,γ (s)xds = e−iγr Sr (seiγ )xds. = Γ(1 − r) Γ(1 − r) 0

0

Hence, i arg z(1−r)

|z|

S1 (z)x = e

0

(|z| − s)−r Sr (sei arg z )xds, z ∈ Σα , x ∈ E, Γ(1 − r)

and there exist Mγ > 0 and ωγ ≥ 0 such that ||S1 (z)|| ≤ Mγ |z|1−r eωγ Rez , z ∈ Σγ . Thanks to Theorem 2.5 and the assertion (i) of this theorem, we have d that A generates an analytic (DSG) G of angle α and that G(δt ) = dt S1 (t), d t > 0. By [61, Proposition 3.2], ( dz Sn (z))z∈Σα is an analytic operator family, T (z1 + z2 ) = T (z1 )T (z2 ), z1 , z2 ∈ Σα and the supposition T (t)x = 0 for all t > 0 implies x = 0. By the uniqueness theorem for analytic functions,

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d S1 (z). Let γ ∈ (0, α) and let it follows that, for every z ∈ Σα , G(δz ) = dz ε ∈ (0, α − γ). Then the Cauchy integral formula gives that, for every z ∈ Σγ with Imz ≥ 0 : ) S1 (λ) r r 1 = G(δ ) dλ z z z 2 2πi (λ − z) |λ−z|=|z| sin(γ+ε−arg z)

1 r ≤ |z| 2π

2π 0

||S1 (z + |z| sin(γ + ε − arg z)eiθ )|| dθ |z| sin(γ + ε − arg z)

1 . sin ε The preceding estimate also holds for every z ∈ Σγ with Imz ≤ 0 and this gives (iii.1). Let A generate an analytic (DSG) G of angle α satisfying that (G(δz ))z∈Σα is an analytic semigroup of growth order r. Put S1 (t)x := t G(δs )xds, t ≥ 0, x ∈ E and assume that Aˆ is the integral generator of 0 (G(δt ))t≥0 . Then it can be simply veriﬁed that A ⊆ Aˆ and that (S1 (t))t≥0 is ˆ By the foregoing, there are an a once integrated semigroup generated by A. n ∈ N\{1} and an analytic function Sn : Σα → L(E) so that A generates an n-times integrated semigroup (Sn (t))t≥0 which satisﬁes Sn (t) = Sn (t), t > 0 dn n ∞ (n) and dt ϕ (t)Sn (t)xdt and n Sn (t) = G(δt ), t > 0. Put G(ϕ)x := (−1) 0 ∞ t H(ϕ)x := − 0 ϕ (t) 0 G(δs )xdsdt, ϕ ∈ D, x ∈ E. Then G, respectively, ˆ and the partial integration H, is a (DSG) generated by A, respectively A, ∞ implies that G(ϕ)x = H(ϕ)x = 0 ϕ(t)G(δt )xdt for all ϕ ∈ D(0,∞) and x ∈ E. So, G(ϕ) = H(ϕ), ϕ ∈ D0 and this forces Aˆ = A. Deﬁne, for every t r−1 iγ x ∈ E, t ≥ 0 and γ ∈ (−α, α) : Sr,γ (t)x =: 0 (t−s) Γ(r) G(δse )xds. Then (G(δt ))t≥0 is a semigroup of growth order r > 0 whose integral generator is A and this implies that, for every γ ∈ (−α, α), (G(δteiγ ))t≥0 is a semigroup of growth order r > 0 whose integral generator is eiγ A and that (Sr,γ (t))t≥0 is an exponentially bounded, r-times integrated semigroup generated by eiγ A. Furthermore, for every γ ∈ (−α, α), there exist Mγ > 0, Mγ > 0 and ωγ ≥ 0 such that t (t − s)r−1 −r ωγ t cos γ s dr ≤ Mγ eωγ t cos γ , t ≥ 0. Sr,γ (t) ≤ Mγ e Γ(r) 1−r

≤2

Mγ+ε eωγ+ε (Rez+|z|)

0

By Lemma 1.4(iv), it follows that A generates an exponentially bounded, analytic r-times integrated semigroup (Sr (t))t≥0 of angle α and the implication (iii.1) ⇒ (iii.2) is proved. The equivalence (iii.2) ⇔ (iii.3) follows from an application of Lemma 1.3(iii) and this completes the proof of theorem. Remark 2.23. (i) Suppose that r ∈ (0, 1) and that A generates a differentiable (DSG) G such that (G(δt ))t>0 is a semigroup of growth order r. Using the same arguments as in the proof of preceding theorem, we easily infer that A generates an exponentially bounded r-times integrated semigroup (Sr (t))t≥0 that is inﬁnitely differentiable in t > 0. It is not

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clear whether the converse statement holds. We refer the reader to [24, Examples 14.4, 14.5], [28, Examples 24.10(b), 24.11(b)], [95,96,108] for examples of non-dense differential operators which generate analytic r-times integrated semigroups of angle π2 . (ii) Using Theorem 2.19(i) and the assertion (i) of Theorem 2.22, one can simply prove that the analyticity of (DSG)’s is preserved under bounded commuting perturbations. Further on, let G be a dense (DSG). Then G∗ is a dense (DSG) in E ∗ [73] and G∗ inherits all considered differential (analytical) properties from G; with [62, Proposition 5.4] in view, the above assertion can be reformulated in the case of dense differentiable (C-DSG)’s. Example. Let E := L2 [0, π], A := −Δ with the Dirichlet or Neumann bound∞ n2 −λ Reλ > 0, λ = n2 , n ∈ ary conditions and let h(λ) := λ12 n2 +λ , n=0

N; h(n2 ) := 0, n ∈ N. Deﬁne h : C\{−n2 : n ∈ N0 } → C by setting: h(λ) =: ∞ 2 h(λ), λ ∈ D(h) and h(λ) := λ12 n=0 nn2 −λ / {−n2 : n ∈ N0 }. It +λ , Re λ ≤ 0, λ ∈ is obvious that the function h is analytic and that there exists an exponen˜ tially bounded, continuous function K such that K(λ) = h(λ), Reλ > 0 and ˜ that 0 ∈ suppK [6]. Furthermore, the function λ → K(λ)R(λ : A), Reλ > 0, h(λ) = 0, can be extended to an analytic function F : {λ ∈ C : Reλ > 0} → L(E) and there exists M > 0 such that ||F (λ)|| ≤

M + λ1 , Reλ > 0. |λ|2

(2.28)

In order to see that the function h(·) is well deﬁned and analytic, one can ∞

ln

n2 −λ

∞

(ln

|n2 −λ|

+i arg( n

2 −λ

))

n2 +λ , the use the equality λ2 h(λ) = e n=0 n2 +λ = e n=0 |n2 +λ| estimates

∞ ∞ ∞ 2|λ| |n2 − λ| 2|λ| ≤ , ln 2 ln 1 + 2 ≤ 2 + λ| |n + λ| |n + λ| |n n=0 n=0 n=0 2

n −λ 2n2 |λ| 2n2 |λ| arg ≤ arctan 4 ≤ 4 , n2 > |λ|, λ ∈ D(h), Reλ ≤ 0, 2 2 n +λ n − |λ| n − |λ|2

and the Weierstrass theorem. Proceeding as in [6], one gets:

h(λ)R(λ : A) ≤

∞ |j 2 − λ| 1 1 , λ ∈ D(h), λ = k 2 , k ∈ N, |λ|2 n=0 |n2 + λ| |j 2 + λ| j =n

(2.29) and a simple computation shows that, for every j ∈ N0 , r > 0 and θ ∈ [ π2 , π) : |

((j 4 − r2 )2 + 4j 4 r2 sin2 θ)1/2 j 2 − reiθ |= 2 iθ j + re j 4 + r2 + 2j 2 r cos θ 1 j 4 + r2 ≤ . ≤ 4 2 j + r + 2j 2 r cos θ 1 + cos θ

(2.30)

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Our goal is to evaluate the right hand side of (2.29). Clearly, for every γ ∈ (0, π2 ) and λ ∈ Σγ+ π2 : √ 2|λ| ∞ ∞ 1 1 1 ≤ + 2 2 2 |n + λ| |n + λ| |n + λ| √ n=0 n=0 ≤

√ 2|λ| n=0

n=

1 + 2 |n + λ|

2|λ|

∞

√

n=

2|λ|

2 n2

2|λ| + 1 π 2 2|λ| + 1 π 2 + ≤ + . ≤ |Imλ| 3 |λ| cos γ 3 1

Suppose now a ∈ ( 12 , 1), ln(1 + t) ≤ ca ta , t ≥ 0, j 4 ≥ 3j 2+ a , j ≥ j0 and γ ∈ (0, π2 ). Keeping in mind (2.30), one obtains that, for every λ = reiθ ∈ Σγ+ π2 with Reλ ≤ 0 : ∞ |j 2 − λ| |j 2 + λ| j=0 ⎞ ⎛ ∞ ∞ |j 2 − λ| j 4 + r2 1 ⎠ ⎝ ln 2 ln ) ≤ exp( ≤ exp |j + λ| 2 j=0 j 4 + r 2 + 2j 2 r cos θ j=0 ⎞ ⎛ r a +j0 ∞ 4 + r2 4 + r2 1 j j 1 ⎠ + ln 4 ln 4 ≤ exp ⎝ 2 j=0 j + r 2 + 2j 2 r cos θ 2 j + r 2 + 2j 2 r cos θ j=r a +j0 ⎞ ⎛ ∞ 2 r| cos γ| 1 1 2j 1 a + )⎠ ln(1 + 4 ≤ exp ⎝ (r + j0 + 2) ln 2 1 + cos( π2 + γ) 2 j + r 2 + 2j 2 r cos θ j=r a +j0 ⎛ ⎞ ∞ 2 1 1 j 1 ≤ exp ⎝ (r a + j0 + 2) ln ca 2a r a ( 4 + )a⎠ 2 1 + cos( π2 + γ) 2 j + r 2 + 2j 2 r cos θ j=r a +j0 ⎞ ⎛ ∞ 1 1 j2 1 a a a a⎠ ⎝ (r +j0 + 2) ln + ) ca 2 r ( ≤exp 1 2 1 + cos( π2 + γ) 2 j 4 + 2j 2 j a cos(γ + π2 ) j=r a +j0 ⎛ ⎞ ∞ 1 a 1 j2 1 a a a⎠ ⎝ ≤ exp ca 2 r ( (r + j0 + 2) ln + ) 1 2 1 + cos( π2 + γ) 2 j=j j 4 + 2j 2 j a cos(γ + π ) 0

2

≤ Mγ,a exp (cγ,a |λ|a ) ,

for appropriate cγ,a > 0 and Mγ,a > 0.The preceding inequality, (2.28)– ∞ 1 (2.29) and the estimation of the term n=0 |n2 +λ| imply that, for every π 2 γ ∈ (0, 2 ) and λ ∈ Σγ+ π2 , λ = k , k ∈ N, ||h(λ)R(λ : A)|| 1 M + |λ| Mγ,a 2|λ| + 1 π 2 a ≤ max + exp(cγ,a |λ| ) . (2.31) , |λ|2 |λ|2 |λ| cos γ 3 ∞ Let Mp = p!s for some s ∈ (1, 2), ωl (λ) = p=1 (1 + plλs ), λ ∈ C and let Kl (t) = L−1 (1/ωl (λ))(t), t ≥ 0, l > 0. By (1.8) and the well known estimates for the associated function of the sequence (p!s ) [53], it follows that there

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˜l (λ)| ≤ e−cl,γ |λ|1/s , λ ∈ Σγ+ π . With exists a constant cl,γ > 0 such that |K 2 (2.31) and Lemma 1.4(ii) in view, we get that, for every l > 0, −Δ generates an exponentially bounded, analytic (Kl ∗0 K)-convoluted semigroup of angle π2 .

3. Differentiable and Analytic Ultradistribution Semigroups We recall the following definition of a (tempered) ultradistribution semigroup and a (tempered) ultradistribution fundamental solution. Definition 3.1. [65] (i) Suppose G ∈ D0∗ (L(E)). If G satisﬁes (U.1) : G(ϕ ∗0 ψ) = G(ϕ)G(ψ), ϕ, ψ ∈ D∗ , then G is called a pre-ultradistribution semigroup of ∗-class, pre-(UDSG) in short. If, additionally, (U.2) : N (G) =: N (G(ϕ)) = {0}, ϕ∈D0∗

then it is said that G is an ultradistribution semigroup of ∗-class, (UDSG) for short. Further on, a pre-(UDSG) G is called dense if the set R(G) := ϕ∈D0∗ R(G(ϕ)) is dense in E. The (inﬁnitesimal) generator A of a (UDSG) G of ∗-class is deﬁned by A := {(x, y) ∈ E × E : G(−ϕ )x = G(ϕ)y for all ϕ ∈ D0∗ }. Let D be another Banach space and let P ∈ D0∗ (L(D, E)). Then we say that G ∈ D0∗ (L(E, D)) is an ultradistribution fundamental solution for P if P ∗G = δ ⊗IE and G∗P = δ ⊗ID . By an ultradistribution fundamental solution (of ∗-class) for a closed linear operator A we mean the ultradistribution fundamental solution for P := δ ⊗ID(A) −δ ⊗A ∈ D0∗ (L([D(A)], E)). (iii) Suppose G is a (UDSG) of ∗-class. Then it is said that G is an exponential ultradistribution semigroup of ∗-class, in short (EUDSG), iﬀ there exists ω ≥ 0 such that e−ωt G ∈ S ∗ (L(E)). (iv) Suppose G is an ultradistribution fundamental solution for a closed linear operator A. Then it is said that G is an exponential ultradistribution fundamental solution of ∗-class for A iﬀ there exists ω ≥ 0 such that e−ωt G ∈ S ∗ (L(E)). (ii)

Due to (U.2), the inﬁnitesimal generator A of a (UDSG) of ∗-class is a function and it can be easily seen that A is a closed linear operator. At this place, it is also worth noting that an ultradistribution fundamental solution for a closed linear operator A is uniquely determined and need not be a (UDSG) of ∗-class (cf. [21, p. 156] and [65]). Furthermore, the following assertions hold: (i) [65] An element G ∈ D0∗ (L(E, [D(A)])) is an ultradistribution fundamental solution for A and satisﬁes N (G) = {0} iﬀ G is a (UDSG) of ∗-class generated by A.

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[65] If G ∈ D0∗ (L(E, [D(A)])) is an ultradistribution fundamental solution for A, then G is a pre-(UDSG) of ∗-class in E. [67] Suppose, in addition, that (Mp ) satisﬁes (M.3) and that A generates a (UDSG) of (Mp )-class. Then the abstract Cauchy problem ⎧ ⎨ u ∈ C ∞ ([0, ∞) : E) ∩ C([0, ∞) : [D(A)]), (ACP ) : u (t) = Au(t), t ≥ 0, ⎩ u(0) = x, has a unique solution for all x ∈ E (Mp ) (A), where the abstract Beurling space of (Mp ) class associated to a closed linear operator A is deﬁned {M } as follows [18,67]: E (Mp ) (A) =: proj limh→+∞ Eh p (A), where ' ( hp ||Ap x|| {M } {M } <∞ . Eh p (A) := x ∈ D∞ (A) : ||x||h p = sup Mp p∈N0 Furthermore, for every compact set K ⊆ [0, ∞) and h > 0, the solution hp ||

dp

u(t)||

dtp u of (ACP ) satisﬁes supt∈K,p∈N0 < ∞. Mp (iv) [67] Suppose, in addition, that (Mp ) satisﬁes (M.3) and that A generates an (EUDSG) of ∗-class. Then there exists an injective operator C ∈ L(E) such that A generates an exponentially bounded C-regularized semigroup (S(t))t≥0 that is inﬁnitely differentiable in t ≥ 0. Furthermore, for every compact set K ⊆ [0, ∞) and for every p

h > 0, supt∈K,p∈N0

d hp || dt p S(t)|| Mp

< ∞.

The following theorem improves the joint result of the author and Pilipovi´c [65] related to exponential ultradistribution semigroups. Theorem 3.2. Suppose A is a closed linear operator. Then there exists an exponential ultradistribution fundamental solution of ∗-class for A iﬀ there exist a ≥ 0, k > 0 and L > 0, in the Beurling case, respectively, there exists a ≥ 0 such that, for every k > 0 there exists Lk > 0, in the Roumieu case, such that: {λ ∈ C : Reλ > a} ⊆ ρ(A) and M (k|λ|)

||R(λ : A)|| ≤ Le

, λ ∈ C, Reλ > a, respectively,

M (k|λ|)

||R(λ : A)|| ≤ Lk e

for all k > 0 and λ ∈ C with Reλ > a.

(3.1) (3.2) (3.3)

Proof. We will prove the assertion only in the Beurling case since the proof is quite similar in the Roumieu case. Suppose ﬁrst that G is an exponential ultradistribution fundamental solution of (Mp )-class for A and that e−ωt G ∈ S (Mp ) (L(E)) for some ω ≥ 0. Let r > 0 and let a function h ∈ E (Mp ) (R) satisfy h(t) = 0, t < −r and h(t) = 1, t ≥ 0. Then the function t → h(t)e(ω−λ)t , t ∈ R belongs to the space S (Mp ) for all λ ∈ C with Reλ > ω. This ˜ implies that the definition of G(λ) := G(h(t)e−λt ) := G(e−ωt (h(t)e(ω−λ)t )) is meaningful for all λ ∈ C with Reλ > ω. Since e−ωt G is an ultradistribution fundamental solution for A − ω, we get that (A − ω)G(e−ωt ϕ)x = G(−e−ωt ϕ )x − ϕ(0)x, x ∈ E, ϕ ∈ D(Mp ) . The continuity of e−ωt G on S (Mp ) , and the denseness of D(Mp ) in S (Mp ) , imply that the previous equality holds for all ϕ ∈ S (Mp ) . Then a simple computation with ϕ(t) = h(t)e(ω−λ)t , t ∈ R,

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˜ ˜ and the obvious inclusion suppG ⊆ [0, ∞), imply AG(λ)x = λG(λ)x − x, x ∈ ˜ E, λ ∈ C, Reλ > ω. Hence, (λ − A)G(λ)x = x, x ∈ E, λ ∈ C, Reλ > ω ˜ ˜ ˜ and, since G(λ)A ⊆ AG(λ), λ ∈ C, Reλ > ω, we have G(λ)(λ − A)x = x, x ∈ D(A), λ ∈ C, Reλ > ω. This implies (3.1) with a = ω and R(λ : A)x = ˜ G(λ)x, x ∈ E, λ ∈ C, Reλ > a. It is obvious that there exist h > 0 and L > 0 ˜ such that ||G(λ)|| = ||(e−ωt G)(h(t)e(ω−λ)t )|| ≤ L||h(t)e(ω−λ)t )||Mp ,h for all λ ∈ C with Reλ > ω. Then the estimate (3.2) follows easily from the preceding inequality, which ends the proof of necessity. To ∞ prove sufﬁciency, suppose β > a, l ≥ max(k, 1), α > 0 and put ϕ(λ) ˜ := −∞ eλt ϕ(t)dt, λ ∈ C, ϕ ∈ 1 D(Mp ) , and G(ϕ) := 2πi ϕ(λ)R(λ ˜ : A)dλ, ϕ ∈ D(Mp ) , where Γl denotes the Γl upwards oriented boundary of the ultra-logarithmic region Λα,β,l . Proceeding as in [18,56] and [86, Sect. 2.3], one gets that G is an ultradistribution fundamental solution for A. By the Paley–Wiener type theorem (cf. [18, Theorem 1.1] and [54]) and Cauchy theorem, we can deform the path of integration Γl into the straight line connected a ¯ − i∞ and a ¯ + i∞, where a ¯ ∈ (a, β). Hence, a¯+i∞ 1 (Mp ) G(ϕ) = 2πi ϕ(λ)R(λ ˜ : A)dλ, ϕ ∈ D . We will prove that e−ωt G ∈ a ¯−i∞ S (Mp ) (L(E)) for all ω ∈ (¯ a, ∞). Suppose ϕ ∈ D(Mp ) , suppϕ ⊆ [a, b], a < 0 (Mp ) and b > 0. Let ζ ∈ D , ζ(t) = 1, t ∈ [−1, 1], ζ(t) = 0, t ≥ 2 and let a function ξ ∈ D(Mp ) satisfy ξ(t) = 1, −1 ≤ t ≤ b + 1, ξ(t) = 0, t ≤ −2 and ξ(t) = ζ(t − b − 1), t ≥ b + 1. Then ϕξ ∈ D(Mp ) , ϕ(t)ξ(t) = ϕ(t), t ≥ 0 and G(ϕ) = G(ϕξ). The partial integration and the proof of implication (III) ⇒ (II) of [86, Theorem 2.3.1] imply that, for every h > 0, n ∈ N0 , β ∈ N0 and λ ∈ C with Reλ = a ¯: −ω· ϕξ(λ)| = |e

(−1)n (λ − ω)n

1 ≤ |λ − ω|n ≤ ≤

1 |λ − ω|n

b

e(λ−ω)t (ϕξ)(n) (t)dt

−2

b

e−(ω−¯a)t |(ϕξ)(n) (t)|dt

−2

b −2

e−(ω−¯a)t (1 +

β

t2 ) 2

dt

||ϕξ||Mp ,h Mn Mβ hn+β

e2(ω−¯a) − e−(ω−¯a)b Mn Mβ ||ϕξ||Mp ,h . (ω − a ¯)|λ − ω|n hn hβ

Noticing that the preceding equality holds for every n ∈ N and that inf

Mn = e−M (h|λ−ω|) , − ω|n

n∈N0 hn |λ

one gets that, for every h > 0, β ∈ N0 and λ ∈ C with Reλ = a ¯, there exists M > 0, independent of a and b, such that: Mβ −M (h|λ−ω|) −ω· ϕξ(λ)| ≤ M |e e ||ϕξ||Mp ,h . hβ

(3.4)

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By the definition of ||·||Mp ,h and the logarithmic convexity Mp+q ≥ Mp Mq , p, q ∈ N0 , one gets that, for every h > 0 : ( ' α+β h 2 β/2 (α) ||ϕξ||Mp ,h = sup (1 + t ) |(ϕξ) (t)| Mα Mβ t∈R, α, β∈N0 * + α

α hα+β ≤ sup (1 + t2 )β/2 |ϕ(α−i) (t)||ξ (i) (t)| i Mα Mβ t∈R, α, β∈N0 i=0 + * α

α hα+β 2 β/2 (α−i) (i) ≤ sup (1 + t ) (t)||ξ (t)| |ϕ i t∈R, α, β∈N0 Mα−i Mi Mβ i=0 * + α

1 hα−i hβ hi α ≤ sup ||ϕ||Mp ,4h |ξ (i) (t)| i Mi (4h)α+β−i t∈R, α, β∈N0 i=0 α

α 1 (2h)i |ξ (i) (t)| 1 ≤ sup ||ϕ||Mp ,4h i α+β−i i 2 Mi 4 t∈R, α, β∈N0 i=0 α

α 1 1 ≤ sup ||ϕ||Mp ,4h ||ξ||Mp ,2h,[−2,b+3] i 2i 4α−i α∈N0 i=0 ≤ ||ϕ||Mp ,4h ||ξ||Mp ,2h,[−2,b+3] . This in combination with (3.4) yields that there exists M > 0, independent of a and b, such that, for every β ∈ N0 and h > 0, Mβ −M (h|λ−ω|) −ω· ϕξ(λ)| ≤ M |e e ||ϕ||Mp ,4h . hβ Now one can apply [12, Lemma 2.1.3] to deduce that there exists h > 0 such that ||e−ωt G(ϕ)|| ≤ Const.||ϕ||Mp ,4h , ϕ ∈ D(Mp ) . The proof of theorem is thereby complete. The relationship between exponential (UDSG)’s of the Beurling class and exponentially bounded convoluted semigroups has been recently discussed in [63]. Notice that the preceding theorem enables one to transfer the assertion of [63, Theorem 3.10] to Roumieu case as well as to remove the density assumption from the formulation of [63, Theorem 3.10(i)]. Corollary 3.3. Suppose A is a closed linear operator and there exists an exponential ultradistribution fundamental solution of ∗-class for A. Then there exists k > 0 such that, for every compact set K ⊆ [0, ∞) and for every l > k, there exists hl,K > 0, in the Beurling case, respectively, for every compact set K ⊆ [0, ∞) and for every l > 0, there exists hl,K > 0, in the Roumieu case, such that A generates an exponentially bounded, Kl -convoluted semigroup (SKl (t))t≥0 such that the mapping t → SKl (t), t ≥ 0 is inﬁnitely p

differentiable and supt∈K,

p∈N0

d hp l,K || dtp SKl (t)|| Mp

< ∞.

Let G be a (UDSG) of ∗-class and let T ∈ E0∗ . Then T ∗ φ ∈ D0∗ , φ ∈ D0∗ and the following definition of the operator G(T ) makes sense: , G(T ) := (x, y) ∈ E × E : G(T ∗ ϕ)x = G(ϕ)y for all ϕ ∈ D0∗ .

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It is worthwhile to mention that (U.2) implies that G(T ) is a closed linear operator for every T ∈ E0∗ . Notice that the inﬁnitesimal generator A of G satisﬁes A = G(−δ ) and that G(δ) = I. Definition 3.4. Suppose that G is a (UDSG) of ∗-class and that α ∈ (0, π2 ]. Then it is said that G is an (inﬁnitely) differentiable ultradistribution semigroup (of ∗-class), respectively an analytic ultradistribution semigroup of angle α (of ∗-class) iﬀ the mapping t → G(δt ) ∈ L(E), t > 0 is (inﬁnitely) differentiable, respectively, iﬀ the mapping t → G(δt ) ∈ L(E), t > 0 can be analytically extended to the sector Σα . Let G be a (DSG), respectively a (UDSG) of ∗-class, generated by A and let the mapping t → G(δt ) ∈ L(E), t > 0 be continuous. Then (G(δt ))t≥0 is a semigroup [73,65] and this implies that there exists ω0 ∈ [−∞, ∞) such t )|| . The asymptotic behavior of (G(δt ))t≥0 in a that ω0 = limt→∞ ln ||G(δ t neighborhood of zero is quite complicated (cf., for example, [27, p. 575]). Theorem 3.5. Suppose that A generates a dense differentiable (UDSG) G of ∗-class. Then there exist h > 0 and ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0, in the Beurling case, respectively, for every h > 0 and σ > 0, there exist Cσ > 0 and Mσ,h > 0 in the Roumieu case, such that (2.2) holds and that, for every λ ∈ Υσ,ω , ||R(λ : A)|| ≤ Mσ eM (h|λ|) , respectively, ||R(λ : A)|| ≤ Mσ,h eM (h|λ|) .

(3.5)

Specifically, G is an (EUDSG) of ∗-class. Proof. We will prove the assertion in the Beurling case and remark the minor changes in the Roumieu case. Suppose ε ∈ (0, 1), ψ ∈ D(Mp ) , ψ(t) = 1, 0 ≤ t ≤ 12 , ψ(t) = 0, t ≥ 1 and ψ(t) = 0, t ≤ −1. Set ψε (t) := ψ( εt ), t ∈ R. By the proof of implication (II) ⇒ (III) of [86, Theorem 2.3.1], it follows that (2.12) is and since the set R(G) is dense in E, we have G(e−λt ψε (t))x = ε valid, −λt ψε (t)G(δt )xdt for all x ∈ E and λ ∈ C (cf. [7, Remarks, p. 416] ε e 2

and the partial integration gives that G(e−λt ψε (t))x = [65]). Then 1 ε −λt d λ ε e dt (ψε (t)G(δt )x)dt for all x ∈ E and λ ∈ C\{0}. This implies 2

d (ψε (t)G(δt ))||. Deﬁne Φε by (2.14). Then (2.13) with Mε = sup 12 ε≤t≤ε || dt 1 −λt ||G(e ψε (t))|| ≤ 2 , λ ∈ Φε , and as in the proof of [7, Theorem 3], one gets that Φε ⊆ ρ(A) and that (2.15) holds. By the continuity of mapping G, we have that there exist h > 0 and C > 0, independent of ε > 0, such that:

||G(e−λt ψε (t))|| ≤ C||e−λt ψε (t)||Mp ,2h ,[0,1] = C||e−λt ψε (t)||Mp ,2h ,[ 2ε ,ε] . (3.6) In the Roumieu case, the previous estimate holds for every h > 0 and a corresponding Ch > 0. By (2.15)–(3.6) and the inequality [86, (2.3.9), Theorem 2.3.1, p. 170], we reveal that, for every λ ∈ Φε : 1

||R(λ : A)|| ≤ 2C||e−λt ψε (t)||Mp ,2h ,[ 2ε ,ε] ≤ 2C||ψε ||Mp ,h ,[ 2ε ,ε] eM ( h )− 2 εReλ . λ

(3.7)

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With (3.7) in view, one obtains that there are an h > 0 and a constant Mε > 0 (in the Roumieu case, such a constant Mε also depends on h ) such that ||R(λ : A)|| ≤ Mε eM ( h ) (1 + |λ|), λ ∈ Φε . λ

Especially, with ε = 12 , one yields that 1 {λ ∈ C : Reλ ≥ max(1, 4| ln( M1/2 )|)} ⊆ ρ(A), 2 which implies by Theorem 3.2 that G is an (EUDSG). Put aε := ε ε )| ). Let Nε > e 2 a1/2 +| ln(εMε )| and let h > h1 . Then h is indemax(1, 2| ln(εM ε pendent of ε ∈ (0, 1) and there exists an appropriate constant Mε > 0 such that the set 1 1 Δε := {λ ∈ C : Reλ ≤ a1/2 , Reλ ≥ − ln |Imλ| + ln Nε } ∪ {λ ∈ C : Reλ ≥ a1/2 } ε ε

belongs to ρ(A) and that ||R(λ : A)|| ≤ Mε eM (h|λ|) , λ ∈ Δε . In the case σ ≥ 1, (3.5) follows with ε = σ1 , ω = a1/2 , Cσ = 1ε ln Nε , Mσ = Mε , and the above chosen h; the case σ ∈ (0, 1) is completely regardless and this ends the proof of theorem. Let a > 0 and b ∈ (0, 1). Set Ka,b (t) := L−1 (e−aλ )(t), t ≥ 0, where 1b = 1. b

Example. (i) [64] Let Mp = p!s , s ∈ (1, 2), E := Lp (R), 1 ≤ p ≤ ∞, D(A) := {f ∈ E : (x + ix2 )f (x) ∈ E} and Af (x) := (x + ix2 )f (x), x ∈ R, f ∈ D(A). Set δ := 1s . Then A generates a global K1,δ -convoluted semigroup since, for every τ > 0, A generates a K1,δ -convoluted semigroup on [0, τ ). To this end, suppose M (λ) ≤ Cs |λ|1/s , λ ∈ C, τ > 0, α > cos( δπ )

2 . It is checked at once that there exists β > 0 such 0 and τ ≤ Cs α1/s that Λα,β,1 ⊆ ρ(A) and that the resolvent of A is bounded on Λα,β,1 , where Λα,β,1 = {λ ∈ C : Reλ ≥ M (αλ) + β}. Put Γ := ∂Λα,β,1 , assume that Γ is upwards oriented and deﬁne δ eλt−λ 1 dλf (x), (Sδ,τ (t)f )(x) := 2πi λ − (x + ix2 )

Γ

f ∈ E, x ∈ R, t ∈ [0, Since |e−λ | ≤ e− cos( δ

δ δπ 2 )|λ|

λt−λδ

|e

cos( δπ 2 ) ). Cs α1/s

, Reλ > 0 and δ βt M (αλ)t−cos( δπ 2 )|λ|

|≤e e

δ βt Cs α1/s |λ|1/s t−cos( δπ 2 )|λ|

≤e e

, λ ∈ Γ,

one can straightforwardly verify that (Sδ,τ (t))t∈[0,τ ) is a local K1,δ convoluted semigroup generated by A. Note that the mapping t → cos( δπ 2 ) ) is inﬁnitely differentiable and that Sδ,τ (t), t ∈ [0, Cs α1/s λp eλt−λδ cos( δπ dp 1 2 ) dtp Sδ,τ (t)f (·) = 2πi Γ λ−(·+i·2 ) dλf (·), f ∈ E, t ∈ [0, Cs α1/s ). This

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implies that A generates a global non-exponentially bounded K1,δ convoluted semigroup (Sδ (t))t≥0 that is inﬁnitely differentiable for t ≥ 0. Furthermore, for every τ > 0, there exists hτ > 0 such that p

supt∈[0,τ ),

d hp τ || dtp Sδ (t)|| < Mp with Lp = p!s/p .

p∈N0 L

∞, and this implies that (Sδ (t))t≥0 is

of class C It can be also proved that there exists τ > 0 such that A generates a local K1, 12 -convoluted semigroup of class C L with Lp = p!2/p . On the other hand, it is obvious that A generates a (UDSG) G of ∗-class (with Mp = p!s ), given by (G(ϕ)f )(x) := ϕ(λ) ˜ 1 ∗ 2πi Γ λ−(x+ix2 ) dλf (x), ϕ ∈ D , f ∈ E, x ∈ R. Let t > 0, f ∈ E and g ∈ E. By the Paley–Wiener theorem and the Cauchy theorem, it follows that (G(ϕ)f )(x) = ϕ(x ˜ + ix2 )f (x), ϕ ∈ D0∗ , f ∈ E, x ∈ R and that (x+ix2 )t G(δt )f = g iﬀ e f (x) = g(x), x ∈ R. Hence, G(δt ) ∈ / L(E) and G is not differentiable. The above example also shows that a large number of (pseudo-)differential operators acting in Lp (Rn ) (the case p = 2 is also allowed) generate (local) convoluted Ka,b -semigroups of class C L . (p)

(ii) [63,67,72] Let E =: {f ∈ C ∞ [0, 1] : f Mp := supp≥0 f Mp∞ < ∞} and A := −d/ds, D(A) := {f ∈ E : f ∈ E, f (0) = 0}. It is well known that A is not stationary dense and that A cannot be the generator of a (DSG). Furthermore, {λ ∈ C : Reλ ≥ 0} ⊆ ρ(A)and R(λ : A) ≤ p ∞ M (˜ l|λ|) , Reλ ≥ 0, for some C > 0 and ˜l > 0 [54,72]. C p=0 |λ| Mp ≤ Ce t Put now (G(ϕ)f )(t) := 0 ϕ(t − s)f (s)ds, ϕ ∈ D(Mp ) , f ∈ E, t ∈ [0, 1]. Then G is a non-dense (EUDSG) generated by A [67] and G(δt ) = 0, t ≥ 1. Suppose now t ∈ (0, 1). Then it can be simply veriﬁed [63,67] that, for ' 0, 0 ≤ s ≤ t, every s ∈ [0, 1] and f ∈ D(G(δt )), (G(δt )f )(s) = f (s − t) , 1 ≥ s > t. In particular, we have G(δt ) ∈ / L(E) and G is not differentiable. Furthermore, for every h > 0, there exists l > ˜l such that A generates a bounded Kl -convoluted semigroup (SKl (t))t≥0 that is inﬁnitely differentiable in t ≥ 0 and satisﬁes supp∈N0 ,t∈K p

d hp || dt p SKl (t)|| Mp

< ∞ for every compact set K ⊆ [0, ∞). This implies that relations between differentiable convoluted semigroups and differentiable (UDSG)’s are more complicated than it looks at ﬁrst sight. Further on, the previous example also shows that the existence of a number h > 0 satisfying that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that (2.2) and (3.5) hold, is not sufﬁcient for the generat tion of differentiable (UDSG)’s. Indeed, put hλ (t) := e−λt 0 eλs f (s)ds, f ∈ E, λ ∈ C. Then we know that R(λ : A)f = hλ if Reλ ≥ 0. Clearly, for every t ∈ [0, 1] and n ∈ N with n ≥ 2 : n−1−k dn dn−1 k d h (t) = f (t) + (−λ) f (t) + (−λ)n hλ (t). λ dtn dtn−1 dtn−1−k n−1

(3.8)

k=1

Let σ > 0 and let Cσ > 0 be arbitrarily chosen. Then the supposition Reλ ≥ −σ ln |Imλ| + Cσ implies that, for every t ∈ [0, 1] :

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eReλ(t−s) ds||f ||L∞ [0,1] ≤ e−Reλ ||f ||L∞ [0,1]

0

≤ e−Cσ |Imλ|σ ||f ||L∞ [0,1] .

By (3.8), the previous inequality and the logarithmic convexity, one gets that there exists Cσ > 0 such that: n

d || dt n hλ (t)||L∞ [0,1] ˜ 1 ≤ Cσ |Imλ|σ eM ((l+ 2 )|λ|) ||f ||, n ∈ N, n ≥ 2. (3.9) Mn Since λhλ (t) + hλ (t) = f (t), t ∈ [0, 1] and hλ (0) = 0, one easily infers that ||hλ (·)||L∞ [0,1] ≤ (1 + e−Cσ |λ||Imλ|σ )||f ||L∞ [0,1] . Keeping in mind (3.9), we obtain that there exists Mσ > 0 such that, for every λ ∈ C with Reλ ≤ 0 and Reλ ≥ −σ ln |Imλ| + Cσ , ||R(λ : A)|| ≤ ˜ Mσ eM ((l+1)|λ|) . Therefore, (2.2) and (3.5) hold but A does not generate a differentiable (UDSG), as claimed. Notice also that ρ(A) = C and that the preceding estimation of the resolvent still holds if one considers S Mp ,1 as the underlying Banach space.

Theorem 3.6. Suppose A generates a dense (UDSG) G of ∗-class. Then the following assertions hold. (i) Suppose h > 0, G is inﬁnitely differentiable and the mapping t → G(δt ), t > 0 satisﬁes that, for every compact set K ⊆ (0, ∞), p

||G(δt )||Mp ,h,K := Put

sup t∈K, p∈N0

d hp || dt p G(δt )|| < ∞. Mp

'

( |λ| λ ∈ C : Reλ ≤ ω, Reλ ≥ Cσ,h − σM H + h1 ∪ {λ ∈ C : Reλ ≥ ω}.

Υhσ,ω :=

Then Υhσ,ω ⊆ ρ(A) and the following holds: ∃ω ≥ 0∃h > 0∀σ > 0∃Cσ,h > 0∃Mσ,h > 0, in the Beurling case, respectively,

(3.10)

∃ω ≥ 0∀h > 0∀σ > 0∃Cσ,h > 0∃Mσ,h,h > 0, in the Roumieu case, such that: ||R(λ : A)|| ≤ Mσ,h e

M ( |λ| )+ 12 M ( h

|λ| H+ 1 h

)

(3.11) , λ ∈ Υh σ,ω ,

in the Beurling case, respectively, ||R(λ : A)|| ≤ Mσ,h,h e

M ( |λ| )+ 12 M ( h

in the Roumieu case.

(ii)

|λ| H+ 1 h

)

(3.12) , λ ∈ Υh σ,ω , (3.13)

Herein H designates the constant appearing in the formulation of (M.2). G is inﬁnitely differentiable and satisﬁes ||G(δt )||Mp ,h,K < ∞ for every compact set K ⊆ (0, ∞) and for every h > 0 iﬀ there exist ω ≥ 0, k > 0 and h > 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0, in the Beurling case, respectively, there exist ω ≥ 0 and k > 0 such

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that, for every h > 0 and σ > 0, there exist Cσ > 0 and Mσ,h > 0, in the Roumieu case, such that

Ξkσ,ω := {λ ∈ C : Reλ ≤ ω, Reλ ≥ Cσ − σM (k |λ|)} ∪ {λ ∈ C : Reλ ≥ ω} ⊆ ρ(A)

(3.14)

and

||R(λ : A)|| ≤ Mσ eM (h |λ|) , λ ∈ Ξkσ,ω , in the Beurling case, respectively, M (h |λ|)

||R(λ : A)|| ≤ Mσ,h e

,λ ∈

Ξkσ,ω ,

(3.15) in the Roumieu case.

(3.16)

Proof. We will employ the terminology given in the proof of Theorem 3.5. To prove (i), notice that the partial integration implies that, for every n ∈ N0 , ε n 1 −λt −λt d ||G(e ψε (t))|| ≤ ||e (ψ (t)G(δt ))||dt |λ|n dtn ε ε 2

1 ≤ |λ|n

ε

e−Reλt

ε 2

n

n 1 dn−i t |ψ (i+1) ( )||| n−i G(δt )|| i+1 i ε ε dt i=0

n

n 1 −Reλt ≤ sup e ||ψ||Mp , 1ε ,[ 12 ,1] n ε i |λ| 2 ≤t≤ε i=0

dn−i G(δt )||L∞ [ 2ε ,ε] Mi+1 , dtn−i and since (M.2) is assumed × ||

n 1 n Mn−i ||G(δt )||Mp ,h,[ 2ε ,ε] i −Reλt ≤ AHM1 ||ψ||Mp , 1 ,[ 1 ,1] n sup e H Mi . ε 2 |λ| 2ε ≤t≤ε hn−i i i=0

By the logarithmic convexity

n

n ||G(δt )||Mp ,h,[ 2ε ,ε] i 1 −Reλt H Mn ≤ AHM1 ||ψ||Mp , 1ε ,[ 12 ,1] n sup e ε i |λ| 2 ≤t≤ε hn−i i=0

≤ AHM1

Mn (H + h1 )n ||ψ||Mp , 1ε ,[ 12 ,1] ||G(δt )||Mp ,h,[ 2ε ,ε] sup e−Reλt . ε |λ|n 2 ≤t≤ε

The previous inequality holds for all n ∈ N0 and this enables one to see that: ||G(e−λt ψε (t))|| −M (

≤ AHM1 e

|λ| H+ 1 h

)

||ψ||Mp , 1ε ,[ 12 ,1] ||G(δt )||Mp ,h,[ 2ε ,ε] sup e−Reλt . ε 2 ≤t≤ε

(3.17) Suppose now that Reλ ≤ 0 and put Mε,h := AHM1 ||ψ||Mp , 1ε ,[ 12 ,1] ||G(δt )||Mp ,h,[ 2ε ,ε] . Then (3.17) implies that, for every λ ∈ C\{0} with Reλ ≤ 0

|λ| −λt and Reλ ≥ − 1ε ln 2M1ε,h − 1ε M ( H+ ψε (t))|| ≤ 12 and 1 ), we have ||G(e h λ ∈ ρ(A). Arguing in the same manner, we have that, for every λ ∈ C with

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|λ| −λt Reλ ≥ 0 and Reλ ≥ − 2ε ln 2M1ε,h − 2ε M ( H+ ψε (t))|| ≤ 12 1 ), one has ||G(e h and λ ∈ ρ(A). Applying the arguments given in the proof of Theorem 3.5, with σ = 1ε , one gets that there exist ω ≥ 0, Cσ,h > 0 and Mσ,h > 0, in the Beurling case, respectively, ω ≥ 0, Cσ,h > 0 and Mσ,h,h > 0, in the Roumieu case, such that Υhσ,ω ⊆ ρ(A) and that (3.12), respectively, (3.13) holds, ﬁnishing the proof of (i). The necessity in (ii) follows from (i) with h = 1. We will prove the sufﬁciency by the following modiﬁcation of the proof of [7, Theorem 3]. Suppose h > 0, σ > 0 and (ρn ) is a regularizing sequence in D∗ . Without loss of generality, one can assume that Cσ > ω and that the mapping By the Paley–Wiener theorem, t → M (t), t ≥ Cσσ−ω is strictly increasing. 1 one can simply prove that G(ϕ)x = 2πi ϕ(λ)R(λ ˜ : A)xdλ, ϕ ∈ D0∗ , x ∈ E, Γσ where Γσ denotes the upwards oriented boundary of Ξkσ,ω . It is clear that there exists a unique number cσ > 0 such that ω = Cσ − σM (k |ω + icσ |). Let Γ1σ := {Cσ − σM (k r) − i(r2 − (Cσ − σM (k r))2 )1/2 : r ≥ ω 2 + c2σ }, Γ2σ := {ω + is : −cσ ≤ s ≤ cσ }, Γ3σ := {Cσ − σM (k r) + i(r2 − (Cσ − σM (k r))2 )1/2 : r ≥ ω 2 + c2σ } and let the curve Γiσ , i = 1, 2, 3 be upwards oriented. Clearly, n1 λt k+1 (k) n1 λt 3 1 Γσ = i=1 Γiσ and ρ. ρ (nt)dt, λ ∈ n (λ) = 0 e nρ(nt)dt = λk 0 e n n k Mk C\{0}, k ∈ N0 , n ∈ N. This implies |. ρn (λ)| ≤ ( |λ| ) hk ||ρ||Mp ,h,[0,1] , λ ∈ C\{0}, k ∈ N0 , h > 0, n ∈ N. Since this inequality holds for every k ∈ N0 , one yields

|. ρn (λ)| ≤ e−M (

h|λ| n )

||ρ||Mp ,h,[0,1] , λ ∈ C\{0}, h > 0, n ∈ N.

(3.18)

On the other hand, dp 1 G(δt ∗ ρn )x = dtp 2πi

eλt R(λ : A)λp ρ. n (λ)xdλ, t > 0, n ∈ N, p ∈ N0 Γσ

(3.19) and dp dp G(δ ∗ ρ )x = G(δt )x, t > 0, p ∈ N0 , x ∈ R(G), t n n→∞ dtp dtp lim

(3.20)

uniformly on every compact of (0, ∞). Then (3.18)–(3.20) and [12, Lemma 2.1.3, (2.11)] together imply the existence of constants Cσ,h > 0 and Cσ,h >0 so that: p

d hp || dt p G(δt ∗ ρn )|| Mp ⎛

⎜Mσ eCσ ≤ ||ρ||Mp ,1,[0,1] ⎝ 2π

Γ1σ ∪Γ3σ

⎞ ⎟ e−σtM (k |λ|) eM (h |λ|) eM (h|λ|) |dλ| + eωt Cσ,h ⎠

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⎛ ⎜ ≤ Cσ,h ⎝eωt + ⎛

⎞

3 h 3 h ⎟ e−σtM (k |λ|)+ 2 ( k +1)M (k |λ|)+ 2 ( k +1)M (k |λ|) |dλ|⎠

Γ1σ ∪Γ3σ

⎜ ≤ Cσ,h ⎝eωt +

545

⎞ (3+ 3h+3h 2k

e

−σt)M (k |λ|)

⎟ |dλ|⎠ < ∞,

Γ1σ ∪Γ3σ

3+ 3h+3h

2k . Taking into account the preceding estimate, (3.20), for every t > σ the arbitrariness of σ > 0, and the denseness of R(G) in E, we infer that the mapping t → G(δt ), t > 0 is inﬁnitely differentiable and that, for every compact subset K of (0, ∞) and h > 0, we have ||G(δt )||Mp ,h,K < ∞, as required.

1/k

Theorem 3.7. Let A be a closed linear operator and let Lk = Mk , k ∈ N0 . Then A generates a (DSG) of class CL iﬀ the following conditions hold: ∃ω ≥ 0∃m ∈ N∃h , k > 0∀σ > 0∃Cσ , Mσ > 0

(3.21)

such that Ξkσ,ω ⊆ ρ(A),

(3.22)

||R(λ : A)|| ≤ Mσ (1 + |λ|) , λ ∈ C, Reλ ≥ ω and m

M (h |λ|)

||R(λ : A)|| ≤ Mσ e

, λ∈

Ξkσ,ω ,

Reλ ≤ ω.

(3.23) (3.24)

Proof. Suppose ﬁrst that A is densely deﬁned. Let A be the generator of a (DSG) G of class CL . Then there exists n ∈ N such that A generates an exponentially bounded n-times integrated semigroup (Sn (t))t≥0 such that the mapping t → Sn (t), t > 0 is inﬁnitely differentiable and that G(ϕ)x = ∞ (−1)n 0 ϕ(n) (t)Sn (t)xdt, ϕ ∈ D, x ∈ E. Let ε > 0, ψ ∈ D(Mp ) , ψ(t) = 1, 0 ≤ t ≤ 12 , ψ(t) = 0, t ≥ 1, ψ(t) = 0, t ≤ −1 and ψε (t) = ψ( εt ), t ∈ R. Then (2.12) holds and the proof of Theorem 3.6(i) implies that, for every |λ| ), we have λ ∈ C\{0} with Reλ ≤ 0 and Reλ ≥ − 1ε ln 2M1ε,1 − 1ε M ( H+1 ||G(e−λt ψε (t))|| ≤ 12 and λ ∈ ρ(A); analogically, for every λ ∈ C with Reλ ≥ 0 |λ| and Reλ ≥ − 2ε ln 2M1ε,1 − 2ε M ( H+1 ), we have ||G(e−λt ψε (t))|| ≤ 12 and λ ∈ ρ(A). The proof of necessity follows from (2.16). To prove the converse, notice ω0 +i∞ 1 that A generates an (EDSG) G given by G(ϕ)x = 2πi ϕ(λ)R(λ ˜ : ω0 −i∞ A)xdλ, x ∈ E, ϕ ∈ D, where ω0 > ω. Let (ρn ) be a regularizing sequence (M ) in D0 p and let Γσ designate the upwards oriented frontier of Ξkσ,ω0 . The Paley–Wiener theorem for ultradifferentiable functions implies that 1 eλt ρ. G(δt ∗ ρn )x = n (λ)R(λ : A)xdλ, x ∈ E, t > 0. 2πi Γσ

Now one can repeat verbatim the arguments given in the proof of the sufﬁciency in Theorem 3.6(ii) to end the proof in the case of densely deﬁned operators. Suppose now that A is not densely deﬁned. If A generates a (DSG) G of class CL , then AF generates a dense (DSG) G|F of class CL in the

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Banach space F = An(A) . The necessity follows from an application of [72, Proposition 2.1(iii)]. Let (3.21)–(3.24) hold. Then A generates an (EDSG) G and, by [72, Proposition 2.1(iii)] and the ﬁrst part of the proof, we have that AF generates a dense (DSG) G|F of class CL in the Banach space F. In particular, G|F is differentiable and AF satisﬁes the spectral properties quoted in the formulation of Theorem 2.5. This, in turn, implies that G must be differentiable and that (2.8) holds. The remnant of the proof follows as in the case of densely deﬁned operators. Having in mind Lema 2.15, Remark 2.16, Theorem 2.14 and the proof of Theorem 2.17, we have the following corollary. Corollary 3.8. (i) Let A be a closed linear operator and let Lk = k!s/k , s > 1, k ∈ N0 . Then A generates a (DSG) of class CL iﬀ there exist ω ≥ 0, m ∈ N, h > 0 and k > 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that: Ξsσ,ω,k := {λ ∈ C : Reλ ≤ ω, Reλ ≥ Cσ − σk|λ|1/s } ∪{λ ∈ C : Reλ ≥ ω} ⊆ ρ(A), ||R(λ : A)|| ≤ Mσ (1 + |λ|)m , λ ∈ C, Reλ ≥ ω and

||R(λ : A)|| ≤ Mσ eh

|λ|1/s

, λ ∈ Ξsσ,ω,k , Reλ ≤ ω.

(3.25) (3.26) (3.27)

1/k Mk , k

(ii) Let α ≥ 0, Lk = ∈ N0 and let A be the generator of a (local) α-times integrated semigroup (Sα (t))t∈[0,τ ) . Then (Sα (t))t∈[0,τ ) is of class CL iﬀ (3.21)–(3.24) hold. (iii) Let α ≥ 0, s > 1, Lk = k!s/k , k ∈ N0 and let A be the generator of a (local) α-times integrated semigroup (Sα (t))t∈[0,τ ) . Then (Sα (t))t∈[0,τ ) is of class CL iﬀ there exist ω ≥ 0, h > 0 and k > 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0 such that (3.25)–(3.27) hold. In the following example, we use the notion given in [2, Chapter 8]. m Example. Let s > 1, k > 0, p ∈ [1, ∞), m > 0, ρ ∈ [0, 1], r > 0, a ∈ Sρ,0 satisﬁes (Hr ),

1 1 m−r−ρ+1 n − < 1, (3.28) 2 p r

that a(Rn ) ∩ and assume that, for every σ > 0, there exists * Cσ > 0 such ≥ n 12 − p1 1+m−ρ , p > 1, r Ξsσ,ω,k = ∅. Let Np be the smallest integer n 1+m−ρ , p = 1. > 2 r Notice that m = r, ρ = 1 and that (3.28) holds, whenever a(·) is an elliptic polynomial of order m. By [2, Proposition 8.3.1, Theorem 8.3.6] and Corollary 3.8, the operator Opp (a) generates an Np -times integrated semigroup on Lp (Rn ) which is of class CL with Lk = k!s/k . Herein we would like to note that the class C L is more appropriate to deal with; for example, consider the elliptic polynomial a(ξ) = −|ξ|2 + i|ξ|4 , ξ ∈ Rn . By Theorem 2.17 (cf. also [43,79,113]), we have that, for every α > n| 12 − p1 |, the operator Opp (a) generates an exponentially bounded α-times integrated semigroup (S(t))t≥0 on Lp (Rn ) which is of class C L with Lk = k!2/k ; by Corollary 3.8, (S(t))t≥0

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is not of class CL . It is of concern to accent that Opp (a) generates a dense (DSG) of class A2 as well as that −Opp (a) generates a dense (UDSG) of (ps )-class iﬀ s ∈ (1, 2], −Opp (a) generates a dense (UDSG) of {ps }-class iﬀ s ∈ (1, 2), and that −Opp (a) does not generate a (DSG). Finally, let us mention that Lp -realizations of some special cases of pseudodifferential operators associated to the linearized Benjamin-Ono-Burgers equation [31,117] likewise generate ρ-hypoanalytic integrated (C-)semigroups. Remark 3.9. (i) A sufﬁcient condition for the generation of dense (C-DSG)’s of class CL can be deduced by means of the arguments already given in the proofs of Theorem 2.2, Theorem 3.6, Theorem 3.7 and [62, Proposition 5.4]. Concerning the proof of Theorem 3.7, we also perceive the following interesting phenomenon. In order to deform the path of integration from the straight line connecting the points ω0 − i∞ and ω0 + i∞ into the upwards oriented frontier of the region Ξkσ,ω0 , we essentially utilize the regularizing sequence (ρn ) in the (M )

space D0 p . Under the assumptions (3.21)–(3.24), we have that A generates an exponentially bounded (m + 2)-times integrated semigroup ω0 +i∞ λt R(λ:A) 1 (Sm+2 (t))t≥0 given by Sm+2 (t) = 2πi e λm+2 dλ, t ≥ 0. On ω0 −i∞ the other hand, the proof of Theorem 3.6(ii) implies that the improper 3+ 3h 1 0 2k eλt R(λ:A) integral 2πi λm+2 dλ =: Sm+2 (t) converges for every t > σ . Γσ Furthermore, for every compact set K ⊆ ( p

h > 0, supt∈K,

p∈N0

0 d hp || dt p Sm+2 (t)|| Mp

3+ 3h 2k σ

, ∞) and for every

< ∞. Unfortunately, it is not clear

3+ 3h

(ii)

0 (t) = Sm+2 (t) for t > σ2k . whether Sm+2 Let A generate a dense inﬁnitely differentiable (UDSG) G and let for every compact set K ⊆ (0, ∞) there exists hK > 0 such that ||G(δt )||Mp ,hK ,K < ∞. By the proof of Theorem 3.6, we have that there exist ω ≥ 0 and h > 0 such that, for every σ > 0, there exist Cσ > 0, Mσ > 0 and kσ > 0 in the Beurling case, respectively, there exists ω ≥ 0 such that, for every h > 0 and σ > 0, there exist Cσ > 0, kσ > 0 σ ⊆ ρ(A) and: and Mσ,h > 0, in the Roumieu case, such that Ξkσ,ω |λ|

1

||R(λ : A)|| ≤ Mσ eM ( h )+ 2 M (kσ |λ|) , λ ∈ Ξσ,kσ , in the Beurling case, respectively, M(

||R(λ : A)|| ≤ Mσ,h e

|λ| )+ 1 M (kσ |λ|) 2 h

in the Roumieu case.

(3.29) , λ ∈ Ξσ,kσ , (3.30)

By the proof of Theorem 3.6(ii), we have that (3.29)–(3.30), together with the additional assumption lim supσ→∞ σkσ = ∞, imply that for every compact set K ⊆ (0, ∞) there exists hK > 0 such that ||G(δt )||Mp ,hK ,K < ∞. At present, we do not know whether the condition lim supσ→∞ σkσ = ∞ can be simply dropped. Theorem 3.10. Suppose A generates a dense (EUDSG) G of ∗-class and α ∈ (0, π2 ].

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(Real analyticity) Suppose that there exist ω ≥ 0 and h > 0, such that for every ε ∈ (0, 1), there exist Cε > 0 and Mε > 0, in the Beurling case, respectively, there exists ω ≥ 0 such that, for every h > 0 and ε ∈ (0, 1), there exist Cε > 0 and Mε,h > 0, in the Roumieu case, so that: Tε := {λ ∈ C : Reλ + Cε |Imλ| ≥ ω} ⊆ ρ(A)

(3.31)

and, for every λ ∈ Tε , ||R(λ : A)|| ≤ Mε eM (h|λ|) eε|Reλ| , respectively,

(3.32)

||R(λ : A)|| ≤ Mε,h eM (h|λ|) eε|Reλ| .

(ii)

Then G is inﬁnitely differentiable and the mapping t → G(δt ), t > 0 is real analytic. (Analyticity) Suppose that the following conditions hold. (ii.1) For every γ ∈ (0, α), there exists ωγ ≥ 0 such that ωγ + Σ π2 +γ ⊆ ρ(A). (ii.2) For every γ ∈ (0, α), there exist hγ > 0 and Cγ > 0, respectively, for every h > 0, there exists Ch,γ > 0 such that ||R(λ : A)|| ≤ Cγ eM (hγ |λ|) , respectively, ||R(λ : A)|| ≤ Ch,γ eM (h|λ|) , λ ∈ ωγ + Σ π2 +γ .

(3.33)

Then G is an analytic (EUDSG) of angle arctan(sin α). Proof. We only consider the Beurling case. To prove differentiability in (i), one can repeat verbatim the argumentation given in the proof of Theorem 3.6(ii). Let us prove that the mapping t → G(δt ), t > 0 is real analytic. By the Pringsheim theorem, it is enough to show that, for every compact hp ||

dp

G(δ )||

t dtp set K ⊆ (0, ∞), there exists h > 0 such that supt∈K,p∈N0 < ∞. p! Denote by Γε the upwards oriented boundary of Tε . Thanks to the Paley– dp Wiener theorem and the proof of Theorem 3.6(ii), it follows that dt p G(δt ∗ 1 λt p ρn )x = 2πi e R(λ : A)λ ρ . (λ)xdλ, t > 0, n ∈ N, p ∈ N , x ∈ R(G) n 0 Γε and that (3.20) holds. Given a compact set K ⊆ (0, ∞), one can ﬁnd ε > 0 ε and h > 0 such that inf K > 2(ε + h 1+C Cε ). Since (Mp ) satisﬁes (M.1), we have M (t + s) ≤ M (2t) + M (2s), t, s ≥ 0 (cf. [12, Lemma 2.1.3]). The next computation involves the last two inequalities, (3.18), (3.20) and (3.32): p

p

d d hp || dt hp || dt p G(δt )|| p G(δt ∗ ρn )|| ≤ sup p! p! n∈N0 ∞ Mε e(γ−Cε s)t eM (h|γ−Cε s+is|) eε|γ−Cε s| hp |γ − Cε s + is|p ds ≤ ||ρ||Mp ,1,[0,1] πp! 0

Mε ||ρ||Mp ,1,[0,1] γ(h+ε)+M (2hγ)+(inf K+sup K)γ e ≤ π ∞ × e−Cε s inf K+M (2h(1+Cε )s)+εCε s+h(1+Cε )s ds 0

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549

≤ Kε eM (inf Kγ)+(inf K+sup K)γ ∞ inf K × e−Cε s inf K+M (inf K(1+Cε )s)+εCε s+( 2 −ε)Cε s ds < ∞, 0

for all p ∈ N0 and t ∈ K, and the proof of (i) automatically follows. By Thedp dp orem 3.6(ii), G is inﬁnitely differentiable and dt p G(δt ) = limn→∞ dtp G(δt ∗ ρn ), t > 0, where (ρn ) is a regularizing sequence in D(Mp ) . Suppose γ ∈ (0, α), t0 > 0, t ∈ C and |t − t0 | < arctan(sin γ). Thanks to the argumenta∞ (t − tion given in the proof of Theorem 3.6(ii), we obtain that the series p=0

p

( d G(δ )) t0 )p drp p!t r=t0

is absolutely convergent. Indeed, let Γγ be the boundary , oriented in such a way that Imλ increases along Γγ , and of 1 + ωγ + Σ let Γ1γ = {λ ∈ Γγ : Imλ ≤ 0} and Γ2γ = {λ ∈ Γγ : Imλ ≥ 0}. Then π π Γ1γ = {1 + ωγ + e−i( 2 +γ) s : s ≥ 0}, Γ2γ = {1 + ωγ + ei( 2 +γ) s : s ≥ 0} and π 2 +γ

∞

p

|t − t0 |p ||

p=0 ∞

d ( dr p G(δr ))r=t0 || p! p

( d p G(δr ∗ ρn ))r=t0 ≤ sup || |t − t0 | || dr p! n∈N0 p=0 ∞ 1 ≤ ||ρ||Mp ,1,[0,1] |t − t0 |p eReλt0 eM (hγ |λ|) |λ|p |dλ| 2πp! p=0 p

Γγ

≤

||ρ||Mp ,1,[0,1] M (2hγ (1+ωγ ))+|t−t0 |(1+ωγ ) e π ∞ π × e(ωγ +s cos( 2 +γ))t0 +M (2hγ s)+|t−t0 |s ds < ∞. 0

This completes the proof of theorem.

Remark 3.11. Assume that the conditions (ii.1) and (ii.2) quoted in the formulation of Theorem 3.10 hold. It is not clear whether there exists β ∈ (arctan(sin α), α] such that G is an analytic (UDSG) of angle β (cf. also [27, Lemma 2.1, Theorem 2.1 and p. 567, l. -9]). The next proposition enquires into the structure of certain classes of exponentially bounded, analytic Ka,b -convoluted semigroups; notice only that (1.8) implies that the assertion (ii) cannot be reformulated in the case of analytic Kl -convoluted semigroups. Proposition 3.12. Suppose α ∈ (0, π2 ], a > 0 and b > 0. (i)

Then A generates an exponentially bounded, analytic Ka,b -convoluted semigroup of angle α iﬀ for every γ ∈ (0, α) there exist ωγ ≥ 0 and

550

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Mγ > 0 such that the following conditions are satisﬁed: ωγ + Σ π2 +γ ⊆ ρ(A),

(3.34)

−a|λ|b cos(b arg λ)

sup

λ∈ωγ +Σ π +γ

||(λ − ωγ )e

R(λ : A)||

2

≤ Mγ , λ ∈ ωγ + Σ π2 +γ and

(3.35)

lim λe−aλ R(λ : A)x = 0, x ∈ E, if D(A) = E. b

λ→+∞

(ii)

If, additionally,

(3.36)

π +α > , (3.37) 2 2 then there does not exist a closed linear operator A which generates an exponentially bounded, analytic Ka,b -convoluted semigroup of angle α. b

π

Proof. The proof of (i) follows immediately from Lemma 1.4 and the obvious b b equality |e−aλ | = e−a|λ| cos(b arg λ) , λ ∈ C\{0}. To prove (ii), suppose to the contrary that (3.37) holds and that a closed linear operator A generates an exponentially bounded, analytic Ka,b -convoluted semigroup of angle α. Then there exist ε ∈ (0, 1) and γ ∈ (0, α) such that b( π2 + γ − ε) > π2 + ε. The ﬁrst part of the proposition yields that there exist Mγ > 0 and ωγ ≥ 0 such that (3.34) and (3.35) hold. Set Aγ := A − ωγ . Then the choice of γ and ε implies that there exists r > 0 such that, for every λ ∈ C with |λ| ≥ r and arg λ ∈ ( π2 + γ − 2ε , π2 + γ) : ||R(λ : Aγ )|| ≤

Mγ a|λ+ωγ |b cos(b( π +γ−ε)) Mγ −a|λ+ωγ |b sin ε 2 e e ≤ . (3.38) |λ| |λ|

An application of [72, Lemma 1.5] gives n(A) ≤ 1. Denote by AF γ the part of F Aγ in the Banach space F = D(A); then Aγ also satisﬁes (3.38). In particular, (3.38) implies that there exists Mγ > 0 such that 1 λ π ε 1 n for all λ ∈ (Mγ , ∞). Hence, ||R(λ : e−i( 2 +γ− 4 ) AF γ ) ||F ≤ λn , λ > Mγ , and ε −i( π +γ− ) F 4 A the Hille–Yosida theorem implies that the operator e 2 γ generates F a C0 -semigroup (T (t))t≥0 in the Banach space F satisfying ||T F (t)||F ≤ eMγ t , t ≥ 0. By [2, Proposition 4.1.3, p. 248], one gets that limλ→∞ λR(λ : π ε e−i( 2 +γ− 4 ) AF γ )x = x for all x ∈ F. On the other hand, (3.38) yields that the above limit equals zero for all x ∈ F, and the contradiction is obvious. i( 2 +γ− 4 ) ||R(λ : e−i( 2 +γ− 4 ) AF R(λei( 2 +γ− 4 ) : AF γ )||F = ||e γ )||F ≤ π

ε

π

ε

π

ε

Differential and analytical properties of hyperfunction fundamental solu¯ tions can be found in the papers of Kochubei [52] and Ouchi [91,92]. Finally, we would like to propose the following questions. Questions. 1. Suppose α ∈ (0, π2 ], γ ∈ (0, α) and A generates an analytic (DSG) of angle α. Do there exist numbers Mγ > 0, ωγ ≥ 0 and nγ ∈ N such that ωγ + Σγ+ π2 ⊆ ρ(A) and that ||R(λ : A)|| ≤ Mγ (1 + |λ|)nγ , λ ∈ ωγ + Σγ+ π2 ?

Vol. 67 (2010) 2.

3.

4.

Differential and Analytical Properties

551

Let α ∈ (0, ∞)\N and let A be the densely deﬁned generator of an α-times integrated semigroup (Sα (t))t≥0 which satisfy that the mapping t → Sα (t), t > 0 is inﬁnitely differentiable. Does there exist ω ≥ 0 such that, for every σ > 0, there exist Cσ > 0 and Mσ > 0, such that Υσ,ω ⊆ ρ(A) and that ||R(λ : A)|| ≤ Mσ |λ|α |Imλ|, λ ∈ Υσ,ω , Reλ ≤ ω? Suppose ρ ≥ 1, γ ≥ 0, A generates a (DSG) of class A , respectively Aγ , B ∈ L(E) and BA ⊆ AB. Denote by GB the (DSG) generated by A + B. Is it true that GB is of class A , respectively Aγ ? Let G be a non-dense differentiable (UDSG) of ∗-class. Must G be exponential (inﬁnitely differentiable)?

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[55] Komatsu, H.: Ultradistributions, III. Vector valued ultradistributions and the theory of kernels. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29, 653–718 (1982) [56] Komatsu, H.: Operational calculus and semi-groups of operators. In: Functional Analysis and Related Topics (Kioto), pp. 213–234. Springer, Berlin (1991) [57] K¯ omura, Y.: Differentiability of nonlinear semigroups. J. Math. Soc. Jpn. 21, 375–402 (1969) [58] Konishi, Y.: On ut = uxx − F (ux ) and the differentiability of the nonlinear semigroups associated with it. Proc. Jpn. Acad. 48, 281–286 (1972) [59] Kosti´c, M.: (a, k)-regularized C-resolvent families: regularity and local properties. Abstr. Appl. Anal., vol. 2009, Article ID 858242, p. 27 (2009) [60] Kosti´c, M.: Distribution cosine functions. Taiwanese J. Math. 10, 739–775 (2006) [61] Kosti´c, M.: On analytic integrated semigroups. Novi Sad J. Math. 35, 127–135 (2005) [62] Kosti´c, M.: C-Distribution semigroups. Studia Math. 185, 201–217 (2008) [63] Kosti´c, M., Pilipovi´c, S.: Global convoluted semigroups. Math. Nachr. 280, 1727–1743 (2007) [64] Kosti´c, M., Pilipovi´c, S.: Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines. J. Math. Anal. Appl. 338, 1224–1242 (2008) [65] Kosti´c, M., Pilipovi´c, S.: Ultradistribution semigroups. Sibirsk. Mat. Zh. (2010) (to appear) [66] Kosti´c, M.: Notes on analytic convoluted C-semigroups. Publ. Inst. Math., Nouv. S´er. (2010) (to appear) [67] Kosti´c, M.: Regularization of ultradistribution semigroups and sines. Publ. Inst. Math., Nouv. S´er. (2010) (to appear) [68] Kosti´c, M.: Local convoluted C-semigroups and cosine functions: spectral properties of subgenerators. Filomat (2010) (to appear) [69] Kosti´c, M.: Perturbation theorems for local convoluted C-semigroups and cosine functions. Bull. Cl. Sci. Math. Nat. Sci. Math. (2010) (to appear) [70] Kovaˇcevi´c, D.: On Integral Transforms and Convolution Equations on the Spaces of Tempered Ultradistributions. Thesis, University of Novi Sad (1992) [71] Kovaˇcevi´c, D., Pilipovi´c, S.: Structural properties of the spaces of tempered ultradistributions. In: Complex Analysis and Generalized Functions, Varna 1991, pp. 169–184. Publ. House Bulgar. Acad. Sci., Soﬁa (1993) [72] Kunstmann, P.C.: Stationary dense operators and generation of non-dense distribution semigroups. J. Oper. Theory 37, 111–120 (1997) [73] Kunstmann, P.C.: Distribution semigroups and abstract Cauchy problems. Trans. Am. Math. Soc. 351, 837–856 (1999) [74] Kunstmann, P.C., Mijatovi´c, M., Pilipovi´c, S.: Classes of distribution semigroups. Studia Math. 187, 37–58 (2008) [75] Kunstmann, P.C.: Banach space valued ultradistributions and applications to abstract Cauchy problems (2010) (Preprint) [76] Kunstmann, P.C.: Regularization of semigroups that are strongly continuous for t > 0. Proc. Am. Math. Soc. 126, 2721–2724 (1998)

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[119] Zheng, Q., Li, Y.: Abstract parabolic systems and regularized semigroups. Paciﬁc J. Math. 182, 183–199 (1998) Marko Kosti´c (B) University of Novi Sad Faculty of Technical Sciences Trg Dositeja Obradovi´ca 6 21125 Novi Sad Serbia e-mail: [email protected] Received: September 2, 2009. Revised: April 16, 2010.

Integr. Equ. Oper. Theory 67 (2010), 559–600 DOI 10.1007/s00020-010-1784-9 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Finite Section Method for a Banach Algebra of Convolution Type Operators on Lp(R) with Symbols Generated by P C and SO Alexei Yu. Karlovich, Helena Mascarenhas and Pedro A. Santos To the memory of Professor Israel Gohberg (23.08.1928–12.10.2009) Abstract. We prove necessary and sufﬁcient conditions for the applicability of the ﬁnite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on Lp (R), 1 < p < ∞. Mathematics Subject Classification (2000). Primary 65R20; Secondary 45E10, 47B35, 47L80, 65J10. Keywords. Finite section method, slowly oscillating function, Fourier multiplier, algebraization, essentialization, homogenization, local principle, two projections theorem.

1. Introduction Given 1 < p < ∞, let B := B(Lp (R)) denote the Banach algebra of all bounded linear operators on the Lebesgue space Lp (R). Let [P C, SO] be the smallest C ∗ -subalgebra of L∞ (R) containing all piecewise continuous (P C) and slowly oscillating (SO) functions, and let [P Cp , SOp ] stand for its Fourier multiplier analogue, which is a Banach subalgebra of Mp , the Banach algebra of all Fourier multipliers on Lp (R). The Fredholm theory for the smallest After this paper was submitted, we received the sad news that Israel Gohberg has passed away on October 12, 2009. Professor Gohberg has left his imprint on many areas of analysis, including the theory of convolution type operators and projection methods for their solution. We would like to dedicate this contribution to him with admiration. This work is partially supported by the grant FCT/FEDER/POCTI/MAT/ 59972/2004.

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Banach subalgebra of B(Lp (R)) that contains all the convolution type operators aF −1 bF where F is the Fourier transform given by 1 eixy ϕ(y) dy (x ∈ R) (1.1) (F ϕ)(x) := √ 2π R

and a ∈ [P C, SO], b ∈ [P Cp , SOp ] is constructed in [1,2] (in those papers, even matrix functions a and b are considered). Let R+ := (0, ∞) and R− := (−∞, 0). For τ ∈ R+ , consider the operators ϕ(t) if |t| < τ, Qτ := I − Pτ (Pτ ϕ)(t) := 0 if |t| > τ, acting on Lp (R) with norm 1. Clearly, Pτ → I and Qτ → 0 strongly as τ → ∞. One says that the finite section method applies to an operator A ∈ B(Lp (R)) if there exists a positive constant τ0 , such that for any τ > τ0 and any f ∈ Lp (R) there exists a unique solution ϕτ of the equation Aτ ϕτ := (Pτ APτ + Qτ )ϕτ = f

(τ ∈ R+ )

(1.2)

and ϕτ converges in the norm of Lp (R) to a solution of the equation Aϕ = f as τ → ∞. We refer to the monographs by Gohberg and Feldman [8], Pr¨ ossdorf and Silbermann [18], Hagen et al. [11,12] for a general theory of projection methods as well as for more speciﬁc issues of the ﬁnite section method for convolution type operators and algebras generated by them. Roch et al. [20] studied the applicability of the ﬁnite section method to an operator in the smallest C ∗ -subalgebra of B(L2 (R)) generated by all aI with a ∈ P C, W 0 (b) with b ∈ P C = P C2 , and the so-called ﬂip operator. For general p = 2, they proved in [22] the applicability of the ﬁnite section method for an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions (P C) and by the convolution operators with piecewise continuous Fourier multipliers (P Cp ). The aim of this paper is to take one more step forward. We prove the applicability of the ﬁnite section method for an arbitrary operator A in the Banach subalgebra of B(Lp (R)), 1 < p < ∞, generated by all operators aI of multiplication by functions a ∈ P C and by all Fourier convolution operators W 0 (b) := F −1 bF with b ∈ [P Cp , SOp ]. Our approach to analyze the applicability of the ﬁnite sections method will follow a general scheme to treat approximation problems. This scheme goes back to Silbermann [25] (see also [11, Section 1.6]), who further developed a brilliant idea of Kozak [13] to consider stability problems in numerical analysis as invertibility problems in suitably chosen Banach algebras. This scheme can be summarized as follows. Let A be a set of (generalized) sequences that contains all sequences of the form (Aτ ) = (Pτ APτ + Qτ ) (τ ∈ R+ ).

(1.3)

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Algebraization: Find a unital Banach algebra E containing A and a closed ideal G of E such that the original problem becomes equivalent to an invertibility problem in the quotient algebra E/G. 2. Essentialization: Find a closed unital subalgebra F of E containing A such that F/G is inverse closed in E/G and a closed ideal J of F containing G such that J can be lifted. The latter means that one has full control about the difference between the invertibility of a coset of a sequence (Aτ ) ∈ F in the algebra F/G and the invertibility of the coset of the same sequence in F/J . This control is usually guaranteed by a lifting theorem. 3. Localization: Find a unital subalgebra L of F such that (a) A, J ⊂ L; (b) L/J is inverse closed in F/J ; (c) the quotient algebra L/J has a large center. Use a local principle to translate the invertibility problem in the algebra L/J to a family of simpler invertibility problems in the local algebras. 4. Identification: Find conditions for the invertibility of the cosets of sequences in A in the local algebras.

1.

The paper is organized as follows. Section 2 contains all necessary properties of piecewise continuous and slowly oscillating Fourier multipliers. In Sect. 3, we formulate several auxiliary results on singular integral operators with piecewise constant coefﬁcients, which will be used in Sects. 10 and 11. In Sect. 4, the algebra E and its ideal G are deﬁned and Kozak’s theorem is formulated. Let A denote the smallest closed subalgebra of E that contains the constant sequences (aI) with a ∈ P C and (W 0 (b)) with b ∈ [P Cp , SOp ] and the sequence (Pτ ). In Sect. 5, we perform the essentialization step: we introduce the algebra F ⊂ E, its ideal J , and the homomorphisms Wi , i ∈ {−1, 0, 1}. It is shown that A ⊂ F. The main result of Sect. 5 (Theorem 5.5) says that for a sequence A = (Aτ ) ∈ F the coset A + G is invertible in F/G if and only if the operators Wi (A), i ∈ {−1, 0, 1}, are invertible and the coset A + J is invertible in F/J . The algebra F/J is still too large for eﬀective studying. In Sect. 6, we introduce the algebra L of sequences of local type such that A ⊂ L ⊂ F and the algebra L/J has a large center. So the latter algebra can be studied with the aid of the Allan–Douglas local principle. According to it, the invertibility of A + J ∈ L/J is equivalent to the invertibility of J the local representatives ΦJ ξ,η (A) in the local algebras Lξ,η , where (ξ, η) ∈ (R × M∞ (SO)) ∪ ({∞} × R) ∪ ({∞} × M∞ (SO)) and M∞ (SO) is the ﬁber of the maximal ideal space of SO over the point ∞ (see Sect. 2.3). In Sect. 7, we introduce the homomorphisms Hη for η ∈ R and show that if a sequence A ∈ A is stable, then all operators Hη (A) for η ∈ R are invertible. In Sects. 8–10, we study the invertibility in the local algebras LJ ξ,η . It turns out that these algebras are too large for a complete description, so

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we will restrict ourselves to studying the invertibility in their subalgebras J AJ ξ,η := Φξ,η (A). Let A ∈ A. In Sect. 8, we prove that the invertibility of the operator J W0 (A) is sufﬁcient for the invertibility of ΦJ ξ,η (A) in the local algebra Lξ,η for (ξ, η) ∈ R × M∞ (SO). Further, in Sect. 9, we obtain that the invertibility of the operator Hη (A) is sufﬁcient for the invertibility of ΦJ ∞,η (A) in the with η ∈ R. local algebra LJ ∞,η J Section 10 is devoted to AJ ∞,η and L∞,η with η ∈ M∞ (SO). We show that the invertibility problem in these algebras can be reduced to the invert± ibility problem in the pairs of simpler algebras A± η and Lη . It turns out that ± the algebras Aη are generated by two idempotents and the identities of the algebras L± η . Applying the two-idempotents theorem (Theorem 10.5), we get necessary and sufﬁcient conditions for the invertibility of an element of A± η in the algebra L± η . These results lead to a criterion of the invertibility of an J element of AJ ∞,η in the algebra L∞,η . In Sect. 11 we gather the results obtained in Sects. 6–10 and prove our main result: a criterion for the stability of a sequence A = (Aτ ) ∈ A. Further, it is speciﬁed for the sequences of the ﬁnite section method (1.3). Finally we illustrate our results for the paired convolution operators A = W 0 (a)χ− I + W 0 (b)χ+ I with a, b ∈ [P Cp , SOp ], where χ− and χ+ are the characteristic functions of the half-axes (−∞, 0) and (0, +∞).

2. Piecewise Continuous and Slowly Oscillating Fourier Multipliers 2.1. Function Algebras Let 1 ≤ p ≤ ∞ and Lp (R) denote the usual Lebesgue space on (−∞, +∞) with the standard norm denoted by · p , let C(R) and denote the spaces of continuous functions on R = [−∞, +∞] and R ∪ {∞}, respectively; ˙ : f (∞) = 0 , Cb (R) := C(R) ∩ L∞ (R); C0 (R) := f ∈ C(R)

R = ˙ C(R) ˙ R =

and P C stand for the set of all functions f : R˙ → C which possess a ﬁnite ˙ left-hand limit f (x− ) and a ﬁnite right-hand limit f (x+ ) at every point x ∈ R. Let V1 (R) be the set of functions a : R → C with the ﬁnite total variation V1 (a) := sup

n

|a(xi ) − a(xi−1 )|,

i=1

where the supremum is taken over all ﬁnite decompositions −∞ ≤ x0 < x1 < · · · < xn ≤ +∞

(n ∈ N)

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of the real line R. It is well known that V1 (R) is a Banach algebra under the norm

a V1 (R) := a ∞ + V1 (a). For a continuous function f : R → C and a set I ⊂ R, let osc(f, I) := sup |f (t) − f (s)|. t,s∈I

Following Power [17] we denote by SO the set of all slowly oscillating functions, SO := f ∈ Cb (R) : lim osc(f, [−2x, −x] ∪ [x, 2x]) = 0 . x→+∞

∗

˙ ⊂ SO. Clearly, SO is a C -subalgebra of L∞ (R) and C(R) Let Cb1 (R) consist of all functions a ∈ Cb (R) with a ∈ Cb (R). The following lemma is, in fact, proved in [1, pp. 154–155]. Lemma 2.1. If a function a is even and lies in 1 := f ∈ Cb1 (R) : lim xf (x) = 0 , SO x→∞

then a ∈ SO. 1

is not contained in SO because its elements can have However, SO different slowly oscillating behavior at −∞ and +∞. Lemma 2.2. ([1, Corollary 2.4]) Every function a ∈ SO can be uniformly approximated by functions in the non-closed algebra 1

∩ SO. SO1 := SO

(2.1)

From now on, we consider 1 < p < ∞. Let B := B(Lp (R)) be the Banach algebra of all bounded linear operators on Lp (R) and K := K(Lp (R)) be the closed two-sided ideal of all compact operators on Lp (R). The Cauchy singular integral operator SR is deﬁned on Lp (R) by f (t) 1 dt (x ∈ R) (SR f )(x) := πi t−x R

where the integral is understood in the sense of principal value. Let F : L2 (R) → L2 (R) denote the Fourier transform given by (1.1), and let F −1 be its inverse. We deﬁne the operator W 0 (a) on L2 (R) ∩ Lp (R) by (W 0 (a)ϕ)(x) := (F −1 aF ϕ)(x), ∞

ϕ ∈ L2 (R) ∩ Lp (R). p

(2.2)

A function a ∈ L (R) is called a Fourier multiplier on L (R) if the operator W 0 (a) given by (2.2) can be extended to a bounded linear operator on Lp (R), which will again be denoted by W 0 (a). By χ+ (resp. χ− ) denote the characteristic function of the semi-axis R+ := (0, +∞) (resp. R− := (−∞, 0)). Then W 0 (χ− ) = PR := (I + SR )/2,

W 0 (χ+ ) = QR := (I − SR )/2

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are two complementary projections on Lp (R) (see e.g. [7, Section 2] or [5, Section 2.5]). The set Mp of all Fourier multipliers on Lp (R) is deﬁned as

Mp := a ∈ L∞ (R) : W 0 (a) ∈ B(Lp (R)) . It is well known that Mp is a Banach algebra with the norm

a Mp := W 0 (a) B(Lp (R)) , and Mp ⊂ M2 = L∞ (R)

for all p ∈ (1, ∞).

(2.3)

According to Stechkin’s inequality (see e.g. [5, Theorem 17.1]), Mp contains all functions a ∈ P C of ﬁnite total variation and

a Mp ≤ SR B ( a ∞ + V1 (a)) . ˙ and Cp (R) stand for the closure in Mp of the set of all functions Let Cp (R) ˙ and C(R), respectively. Denote by P Cp with ﬁnite total variation in C(R) the closure in Mp of the set of all piecewise constant functions on R which have at most ﬁnite sets of jumps. Lemma 2.3. ([26, Lemma 1.1]) If 1 < p < ∞, then the algebra Cp (R) is generated by the functions f (x) = 1 and g(x) = tanh x. In view of (2.1), every function a ∈ SO1 has the properties a ∈ Cb (R) and Da ∈ Cb (R) where (Da)(x) = xa (x). Therefore, by Mikhlin’s theorem (see e.g. [10, Theorem 5.2.7(a)]), every function a ∈ SO1 belongs to all spaces ˙ and taking into account Mp for p ∈ (1, ∞). Hence, by analogy with Cp (R) Lemma 2.2, one can deﬁne the set SOp of slowly oscillating Fourier multipliers as the closure in Mp of the set SO1 . Clearly, SOp is a Banach subalgebra of Mp . The set SOp was introduced in [1,2]. From (2.3) it follows that SO1 ⊂ SOp ⊂ SO2 = SO

for all 1 < p < ∞.

(2.4)

For a ∈ L∞ (R) by a denote the function a(t) := a(t), where the bar denotes the complex conjugation. Lemma 2.4. Let 1 < p < ∞ and 1/p + 1/q = 1. If a ∈ SOp , then a ∈ SOq . 1 Proof. Let a be the limit of a sequence (an )∞ n=1 ⊂ SO in the norm of Mp . Obviously, an ∈ SO1 ⊂ SOq . Since a − an ∈ Mp , from [6, Section 9.3(b)] we see that a − an ∈ Mq and a − an Mq = a − an Mp → 0 as n → ∞. This means that a ∈ SOq .

We denote by [P C, SO] the smallest C ∗ -subalgebra of L∞ (R) that contains P C and SO. Similarly, for every p ∈ (1, ∞), we introduce the Banach subalgebra [P Cp , SOp ] of Mp generated by P Cp and SOp . Obviously, [P Cp , SOp ] ⊂ [P C2 , SO2 ] = [P C, SO]

for all 1 < p < ∞.

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2.2. Compactness of Hankel Type Operators and Commutators Put C+ := {z ∈ C : Im z > 0},

C− := {z ∈ C : Im z < 0},

and denote by H ∞ (C± ) the set of all bounded and analytic functions in C± . Fatou’s theorem says that functions in H ∞ (C± ) have non-tangential limits ∞ the set of all a ∈ L∞ (R) on R almost everywhere, and we denote by H± ∞ that are non-tangential limits of functions in H (C± ). It is well known that ∞ are closed subalgebras of L∞ (R). Sarason discovered (see e.g. [6, p. 107]) H± ˙ and that the smallest closed subalgebra of L∞ (R) that contains both C(R) ∞ ∞ ˙ H± coincides with the sum of C(R) and H± . Lemma 2.5. Let 1 < p < ∞. If b ∈ SOp , then χ+ W 0 (b)χ− I and χ− W 0 (b)χ+ I are compact on Lp (R). Proof. From [6, Sections 9.11, 9.35, and 2.80] it follows that ˙ + H ∞ ). ˙ + H ∞ ) ∩ (C(R) SO ⊂ (C(R) + −

(2.5)

By Hartman’s theorem (see e.g. [5, Theorem 2.18] and also [6, Section 2.54]), ˙ + H ∞ ), then the operator χ+ W 0 (b)χ− I ˙ + H ∞ (resp. b ∈ C(R) if b ∈ C(R) − + (resp. χ− W 0 (b)χ+ I) is compact on L2 (R). Let b ∈ SOp be the limit in the norm of Mp of a sequence bn ∈ SO1 . From the above results and (2.4)–(2.5) it follows that χ± W 0 (bn )χ∓ I are compact on L2 (R). Moreover, these operators are bounded on every Lp (R) with p ∈ (1, ∞). By the Krasnosel’skii interpolation theorem [14] (see also [15, Theorem 3.10]), the operators χ± W 0 (bn )χ∓ I are compact on Lp (R) for every p ∈ (1, ∞). From

χ± W 0 (b)χ∓ I − χ± W 0 (bn )χ∓ I B(Lp (R)) ≤ W 0 (b − bn ) B(Lp (R)) = b − bn Mp = o(1) as n → ∞ it follows that the operators χ± W 0 (b)χ∓ I are compact on Lp (R). Lemma 2.6. ([7, Lemma 7.4]) Let 1 < p < ∞. If a ∈ C(R) and b ∈ Cp (R), then the operator aW 0 (b) − W 0 (b)aI is compact on the space Lp (R). Lemma 2.7. ([1, Theorem 4.2]) Let 1 < p < ∞. If a ∈ [P C, SO], b ∈ SOp or a ∈ SO, b ∈ [P Cp , SOp ], then the operator aW 0 (b) − W 0 (b)aI is compact on the space Lp (R). 2.3. Maximal Ideal Spaces of Some Commutative Banach Algebras Let M (SO) denote the maximal ideal space of SO. Identifying points t ∈ R˙ with the evaluation functionals at t, one can identify the ﬁber of M (SO) over t ∈ R˙ by Mt (SO) = η ∈ M (SO) : η|C(R) ˙ =t .

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If t ∈ R, then the ﬁber Mt (SO) consists of the only evaluation functional at t, and thus M (SO) = Mt (SO) = R ∪ M∞ (SO). ˙ t∈R

According to [1, Proposition 3.1], the ﬁber M∞ (SO) has the form M∞ (SO) = (closSO∗ R)\R where closSO∗ R is the weak-star closure of R in SO∗ , the dual space of SO. Thus, for every functional η ∈ M∞ (SO) there exists a net tω ∈ R that tends to ∞ in the usual topology of R and such that η is the limit of tω in the Gelfand topology, that is, η(a) = lim a(tω ) for every a ∈ SO. The next lemma ω shows, in particular, that for a ﬁxed a ∈ SO, the net tω can be replaced by a sequence τn → +∞. Lemma 2.8. ([1, Corollary 3.3]) If (ak )∞ k=1 is a countable subset of SO and η is an element of M∞ (SO), then there exists a sequence (τn )∞ n=1 ⊂ R+ such that τn > 1, τn → +∞ as n → ∞, and for every x ∈ R\{0}, η(ak ) = lim ak (τn x) n→∞

(k ∈ N).

In what follows we write a(η) := η(a) for every a ∈ SO and every η ∈ M (SO). A unital Banach algebra A is called inverse closed in a unital Banach algebra B ⊃ A with the same unit if for any a ∈ A invertible in B its inverse a−1 belongs to A. From [1, Section 3] it follows that the Banach algebras SOp and [P Cp , SOp ] are inverse closed in the C ∗ -algebras SO and [P C, SO], respectively, and their maximal ideal spaces coincide as sets: M (SOp ) = M (SO),

M ([P Cp , SOp ]) = M ([P C, SO]).

It is well known that M∞ (P C) = {±∞}. According to [17, Section 1], the ﬁber M∞ ([P C, SO]) is homeomorphic to the product M∞ (P C) × M∞ (SO) = {±∞} × M∞ (SO) and the homeomorphism is given by the restriction map β → (β|P C , β|SO ). Thus, every β ∈ M∞ ([P C, SO]) can be viewed as a functional of the form either (+∞, η) or (−∞, η) with η ∈ M∞ (SO). Fix η ∈ M∞ (SO). Then there exists a homomorphism αη : [P C, SO] → P C|{±∞} ,

(αη a)(±∞) = aη (±∞) := (±∞, η)a. (2.6)

As the values aη (±∞) are uniquely deﬁned, for every η ∈ M∞ (SO) we get the homomorphism γη : [P C, SO] → P C,

γη a = aη (−∞)χ− + aη (+∞)χ+ ,

(2.7)

where aη (±∞) are deﬁned by (2.6). As M∞ (P Cp ) = M∞ (P C) = {±∞} for every p ∈ (1, ∞), we infer that the restriction of the homomorphism αη to [P Cp , SOp ] sends this algebra

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to P Cp |M∞ (P C) = P C|M∞ (P C) according to (2.6). Therefore, for all η ∈ M∞ (SO), the homomorphisms γη given by (2.7) map [P Cp , SOp ] into P Cp . To have an idea how γη acts, we give the following example. Example. Let k ∈ N and ck , dk ∈ C. Put

t2 2k exp i log(t + 1) fk (t) := ck χ− (t) + dk χ+ (t), gk (t) := 2 t +1

(t ∈ R).

The functions fk are piecewise constant with the only jumps at the origin and inﬁnity. So fk ∈ P Cp for every p ∈ (1, ∞). It is obvious that the functions gk 1 \C(R) because |gk (t)| < 1 for all t ∈ R, belong to SO kt2k i 2 + · lim tgk (t) = lim gk (t) = 0 t→∞ t→∞ t2 + 1 t2k + 1 log(t2k + 1) and gk do not have limits as t → ±∞. Since gk are even, from Lemmas 2.1–2.2 it follows that gk ∈ SO1 ⊂ SOp for all p ∈ (1, ∞). By Lemma 2.8, for every functional η ∈ M∞ (SO) there exists a sequence τn > 1, τn → +∞ such that η(gk ) = lim gk (τn ) n→∞

for all k ∈ N.

Notice that η(gk ) ∈ T := {z ∈ C : |z| = 1}. For every m ∈ N, the function a=

m

fk gk

k=1

belongs to the algebra [P Cp , SOp ] and (αη a)(−∞) = (αη a)(+∞) =

m

(αη fk )(−∞) · (αη gk )(−∞) =

m

k=1 m

k=1 m

k=1

k=1

(αη fk )(+∞) · (αη gk )(+∞) =

ck η(gk ), dk η(gk ).

Thus, γη a is the piecewise constant function γη a =

m

(ck χ− + dk χ+ )η(gk ) ∈ P Cp .

k=1

Let C π denote the smallest closed subalgebra of the Calkin algebra B/K ˙ and W 0 (b) + K with that contains all the cosets aI + K with a ∈ C(R) b ∈ SOp . Lemma 2.9. The algebra C π is commutative. The maximal ideal space M (C π ) of C π is homeomorphic to the set Ω := (R × M∞ (SO)) ∪ ({∞} × R) ∪ ({∞} × M∞ (SO)) .

(2.8)

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Proof. Lemma 2.7 implies that the algebra C π is commutative. Then in the same way as in [1, Lemma 5.1] (see also [23, Proposition 14.1]) one can prove that its maximal ideal space M (C π ) is homeomorphic to Ω.

3. Singular Integral Operators on Connected Subsets of R 3.1. Circular Arcs In this subsection we will follow [9, Section 9.1] and [4, Section 7.4]. Given two points z1 , z2 ∈ C and a number s ∈ (1, ∞) one can deﬁne the circular arc As (z1 , z2 ) between z1 and z2 by z − z1 2π + 2πZ ∪ {z1 , z2 }. ∈ As (z1 , z2 ) := z ∈ C\{0, 1} : arg z − z2 s If z1 = z2 =: z, then As (z1 , z2 ) is simply {z}. The set A2 (z1 , z2 ) is the segment [z1 , z2 ], if s > 2 (resp. 1 < s < 2), then As (z1 , z2 ) is the circular arc at the points of which the segment [z1 , z2 ] is seen at the angle 2π/s (resp. 2π − 2π/s) and which lies on the right (resp. left) of the straight line passing ﬁrst z1 and then z2 . The arc As (z1 , z2 ) can be analytically represented by z = z1 [1 − fs (μ)] + z2 fs (μ)

(μ ∈ [0, 1],

where fs : [0, 1] → C is deﬁned by ⎧ ⎨μ fs (μ) := sin(πμ − 2πμ/s) i(π−2π/s)(μ−1) e ⎩ sin(π − 2π/s)

z ∈ As (z1 , z2 )), if s = 2, if s ∈ (1, 2) ∪ (2, ∞).

3.2. The Gohberg–Krupnik Theorem Let −∞ ≤ α < β ≤ +∞ and (α, β) = R. For ϕ ∈ L1 (α, β), consider the Cauchy singular integral operator S(α,β) given by 1 (S(α,β) ϕ)(t) := πi

β α

ϕ(τ ) dτ, τ −t

where the integral is understood in the principal value sense. Put P(α,β) := (I + S(α,β) )/2,

Q(α,β) := (I − S(α,β) )/2.

It is well known that these operators are bounded on Lp (α, β) for 1 < p < ∞ (see e.g. [9, Chap. 1, Theorem 3.1]). By P C[α, β] denote the collection of all elements of P C with a ﬁnite number of jumps restricted to [α, β]. For a ∈ P C[α, β], p ∈ (1, ∞), and μ ∈ [0, 1], put ⎧ if t = α, ⎨ [1 − fp (μ)] + a(α+ )fp (μ) − + )[1 − f (μ)] + a(t )f (μ) if t ∈ (α, β), a(t ap (t, μ) := p p ⎩ if t = β. a(β − )[1 − fp (μ)] + fp (μ) The range of this function is a closed continuous curve obtained from the + range of the function a by adding the arcs Ap (a(t− k ), a(tk )) for all jumps tk − + of a and the arcs Ap (a(β ), 1) and Ap (1, a(α )). It can be oriented in the natural manner: on the intervals of continuity of a, the motion along this curve

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agrees with the increment of t, while the supplementary arcs Ap (·, ·) are oriented from the point on the ﬁrst position to the point on the second position in the definition of the arc Ap (·, ·). If ap (t, μ) = 0 for all (t, μ) ∈ [α, β] × [0, 1], then by wind ap we denote the winding number of this curve about the origin. The Gohberg–Krupnik one-sided invertibility criteria for singular integral operators with piecewise continuous coefﬁcients over the segment [α, β] read as follows (see [9, Chap. 9, Theorem 4.1] and also [16, Chap. IV, Theorems 5.1 and 6.1]). Theorem 3.1. (Gohberg–Krupnik) Let −∞ ≤ α < β ≤ +∞ and (α, β) = R. Suppose 1 < p < ∞ and c, d ∈ P C[α, β]. The operator A = P(α,β) cI + Q(α,β) dI is at least one-sided invertible on Lp (α, β) if and only if the following conditions are satisfied for all μ ∈ [0, 1]: ⎧ ⎨ c(α+ )fp (μ) + d(α+ )[1 − fp (μ)] = 0, c(t+ )d(t− )fp (μ) + c(t− )d(t+ )[1 − fp (μ)] = 0 (t ∈ (α, β)), (3.1) ⎩ c(β − )[1 − fp (μ)] + d(β − )fp (μ) = 0. If these conditions are fulfilled, then the operator A is invertible, invertible only from the left, invertible only from the right depending on whether the number windp (c/d) is equal to zero, positive, or negative, respectively. 3.3. Spectra of the Operators P(α,β) and Q(α,β) For p ∈ (1, ∞), put q := p/(p − 1) and deﬁne the lentiform domain Lp by Lp := {z ∈ As (0, 1) : min{p, q} ≤ s ≤ max{p, q}} .

(3.2)

By B(Lp (α, β)) denote the Banach algebra of all bounded linear operators on Lp (α, β). The spectrum of an element a in a unital Banach algebra B will be denoted by spB (a). Theorem 3.2. Let 1 < p < ∞ and Lp be the lentiform domain given by (3.2). If −∞ ≤ α < β ≤ +∞ and (α, β) = R, then spB(Lp (α,β)) (P(α,β) ) = spB(Lp (α,β)) (Q(α,β) ) = Lp . Proof. Let λ ∈ C. Then for the operator Aλ := P(α,β) − λI = P(α,β) (1 − λ)I + Q(α,β) (−λ)I conditions (3.1) with c := 1 − λ and d := −λ have the form, for μ ∈ [0, 1], c(α+ )fp (μ) + d(α+ )[1 − fp (μ)] = (1 − λ)fp (μ) − λ[1 − fp (μ)] = fp (μ) − λ = 0; c(t+ )d(t− )fp (μ) + c(t− )d(t+ )[1 − fp (μ)] = (1 − λ)(−λ)fp (μ) + (1 − λ)(−λ)[1 − fp (μ)] = λ(λ − 1) = 0 whenever t ∈ (α, β); and c(β − )[1 − fp (μ)] + d(β − )fp (μ) = (1 − λ)[1 − fp (μ)] + (−λ)fp (μ) = 1 − λ − fp (μ) = 0.

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These conditions are equivalent to 1−λ∈ / Ap (0, 1).

λ∈ / Ap (0, 1),

(3.3)

By Theorem 3.1, the operator Aλ is one-sided invertible if and only if (3.3) is fulﬁlled. Further, if (3.3) holds, then ⎧ λ−1 ⎪ ⎪ if t = α, ⎪ λ fp (μ) + [1 − fp (μ)] ⎪ ⎨ c λ−1 (t, μ) = if t ∈ (α, β), ⎪ d p λ ⎪ ⎪ ⎪ ⎩ λ − 1 [1 − fp (μ)] + fp (μ) if t = β. λ So, the range of (c/d)p coincides with the closed curve λ−1 λ−1 ,1 . Ap 1, ∪ Ap λ λ It is easy to see that wind Ap

λ−1 1, λ

∪ Ap

λ−1 ,1 = 0 λ

if and only if the origin lies outside the convex λ−1 lentiform domain Mp (λ) λ−1 bounded by the arcs Ap 1, λ = Aq λ , 1 , where 1/p + 1/q = 1, and Ap λ−1 λ ,1 . Let Ip := [min{p, q}, max{p, q}]. By Theorem 3.1, the operator Aλ is / Mp (λ). Then invertible on Lp (α, β) if and only if (3.3) is fulﬁlled and 0 ∈ the spectrum of P(α,β) is equal to σ1 ∪ σ2 ∪ σ3 , where σ1 := {λ ∈ C : λ ∈ Ap (0, 1)} = Ap (0, 1), σ2 := {λ ∈ C : 1 − λ ∈ Ap (0, 1)} = Ap (0, 1), and σ3 := {λ ∈ C\(σ1 ∪ σ2 ) : 0 ∈ Mp (λ)} ⎫ ⎧ ⎬ ⎨ λ−1 ,1 = λ ∈ C\Ap (0, 1) : 0 ∈ As ⎭ ⎩ λ s∈Ip λ−1 = λ ∈ C\Ap (0, 1) : [1 − fs (μ)] + fs (μ) = 0 for some s ∈ Ip λ = {λ ∈ C\Ap (0, 1) : 1 − λ ∈ As (0, 1) for some s ∈ Ip } = {λ ∈ C\Ap (0, 1) : 1 − λ ∈ Lp (0, 1)} = {λ ∈ C\Ap (0, 1) : λ ∈ Lp (0, 1)} = Lp \Ap (0, 1). Thus spB(Lp (α,β)) (P(α,β) ) = σ1 ∪ σ2 ∪ σ3 = Ap (0, 1) ∪ Ap (0, 1) ∪ (Lp \Ap (0, 1)) = Lp . The proof of the equality spB(Lp (α,β)) (Q(α,β) ) = Lp is analogous. Let χE denote the characteristic function of a set E ⊂ R.

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Corollary 3.3. Let 1 < p < ∞ and Lp be the lentiform domain given by (3.2). Then I = Lp ; (a) spB χ(−1,0) PR χ(−1,0) I + χ(−∞,−1) QR χ(−∞,−1) (b) spB χ(1,∞) PR χ(1,∞) I + χ(0,1) QR χ(0,1) I = Lp ; (c) spB (χ+ PR χ+ I + χ− QR χ− I) = Lp . Proof. Let us prove equality (a). The operator χ(−1,0) PR χ(−1,0) I + χ(−∞,−1) QR χ(−∞,−1) I − λI = χ(−∞,−1) (QR − λI)χ(−∞,−1) I + χ(−1,0) (PR − λI)χ(−1,0) I − λχ+ I is represented in the direct sum ·

·

Lp (−∞, −1) + Lp (−1, 0) + Lp (R+ ) = Lp (R) by the matrix

⎡

Q(−∞,−1) − λI ⎣0 0

0 P(−1,0) − λI 0

⎤ 0 ⎦. 0 −λI

Hence, by Theorem 3.2, spB χ(−1,0) PR χ(−1,0) I + χ(−∞,−1) QR χ(−∞,−1) I = spB(Lp (−∞,−1)) (Q(−∞,−1) ) ∪ spB(Lp (−1,0)) (P(−1,0) ) ∪ spB(Lp (R+ )) (0) = Lp ∪ Lp ∪ {0} = Lp . The proof of equalities (b)–(c) is analogous.

3.4. Singular Integral Operator with Piecewise Constant Coefficients Let z1 , z2 , z3 be an ordered triple of points in the complex plane. It is clear that Ap (z1 , z2 ) ∪ Ap (z2 , z3 ) ∪ Ap (z3 , z1 ) =: Cp (z1 , z2 , z3 ) is a closed curve in the complex plane (which degenerates to the point z if z = z1 = z2 = z3 ). Each arc Ap (z1 , z2 ), Ap (z2 , z3 ), and Ap (z3 , z1 ) can be naturally oriented starting from the point on the ﬁrst position and terminating at the point on the second position in the definition of the arc Ap (·, ·). This orientation induces the orientation of the curve Cp (z1 , z2 , z3 ). If 0∈ / Cp (z1 , z2 , z3 ), then by wind Cp (z1 , z2 , z3 ) we denote the winding number of the curve Cp (z1 , z2 , z3 ) about the origin. By definition, wind Cp (z, z, z) = 0. Lemma 3.4. Let 1 < p < ∞. Suppose a− , a+ , b− , b+ ∈ C. The singular integral operator P(−1,1) (a− χ(−1,0) + b− χ(0,1) )I + Q(−1,1) (a+ χ(−1,0) + b+ χ(0,1) )I is invertible on the space Lp (−1, 1) if and only if a− b− , 0∈ / {a− , a+ , b− , b+ }, wind Cp 1, = 0. a+ b+

(3.4)

This lemma follows from Theorem 3.1. Its proof is similar to the proof of Theorem 3.2 and it is omitted here.

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4. Algebraization 4.1. Stability of the Finite Section Method Let E be the set formed by all the sequences (Aτ ) (depending on a parameter τ ∈ R+ ) of operators Aτ ∈ B such that sup Aτ B < ∞.

τ ∈R+

One says that the sequence (Aτ ) ∈ E is stable if there exists a positive constant τ0 such that the operator Aτ ∈ B is invertible for any τ > τ0 and sup A−1 τ B < ∞.

τ >τ0

The following result is well known (see e.g. [11, Proposition 1.1] and also [6, Proposition 7.3]). Theorem 4.1. (Polski) The finite section method applies to an operator A ∈ B if and only if A is invertible and the sequence (Pτ APτ + Qτ ) is stable. 4.2. Algebra E and its Ideal G The following statement is proved as [12, Proposition 1.13]. Lemma 4.2. The set E with the operations (Aτ )+(Bτ ) := (Aτ + Bτ ), (Aτ )(Bτ ) := (Aτ Bτ ), λ(Aτ ) := (λAτ )

(λ ∈ C),

the identity element (I), and the norm

(Aτ ) E := sup Aτ B τ ∈R+

forms a unital Banach algebra. Note that the constant sequences (A) are included in E for any A ∈ B. Definition 4.3. By A denote the smallest Banach subalgebra of the algebra E that contains all constant sequences (aI) with a ∈ P C and (W 0 (b)) with b ∈ [P Cp , SOp ] and the sequence (Pτ ). We will be interested in conditions for the stability of any sequence in A. Clearly, A contains the sequences (1.3). Let G be the set of all sequences (Aτ ) ∈ E satisfying lim Aτ B = 0.

τ →∞

Lemma 4.4. The set G is a closed two-sided ideal of the algebra E. The proof is similar to the proof of [12, Proposition 1.14]. Theorem 4.5. (Kozak) Let (Aτ ) ∈ E. The sequence (Aτ ) is stable if and only if the coset (Aτ ) + G is invertible in the quotient algebra E/G. This result is well known (see e.g. [6, Proposition 7.3], [11, Proposition 1.2], [12, Theorem 1.15]).

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5. Essentialization 5.1. Algebra F and its Ideal G A sequence of operators on a Banach space is said to converge *-strongly if it converges strongly and the sequence of the adjoint operators converges strongly on the dual space. Let τ ∈ R+ . By Vτ denote the additive shift operator given by (Vτ f )(x) := f (x − τ )

(x ∈ R).

It is clear that Vτ is bounded and invertible on Lp (R) and Vτ−1 = V−τ . Moreover, Vτ B = 1 for all τ ∈ R. Let F denote the set of all sequences A := (Aτ ) ∈ E such that the sequences (Aτ ), (V−τ Aτ Vτ ), and (Vτ Aτ V−τ ) are *-strongly convergent as τ → +∞. Lemma 5.1. (a) The set F is a closed unital subalgebra of the algebra E. (b) Let i ∈ {−1, 0, 1}. The mappings Wi : F → B given by W−1 (A) := s- lim Vτ Aτ V−τ , τ →+∞

W0 (A) := s-lim Aτ , τ →+∞

W1 (A) := s- lim V−τ Aτ Vτ τ →+∞

for A = (Aτ ) ∈ F are bounded unital homomorphisms with the norms

W−1 = W0 = W1 = 1. (c) The set G is a closed two-sided ideal of the algebra F. (d) The ideal G lies in the kernel of the homomorphism Wi for i ∈ {−1, 0, 1}. (e) The algebra F is inverse closed in the algebra E and the algebra F/G is inverse closed in the algebra E/G. The proof follows the proof of [22, Proposition 4.1]. Notice that the algebra F in the present paper is larger than the algebra considered in [22] and also denoted there by F (see also Remark 7.3). 5.2. The Algebra A is Contained in the Algebra F Proposition 5.2. (a) If P = (Pτ ), then P ∈ F and W−1 (P) = χ+ I,

W0 (P) = I,

W1 (P) = χ− I.

(5.1)

(b) If A = (aI) with a ∈ P C, then A ∈ F and W−1 (A) = a(−∞)I,

W0 (A) = aI,

W1 (A) = a(+∞)I.

(5.2)

(c) If B = (W 0 (b)) with b ∈ [P Cp , SOp ], then B ∈ F and W−1 (B) = W 0 (b),

W0 (B) = W 0 (b),

W1 (B) = W 0 (b).

(5.3)

Proof. The proof of equalities (5.1) and (5.2) is straightforward, equalities in (5.3) are trivial because the convolution operator W 0 (b) is translation-invariant, that is, V±τ W 0 (b)V∓τ = W 0 (b) for every b ∈ Mp , in particular, for every b ∈ [P Cp , SOp ]. To ﬁnish the proof, it remains to note that [aI]∗ = aI ∈ B(Lq (R)),

[W 0 (b)]∗ = W 0 (b) ∈ B(Lq (R)),

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and Pτ∗ = Pτ ∈ B(Lq (R)), where 1/p + 1/q = 1. Therefore the existence of the strong limits for the adjoint operators in the definition of the algebra F can be obtained in the same way as the existence of the strong limits in (5.1)–(5.3). Corollary 5.3. The algebra A is a closed unital subalgebra of the algebra F. 5.3. Ideals J−1 , J0 , J1 , and J of the Algebra F Recall that K denotes the ideal of the compact operators on Lp (R). Put J−1 := {(V−τ KVτ ) + (Gτ ) : K ∈ K, (Gτ ) ∈ G} , J0 := {(K) + (Gτ ) : K ∈ K, (Gτ ) ∈ G} , J1 := {(Vτ KV−τ ) + (Gτ ) : K ∈ K, (Gτ ) ∈ G} , and J := {(Vτ K1 V−τ )+(K0 )+(V−τ K−1 Vτ )+(Gτ ) : K−1 , K0 , K1 ∈ K, (Gτ ) ∈ G}. Lemma 5.4. The sets Ji with i ∈ {−1, 0, 1} and J are closed two-sided ideals of the algebra F. Proof. First we note that (V±τ ) converges weakly to zero as τ → +∞. Let us show that J−1 ⊂ F. Let J = (Jτ ) = (V−τ KVτ ) + (Gτ ) for some K ∈ K 2 = V±2τ , we obtain that (Jτ ) and (Gτ ) ∈ G. Since K is compact and V±τ is *-strongly convergent to zero, (V−τ Jτ Vτ ) is *-strongly convergent to zero, and (Vτ Jτ V−τ ) is *-strongly convergent to K. Thus J−1 ⊂ F. Let A = (Aτ ) ∈ F. Then AJ = (Aτ V−τ KVτ ) + (Aτ Gτ ) = (V−τ Vτ Aτ V−τ KVτ ) + (Aτ Gτ ) = (V−τ W−1 (A)KVτ ) + (V−τ [Vτ Aτ V−τ K − W−1 (A)K]Vτ + Aτ Gτ ) =: (V−τ W−1 (A)KVτ ) + (Gτ ). Since the sequence (Vτ Aτ V−τ − W−1 (A)) converges strongly to zero and K ∈ K, the sequence (Vτ Aτ V−τ K − W−1 (A)K) converges uniformly to zero. Thus (Gτ ) := (V−τ [Vτ Aτ V−τ K − W−1 (A)K]V−τ + Aτ Gτ ) ∈ G and AJ ∈ J−1 . Analogously one can show that JA ∈ J−1 . This means that J−1 is a two-sided ideal of F. Let us show that J−1 is closed. Suppose (Jk )k∈N is a Cauchy sequence in J−1 . Since F is closed in E by Lemma 5.1(a), the sequence (Jk )k∈N is convergent to some J = (Jτ ) ∈ F. We claim that J ∈ J−1 . By definition of (j) J−1 , there exist sequences (K (j) )j∈N ⊂ K and (Gτ )j∈N ⊂ G such that Jj = (Jτ(j) ) = (V−τ K (j) Vτ ) + (Gτ(j) ). From Lemma 5.1(b) it follows that for all j, k ∈ N,

K (j) − K (k) B = W−1 (Jj ) − W−1 (Jk ) B ≤ Jj − Jk E . Then (K (j) )j∈N is a Cauchy sequence in B. Let K ∈ K be its limit. Put G := (Gτ ) := (Jτ ) − (V−τ KVτ ).

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Taking into account that V±τ B = 1, we obtain

(Gτ ) − (Gτ(j) ) E ≤ (Jτ ) − (Jτ(j) ) E + (V−τ KVτ ) − (V−τ K (j) Vτ ) E ≤ J − Jj E + K − K (j) B . (j)

Hence G is the limit of (Gj )j∈N with Gj := (Gτ ) ∈ G in the norm of E. By Lemma 5.1(c), G ∈ G. Thus J ∈ J−1 and J−1 is closed. Analogously it can be shown that J0 , J−1 , and J are closed two-sided ideals of the algebra F. 5.4. Lifting The main result of this section is the following lifting theorem adapted for our purposes from [11, Theorem 1.8]. Theorem 5.5. Let A = (Aτ ) ∈ F. The coset A + G is invertible in the quotient algebra F/G if and only if the operators W−1 (A), W0 (A), and W1 (A) are invertible in B and the coset A + J is invertible in the quotient algebra F/J . Proof. The proof is analogous to the proof of the abstract lifting theorem [11, Theorem 1.8], although we consider the invertibility of Wi (A) in B, while there it is considered in the images of the homomorphisms Wi . Necessity. If A + G is invertible in F/G, then there exist sequences B ∈ F and G1 , G2 ∈ G such that AB = I + G1 ,

BA = I + G2 .

(5.4)

Let i ∈ {−1, 0, 1}. From Lemma 5.1(b),(d) we know that Wi : F → B is a bounded unital homomorphism and Wi (G1 ) = Wi (G2 ) = 0. Applying this homomorphism to the equalities in (5.4), we conclude that the operator Wi (A) is invertible in B and its inverse is Wi (B). From the definition of the ideal J it follows that G1 , G2 ∈ J . Thus the equalities (5.4) imply also that A + J is invertible in F/J . The necessity part is proved. Sufficiency. Assume that the operators W−1 (A), W0 (A), W1 (A) are invertible in B and the coset A + J is invertible in the quotient algebra F/J . Then there exist sequences B ∈ F and J ∈ J such that BA = I + J. From the definition of J we conclude that there exist elements Ji ∈ Ji such that J = J−1 + J0 + J1 . Taking into account that, by Lemma 5.1(d), W−1 [(V−τ KVτ ) + (Gτ )] = K, W0 [(K) + (Gτ )] = K,

(5.5)

W1 [(Vτ KV−τ ) + (Gτ )] = K for every K ∈ K and every (Gτ ) ∈ G, we have that Wi maps the ideal Ji onto the ideal K for i ∈ {−1, 0, 1}. Since Wi (Ji ) ∈ K, we obviously have Wi (Ji )Wi (A)−1 ∈ K. Consequently for every i ∈ {−1, 0, 1}, there exists Ji ∈ Ji such that Wi (Ji ) = Wi (Ji )Wi (A)−1 .

(5.6)

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Put B := B − J−1 − J0 − J1 . Then B + J = B + J and B A = I + J − J−1 A − J0 A − J1 A

= I + (J−1 − J−1 A) + (J0 − J0 A) + (J1 − J1 A).

(5.7)

From (5.6) it follows that Wi (Ji ) = Wi (Ji )Wi (A) = Wi (Ji A). Hence Wi (Ji − Ji A) = 0. From this observation and (5.5) we conclude that Ji − Ji A ∈ G for i ∈ {−1, 0, 1}. Thus (5.7) can be written as B A = I + G with G = (J−1 − J−1 A) + (J0 − J0 A) + (J1 − J1 A) ∈ G. This means that A + G is left-invertible in F/G. Analogously it can be shown that A + G is right-invertible in F/G. Remark 5.6. A detailed discussion of various versions of abstract lifting theorems and their applications in the setting of Banach algebras and C ∗ -algebras is contained in [21, Chap. 6]. In particular, the above theorem follows from the Banach algebra inverse closed lifting theorem (see [21, Theorem 6.2.8]).

6. Localization 6.1. Allan–Douglas Local Principle Recall that the center of an algebra A consists of all elements a ∈ A such that ab = ba for all b ∈ A. Let A be a Banach algebra with identity. By a central subalgebra C of A one means a subalgebra of the center of A. Theorem 6.1. ([6, Theorem 1.35(a)] or [11, Theorem 1.5]) Let A be a Banach algebra with identity e and let C be closed central subalgebra of A containing e. Let M (C) be the maximal ideal space of C, and for ω ∈ M (C), let Iω refer to the smallest closed two-sided ideal of A containing the ideal ω. Then an element a ∈ A is invertible in A if and only if a + Iω is invertible in the quotient algebra A/Iω for all ω ∈ M (C). 6.2. Sequences of Local Type We say that a sequence (Aτ ) ∈ F is of local type if (Aτ )(f I) − (f I)(Aτ ) ∈ J ,

(Aτ )(W 0 (g)) − (W 0 (g))(Aτ ) ∈ J

˙ and all g ∈ SOp . Let L denote the set of all sequences of for all f ∈ C(R) local type. Lemma 6.2. (a) The set L is a closed unital subalgebra of the algebra F. (b) The set J is a closed two-sided ideal of the algebra L. (c) The algebra L/J is inverse closed in the algebra F/J . (d) The algebra L is inverse closed in the algebra F.

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Proof. (a) By definition, L ⊂ F. It is clear that the sequence (I) is the identity of L. Hence L is a unital subalgebra of F. Let us show that L is (j) closed. Suppose (Aj )j∈N with Aj = (Aτ ) ∈ L is a Cauchy sequence in L. Let A = (Aτ ) ∈ F be its limit. We show that A ∈ L. By definition of L, Aj (f I) − (f I)Aj ∈ J ,

Aj (W 0 (g)) − (W 0 (g))Aj ∈ J

(6.1)

˙ and g ∈ SOp . Passing to the limit in (6.1) as for all j ∈ N, f ∈ C(R), j → ∞ and taking into account that J is closed in F, we conclude that ˙ and A(f I) − (f I)A ∈ J and A(W 0 (g)) − (W 0 (g))A ∈ J for all f ∈ C(R) g ∈ SOp . Thus A ∈ L. Part (a) is proved. (b) Since J ⊂ L ⊂ F and J is a closed two-sided ideal of the algebra F, we also observe that J is a closed two-sided ideal of L. Part (b) is proved. (c) Let A = (Aτ ) ∈ L and A + J be invertible in F/J . Then there (1) (2) exist sequences B = (Bτ ) ∈ F and J1 = (Jτ ), J2 = (Jτ ) belonging to J and such that I − Aτ Bτ = Jτ(1) ,

I − Bτ Aτ = Jτ(2) .

˙ or W 0 (g) with g ∈ SOp . Let C be one of the operators f I with f ∈ C(R) Then C = CAτ Bτ + CJτ(1) = Aτ CBτ + [CAτ − Aτ C]Bτ + CJτ(1) and Bτ C − CBτ = Bτ Aτ CBτ + Bτ [CAτ − Aτ C]Bτ + Bτ CJτ(1) − CBτ = [CBτ − Jτ(2) CBτ ] + Bτ [CAτ − Aτ C]Bτ + Bτ CJτ(1) − CBτ = Bτ [CAτ − Aτ C]Bτ + [Bτ CJτ(1) − Jτ(2) CBτ ]. If we put C = (C), then BC−CB = B[CA−AC]B+J, where CA−AC ∈ J and J := BCJ1 − J2 CB ∈ J . Thus BC − CB ∈ J and B + J ∈ L/J . Part (c) is proved. (d) Part (d) follows from part (c). Remark 6.3. The algebra L is larger than the algebra of sequences of local type considered in [22] and denoted there by F0 (see also Remark 7.3). 6.3. Sequences in A are of Local Type Theorem 6.4. The algebra A is a closed unital subalgebra of L. Proof. The idea of the proof is borrowed from [22, Proposition 4.11]. From ˙ and g ∈ SOp . It is Corollary 5.3 we know that A ⊂ F. Suppose f ∈ C(R) sufﬁcient to prove that (aI) with a ∈ P C, (W 0 (b)) with b ∈ [P Cp , SOp ], and (Pτ ) commute with (f I) and (W 0 (g)) modulo the ideal J . Obviously aI commutes with f I. By Lemma 2.7, aW 0 (g) − W 0 (g)aI is compact. Therefore (aI)(f I) − (f I)(aI) ∈ J , Hence (aI) ∈ L.

(aI)(W 0 (g)) − (W 0 (g))(aI) ∈ J .

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It is clear that W 0 (b) commutes with W 0 (g). Applying Lemma 2.7 once again, we see that f W 0 (b) − W 0 (b)f I is compact. Then (W 0 (b))(f I) − (f I)(W 0 (b)) ∈ J ,

(W 0 (b))(W 0 (g)) − (W 0 (g))(W 0 (b)) ∈ J .

Thus (W 0 (b)) ∈ L. Since Pτ commutes with f I, it is trivial that (Pτ )(f I) − (f I)(Pτ ) ∈ J . Consider (Pτ W 0 (g) − W 0 (g)Pτ ) = (Pτ W 0 (g)Qτ − Qτ W 0 (g)Pτ ) = (Pτ χ+ W 0 (g)χ+ Qτ − Qτ χ+ W 0 (g)χ+ Pτ ) +(Pτ χ+ W 0 (g)χ− Qτ − Qτ χ+ W 0 (g)χ− Pτ ) +(Pτ χ− W 0 (g)χ+ Qτ − Qτ χ− W 0 (g)χ+ Pτ ) +(Pτ χ− W 0 (g)χ− Qτ − Qτ χ− W 0 (g)χ− Pτ ). (6.2) 0

0

By Lemma 2.5, the operators χ+ W (g)χ− I and χ− W (g)χ+ I are compact. Since (Qτ ) converges *-strongly to zero, we conclude that the sequences of the second and the third line on the right-hand side of (6.2) belong to G and thus to J . It remains to show that (Pτ χ+ W 0 (g)χ+ Qτ − Qτ χ+ W 0 (g)χ+ Pτ ) ∈ J , 0

0

(Pτ χ− W (g)χ− Qτ − Qτ χ− W (g)χ− Pτ ) ∈ J .

(6.3) (6.4)

It is easy to see that V−τ Pτ χ+ I = χ(−τ,0) V−τ ,

χ+ Qτ Vτ = Vτ χ+ I,

V−τ Qτ χ+ I = χ+ V−τ ,

χ+ Pτ Vτ = Vτ χ(−τ,0) I,

where χ(−τ,0) denotes the characteristic function of the interval (−τ, 0), and Vτ V−τ = I,

V−τ W 0 (g)Vτ = W 0 (g).

Then (Pτ χ+ W 0 (g)χ+ Qτ − Qτ χ+ W 0 (g)χ+ Pτ ) = (Vτ V−τ [Pτ χ+ W 0 (g)χ+ Qτ − Qτ χ+ W 0 (g)χ+ Pτ ]Vτ V−τ ) = (Vτ [χ(−τ,0) V−τ W 0 (g)Vτ χ+ I − χ+ V−τ W 0 (g)Vτ χ(−τ,0) I]V−τ ) = (Vτ [χ− W 0 (g)χ+ I − χ+ W 0 (g)χ− I]V−τ ) +(Vτ [χ(−τ,0) I − χ− I]χ− W 0 (g)χ+ V−τ ) −(Vτ χ+ W 0 (g)χ− [χ(−τ,0) I − χ− I]V−τ ). 0

(6.5)

0

The operators χ− W (g)χ+ I and χ+ W (g)χ− I are compact by Lemma 2.5. It is easy to see that χ(−τ,0) I is *-strongly convergent to χ− I. Thus the (1)

sequence on the right hand side of (6.5) has the form (Vτ K1 V−τ ) + (Gτ ), (1) where K1 ∈ K and (Gτ ) ∈ G. This proves (6.3). Analogously, it can be shown that (Pτ χ− W 0 (g)χ− Qτ − Qτ χ− W 0 (g)χ− Pτ ) = (V−τ K−1 Vτ ) + (G(2) τ ), (2)

where K−1 ∈ K and (Gτ ) ∈ G. This proves (6.4) and ﬁnishes the proof of the theorem.

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6.4. Central Subalgebra of the Algebra L/J and its Maximal Ideal Space Let C be the smallest closed subalgebra of L that contains all sequences (f I) ˙ and (W 0 (g)) with g ∈ SOp . From the proof of Theorem 6.4 with f ∈ C(R) it follows that C is not trivial. Put C J := (C + J )/J ,

LJ := L/J .

Lemma 6.5. The set C J is a closed central subalgebra of the algebra LJ . Proof. This fact follows immediately from the definition of the algebras L, LJ , and C J . Theorem 6.6. The maximal ideal space M (C J ) of the commutative Banach algebra C J is homeomorphic to the set Ω given by (2.8). Proof. Since (V±τ ) converge weakly to zero as τ → +∞, it is easy to see that W0 (J) ∈ K for every J ∈ J . Hence Φ : F/J → B/K,

A + J → W0 (A) + K

˙ and is a well deﬁned homomorphism. Clearly Φ(J ) = K and for f ∈ C(R) g ∈ SOp , Φ((f I) + J ) = f I + K,

Φ((W 0 (g)) + J ) = W 0 (g) + K.

Hence the mapping Φ|C J : C J → C π is injective and onto, where the algebra C π is deﬁned just before Lemma 2.9. Therefore C J and C π are isomorphic. Then the maximal ideal spaces of C J and C π are homeomorphic. Thus, from Lemma 2.9 it follows that M (C J ) is homeomorphic to the set Ω given by (2.8). 6.5. Localization via the Allan–Douglas Local Principle J With every point (ξ, η) ∈ Ω we associate the closed two-sided ideal Nξ,η of J the algebra C generated by all the cosets

(f I) + J

˙ f (ξ) = 0), (f ∈ C(R),

(W 0 (g)) + J

(g ∈ SOp , g(η) = 0).

J be the smallest closed two-sided ideal of LJ that contains the maxLet Iξ,η J imal ideal Nξ,η and let J J J J ΦJ ξ,η : L → L /Iξ,η =: Lξ,η

be the canonical homomorphism of LJ onto the quotient algebra LJ ξ,η . Now we are in a position to apply the Allan–Douglas local principle. Summarizing the results obtained so far we arrive at the following. Theorem 6.7. A sequence A = (Aτ ) ∈ L is stable if and only if the operaJ tors W−1 (A), W0 (A), and W1 (A) are invertible in B and the coset A + Iξ,η is invertible in the local algebra LJ ξ,η for every (ξ, η) ∈ Ω, where Ω is given by (2.8).

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Proof. From Lemma 5.1(a) and the definition of L we know that L ⊂ F ⊂ E. In view of Theorem 4.5, A is stable if and only if A + G is invertible in E/G. By Lemma 5.1(e), the latter is equivalent to the invertibility of A+G in F/G. By the lifting theorem (Theorem 5.5), this is equivalent to the invertibility of the operators W−1 (A), W0 (A), and W1 (A) in the algebra B and the invertibility of the coset A + J in the quotient algebra F/J . From Lemma 6.2(c) we conclude that A + J is invertible in F/J if and only if it is invertible in L/J . The latter assertion is equivalent to the invertibility of the cosets J A + Iξ,η in the local algebras LJ ξ,η for all (ξ, η) ∈ Ω by the Allan–Douglas local principle (Theorem 6.1). Our next aim is to study the invertibility of the elements J ΦJ ξ,η (A) := A + Iξ,η

(A ∈ L)

J in the local algebras LJ ξ,η . It turns out that the algebras Lξ,η are too large in order to obtain via Theorem 6.7 eﬀectively veriﬁable stability conditions for an arbitrary A ∈ L. So we restrict ourselves to the case of sequences belonging to A (recall that A ⊂ L by Theorem 6.4). We denote by AJ the smallest closed subalgebra of LJ that contains the cosets A+J for all A = (Aτ ) ∈ A. J The canonical homomorphism ΦJ onto ξ,η sends A J J J AJ ξ,η := Φξ,η (A ) ⊂ Lξ,η .

In Sects. 8–10 we obtain sufﬁcient conditions for the invertibility of the eleJ ments ΦJ ξ,η (A) in the local algebras Lξ,η for A ∈ A and (ξ, η) ∈ Ω.

7. Homogenization 7.1. Algebra of Sequences Generated by Homogeneous Operators and (Pτ ) Given any positive real number τ , deﬁne the operator Zτ : Lp (R) → Lp (R),

(Zτ f )(x) = τ −1/p f (x/τ ).

Obviously Zτ B = 1. It is also clear that Zτ is invertible and Zτ−1 = Z1/τ . An operator A ∈ B is called homogeneous (or dilation invariant) if Zτ−1 AZτ = A for each τ ∈ R+ . It is easy to see that the operators χ± I and W 0 (χ± ) are homogeneous operators. Let H be the smallest closed subalgebra of E that contains the sequence (Pτ ) and the constant sequences (H), where H is a homogeneous operator. Lemma 7.1. Let A = (Aτ ) ∈ H ∩ L. Then the coset A + G is invertible in the algebra L/G if and only if the operator A1 is invertible. Proof. The idea of the proof is borrowed from [19, Proposition 1]. Necessity. If A + G is invertible in L/G, then obviously it is invertible in E/G. Therefore, by Theorem 4.5, A is stable. Hence the operators Aτ are invertible for all sufﬁciently large τ . Since Zτ−1 Aτ Zτ = A1

(τ ∈ R+ ),

(7.1)

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we conclude that the operator A1 is invertible. The necessity portion is proved. = Zτ A−1 Sufficiency. If A1 is invertible, then (7.1) implies that A−1 τ 1 −1 −1 −1 Zτ . Since Zτ B Zτ B = 1, the norms of Aτ are uniformly bounded. That is A = (Aτ ) is stable. By Theorem 4.5, A + G is invertible in E/G. But L/G is inverse closed in F/G (Lemma 6.2(c)) and F/G is inverse closed in E/G (Lemma 5.1(e)). Thus A + G is invertible in L/G. 7.2. Algebras Hη and the Ideal G For η ∈ R, put Uη : Lp (R) → Lp (R),

(Uη f )(x) = eiηx f (x).

It is clear that Uη−1 = U−η and Uη±1 B = 1. Let Hη denote the set of all sequences A = (Aτ ) ∈ E such that the sequence (Zτ−1 Uη Aτ Uη−1 Zτ ) is *-strongly convergent as τ → +∞. Lemma 7.2. Suppose η ∈ R. (a) The set Hη is a closed unital subalgebra of the algebra E. (b) The mapping Hη : Hη → B given by Hη (A) := s-lim Zτ−1 Uη Aτ Uη−1 Zτ τ →+∞

for A = (Aτ ) ∈ Hη is a bounded unital homomorphism with the norm

Hη = 1. (c) The set G is a closed two-sided ideal of the algebra Hη . (d) The ideal G lies in the kernel of the homomorphism Hη . (e) The algebra Hη /G is inverse closed in the algebra E/G. The proof is analogous to the proof of [22, Proposition 4.1]. Remark 7.3. Let F! be the algebra denoted by F in the paper [22]. Then F! ⊂ F ∩ (∩η∈R Hη ) . We conjecture that this inclusion is proper. 7.3. The Algebra A is Contained in the Algebras Hη Proposition 7.4. Suppose η ∈ R. (a) If P = (Pτ ), then P ∈ Hη and Hη (P) = P1 . (b) If A = (aI) with a ∈ P C, then A ∈ Hη and Hη (A) = a(−∞)χ− I + a(+∞)χ+ I. 0

(c) If B = (W (b)) with b ∈ [P Cp , SOp ], then B ∈ Hη and Hη (B) = b(η − )W 0 (χ− ) + b(η + )W 0 (χ+ ). Proof. (a) It is easy to see that for every τ ∈ R+ one has Uη Pτ Uη−1 = Pτ and Zτ−1 Pτ Zτ = P1 . From these equalities it follows immediately that Hη (P) = P1 . Taking into account that Zτ∗ = Z1/τ ,

Uη∗ = U−η ,

Pτ∗ = Pτ ,

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from above we conclude that (Zτ−1 Uη Pτ Uη−1 Zτ )∗ converges strongly to P1∗ . Thus P ∈ Hη . Part (a) is proved. Part (b) is proved in [23, Proposition 13.1(b)] and the proof of part (c) can be developed by analogy with [2, Theorem 4.2(i)]. Remark 7.5. For A ∈ A, each operator Hη (A) is homogeneous. So the passage from a sequence A ∈ A to its image under the homomorphism Hη can be naturally called homogenization. Corollary 7.6. For every η ∈ R, the algebra A is a closed unital subalgebra of the algebra Hη . 7.4. Necessary Condition for the Stability of A ∈ A Now we are in a position to prove the main result of this section. Theorem 7.7. If a sequence A ∈ A is stable, then the operators Hη (A) are invertible in the algebra B for all η ∈ R. Proof. If A ∈ A is stable, then by Theorem 4.5, the coset A + G is invertible in the quotient algebra E/G. Then from Lemma 7.2(e) it follows that A + G is invertible in the quotient algebra Hη /G for every η ∈ R. This means that there exist B ∈ Hη and G1 , G2 ∈ G such that AB = I + G1 ,

BA = I + G2 .

(7.2)

From Lemma 7.2(b),(d) we know that Hη : Hη → B is a unital homomorphism and Hη (G1 ) = Hη (G2 ) = 0. Applying this homomorphism to the equalities in (7.2), we obtain that Hη (A) is invertible in B and its inverse is equal to Hη (B). Remark 7.8. The proof of Theorem 7.7 given above does not use the results of Sects. 5 and 6. 7.5. Strong Convergence of Families of Sequences Associated to M∞ (SO) The following statement can be considered as a counterpart of Proposition 7.4(c) for the ﬁber M∞ (SO). It will be used in Sect. 10 and can be proved by analogy with [2, Theorem 4.2(ii)] with the aid of Lemma 2.8. Lemma 7.9. Suppose η ∈ M∞ (SO) and g1 , . . . , gm ∈ SOp is a finite family such that η(g1 ) = · · · = η(gm ) = 0. Then there exists a sequence (τn )∞ n=1 ⊂ R+ such that τn → +∞ as n → ∞ and =0 s-lim Zτn W 0 (gj )Zτ−1 n n→∞

for all j = 1, . . . , m.

8. Invertibility in the Local Algebra LJ ξ,η with (ξ, η) ∈ R × M∞ (SO) 8.1. Local Images of Elements of A Lemma 8.1. Suppose A ∈ A and (ξ, η) ∈ R × M∞ (SO). Then the constant sequence A0 := (W0 (A)) belongs to A and J ΦJ ξ,η (A) = Φξ,η (A0 ).

(8.1)

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Proof. This statement is proved by analogy with [20, Proposition 2.14]. If A = (aI) with a ∈ P C or A = (W 0 (b)) with b ∈ [P Cp , SOp ], then obviously A0 = A ∈ A. For P = (Pτ ) ∈ A, we have W0 (P) = I and P0 = (I) =: I ∈ A. These facts imply that A0 ∈ A. For all constant sequences A ∈ A, we have A = A0 ∈ A. This implies J (8.1) for such sequences. It remains to show that ΦJ ξ,η (P) = Φξ,η (I). ˙ such that Assume that y > |ξ| and consider a function fξ ∈ C(R) J it fξ (ξ) = 1 and supp fξ ⊂ (−y, y). From the definition of the ideal Iξ,η follows that J (fξ I) − (I) + J = ((fξ − 1)I) + J ∈ Iξ,η . J J Hence ΦJ ξ,η [(fξ I)] = Φξ,η (I) is the identity in the local algebra Lξ,η . Put Q := (Qτ ). Then J J J ΦJ ξ,η (Q) = Φξ,η [(fξ I)]Φξ,η (Q) = Φξ,η [(fξ Qτ )].

Since supp fξ ⊂ (−y, y), we have fξ Qτ B → 0 as τ → ∞. Then (fξ Qτ ) ∈ J J G ⊂ J . This means that ΦJ ξ,η (Q) is the zero element in Lξ,η . Hence Φξ,η (P) = J Φξ,η (I). 8.2. Sufficient Conditions for the Invertibility in LJ ξ,η Theorem 8.2. Let (ξ, η) ∈ R × M∞ (SO). If A ∈ A and the operator W0 (A) J is invertible, then the coset ΦJ ξ,η (A) is invertible in the local algebra Lξ,η . Proof. If W0 (A) is invertible, then the constant sequence A0 = (W0 (A)) ∈ A is stable. From Theorem 4.5 and Lemma 5.1(e) we obtain that A0 + G is invertible in F/G. Therefore, A0 + J is invertible in the quotient algebra F/J because G ⊂ J . From Lemma 6.2(c) it follows that the latter fact is equivalent to the invertibility of A0 + J in the quotient algebra L/J . By J Theorem 6.1, this implies that ΦJ ξ,η (A0 ) is invertible in the local algebra Lξ,η . J It remains to recall that ΦJ ξ,η (A) = Φξ,η (A0 ) by Lemma 8.1(b).

9. Invertibility in the Local Algebra LJ ∞,η with η ∈ R 9.1. Two Auxiliary Lemmas + Given η ∈ R, let χ− η and χη denote the characteristic functions of (−∞, η) + and (η, +∞), respectively. Clearly, χ− 0 = χ− and χ0 = χ+ . It is easy to see that −1 0 W 0 (χ− η ) = Uη W (χ− )Uη ,

−1 0 W 0 (χ+ η ) = Uη W (χ+ )Uη .

Consider the constant sequences X− := (χ− I),

X+ := (χ+ I),

η W− := (W 0 (χ− η )),

η W+ := (W 0 (χ+ η )).

Lemma 9.1. Let η ∈ R. (a) If a ∈ P C and A = (aI), then J J ΦJ ∞,η (A) = a(−∞)Φ∞,η (X− ) + a(+∞)Φ∞,η (X+ ).

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(b) If b ∈ [P Cp , SOp ] and B = (W 0 (b)), then η η − J + J ΦJ ∞,η (B) = b(η )Φ∞,η (W− ) + b(η )Φ∞,η (W+ ).

Proof. (a) Consider a0 := a − [a(−∞)χ− + a(+∞)χ+ ]. Clearly a0 ∈ P C and J . Hence a0 (−∞) = a0 (+∞) = 0. Therefore (a0 I) + J ∈ I∞,η J J J 0 = ΦJ ∞,η [(a0 I)] = Φ∞,η (A) − [a(−∞)Φ∞,η (X− ) + a(+∞)Φ∞,η (X+ )].

Part (a) is proved. + + (b) The function b0 := b − [b(η − )χ− η + b(η )χη ] belongs to [P Cp , SOp ] and, moreover, it is continuous and vanishing at the point η ∈ R. Hence the J . Therefore coset (W 0 (b0 )) + J belongs to the ideal I∞,η η η 0 J − J + J 0 = ΦJ ∞,η [(W (b0 I))] = Φ∞,η (B) − [b(η )Φ∞,η (W− ) + b(η )Φ∞,η (W+ )].

This completes the proof of part (b). Lemma 9.2. Suppose A ∈ A and η ∈ R. Then the sequence Aη given by Aη := (Uη−1 Zτ Hη (A)Zτ−1 Uη ) belongs to A and J ΦJ ∞,η (A) = Φ∞,η (Aη ).

(9.1)

Proof. From Proposition 7.4(a) it follows that Uη−1 Zτ Hη (P)Zτ−1 Uη = Uη−1 Zτ P1 Zτ−1 Uη = Pτ .

(9.2)

By Proposition 7.4(b), for A = (aI) with a ∈ P C we have Uη−1 Zτ Hη (A)Zτ−1 Uη = Uη−1 Zτ [a(−∞)χ− I + a(+∞)χ+ I]Zτ−1 Uη = a(−∞)Zτ χ− Zτ−1 + a(+∞)Zτ χ+ Zτ−1 = a(−∞)χ− I + a(+∞)χ+ I.

(9.3)

0

Assume that b ∈ [P Cp , SOp ] and A = (W (b)). By Proposition 7.4(c), Uη−1 Zτ Hη (A)Zτ−1 Uη = Uη−1 Zτ [b(η − )W 0 (χ− ) + b(η + )W 0 (χ+ )]Zτ−1 Uη = b(η − )Uη−1 Zτ W 0 (χ− )Zτ−1 Uη +b(η + )Uη−1 Zτ W 0 (χ+ )Zτ−1 Uη = b(η − )Uη−1 W 0 (χ− )Uη + b(η + )Uη−1 W 0 (χ+ )Uη + 0 + = b(η − )W 0 (χ− η ) + b(η )W (χη ).

(9.4)

Equalities (9.2)–(9.4) imply that Aη ∈ A for every A ∈ A. Obviously, it is sufﬁcient to check identity (9.1) on the generators of the algebra A. From (9.2) we get J ΦJ ∞,η (Aη ) = Φ∞,η (A)

for A = P. From Lemma 9.1(a) and equality (9.3) it follows that J J J ΦJ ∞,η (Aη ) = a(−∞)Φ∞,η (X− ) + a(+∞)Φ∞,η (X+ ) = Φ∞,η (A)

for A = (aI) with a ∈ P C. Equality (9.4) and Lemma 9.1(b) yield η η − J + J J ΦJ ∞,η (Aη ) = b(η )Φ∞,η (W− ) + b(η )Φ∞,η (W+ ) = Φ∞,η (A)

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with A = (W 0 (b)) and b ∈ [P Cp , SOp ].

9.2. Sufficient Condition for the Invertibility in LJ ∞,η Theorem 9.3. Let η ∈ R and A ∈ A. If the operator Hη (A) is invertible in J the algebra B, then the coset ΦJ ∞,η (A) is invertible in the local algebra L∞,η . Proof. Let B ∈ B be the inverse of Hη (A). Multiplying the equalities BHη (A) = I = Hη (A)B from the left by Uη−1 Zτ and from the right by Zτ−1 Uη , we obtain B(η) Aη = I = Aη B(η) , where B(η) := (Uη−1 Zτ BZτ−1 Uη ) ∈ E. It is easy to see that B(η) E ≤ B B . Hence all operators Uη−1 Zτ Hη (A)Zτ−1 Uη are invertible and the norms of their inverses are uniformly bounded. Thus the sequence Aη is stable. From Lemma 9.2 and Theorem 6.4 we conclude that Aη ∈ L. Then the stability of Aη implies the invertibility of the coset ΦJ ∞,η (Aη ) in the local algebra J LJ due to Theorem 6.7. It remains to recall that ΦJ ∞,η ∞,η (Aη ) = Φ∞,η (A) by Lemma 9.2.

10. Invertibility in the Local Algebra LJ ∞,η with η ∈ M∞ (SO) 10.1. On the Center of the Algebra AJ ∞,η Let P := (Pτ ) and Q := (Qτ ). From Lemma 2.3 it follows that the functions u± (x) := (1 ± tanh x)/2

(x ∈ R)

belong to Cp (R). Hence the convolution operators W 0 (u± ) are well deﬁned and bounded on Lp (R). Consider the following constant sequences U− := (u− I),

U+ := (u+ I),

V− := (W 0 (u− )),

V+ := (W 0 (u+ )),

X− := (χ− I),

X+ := (χ+ I),

W− := (W 0 (χ− )),

W+ := (W 0 (χ+ )).

Lemma 10.1. Let η ∈ M∞ (SO). (a) We have J ΦJ ∞,η (X− ) = Φ∞,η (U− ),

J ΦJ ∞,η (X+ ) = Φ∞,η (U+ ),

J ΦJ ∞,η (W− ) = Φ∞,η (V− ),

J ΦJ ∞,η (W+ ) = Φ∞,η (V+ ).

(b) If a ∈ P C and A = (aI), then J J ΦJ ∞,η (A) = a(−∞)Φ∞,η (X− ) + a(+∞)Φ∞,η (X+ ).

(c) If b ∈ [P Cp , SOp ] and B = (W 0 (b)), then J J ΦJ ∞,η (B) = bη (−∞)Φ∞,η (W− ) + bη (+∞)Φ∞,η (W+ ).

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Proof. (a) Since χ− (−∞) = u− (−∞) and χ− (+∞) = u− (+∞), the function ˙ Hence the χ− − u− ∈ P Cp is continuous and vanishing at the point ∞ of R. J cosets X− − U− + J and W− − V− + J belong to the ideal I∞,η . Therefore J J J ΦJ ∞,η (X− ) = Φ∞,η (U− ) and Φ∞,η (W− ) = Φ∞,η (V− ). Analogously one can J J J show that ΦJ ∞,η (X+ ) = Φ∞,η (U+ ) and Φ∞,η (W+ ) = Φ∞,η (V+ ). (b) The proof coincides with the proof of Lemma 9.1(a). (c) Consider the function !b := b−bη (−∞)χ− −bη (+∞)χ+ ∈ [P Cp , SOp ]. Then (αη!b)(−∞) = (αη!b)(+∞) = 0. In particular, if b ∈ SOp , then !b = b − J b(η) and !b(η) = 0. This implies that (W 0 (!b)) + J ∈ I∞,η for b ∈ [P Cp , SOp ]. Thus 0 ! J J J 0 = ΦJ ∞,η [(W (b))] = Φ∞,η (B) − bη (−∞)Φ∞,η (X− ) − bη (+∞)Φ∞,η (X+ ),

which ﬁnishes the proof of part (c).

J Lemma 10.2. Let η ∈ M∞ (SO). Then the cosets ΦJ ∞,η (X− ) and Φ∞,η (X+ ) J commute with any element of the local algebra A∞,η . J Proof. We will prove that ΦJ ∞,η (X+ ) commutes with the elements of A∞,η . It is sufﬁcient to prove this statement for the generators of AJ ∞,η . It is obvious that χ+ Pτ = Pτ χ+ I and χ+ a = aχ+ for all a ∈ P C. As usual, denote P = (Pτ ) and A = (aI). Then J J J ΦJ ∞,η (X+ )Φ∞,η (P) = Φ∞,η (P)Φ∞,η (X+ ), J J J ΦJ ∞,η (X+ )Φ∞,η (A) = Φ∞,η (A)Φ∞,η (X+ ).

By Lemma 10.1(a),(c), J ΦJ ∞,η (X+ ) = Φ∞,η (U+ ),

ΦJ ∞,η (B)

=

bη (−∞)ΦJ ∞,η (V− )

(10.1) +

bη (+∞)ΦJ ∞,η (V+ ).

(10.2)

In view of Lemma 2.6, u+ W 0 (u− ) − W 0 (u− )u+ I ∈ K. Hence U+ V− − V− U+ ∈ J and therefore J ΦJ ∞,η (U+ V− ) = Φ∞,η (V− U+ ).

(10.3)

J ΦJ ∞,η (U+ V+ ) = Φ∞,η (V+ U+ ).

(10.4)

Analogously,

Combining (10.1)–(10.4), we get

" # J J J J ΦJ ∞,η (X+ )Φ∞,η (B) = Φ∞,η (U+ ) bη (−∞)Φ∞,η (V− ) + bη (+∞)Φ∞,η (V+ ) J = bη (−∞)ΦJ ∞,η (U+ V− ) + bη (+∞)Φ∞,η (U+ V+ ) J = bη (−∞)ΦJ ∞,η (V− U+ ) + bη (+∞)Φ∞,η (V+ U+ ) # J " J = bη (−∞)ΦJ ∞,η (V− ) + bη (+∞)Φ∞,η (V+ ) Φ∞,η (U+ ) J = ΦJ ∞,η (B)Φ∞,η (X+ ),

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which ﬁnishes the proof of the statement for ΦJ ∞,η (X+ ). This immediately implies that ΦJ (X ) commutes with the elements of AJ − ∞,η ∞,η because J J ΦJ ∞,η (X− ) = Φ∞,η (I) − Φ∞,η (X+ ) J and ΦJ ∞,η (I) is the identity of A∞,η .

10.2. Reduction to Algebras Generated by Two Idempotents An element p of a Banach algebra B is said to be idempotent if p2 = p. If B is a unital Banach algebra with identity e, then e − p is also an idempotent and pBp := {pap : a ∈ B} ,

(e − p)B(e − p) := {(e − p)a(e − p) : a ∈ B}

are unital Banach algebras with the identities p and e − p, respectively. Lemma 10.3. Let B be a Banach algebra with identity e and let p = e be an idempotent element of B. Suppose A is a closed subalgebra of B that contains e and p. If p commutes with any element of A, then (a) an element a ∈ A is invertible in the algebra B if and only if the element pap is invertible in the algebra pBp and the element (e − p)a(e − p) is invertible in the algebra (e − p)B(e − p); (b) the algebra A is inverse closed in the algebra B if and only if the algebra pAp is inverse closed in the algebra pBp and the algebra (e − p)A(e − p) is inverse closed in the algebra (e − p)B(e − p). The proof of this lemma is straightforward and therefore it is omitted. J The above results allow us to split the algebras AJ ∞,η and L∞,η into ± ± − + pairs of simpler algebras Aη and Lη . The algebras Aη and Aη have a nice algebraic structure: they are generated by two idempotents and the identity. Lemma 10.4. Let η ∈ M∞ (SO) and J J J A− η := Φ∞,η (X− )A∞,η Φ∞,η (X− ),

J J J A+ η := Φ∞,η (X+ )A∞,η Φ∞,η (X+ ),

J J J L− η := Φ∞,η (X− )L∞,η Φ∞,η (X− ),

J J J L+ η := Φ∞,η (X+ )L∞,η Φ∞,η (X+ ).

(a) The elements p− := ΦJ ∞,η (PX− ),

r− := ΦJ ∞,η (W− X− )

− are idempotents in the algebra L− η and the algebra Aη is the smallest − closed subalgebra of Lη that contains p− , r− , and the identity

e− := ΦJ ∞,η (X− ). (b) The elements p+ := ΦJ ∞,η (PX+ ),

r+ := ΦJ ∞,η (W+ X+ )

+ are idempotents in the algebra L+ η and the algebra Aη is the smallest + closed subalgebra of Lη that contains p+ , r+ , and the identity

e+ := ΦJ ∞,η (X+ ).

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J Proof. (a) Since the element ΦJ ∞,η (X− ) commutes with any element of A∞,η 2 by Lemma 10.2 and X− = X− , we have J p− = ΦJ ∞,η (PX− ) = Φ∞,η (X− PX− ) J J − = ΦJ ∞,η (X− )Φ∞,η (P)Φ∞,η (X− ) ∈ Aη

(10.5)

and taking into account that P2 = P, we have J J 2 J 2 p2− = ΦJ ∞,η (PX− )Φ∞,η (PX− ) = Φ∞,η (P )Φ∞,η (X− ) J = ΦJ ∞,η (P)Φ∞,η (X− ) = p− . 2 Analogously, taking into account that W− = W− , one can verify that r− − 2 belongs to Aη and r− = r− . It remains to show that A− η is generated by p− , r− , and e− . In view J of (10.5) it is sufﬁcient to show that ΦJ ∞,η (X− AX− ) and Φ∞,η (X− BX− ), 0 where A = (aI) with a ∈ P C and B = (W (b)) with b ∈ [P Cp , SOp ], are represented as linear combinations of p− , r− , and e− . From Lemmas 10.1 and 10.2 it follows that

ΦJ ∞,η (X− AX− ) " # J J J = ΦJ ∞,η (X− ) a(−∞)Φ∞,η (X− ) + a(+∞)Φ∞,η (X+ ) Φ∞,η (X− ) J J = a(−∞)ΦJ ∞,η (X− )Φ∞,η (X− )Φ∞,η (X− ) J J +a(+∞)ΦJ ∞,η (X− )Φ∞,η (X+ )Φ∞,η (X− )

= a(−∞)e−

(10.6)

and ΦJ ∞,η (X− BX− ) " # J J J = ΦJ ∞,η (X− ) bη (−∞)Φ∞,η (W− ) + bη (+∞)Φ∞,η (W+ ) Φ∞,η (X− ) J J = bη (−∞)ΦJ ∞,η (X− )Φ∞,η (W− )Φ∞,η (X− ) J J J +bη (+∞)ΦJ ∞,η (X− )[Φ∞,η (I) − Φ∞,η (W− )]Φ∞,η (X− )

= bη (−∞)r− + bη (+∞)(e− − r− ),

(10.7)

which ﬁnishes the proof of part (a). (b) The proof of part (b) is similar. We only note that ΦJ ∞,η (X+ PX+ ) = p+ ,

(10.8)

ΦJ ∞,η (X+ AX+ ) ΦJ ∞,η (X+ BX+ )

(10.9)

for further references.

= a(+∞)e+ , = bη (−∞)(e+ − r+ ) + bη (+∞)r+

(10.10)

10.3. The Two Idempotents Theorem + The algebras A− η and Aη introduced in the previous subsection are generated by two idempotent elements and the identity. Invertibility of elements of such algebras can be described with the aid of the so-called two idempotents theorem presented below.

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Theorem 10.5 ([4, Theorem 8.7]). Let B be a Banach algebra with identity e and let p and r be idempotents in B. Let further A stand for the smallest closed subalgebra of B containing e, p, and r. Put X := prp + (e − p)(e − r)(e − p) and suppose the points 0 and 1 are cluster points of spB (X). For define the map σx : {e, p, r} → C2×2 by $ % $ % 1 0 1 0 σx (e) = , σx (p) = , 0 1 0 0 $ % x x(1 − x) σx (r) = , x(1 − x) 1−x where x(1 − x) denotes any complex number such that its square to x(1 − x).

(10.11) x ∈ C,

(10.12) (10.13) is equal

(a) For each x ∈ spB (X) the map σx extends to a Banach algebra homomorphism σx of A onto C2×2 . (b) An element a ∈ A is invertible in B if and only if σx (a) is invertible in C2×2 for every x ∈ spB (X). (c) An element a ∈ A is invertible in A if and only if σx (a) is invertible in C2×2 for every x ∈ spA (X). 10.4. Spectra of Two Elements of the Algebra LJ ∞,η ± To apply the two idempotents theorem to the algebras A± η and Lη , we need to ﬁnd the spectrum of the canonical element given by (10.11) in these algebras. This requires the following auxiliary result.

Lemma 10.6. Let η ∈ M∞ (SO) and Lp be the lentiform domain defined by (3.2). Then spLJ∞,η ΦJ (10.14) ∞,η (PX− W− X− P + QX− W+ X− Q) = Lp , J spLJ∞,η Φ∞,η (PX+ W+ X+ P + QX+ W− X+ Q) = Lp . (10.15) Proof. Let us prove equality (10.14). It is easy to see that PX− W− X− P + QX− W+ X− Q − λI ∈ H ∩ L, where the algebra H is deﬁned in Sect. 7.1. By Lemma 7.1, the coset PX− W− X− P + QX− W+ X− Q − λI + G is invertible in the algebra L/G if and only if the operator P1 χ− W 0 (χ− )χ− P1 + Q1 χ− W 0 (χ+ )χ− Q1 − λI = χ(−1,0) PR χ(−1,0) I + χ(−∞,−1) QR χ(−∞,−1) I − λI is invertible on Lp (R). Hence, taking into account Corollary 3.3(a), we obtain spL/G (PX− W− X− P + QX− W+ X− Q + G) = spB χ(−1,0) PR χ(−1,0) I + χ(−∞,−1) QR χ(−∞,−1) I = Lp .

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If λ ∈ / Lp , then there exists a sequence A ∈ L such that (PX− W− X− P + QX− W+ X− Q − λI + G)(A + G) = I + G, (A + G)(PX− W− X− P + QX− W+ X− Q − λI + G) = I + G. Obviously, the same equalities are true with G replaced by the larger J ideal J . Applying the homomorphism ΦJ → LJ ∞,η : L ∞,η to those equalities, we obtain J J ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q − λI)Φ∞,η (A) = Φ∞,η (I), J J ΦJ ∞,η (A)Φ∞,η (PX− W− X− P + QX− W+ X− Q − λI) = Φ∞,η (I). Then λ ∈ / spLJ∞,η ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q) . Thus

spLJ∞,η ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q) ⊂ Lp .

(10.16)

Assume now that λ∈ / spLJ∞,η ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q) . Then there exists a sequence B ∈ L such that J B(PX− W− X− P + QX− W+ X− Q − λI) − I + J ∈ I∞,η , J , (PX− W− X− P + QX− W+ X− Q − λI)B − I + J ∈ I∞,η J is deﬁned in Sect. 6.5. From that definition and the above relawhere I∞,η tions it follows that without loss of generality we may assume that there exist ﬁnite sums n

Ci (fi I) +

i=1 k

m

J Dj (W 0 (gj )) + J ∈ I∞,η ,

j=1

(hi I)Ei +

i=1

l

J (W 0 (vj ))Fj + J ∈ I∞,η

j=1

with Ci , Dj , Ei , Fj ∈ L and ˙ fi , hi ∈ C(R),

fi (∞) = hi (∞) = 0,

gj , vj ∈ SOp ,

gj (η) = vj (η) = 0

for all i, j and elements J1 , J2 ∈ J such that B(PX− W− X− P + QX− W+ X− Q − λI) − I n m = J1 + Ci (fi I) + Dj (W 0 (gj )), i=1

j=1

(PX− W− X− P + QX− W+ X− Q − λI)B − I = J2 +

k i=1

(hi I)Ei +

l j=1

(W 0 (vj ))Fj .

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Applying the homomorphism W−1 to these equalities, taking into account Proposition 5.2 and that W−1 (J1 ) = K1 ∈ K, W−1 (J2 ) = K2 ∈ K, we obtain W−1 (B)Hλ − I = K1 +

m

W−1 (Dj )W 0 (gj ),

(10.17)

W 0 (vj )W−1 (Fj ),

(10.18)

j=1

Hλ W−1 (B) − I = K2 +

l j=1

where Hλ := χ+ W 0 (χ− )χ+ I +χ− W 0 (χ+ )χ− I −λI = χ+ PR χ+ I +χ− QR χ− I − λI. By Lemma 7.9, for g1 , . . . , gm there exists a sequence (τn )∞ n=1 ⊂ R+ such that τn → +∞ and s-lim Zτn W 0 (gj )Zτ−1 = 0 on Lp (R) n n→∞

(10.19)

for all j = 1, . . . , m. From Lemma 2.4 it follows that v1 , . . . , vl ∈ SOq . So, applying Lemma 7.9 to the space Lq (R), we can guarantee that there exists ∗ a sequence (τn∗ )∞ n=1 ⊂ R+ such that τn → +∞ and s-lim Zτn∗ W 0 (vj )Zτ−1 on ∗ = 0 n n→∞

Lq (R)

(10.20)

for all j = 1, . . . , l. Since K1 , K2 are compact and (Zτ±1 )∞ converge weakly to zero on n n=1 ∞ converge weakly to zero on Lq (R) as n → ∞, we conLp (R) and (Zτ±1 ∗ ) n n=1 clude that s-lim Zτn K1 Zτ−1 = 0 on Lp (R), n

(10.21)

s-lim Zτn∗ K2∗ Zτ−1 on Lq (R). ∗ = 0 n

(10.22)

n→∞ n→∞

and Multiplying (10.17) from the left by Zτn and from the right by Zτ−1 n taking into account that the operator Hλ is homogeneous, we obtain (Zτn W−1 (B)Zτ−1 )Hλ − I n m + (Zτn W−1 (Dj )Zτ−1 )(Zτn W 0 (gj )Zτ−1 ) =: An . (10.23) = Zτn K1 Zτ−1 n n n j=1

B

Zτn W−1 (Dj )Zτ−1 n

≤ W−1 (Dj ) B , from (10.19) and (10.21) we conSince clude that the sequence (An )∞ n=1 converges strongly to zero as n → ∞. For u ∈ Lp (R) from (10.23) it follows that

B Hλ u p + An u p

u p ≤ Zτn W−1 (B)Zτ−1 n ≤ W−1 (B) B Hλ u p + An u p . Passing to the limit as n → ∞, we obtain

u p ≤ W−1 (B) B Hλ u p for all u ∈ Lp (R). This means that the kernel of Hλ is trivial and the range of Hλ is closed.

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Analogously, from (10.18) it follows that ∗ (Zτn∗ W−1 (B)∗ Zτ−1 ∗ )Hλ − I n

= Zτn∗ K2∗ Zτ−1 ∗ + n

l j=1

−1 0 (Zτn∗ W−1 (Fj )∗ Zτ−1 ∗ )(Zτ ∗ W (vj )Zτ ∗ ) =: Bn . n n n

As above, taking into account (10.20) and (10.22), we see that (Bn )∞ n=1 converges strongly to zero on Lq (R) and therefore

w q ≤ W−1 (B) B Hλ∗ w q for every w ∈ Lq (R). Hence the kernel of Hλ∗ is trivial, too. Thus Hλ is / Lp . Thus invertible on Lp (R). From Corollary 3.3(c) we conclude that λ ∈ J Lp ⊂ spLJ∞,η Φ∞,η (PX− W− X− P + QX− W+ X− Q) . (10.24) Combining (10.16) and (10.24), we arrive at (10.14). To prove equality (10.15), we argue similarly, applying Corollary 3.3(b) instead of Corollary 3.3(a) and the homomorphism W1 instead of W−1 . ± 10.5. Spectra of the Canonical Elements in the Algebras A± η and Lη Now we are ready to calculate the spectra of the canonical elements required in the two idempotents theorem.

Lemma 10.7. Let η ∈ M∞ (SO) and Lp be the lentiform domain defined by (3.2). (a) If X− := p− r− p− + (e− − p− )(e− − r− )(e− − p− ), then spA− (X− ) = spL− (X− ) = Lp . η η (b) If X+ := p+ r+ p+ + (e+ − p+ )(e+ − r+ )(e+ − p+ ), then spA+ (X+ ) = spL+ (X+ ) = Lp . η η Proof. (a) Let λ ∈ C. From Lemmas 10.3(a) and 10.2 it follows that the J element ΦJ ∞,η (PX− W− X− P + QX− W+ X− Q − λI) is invertible in L∞,η if and only if J J ΦJ ∞,η (X− )Φ∞,η (PX− W− X− P+QX− W+ X− Q − λI)Φ∞,η (X− ) = X− − λe−

is invertible in L− η and J J ΦJ ∞,η (X+ )Φ∞,η (PX− W− X− P + QX− W+ X− Q − λI)Φ∞,η (X+ ) = −λe+

is invertible in L+ η . Therefore J (X− ) ∪ {0}. spLJ∞,η Φ∞,η (PX− W− X− P + QX− W+ X− Q) = spL− η In view of (10.14), the above equality means that spL− (X− ) ∪ {0} = Lp . But η 0 is not an isolated point of Lp . Due to the compactness of the spectrum (X− ) we conclude that 0 ∈ spL− (X− ). Hence spL− η η (X− ) = Lp . spL− η

(10.25)

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− Recall that the identity element in the unital algebras A− η ⊂ Lη is the same. Since the lentiform domain Lp does not separate the complex plane, that is, Lp and C\Lp are connected sets, from [24, Corollary of Theorem 10.18] (X− ) = Lp . Part (a) is proved. and equality (10.25) it follows that spA− η The proof of part (b) is analogous.

10.6. Invertibility in the Local Algebra LJ ∞,η Now we are in a position to describe the local algebras AJ ∞,η with the aid of the two idempotents theorem. Theorem 10.8. Let η ∈ M∞ (SO) and Lp be the lentiform domain defined by (3.2). Suppose x ∈ Lp . (a) The maps

J

J

+ Σ− x , Σx : Φ∞,η (P) ∪ Φ∞,η [(aI)] : a ∈ P C " 0 #

2×2 ∪ ΦJ ∞,η (W (b)) : b ∈ [P Cp , SOp ] → C

given by J Σ− x (Φ∞,η (P))

$ :=

1 0

% $ 0 a(−∞) − J , Σx (Φ∞,η [(aI)]) := 0 0

% 0 , a(−∞) (10.26)

J 0 Σ− x (Φ∞,η [(W (b)])

$ bη (−∞)x + bη (+∞)(1 − x) := [bη (−∞) − bη (+∞)] x(1 − x)

% [bη (−∞) − bη (+∞)] x(1 − x) bη (−∞)(1 − x) + bη (+∞)x (10.27)

and J Σ+ x (Φ∞,η (P))

$ :=

1 0

% $ 0 a(+∞) + J , Σx (Φ∞,η [(aI)]) := 0 0

% 0 , a(+∞) (10.28)

J 0 Σ+ x (Φ∞,η [(W (b)])

$ bη (+∞)x + bη (−∞)(1 − x) := [bη (+∞) − bη (−∞)] x(1 − x)

% [bη (+∞) − bη (−∞)] x(1 − x) , bη (+∞)(1 − x) + bη (−∞)x

(10.29) where x(1 − x) denotes any complex number such that its square is equal to x(1 − x), extend to Banach algebra homomorphisms J 2×2 Σ− , x : A∞,η → C

J 2×2 Σ+ . x : A∞,η → C

J (b) Let A ∈ A. The coset ΦJ ∞,η (A) is invertible in the local algebra L∞,η if and only if J det Σ− x (Φ∞,η (A)) = 0,

J det Σ+ x (Φ∞,η (A)) = 0

for all x ∈ Lp . J (c) The algebra AJ ∞,η is inverse closed in the algebra L∞,η .

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+ Proof. From Lemma 10.4 we obtain that the algebras A− η and Aη are subject to the two idempotents theorem. Let x ∈ C. Deﬁne the maps

σx± : {e± , p± , r± } → C2×2 by equalities (10.12)–(10.13) with e± , p± , r± and σx± in place of e, p, r, and σx , respectively. By Lemma 10.7, the spectra spA− (X− ), η

spL− (X− ), η

spA+ (X+ ), η

spL+ (X+ ) η

all coincide and are equal to the lentiform domain Lp given by (3.2). Then, by Theorem 10.5(a), for x ∈ Lp the maps σx− and σx+ extend to Banach algebra homomorphisms 2×2 σx− : A− , η →C

2×2 σx+ : A+ . η →C

Let A ∈ A. From Theorem 10.5(b), (c) we know that ΦJ ∞,η (X− AX− ) (respecJ − tively, Φ∞,η (X+ AX+ )) is invertible in both algebras A− η and Lη (respec+ + − J tively, in both Aη and Lη ) if and only if the element σx [Φ∞,η (X− AX− )] 2×2 (respectively, σx+ [ΦJ . In particular, the ∞,η (X+ AX+ )]) is invertible in C − is inverse closed in the algebra L and the algebra A+ algebra A− η η η is inverse + closed in the algebra Lη . Further, for x ∈ Lp the maps J 2×2 , Σ− x : A∞,η → C

J − J Σ− x [Φ∞,η (A)] = σx [Φ∞,η (X− AX− )],

J 2×2 , Σ+ x : A∞,η → C

J + J Σ+ x [Φ∞,η (A)] = σx [Φ∞,η (X+ AX+ )]

are Banach algebra homomorphisms. From (10.12)–(10.13) and (10.5)–(10.10) + J it follows that Σ− x and Σx are deﬁned on the generators of A∞,η by formulas (10.26)–(10.29). This ﬁnishes the proof of parts (a) and (b). − + Taking into account that A− η is inverse closed in Lη and Aη is inverse + J J closed in Lη , we obtain that A∞,η is inverse closed in L∞,η in view of Lemma 10.3(b). Part (c) is proved. Let R : C\ ((−∞, 0) ∪ (1, +∞)) → C be a continuous branch of the function x → x(1 − x). Since Lp ∩ ((−∞, 0) ∪ (1, +∞)) = ∅, the function R is continuous on Lp . Corollary 10.9. Let η ∈ M∞ (SO) and Lp be the lentiform domain defined by (3.2). The maps 0

+ N− η , Nη : {P} ∪ {(aI) : a ∈ P C} ∪ (W (b)) : b ∈ [P Cp , SOp ] → C(Lp , C2×2 )

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given for x ∈ Lp by

N− η (P)

$

1 (x) := 0

% $ − 0 a(−∞) , Nη [(aI)] (x) := 0 0

595

% 0 , a(−∞) (10.30)

0 N− η [(W (b)]

(x) $ b (−∞)x + bη (+∞)(1 − x) := η [bη (−∞) − bη (+∞)]R(x)

[bη (−∞) − bη (+∞)]R(x) bη (−∞)(1 − x) + bη (+∞)x

% (10.31)

and + Nη (P) (x) :=

$

1 0

% $ + 0 a(+∞) , Nη [(aI)] (x) := 0 0

% 0 , a(+∞) (10.32)

0 N+ η [(W (b)]

(x) $ b (+∞)x + bη (−∞)(1 − x) := η [bη (+∞) − bη (−∞)]R(x)

% [bη (+∞) − bη (−∞)]R(x) , bη (+∞)(1 − x) + bη (−∞)x (10.33)

extend to Banach algebra homomorphisms 2×2 N− ), η : A → C(Lp , C

2×2 N+ ). η : A → C(Lp , C

+ Proof. It is clear that N− η (resp. Nη ) is the composition of the canonical J Banach algebra homomorphism Φ∞,η |A : A → AJ ∞,η and the Banach algebra − + homomorphism Σx (resp. Σx ) deﬁned in Theorem 10.8(a). Note that the J function R was chosen so that Σ± x (Φ∞,η (A)) are continuous in x ∈ Lp for A ∈ A.

11. Main Result and Some Corollaries 11.1. Main Result Combining the results of Sects. 6–10, we arrive at the main result of the paper. Theorem 11.1. Let Lp be the lentiform domain defined by (3.2) and A be the algebra from Definition 4.3. A sequence A ∈ A is stable if and only if the following three conditions are fulfilled: (a) the operators W−1 (A), W0 (A), and W1 (A) are invertible in B; (b) the operators Hη (A) are invertible in B for all η ∈ R; (c) for every η ∈ M∞ (SO) and every x ∈ Lp , det N+ det N− η (A) (x) = 0, η (A) (x) = 0, + where the 2 × 2 matrix functions N− η (A) and Nη (A) are images of the homomorphisms defined in Corollary 10.9.

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Proof. Necessity. If A ∈ A is stable, then condition (b) holds due to Theorem 7.7. Further, in view of Theorem 6.7, condition (a) is fulﬁlled and the J coset ΦJ ξ,η (A) is invertible in the local algebra Lξ,η for every pair (ξ, η) ∈ Ω, where Ω is given by (2.8). If (ξ, η) ∈ {∞} × M∞ (SO), then applying Theorem 10.8(b) we obtain for all x ∈ Lp , J det Σ− x (Φ∞,η (A)) = 0,

J det Σ+ x (Φ∞,η (A)) = 0.

− J + + J Since (N− η (A))(x) = Σx (Φ∞,η (A)) and (Nη (A))(x) = Σx (Φ∞,η (A)), this immediately implies condition (c). The necessity portion is proved. Sufficiency. If condition (a) is fulﬁlled, then from Theorem 8.2 it follows J that ΦJ ξ,η (A) is invertible in the local algebra Lξ,η with (ξ, η) ∈ R×M∞ (SO). From condition (b) and Theorem 9.3 we infer that the coset ΦJ ∞,η (A) is invertible in the local algebra LJ with η ∈ R. From condition (c) in view ∞,η J of Theorem 10.8(b) and Corollary 10.9 it follows that the coset Φ∞,η (A) is invertible in the local algebra LJ ∞,η with η ∈ M∞ (SO). Thus the coset ΦJ (A) is invertible in the local algebra LJ ξ,η ξ,η for every (ξ, η) ∈ Ω, where Ω is given by (2.8). Combining this fact with condition (a) we see that A is stable by Theorem 6.7.

11.2. Applicability of the Finite Section Method It is possible now to specify Theorem 11.1 to the ﬁnite section method applied to an operator acting on Lp (R). The theorem below generalizes results going back to B¨ ottcher [3] (see also [6, Sections 9.38–9.49]), who observed for the ﬁrst time the importance of the lentiform domain Lp in the theory of the ﬁnite section method for convolution type operators. Theorem 11.2. Suppose Lp is a lentiform domain defined by (3.2). Let A be an operator in the smallest closed subalgebra of the algebra B that contains the operators aI with a ∈ P C and W 0 (b) with b ∈ [P Cp , SOp ]. Then the finite section method (1.2) applies to the operator A if and only if the following three conditions hold with A = (A): (a) the operators χ+ W−1 (A)χ+ I + χ− I,

χ− W1 (A)χ− I + χ+ I,

and A are invertible in B; (b) the operators P1 Hη (A)P1 + Q1 are invertible in B for all η ∈ R; (c) for every η ∈ M∞ (SO) and every x ∈ Lp , [N− η (A)]11 (x) = 0,

[N+ η (A)]11 (x) = 0,

where [f ]11 denotes the (1, 1)-entry of a 2 × 2 matrix function f . Proof. From Proposition 5.2(a) it follows that W−1 (PAP + Q) = χ+ W−1 (A)χ+ I + χ− I, W0 (PAP + Q) = A, W1 (PAP + Q) = χ− W1 (A)χ− I + χ+ I. Hence, condition (a) of Theorem 11.1 for the sequence PAP + Q is just condition (a) of the present theorem.

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From Proposition 7.4(a) we obtain for every η ∈ R, Hη (PAP + Q) = P1 Hη (A)P1 + Q1 . This means that condition (b) of Theorem 11.1 speciﬁes to condition (b) of the present theorem for the sequence PAP + Q. Finally from (10.30) and (10.32) we get for every η ∈ M∞ (SO) and x ∈ Lp , $ % $ % $ % − 1 0 1 0 0 0 − det Nη (PAP + Q) (x) = det Nη (A)(x) + 0 0 0 0 0 1 $ − % [Nη (A)]11 (x) 0 = det = [N− η (A)]11 (x) (11.1) 0 1 and analogously

+ det N+ η (PAP + Q) (x) = [Nη (A)]11 (x).

(11.2)

Equalities (11.1) and (11.2) imply that condition (c) of Theorem 11.1 for the sequence PAP + Q coincides with condition (c) of the present theorem. 11.3. Applicability of the Finite Section Method to Convolution Operators To illustrate Theorem 11.2, we apply it to the so-called paired convolution operator A = W 0 (a)χ− I + W 0 (b)χ+ I

(a, b ∈ [P Cp , SOp ]).

(11.3)

For g ∈ Mp , the Wiener–Hopf operator on Lp (R+ ) with symbol g is the operator W (g) deﬁned by W (g) : Lp (R+ ) → Lp (R+ ),

ϕ → χ+ W 0 (g)χ+ ϕ.

Let J denote the ﬂip operator on the space Lp (R) deﬁned by (Jϕ)(x) = ϕ(−x). For g ∈ Mp , put g!(x) := g(−x). Corollary 11.3. The finite section method (1.2) applies to the paired convolution operator given by (11.3) if and only if the following three conditions are fulfilled: (a) the operator (11.3) is invertible on the space Lp (R) and the Wiener–Hopf operators W (a) and W (!b) are invertible on the space Lp (R+ ); (b) for every η ∈ R, a(η − ) b(η − ) ± ± , a(η ) = 0, b(η ) = 0, wind Cp 1, = 0, a(η + ) b(η + ) where wind Cp (·, ·, ·) is defined in Sect. 3.4; (c) for every η ∈ M∞ (SO) and every x ∈ Lp , aη (−∞)x + aη (+∞)(1 − x) = 0,

bη (+∞)x + bη (−∞)(1 − x) = 0.

Proof. Let A = (W 0 (a)χ− I + W 0 (b)χ+ I). Taking into account Proposition 5.2(b)–(c) we see that the operator χ+ W−1 (A)χ+ I + χ− I = χ+ W 0 (a)χ+ I + χ− I

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is invertible on Lp (R) if and only if the Wiener–Hopf operator W (a) is invertible on Lp (R+ ) and that the operator χ− W1 (A)χ− I + χ+ I = χ− W 0 (b)χ− I + χ+ I is invertible on Lp (R) if and only if the operator Jχ− W 0 (b)χ− J = χ+ JW 0 (b)Jχ+ I = W (!b) is invertible on Lp (R+ ). This implies that condition (a) of Theorem 11.2 for the constant sequence A is nothing but condition (a) of the present corollary. From Proposition 7.4(b)–(c) it follows that for every η ∈ R, " P1 Hη (A)P1 + Q1 = P1 (a(η − )W 0 (χ− ) + a(η + )W 0 (χ+ ))χ− I # + (b(η − )W 0 (χ− ) + b(η + )W 0 (χ+ ))χ+ I P1 + Q1 = χ(−1,1) a(η − )PR + a(η + )QR χ(−1,0) I + χ(−1,1) b(η − )PR + b(η + )QR χ(0,1) I + χR\(−1,1) I. Obviously this operator is invertible on Lp (R) if and only if the operator P(−1,1) (a(η − )χ(−1,0) + b(η − )χ(0,1) )I + Q(−1,1) (a(η + )χ(−1,0) + b(η + )χ(0,1) )I is invertible on Lp (−1, 1). By Lemma 3.4, this operator is invertible on Lp (−1, 1) if and only if condition (b) is fulﬁlled. This means that for A condition (b) of Theorem 11.2 is equivalent to condition (b) of the present corollary. From formulas (10.30)–(10.33) it follows that [N− η (A)]11 (x) = aη (−∞)x + aη (+∞)(1 − x), [N+ η (A)]11 (x) = bη (+∞)x + bη (−∞)(1 − x) for x ∈ Lp and η ∈ M∞ (SO). This means that condition (c) of Theorem 11.2 speciﬁes to condition (c) of the present corollary.

References [1] Bastos, M.A., Bravo, A., Karlovich, Y. I.: Convolution type operators with symbols generated by slowly oscillating and piecewise continuous matrix functions. In: “Operator Theoretical Methods and Applications to Mathematical Physics. The Erhard Meister Memorial Volume”. Operator Theory: Advances and Applications, vol. 147, pp. 151–174 (2004) [2] Bastos, M.A., Bravo, A., Karlovich, Y.I.: Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data. Math. Nachr. 269–270, 11–38 (2004) [3] B¨ ottcher, A.: The ﬁnite section method for the Wiener-Hopf integral operator. PhD thesis, Rostov-on-Don State University (1984, in Russian) [4] B¨ ottcher, A., Karlovich, Y.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Birkh¨ auser, Basel (1987) [5] B¨ ottcher, A., Karlovich, Y.I., Spitkovsky, I.M.: Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Applications, vol. 131. Birkh¨ auser Verlag, Basel (2002)

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[6] B¨ ottcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006) [7] Duduchava, R.: Integral Equations with Fixed Singularities. Teubner, Leipzig (1979) [8] Gohberg, I., Feldman, I.A.: Convolution Equations and Projection Methods for Their Solution. AMS, Providence (1974) [9] Gohberg, I., Krupnik, N.: One-Dimensional Linear Singular Integral Equations, Vols. 1 and 2. Operator Theory: Advances and Applications, Vols. 53–54. Birkh¨ auser, Basel (1992) [10] Grafakos, L.: Classical Fourier Analysis. Springer, New York (2008) [11] Hagen, R., Roch, S., Silbermann, B.: Spectral Theory of Approximation Methods for Convolution Equations. Operator Theory: Advances and Applications, vol. 74. Birkh¨ auser, Basel (1995) [12] Hagen, R., Roch, S., Silbermann, B.: C ∗ -Algebras and Numerical Analysis. Monographs and Textbooks in Pure and Applied Mathematics, vol. 236. Marcel Dekker, Inc., New York (2001) [13] Kozak, A.V.: A local principle in the theory of projection methods. Sov. Math. Dokl. 14, 1580–1583 (1973) [14] Krasnosel’skii, M.A.: On a theorem of M. Riesz. Sov. Math. Dokl. 1, 229– 231 (1960) [15] Krasnosel’skii, M.A., Zabreiko, P.P., Pustyl’nik, E.I., Sobolevskii, P.E.: Integral Operators in Spaces of Summable Functions. Noordhoﬀ International Publishing, Leyden (1976) [16] Mikhlin, S.G., Pr¨ ossdorf, S.: Singular Integral Operators. Springer, Berlin (1986) [17] Power, S.C.: Fredholm Toeplitz operators and slow oscillation. Can. J. Math. 32, 1058–1071 (1980) [18] Pr¨ ossdorf, S., Silbermann, B.: Numerical Analysis for Integral and Related Operator Equations. Operator Theory: Advances and Applications, vol. 52. Birkh¨ auser, Basel (1991) [19] Roch, S.: Finite sections of operators generated by convolutions. In: Seminar Analysis (Berlin, 1987/1988), pp. 118–138. Akademie-Verlag, Berlin (1988) [20] Roch, S., Santos, P.A., Silbermann, B.: Finite section method in some algebras of multiplication and convolution operators and a ﬂip. Z. Anal. Anwendungen 16, 575–606 (1997) [21] Roch, S., Santos, P.A., Silbermann, B.: Non-commutative Gelfand Theories. A Tool-Kit for Operator Theorists and Numerical Analysts. Springer, Berlin (2010, to appear) [22] Roch, S., Santos, P.A., Silbermann, B.: A sequence algebra of ﬁnite sections, convolution and multiplication operators on Lp (R). Numer. Funct. Anal. Optim. 31, 45–77 (2010) [23] Roch, S., Silbermann, B.: Algebras of convolution operators and their image in the Calkin algebra. Report MATH, 90-05. Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut f¨ ur Mathematik, Berlin (1990) [24] Rudin, W.: Functional Analysis. 2nd edn. McGraw-Hill, New York (1991) [25] Silbermann, B.: Lokale Theorie des Reduktionsverfahrens f¨ ur Toeplitzoperatoren. Math. Nachr. 104, 137–146 (1981)

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[26] Simonenko, I.B., Min, C.N.: Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewsie Continuous Coefﬁcients. Noetherity. University Press, Rostov on Don (1986, in Russian) Alexei Yu. Karlovich Departamento de Matem´ atica Faculdade de Ciˆencias e Tecnologia Universidade Nova de Lisboa Quinta da Torre 2829-516 Caparica Portugal e-mail: [email protected] Helena Mascarenhas Departamento de Mathem´ atica Instituto Superior T´ecnico Universidade T´ecnica de Lisboa Av. Rovisco Pais 1049-001 Lisbon Portugal e-mail: [email protected] Pedro A. Santos Departamento de Mathem´ atica Instituto Superior T´ecnico Universidade T´ecnica de Lisboa Av. Rovisco Pais 1049-001 Lisbon Portugal e-mail: [email protected] Received: September 21, 2009. Revised: February 12, 2010.

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