Integral Equations and Operator Theory - Volume 63

Integr. equ. oper. theory 63 (2009), 1–16 c 2008 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-16, published o...

Author: I. Gohberg (Chief Editor)

28 downloads 863 Views 9MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Integr. equ. oper. theory 63 (2009), 1–16 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010001-16, published online November 10, 2008 DOI 10.1007/s00020-008-1633-2

Integral Equations and Operator Theory

Conorm and Essential Conorm in C*-Algebras Mar´ıa Burgos and Antonio M. Peralta Abstract. Let A be a unital C*-algebra with non-zero socle (soc(A)). We introduce the essential conorm of an element a in A (denoted by γe (a)), as the conorm of the element π(a), where π denotes the canonical projection of A onto A/soc(A). It is n established that for every o von Neumann regular element a ∈ A, γe (a) = max γ(a + k) : k ∈ soc(A) . We characterize the continuity

points of the conorm and essential conorm for extremally rich C*-algebras. Some formulae for the distance from zero to the generalized spectrum and Atkinson spectrum are also obtained. Mathematics Subject Classification (2000). Primary 47B49; Secondary 47A10. Keywords. C*-algebra, conorm, essential conorm, extremally rich, socle, generalized spectrum, Atkinson and Fredholm elements.

1. Introduction Let L(X) denote the Banach algebra of all bounded linear operators T on a Banach space X. The reduced minimum modulus of T is defined by γ(T ) = inf{kT xk : dist(x, Ker(T )) = 1}. It is well known that γ(T ) > 0 if and only if T has closed range. The reduced minimum modulus of an operator plays a very important role in the study of spectral properties of operators and generalized inverses (see [1], [16], [3], [4] and [10]). Let A be a unital (complex) Banach algebra. An element a in A is called (von Neumann) regular if it has a generalized inverse, that is, if there exists b in A such that a = aba and b = bab. Observe that the first equality a = aba is a necessary and sufficient condition for a to be regular, and that, if a has generalized inverse b, then p = ab and q = ba are idempotents in A satisfying aA = pA and Aa = Aq. Authors partially supported by I+D MEC projects no. MTM2005-02541 and MTM2007-65959, and Junta de Andaluc´ıa grants FQM0199 and FQM1215.

2

Burgos and Peralta

IEOT

The generalized inverse of a regular element a is not unique. For an element a in A let us consider the left and right multiplication operators La : x 7→ ax and Ra : x 7→ xa, respectively. If a is regular, then so are La and Ra , and thus their ranges aA = La (A) and Aa = Ra (A) are both closed. In [12], the conorm, γ(a), of an element a ∈ A is defined as the reduced minimum modulus of the left multiplication operator by a, γ(a) = γ(La ). If b is a generalized inverse of a, with a 6= 0, it follows by [12, Theorem 2] that kbk−1 ≤ γ(a) ≤ kbak kabk kbk−1 .

(1.1)

Regular elements in unital C*-algebras have been studied by Harte and Mbekhta in [11] and [12]. The main result in [11] states that an element a in a C*-algebra A is regular if and only if aA is closed, or equivalently γ(a) > 0. Throughout the paper, A will denote a C*-algebra. The symbol Ar will stand for the set of all regular elements in A. Given a and b in A, we shall say that b is a Moore-Penrose inverse of a if b is a generalized inverse of a and the associated idempotents ab and ba are self-adjoint. It is known that every regular element a in A has a unique Moore-Penrose inverse that will be denoted by a† . It was established by Harte and Mbekhta (see [12, Theorem 2]) that for each a ∈ Ar we have γ(a) = ka† k−1 . (1.2) Theorem 4 in [12] also gives γ(a) = inf {σ(|a|) \ {0}} ,

(1.3)

where, for each x ∈ A, σ(x) stands for the spectrum of the element x and |x| = (x∗ x)1/2 . Let H be a Hilbert space. Denote by K(H) the ideal of compact operators and by C(H) = L(H)/K(H) the Calkin algebra. Let π : L(H) → C(H) denote the quotient homomorphism. In [19], the essential conorm, γe (T ), of a bounded linear operator T on H is defined as the conorm of the element π(T ) in the Calkin algebra. One of the main results of [19] states that γe (T ) = max{γ(T + K) : K ∈ K(H)}. More recently, Xue shows in [26] that the conorm of a non zero element a in a unital C*-algebra A is given by γ(a) = inf{ka − bk : Ker(La )

Ker(Lb )}.

Under the additional hypothesis of A being of real rank zero (i.e. the set of all real linear combinations of orthogonal projections is dense in the set of all hermitian elements of A, [6]), then the equality γA/I (a + I) = sup{γ(a + k) : k ∈ I}, holds for every closed ideal I of A. Besides, the supremum is attained whenever I is a σ-unital essential ideal.

Vol. 63 (2009)

Conorm and Essential Conorm

3

We will focus our attention in the particular case of I being the ideal of compact elements in a unital C*-algebra A. We recall that an element x of A is said to be finite (respectively, compact) in A, if the wedge operator x ∧ x : A → A, given by x∧x(a) = xax, is a finite rank (respectively, compact) operator on A. It is known that the ideal F(A) of finite rank elements in A coincides with soc(A), the socle of A, that is, the sum of all minimal right (equivalently left) ideals of A, and that K(A) = soc(A) is the ideal of compact elements in A (see [5, Section C*.1]). Moreover, if H is a Hilbert space, then F(L(H)) = F(H) and K(L(H)) = K(H) are the ideals of finite rank and compact operators on H, respectively. By analogy we call C(A) = A/K(A) the generalized Calkin algebra of A. An element a in A is called Fredholm if it is invertible modulo soc(A). We say that a in A is Atkinson if it is left or right invertible modulo soc(A). Denote by Φ(A) and A(A) the set of Fredholm and Atkinson elements in A, respectively. Another important achievement in [19] is that, a bounded linear operator T on a Hilbert space H is a continuity point of the essential conorm if and only if γe (T ) = 0 or T is a semi-Fredholm operator. Recall that an operator T ∈ L(H) is Fredholm if it has closed range and its kernel and cokernel are both finitedimensional; T is semi-Fredholm if it has closed range and either its kernel or its cokernel is finite-dimensional. Moreover, Φ(L(H)) = Φ(H) is the set of Fredholm operators on H and that A(L(H)) = A(H) is the set of semi-Fredholm operators on H. The set σSF (T ) = {λ ∈ C : T − λ is not semi-Fredholm} is called the semi-Fredholm spectrum of T . Recall from [1] (see also [16]) that the generalized spectrum σg (T ) of T , σg (T ) = λ ∈ C : lim γ(a − µ) = 0 , µ→λ

is the complement of the minimal open set where T − λ has an analytic generalized inverse. In [17] Mbekhta obtains some formulae of distance from zero to the generalized spectrum and to the semi-Fredholm spectrum by means of the conorm and the essential conorm of the conjugation orbits of T , respectively. This paper is devoted to the study of the conorm and essential conorm in a unital C*-algebra A. In Section 3 we prove that for every regular element a in A, γe (a) = max{γ(a + k) : k ∈ K(A)}. We introduce the generalized spectrum and the Atkinson spectrum. In analogy with [17], we shall establish appropriate formulae of distance from zero to the above spectra. Section 4 is concerned with the study of the continuity points of the conorm in C*-algebras with extremal richness, in the sense that, the closed unit ball of the algebra equals the convex hull of the set of its extreme points. Since the algebra of all bounded linear operators on a Hilbert space is a extremally rich C*-algebra, the results appearing in this section generalize those obtained in [19].

4

Burgos and Peralta

IEOT

2. Preliminary results Let A be a unital C*-algebra having non-zero socle. Throughout this paper π : A → C(A) will denote the natural quotient homomorphism. A closed ideal I of A is said to be essential if for any x ∈ A, the condition xI = Ix = 0 implies x = 0. The set of all primitive ideals of A will be denoted by ΠA . Definition 2.20 in [22] together with Lemma C*.2.4 in [5] imply that (left, right) invertibility modulo F(A) = soc(A) is equivalent to (left, right) invertibility modulo K(A). As the latter is a closed ideal of A, Φ(A) and A(A) are open multiplicative semigroups of A which are stable under compact perturbations. We refer to [5], [14], [22], [23], [24] and [25] as basic references on Atkinson and Fredholm theory in Banach algebras and C*-algebras. We recall from [5, Definition F.1.9] that an element x in A admits a left (respectively, right) Barnes idempotent if there exists an idempotent p ∈ A satisfying that xA = (1 − p)A (respectively, Ax = A(1 − p)). Notice that if p is a right (respectively, left) Barnes idempotent for x then, Ker(Lx ) = pA (respectively, Ker(Rx ) = Ap). It is also known that x is left (respectively, right) invertible modulo K(A) (equivalently, modulo soc(A)) if and only if it admits a right (respectively, left) Barnes idempotent p in soc(A) (compare [5, Theorem F.1.10], [22, Definition 2.20] and [5, Lemma C*.2.4]). A non-zero projection e ∈ A is said to be minimal if eAe = Ce. Let us assume that e is a minimal projection in a unital primitive C*-algebra A. Then the minimal left ideal Ae can be endowed with an inner product, h x, y ie = y ∗ x, (for all x, y ∈ Ae), under which Ae becomes a Hilbert space in the algebra norm. Let ρ : A → L(Ae) be the left regular representation on Ae, given by ρ(a)(x) = ax. The mapping ρ is an isometric irreducible ∗-representation, satisfying: ρ(F(A)) = F(Ae), ρ(K(A)) = K(Ae), ρ(Φ(A)) = Φ(Ae) ∩ ρ(A) (see [5, Section F.4, Theorem C*.4.3]). It is also known that for each x ∈ A, dim(ρ(x)(Ae)), dim(Ker(ρ(x))), and codim(ρ(x)(Ae)) are independent of the choice of e (compare [5, Lemma F.2.1]). Consequently, for each x ∈ A, we define the rank, nullity and defect of x by the expressions rank(x) := dim(ρ(x)(Ae)) = dim(xAe), nul(x) := dim(Ker(ρ(x))), and def(x) := codim(ρ(x)(Ae)) = dim(Ae/xAe), respectively. According to the above terminology F(A) = soc(A) = {x ∈ A : rank(x) < ∞}.

Vol. 63 (2009)

Conorm and Essential Conorm

5

The index function ind : Φ(A) → Z is defined by ind(x) = nul(x) − def(x), for all x ∈ Φ(A). The index function in a unital primitive C*-algebra A satisfies the familiar properties of the classical index for Fredholm operators on a Hilbert space. In particular (see [5, Sections F.2 and R.2]), the index is a continuous map fulfilling the following conditions: i) ind(xy) = ind(x) + ind(y), for all x, y ∈ Φ(A), ii) ind(x) = 0 if and only if x ∈ A−1 + K(A), where A−1 denotes the group of all invertible elements in A, iii) ind(x + z) = ind(x), for all x ∈ Φ(A) and z ∈ K(A), iv) x ∈ Φ(A) if and only if x∗ ∈ Φ(A) and in such a case, ind(x∗ ) = −ind(x). The index can be extended to a unital C*-algebra A. Following [5, Theorem F.3.4], given x ∈ Φ(A), there exists a finite subset Ω of ΠA such that x + P ∈ Φ(A/P ) for all P ∈ Ω, and x + P is invertible (that is, nul(x + P ) = 0 and def(x + P ) = 0), for all P ∈ ΠA \ Ω. The nullity, defect and index of x ∈ Φ(A), are respectively defined by P nul(x) := P ∈ΠA nul(x + P ), P def(x) := P ∈ΠA def(x + P ), ind(x) := nul(x) − def(x). For an Atkinson element x in A, we define nul(x) = ∞ when a is not left invertible modulo soc(A), def(x) = ∞ when a is not right invertible modulo soc(A), and ind(x) = nul(x)−def(x). The above properties i) − iv) remain valid, whenever they make sense, for Atkinson elements (see [22]). The following connections between regular and Atkinson elements in a unital C*-algebra with non-zero socle A can be deduced from [25, Proposition 2.5]). (LFR) x is left invertible modulo K(A) if and only if x ∈ Ar and Ker(Lx ) is contained in K(A). (RFR) x is right invertible modulo K(A) if and only if x ∈ Ar and Ker(Rx ) is contained in K(A). Remark 2.1. If a is left (respectively, right) invertible modulo K(A) and Ker(La ) = {0} (respectively, Ker(Ra ) = {0}), then a is left (respectively, right) invertible.

3. The essential conorm and some distance formulae Let A be a unital C*-algebra having non-zero socle. Denote by σe (a) the essential spectrum of an element a ∈ A, that is σe (a) = σ(π(a)) = {λ ∈ C : a − λ is not Fredholm} . We say that a ∈ A is essentially regular if π(a) is regular and we call γe (a) := γ(π(a)) the essential conorm of a. Clearly, γ(a+k) ≤ γe (a+k) = γe (a), for

6

Burgos and Peralta

IEOT

all k ∈ K(A). Moreover, a is essentially regular if and only if γe (a) > 0. In particular, every regular element is essentially regular and if a ∈ Ar then π(a† ) = π(a)† . Aress will denote the set of essentially regular elements of A. The following result improves [19, Theorem 7]. Theorem 3.1. Let A be a unital C*-algebra having non-zero socle and let a be a regular element in A. Then there exists k in soc(A) such that γe (a) = γ(a + k). Proof. Since γ(a) > 0, 0 is an isolated point of σ(|a|), or 0 ∈ / σ(|a|). If ]0 , γe (a)[⊆ C \ σ(|a|) then γe (a) = γ(a) and thus we can chose k = 0. We may then assume that ]0 , γe (a)[∩σ(|a|) 6= ∅. Hence, by virtue of [5, Theorem R.2.7], we have ]0 , γe (a)[∩σ(|a|) = {λn }n≥1 , where λm 6= λn , whenever n 6= m, and the only possible accumulation point of {λn }n≥1 is γe (a). We assume that γ(a) < γe (a) (otherwise, we can take k = 0). Then, the set Λ = {n ≥ 1 : λn ≤ γ(a)} is finite. For every n ∈ Λ, |a|−λn is a selfadjoint Fredholm element and, in particular, it admits a right Barnes idempotent pn ∈ soc(A), that is A(|a| − λn ) = A(1 − pn ), Ker(L|a|−λn ) = pn A, and A = A(1 − pn ) ⊕ Apn = A(|a| − λn ) ⊕ Apn . By [5, BA.4.3] and [25, Proposition 2.3], we may assume that p∗n = pn , for all n. Consequently, (|a| − λn )pn = 0 gives |a|pn = λn pn = pn |a|. Moreover, for every n, m ∈ Λ with n 6= m, pn A = Ker(L|a|−λn ) and Ker(L|a|−λn ) ⊆ (|a| − λm )A. Indeed, writing |a| − λn = |a| − λm − (λn − λm ), if x ∈ Ker(L|a|−λn ), as λn 6= λm , then 1 x, λn − λm which shows that x ∈ (|a| − λm )A. Thus, pn = (|a| − λm )x, for a suitable x ∈ A. Therefore pn = p∗n = x∗ (|a| − λm ) ∈ A(1 − pm ), and hence pn pm = pm pn = 0. Let P p = n∈Λ pn . By the above comments, it is clear that p = p∗ = p2 ∈ soc(A) and p|a| = |a|p. We claim that ]0, γe (a)[⊆ C \ σ(|a|(1 − p)). (3.1) Indeed, let λ ∈]0, γe (a)[. Then |a| − λ is a Fredholm element. Since |a|p belongs to soc(A), |a|(1 − p) − λ = |a| − λ − |a|p is also Fredholm. Noticing that |a|(1 − p) − λ is a selfadjoint element, Remark 2.1 guarantees that in order to deduce (3.1) it is enough to show that Ker(L|a|(1−p)−λ ) = {0}. Let us fix an arbitrary x ∈ Ker(L|a|(1−p)−λ ). Then, x = (|a| − λm )

(|a| − λ)x = |a|px = p|a|x, x = λ−1 (1 − p)|a|x, which shows that (|a| − λ)x = p|a|x = |a|px = λ−1 |a|p(1 − p)|a|x = 0.

Vol. 63 (2009)

Conorm and Essential Conorm

7

That is, x ∈ Ker(L|a|−λ ). If λ 6= λk for all k ∈ Λ, then, |a| −λ is invertible, and thus, x ∈ Ker(L|a|−λ ) = {0}. If λ = λk for some k ∈ Λ, then x ∈ Ker(L|a|(1−p)−λk ) ⊆ Ker(L|a|−λk ) = pk A. Thus pk x = x, λk x = |a|(1 − p)x, and λk x = |a|(1 − p)x = |a|(1 − p)pk x = 0. Noticing that |a|(1 − p) = |a(1 − p)|, we have just proved that ]0 , γe (a)[⊆ C \ σ(|a(1 − p)|). In particular, from this it is clear that γ(a(1 − p)) = inf σ(|a(1 − p)|) \ {0} > 0. Therefore, [γ(a(1 − p)) , γe (a(1 − p))[⊆]0 , γe (a)[⊆ C \ σ(|a(1 − p)|), which proves that γe (a) = γe (a − ap) = γ(a − ap).

Corollary 3.2. Let A be a unital C*-algebra having non-zero socle. If A is prime or A has real rank zero, then for each a in A there exists k ∈ K(A) such that γe (a) = γ(a + k). Proof. Let a ∈ A. If a ∈ K(A) it suffices to take k = −a. Suppose that a ∈ / K(A) and γe (a) > 0 (otherwise is trivial). First consider the case when A is prime, and thus (since A has non-zero socle) primitive (compare Remark 1 in page 277 and comments preceding Lemma 2.5 in [15]). Let e be a minimal projection in A and let ρ : A → L(Ae) be the left regular representation on Ae, introduced in the previous section. Since ρ is an isometric ∗-representation, it is clear that, for every x ∈ A, γ(x) = γ(ρ(x)). Moreover, since ρ(K(A)) = K(Ae), we also have γe (x) = γe (ρ(x)), for all x ∈ A. Then, γe (ρ(a)) = γe (a) > 0 and thus, by [19, Theorem 2], there exists K ∈ K(Ae) such that γe (ρ(a)) = γ(ρ(a) + K). Let k ∈ K(A) be such that ρ(k) = K. Then, γe (a) = γe (ρ(a)) = γ(ρ(a + k)) = γ(a + k). When A has real rank zero, it follows by [26, Theorem 3.3] that γe (a) = sup γ(a + k). k∈K(A)

Therefore, there exists k0 ∈ K(A) such that γ(a + k0 ) > 0. Theorem 3.1 guarantees the existence of k1 ∈ soc(A) satisfying γe (a + k0 ) = γ(a + k0 + k1 ). Taking k = k0 + k1 ∈ K(A), we have γe (a) = γe (a + k0 ) = γ(a + k). The following result follows directly from the above corollary and (1.2).

8

Burgos and Peralta

IEOT

Corollary 3.3. Let A be a unital C*-algebra having non-zero socle. If A is prime or A has real rank zero, then C(A)r = π(Ar ). Moreover, if a ∈ Aress there exists k ∈ K(A) such that kπ(a)† k = k(a + k)† k.

Remark 3.4. The last statement in the above corollary remains true for every regular element in a unital C*-algebra A having non-zero socle. Indeed, by Theorem 3.1, for every a ∈ Ar there exists k ∈ soc(A) such that γe (a) = γ(a + k). Now (1.2) gives that kπ(a)† k = k(a + k)† k. The next result is well known for algebras of bounded linear operators on a Hilbert space. Lemma 3.5. Let A be a unital C*-algebra having non-zero socle, a ∈ A(A) and x ∈ A. If ka − xk < γ(a) then x ∈ A(A) and ind(a) = ind(x). Proof. Assume that a is left invertible modulo K(A) (otherwise the reasoning is analogous). In this case, (LFR) implies that a is regular. Let π(b) be the unique element in C(A) such that π(b)π(a) = π(1) and (π(a)π(b))∗ = π(a)π(b). If a† is the Moore-Penrose inverse of a, then we have π(a)π(a† )π(a) = π(a), (π(a)π(a† ))∗ = π(a)π(a† ), and (π(a† )π(a))∗ = π(a† )π(a). Multiplying the first identity on the left by π(b), we get π(a† )π(a) = π(1). The uniqueness of π(b) implies that π(a† ) = π(b) = π(a)† . Therefore 1 − a† a ∈ K(A). Let us suppose that ka − xk < γ(a) = ka† k−1 . Then the inequality ka† a − a† xk < 1, shows that 1 − a† a + a† x is invertible. Since 1 − a† a ∈ K(A), it follows that a† x is Fredholm of zero index. Hence, x is left invertible modulo K(A) and, by [23, Theorem 4.7] ind(x) = ind(a† x) − ind(a† ) = −ind(a† ) = ind(a).

Remark 3.6. Let a be a left (respectively, right) invertible element in a unital C*-algebra A. Given b ∈ A, if ka − bk < γ(a) then b is left (respectively, right) invertible in A. Indeed, assume that a is left invertible and let a† be the unique element in A such that a† a = 1 and (aa† )∗ = aa† .

Vol. 63 (2009)

Conorm and Essential Conorm

9

Then ka − bk < γ(a) = ka† k−1 and thus k1 − a† bk < 1. This proves that a† b is invertible, and consequently that b is left invertible. In the next corollary we generalize [19, Corollary 5] to the more general setting of unital C*-algebras. Corollary 3.7. Let A be a unital C*-algebra having non-zero socle, a ∈ A(A) and x ∈ A. If ka − xk < γe (a) then x ∈ A(A) and ind(a) = ind(x). Proof. Let us suppose that a is left invertible modulo K(A). Again (LFR) implies that a is regular (and hence 0 < γ(a) ≤ γe (a)). Since kπ(a) − π(x)k ≤ ka − xk < γe (a), it follows from the above remark that x is left invertible modulo K(A). Now, Theorem 3.1 implies the existence of a finite rank element k in A such that γe (a) = γ(a + k). Then, a + k is left invertible modulo K(A) and k(a + k) − (x + k)k < γ(a + k). Finally, we apply the preceding lemma and the stability properties of the index to conclude that x is left invertible modulo K(A) with ind(x) = ind(x + k) = ind(a + k) = ind(a).

Let A be a unital C*-algebra. For a ∈ A, reg(a) denotes the regular set of a, that is, the set of all λ ∈ C such that there exists a neighborhood Uλ of λ and an analytic function b : Uλ → A such that b(µ) is a generalized inverse of a − µ1 for any µ ∈ Uλ . The complement σg (a) = C \ reg(a) of reg(a) in C is called the generalized spectrum of a. Remark 3.8. The following properties of the generalized spectrum are well known (compare [12], [18] and [20, Sections 12 and 13]): (1) S 0 belongs to reg(a) if and only if a has a generalized inverse and n n≥1 Ker(La ) ⊆ aA. P∞ (2) If 0 ∈ reg(a), and b is a generalized inverse for a, then b(λ) = k=0 λk bk+1 is a generalized inverse of a − λ, depending analytically on λ for |λ| < kbk−1 . (3) σg (a∗ ) = σg (a). (4) σg (a) = {λ ∈ C : limµ→λ γ(a − µ) = 0}. Now we are interested in to obtain some formulae for the distance from zero to the generalized spectrum and the Atkinson spectrum in line with [17]. Let A be a unital C*-algebra. Take a, b ∈ A such that 0 ∈ / reg(a) and aba = a. By [13] (see also [20, Theorem 12.26]) and the preceding remark it follows that dist(0, σg (a)) = lim γ(an )1/n = sup γ(an )1/n . n→∞

n∈N

10

Burgos and Peralta

IEOT

From [24, Proposition 1.3], an+1 bn (1 − ba) = 0, that is, an+1 bn = an+1 bn+1 a, for all n ∈ N. Then an+1 bn an = an+1 bn+1 an+1 , for all n ∈ N. Arguing by induction, it can be easily proved that an bn an = an , for all n ∈ N. Therefore, by equation (1.1), we have γ(an ) ≥ kbn k−1 , for all n ∈ N, and thus lim γ(an )1/n = sup γ(an )1/n ≥ r(b)−1 ,

n→∞

n∈N

where r(x) denotes the spectral radius of x ∈ A. Hence, dist(0, σg (a)) ≥ sup{r(b)−1 : aba = a}. In fact, analogous arguments to those employed in [4, Theorem 2.3], show that the equality holds. Theorem 3.9. Let A be a unital C*-algebra and let a in A such that 0 ∈ reg(a). Then, dist(0, σg (a)) = lim γ(an )1/n = sup{r(b)−1 : aba = a}. n→∞

Given a unital C*-algebra A, for an element a in A the set σge (a) := σg (π(a)) will be called the generalized essential spectrum of a. The set σA (a) := {λ ∈ C : a − λ is not Atkinson} will be termed the Atkinson spectrum of a. According to the above notation σge (a) ⊆ σA (a). Denote by re (a) = r(π(a)) the essential spectral radius of x ∈ A. We recall here the well known spectral radius formula re (a) = inf{r(a + k) : k ∈ K(A)},

(∀a ∈ A).

Theorem 3.10. Let A be a unital C*-algebra with non-zero socle and let a ∈ A(A). Then, dist(0, σA (a)) = lim γe (an )1/n = sup{re (b)−1 : aba − a ∈ K(A)}. n→∞

Proof. Since a ∈ A(A), we have 0 ∈ reg(π(a)). It follows from the above comments and Theorem 3.9 that dist(0, σA (a)) ≤ dist(0, σge (a)) = dist(0, σg (π(a))) = lim γe (an )1/n = sup{re (b)−1 : aba − a ∈ K(A)}. n→∞

Besides, from [5, Theorem C*.4.3] there exists an isometric ∗-representation, % of A on a Hilbert space H, such that %(K(A)) = K(H) ∩ %(A). It follows that γe (x) ≤ γe (%(x)), for all x ∈ A. From [22, Theorem 7.2], we get that x ∈ A is

Vol. 63 (2009)

Conorm and Essential Conorm

11

Atkinson if and only if %(x) is an Atkinson (equivalently, semi-Fredholm) operator on H and thus σA (x) = σA (%(x)), for all x ∈ A. Now, by [19, Theorem 6] dist(0, σA (a)) = dist(0, σA (%(a))) = lim γe (%(a)n )1/n . n→∞

Therefore, dist(0, σA (a)) ≤ ≤

lim γe (an )1/n = sup{re (b)−1 : aba − a ∈ K(A)}

n→∞

lim γe (%(a)n )1/n = dist(0, σA (a)),

n→∞

which concludes the proof.

We shall finish this section with two theorems which are generalizations of the main results in [17] to the wider scope of unital C*-algebras. Theorem 3.11. Let A be a unital C*-algebra and let a in A such that 0 ∈ reg(a). Then dist(0, σg (a))

= sup{γ(uau−1 ) : u ∈ A−1 } =

sup{γ(uau−1 ) : 0 ≤ u ∈ A−1 }

=

sup{γ(eh ae−h ) : h = h∗ ∈ A}.

Proof. It is clear that σg (x) = σg (uxu−1 ) for all x ∈ A and u ∈ A−1 . Moreover, arguing as in [17, Lemma 2.3] it can be shown that if |λ| < γ(a) then λ ∈ reg(a). Thus dist(0, σg (a)) ≥ sup{γ(uau−1 ) : u ∈ A−1 } ≥ sup{γ(uau−1 ) : 0 ≤ u ∈ A−1 } ≥ sup{γ(eh ae−h ) : h = h∗ ∈ A}. Let b ∈ A be such that aba = a, and h = h∗ ∈ A. Then, γ(eh ae−h ) ≥ keh be−h k−1 . Since, by [21], r(b) = inf{keh be−h k : h = h∗ ∈ A}, for each ε > 0 there exists h = h∗ ∈ A such that keh be−h k < r(b) + ε. Therefore, γ(eh ae−h ) ≥ keh be−h k−1 > (r(b) + ε)−1 . It follows that sup{γ(eh ae−h ) : h = h∗ ∈ A} > (r(b) + ε)−1 . Finally, the arbitrariness of ε and Theorem 3.9 give us sup{γ(eh ae−h ) : h = h∗ ∈ A} ≥ sup{r(b)−1 : aba = a} = dist(0, σg (a)).

12

Burgos and Peralta

IEOT

Theorem 3.12. Let A be a unital C*-algebra having non-zero socle, and let a be an Atkinson element in A. Then dist(0, σA (a))

= sup{γe (uau−1 ) : u ∈ A−1 } =

sup{γe (uau−1 ) : 0 ≤ u ∈ A−1 }

=

sup{γe (eh ae−h ) : h = h∗ ∈ A}.

Proof. It is easy to check that σA (x) = σA (uxu−1 ), for all x ∈ A and u ∈ A−1 . Thus, by Corollary 3.7, if |λ| < γe (a) then λ ∈ / σA (a), and hence dist(0, σA (a)) ≥ sup{γe (uau−1 ) : u ∈ A−1 } ≥ sup{γe (uau−1 ) : 0 ≤ u ∈ A−1 } ≥ sup{γe (eh ae−h ) : h = h∗ ∈ A}. Moreover, dist(0, σA (a)) ≤ dist(0, σge (a)) = sup{γ(eπ(h) π(a)e−π(h) ) : π(h) = π(h)∗ ∈ C(A)}. Finally, we notice that every element eπ(h) with π(h) = π(h)∗ in C(A), can be lifted to an element eh with h = h∗ in A. Therefore, dist(0, σA (a)) ≤ dist(0, σge (a)) = sup{γe (eh ae−h ) : h = h∗ ∈ A} ≤ dist(0, σA (a)), which gives the desired statement.

4. Extremally rich C*-algebras and continuity points of the conorm For a unital C*-algebra A, denote by E(A) the set of extreme points in the closed unit ball BA of A, that is, the set of all partial isometries v in A such that (1 − v ∗ v)A(1 − vv ∗ ) = {0}. The elements in E(A) are also called complete partial := A−1 E(A)A−1 will denote the set of quasiisometries. According to [7] A−1 q invertible elements in A. are either left or right Notice that when A is prime, the elements in A−1 q invertible. The C*-algebra A is called extremally rich if the set A−1 is dense q in A. Equivalently (see [8]), A is extremally rich if the convex hull of the set E(A) is equal to the closed unit ball. The class of C*-algebras with extremal richness was introduced with the objective of extending the theory and results of finite C*-algebras to the infinite case. The class of extremally rich C*-algebras includes stable rank one algebras, von Neumann algebras and purely infinite simple C*- algebras. Remark 4.1. Let a be an element in a unital C*-algebra A. It is not hard to see that γ(a) ≤ γ(a + P ), for all P ∈ ΠA . Besides, since γ(a) = inf σ(|a|) \ {0} and [ [ σ(|a|) = σ(|a| + P ) = σ(|a + P |), P ∈ΠA

P ∈ΠA

Vol. 63 (2009)

Conorm and Essential Conorm

13

it follows that γ(a) = inf γ(a + P ). P ∈ΠA

Lemma 4.2. Let A be a unital C*-algebra. Let a ∈ A−1 and b ∈ A, satisfying that q ka − bk ≤ γ(a) then |γ(a) − γ(b)| ≤ ka − bk. Proof. From (i) ⇒ (v) or (i) ⇒ (vi) in [7, Theorem 1.1] it is clear that every quasi-invertible element is regular and thus γ(a) > 0. We claim that the condition ka − bk ≤ γ(a) implies that b is also quasiinvertible. Indeed, as a ∈ A−1 q , from [7, Theorem 1.1], there exists an orthogonal pair of closed ideals I, J of A, such that a + I is left invertible in A/I and a + J is right invertible in A/J. Therefore, ka − b + Ik ≤ ka − bk ≤ γ(a) ≤ γ(a + I) which shows (by Remark 3.6) that b + I is left invertible. Similarly it can be proved that b + J is right invertible. Then, by [7, Theorem 1.1] b ∈ A−1 q . Let P ∈ ΠA . As I ∩ J = {0} ⊆ P , either I ⊆ P or J ⊆ P . Suppose that I ⊆ P . Then a and b are left invertible modulo P . For all x ∈ A \ P , γ(a + P ) ≤

k(a − b)x + bx + P k kbx + P k kax + P k = ≤ ka − b + P k + . kx + P k kx + P k kx + P k

Therefore, γ(a + P ) − ka − b + P k ≤ γ(b + P ). This proves that for all P ∈ ΠA , γ(a) − ka − bk ≤ γ(b + P ). From the above remark, we obtain that γ(a) − ka − bk ≤ γ(b). In the same way γ(b) − ka − bk ≤ γ(a), and hence |γ(a) − γ(b)| ≤ ka − bk.

It is known that the conorm is upper semi-continuous in every unital C*-algebra (see [12, Theorem 7]) and that the only non-trivial continuity points of the conorm for the algebra of bounded linear operators on a Hilbert space are the one-sided invertible operators (see [12, Theorem 9]). We recover this result as a direct consequence of the following theorem. Theorem 4.3. Let A be an extremally rich unital C*-algebra and a ∈ Ar . Then a ∈ A−1 if and only if γ(.) is continuous at a. q Proof. If a ∈ A−1 q , the continuity of γ(.) in a follows directly from the preceding lemma. Suppose that a is a continuity point of γ(.). There exists δ > 0 such that −1 γ(b) > γ(a)/2 whenever ka − bk < δ. As A = A−1 such that q , there is x ∈ Aq γ(a) γ(a) ka − xk < min{δ, 2 }. In particular ka − xk < δ, that is, γ(x) ≥ 2 > ka − xk, which shows, as in the previous proof, that a ∈ A−1 q . Corollary 4.4. Let A be an extremally rich prime unital C*-algebra. A regular element a ∈ A is a continuity point of the conorm if and only if a is either left or right invertible.

14

Burgos and Peralta

IEOT

Let A be a unital C*-algebra with non-zero socle. Denote by −1 Aeq := π −1 (C(A)−1 q ) the set of all the elements a ∈ A such that π(a) is quasir invertible in the generalized Calkin algebra. Clearly A−1 eq ⊆ Aess . The following lemma generalizes [19, Lemma 10]. Lemma 4.5. Let A be a unital C*-algebra with non-zero socle. Given a ∈ A−1 eq and b ∈ A satisfying that ka − bk ≤ γe (a) then we have |γe (a) − γe (b)| ≤ ka − bk. Proof. Let b ∈ A be an element satisfying ka − bk ≤ γe (a). Then, kπ(a) − π(b)k ≤ ka − bk ≤ γe (a) = γ(π(a)). By Lemma 4.2, b ∈ A−1 eq and |γe (a) − γe (b)| = |γ(π(a)) − γ(π(b))| ≤ kπ(a) − π(b)k ≤ ka − bk.

Theorem 4.6. Let A be an extremally rich unital C*-algebra with non-zero socle and a ∈ Aress . Then a ∈ A−1 eq if and only if γe (.) is continuous in a. Proof. If a ∈ A−1 eq , the continuity of γe (.) in a follows from the above lemma. Suppose that a is a continuity point of γe (.). We firstly note that, since every quotient of an extremally rich C*-algebra is extremally rich (see [7, Theorem 3.5]), then so is the quotient C(A) and therefore −1 Aeq is dense in C(A). Following the arguments given in the proof of Theorem 4.3, there exists δ > 0 such that γe (b) > γe (a)/2 whenever ka − bk < δ. Now, let x ∈ A−1 eq such that ka − xk < min{δ, γe2(a) }. Then γe (x) ≥ γe2(a) > ka − xk ≥ kπ(a) − π(x)k, which shows that a ∈ A−1 eq . Just by quoting that for every (von Neumann) factor A, C(A) is an extremally rich prime C*-algebra (recall that each closed ideal in a factor is a prime ideal), the next corollary follows directly from Theorem 4.6 and extends [19, Theorem 8]. Corollary 4.7. Let A be a factor. An essentially regular element a ∈ A is a continuity point of the essential conorm if and only if a is Atkinson.

References [1] C. Apostol, The reduced minimum modulus. Michigan Math. J. 32 (1985), no. 3, 279–294. [2] B. Aupetit, A primer on spectral theory. Springer-Verlag, Mew York, 1991. [3] C. Badea and M. Mbekhta, Generalized inverses and the maximal radius of regularity of a Fredholm operator. Integr. Equ. Oper. Theory 28 (1997), 133–146. [4] C. Badea and M. Mbekhta, Compressions of resolvents and maximal radius of regularity. Trans. Amer. Math. Soc. 351 (1999), 2949–2960. [5] B.A. Barnes, G.J. Murphy, M.R.F. Smyth and T.T. West, Riesz and Fredholm theory in Banach algebras. London. Pitman, 1982.

Vol. 63 (2009)

Conorm and Essential Conorm

15

[6] L.G. Brown and G.K. Pedersen, C ∗ -algebras of real rank zero. J. Funct. Anal. 99 (1991), 131–149. [7] L.G. Brown and G.K. Pedersen, On the geometry of the unit ball of a C ∗ -algebra. J. Reine Angew. Math. 469 (1995), 113–147. [8] L.G. Brown and G.K. Pedersen, Approximation and convex decomposition by extremals in C*-algebras. Math. Scand. 81 (1997), 69–85. [9] L.G. Brown and G.K. Pedersen, Extremal K-theory and index for C ∗ -algebras. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part III. K-Theory 20 (2000), 201–241. [10] M. Burgos, A. Kaidi, A. Morales Campoy, A. Peralta, and M. Ram´ırez, Von Neumann Regularity and Quadratic Conorms in JB*-triples and C*-algebras. Acta Math. Sin. 24 (2008), no. 2, 185–200. [11] R. Harte and M. Mbekhta, On generalized inverses in C*-algebras. Studia Math. 103 (1992), 71–77. [12] R. Harte and M. Mbekhta, Generalized inverses in C*-algebras II. Studia Math. 106 (1993), 129–138. [13] V. Kordula and V. M¨ uller, The distance from the Apostol spectrum. Proc. Amer. Math. Soc. 124 (1996), 3055–3061. [14] D. Maennle and Ch. Schmoeger, Generalized Fredholm theory in semisimple algebras. Rend. di Mat. 19 (1999), 583–613. [15] M. Mathieu, Elementary operators on prime C*-algebras II. Glasgow Math. J. 30 (1988), 275–284. [16] M. Mbekhta, R´esolvant g´en´eralis´e et th´eorie spectrale. J. Operator Theory 21 (1989), 69–105. [17] M. Mbekhta, Formules de distance au spectre g´en´eralis´e et au spectre semi-Fredholm. J. Funct. Anal. 194 (2002), 231–247. [18] M. Mbekhta and A. Ouahab, Op´erateur s-r´egulier dans un espace de Banach et th´eorie spectrale. Acta Sci. Math. (Szeged) 59 (1994), 525–543. [19] M. Mbekhta and R. Paul, Sur la conorme essentielle. Studia Math. 117 (1996), 243–252. [20] V. M¨ uller, Spectral theory of linear operators and spectral systems in Banach algebras. Operator Theory: Advances and Applications, 139. Birkh¨ auser Verlag, Basel, 2003. [21] G.J. Murphy and T.T. West, Spectral radius formulae. Proc. Edinburgh Math. Soc. (2) 22 (1979), 271–275. [22] J.W. Rowell, Unilateral Fredholm Theory and Unilateral Spectra. Proc. Roy. Irish. Acad. 84A (1984), 69–85. [23] Ch. Schmoeger, Atkinson theory and holomorphic functions in Banach algebras. Proc. Roy. Irish. Acad. 91A (1991), 113–127. [24] Ch. Schmoeger, The punctured neighbourhood theorem in Banach algebras. Proc. Roy. Irish Acad. Sect. A. 91 (1991), 205–218. [25] Ch. Schmoeger, Ascent, descent, and the Atkinson region in Banach algebras I. Ricerche di Matematica XLII (1993), 123–143. [26] Y. Xue, The Reduced Minimum Modulus in C*-algebras. Integral Equations and Operator Theory 59 (2007), 269–280.

16

Burgos and Peralta

Mar´ıa Burgos Departamento de An´ alisis Matem´ atico Facultad de Ciencias Universidad de Granada 18071 Granada Spain e-mail: [email protected] Antonio M. Peralta Departamento de An´ alisis Matem´ atico Facultad de Ciencias Universidad de Granada 18071 Granada Spain e-mail: [email protected] Submitted: January 25, 2008. Revised: August 26, 2008.

IEOT

Integr. equ. oper. theory 63 (2009), 17–28 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010017-12, published online December 22, 2008 DOI 10.1007/s00020-008-1648-8

Integral Equations and Operator Theory

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems B.P. Duggal Abstract. A Banach space operator T ∈ B(X ) is polaroid if points λ ∈ iso σ(T ) are poles of the resolvent of T . Let σa (T ), σw (T ), σaw (T ), σSF+ (T ) and σSF− (T ) denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T .For A, B and C ∈ B(X ), let MC denote the operator ma A C trix . If A is polaroid on π0 (MC ) = {λ ∈ iso σ(MC ) : 0 < 0 B −1 dim(MC − λ) (0) < ∞}, M0 satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B has SVEP at points µ ∈ σw (M0 ) \ σSF− (B), or, (ii) both A and A∗ have SVEP at points λ ∈ σw (M0 ) \ σSF+ (A), or, (iii) A∗ has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B ∗ has SVEP at points µ ∈ σw (M0 ) \ σSF− (B), then σ(MC ) \ σw (MC ) = π0 (MC ). Here the hypothesis that λ ∈ π0 (MC ) are poles of the resolvent of A can not be replaced by the hypothesis λ ∈ π0 (A) are poles of the resolvent of A. For an operator T ∈ B(X ), let π0a (T ) = {λ : λ ∈ iso σa (T ), 0 < dim(T − −1 λ) (0) < ∞}. We prove that if A∗ and B ∗ have SVEP, A is polaroid on π0a (MC ) and B is polaroid on π0a (B) , then σa (MC ) \ σaw (MC ) = π0a (MC ). Mathematics Subject Classification (2000). Primary 47B47, 47A10, 47A11. Keywords. Banach space, operator matrix, Browder spectrum, Weyl spectrum, single valued extension property, polaroid operator, Browder and Weyl theorems.

1. Introduction A Banach space operator A, A ∈ B(X ), is upper semi-Fredholm (resp., lower semi-Fredholm) at a complex number λ ∈ C if the range (A − λ)X is closed and α(A − λ) = dim(A − λ)−1 (0) < ∞ (resp., β(A − λ) = dim(X /(A − λ)X ) < ∞). Let λ ∈ Φ+ (A) (resp., λ ∈ Φ− (A)) denote that A is upper semi-Fredholm (resp.,

18

B.P. Duggal

IEOT

lower semi-Fredholm) at λ. The operator A is Fredholm at λ, denoted λ ∈ Φ(A), if λ ∈ Φ+ (A) ∩ Φ− (A). A is Browder (resp., Weyl) at λ if λ ∈ Φ(A) and asc(A − λ) = dsc(A − λ) < ∞ (resp., if λ ∈ Φ(A) and ind(A − λ) = 0). Here ind(A − λ) = α(A − λ) − β(A − λ) denotes the Fredholm index of A − λ, asc(A − λ) denotes the ascent of A − λ (= the least non-negative integer n such that (A − λ)−n (0) = (A − λ)−(n+1) (0)) and dsc(A − λ) denotes the descent of A − λ (= the least non-negative integer n such that (A − λ)n X = (A − λ)n+1 X ). Let σ(A) denote the spectrum, σa (A) the approximate point spectrum, iso σ(A) the set of isolated points of σ(A), π0 (A) = {λ ∈ iso σ(A) : 0 < α(A − λ) < ∞}, π0a (A) = {λ ∈ iso σa (A) : 0 < α(A − λ) < ∞}, and p0 (A) the set of finite rank poles (of the resolvent) of A. The Browder spectrum σb (A) of A is the set {λ ∈ C : A − λ is not Browder}, the Weyl spectrum σw (A) of A is the set {λ ∈ C : A − λ is not Weyl}, the Browder essential approximate spectrum σab (A) of A is the set {λ ∈ C : λ ∈ / Φ+ (A) or asc(A − λ) 6< ∞}, and the Weyl essential approximate spectrum σaw (A) of A is the set {λ ∈ C : λ ∈ / Φ+ (A) or ind(A − λ) 6≤ 0}. Following current terminology, the operator A satisfies: Browder’s theorem, or Bt, if σw (A) = σb (A) (equivalently, σ(A) \ σw (A) = p0 (A)); Weyl’s theorem, or W t, if σ(A) \ σw (A) = π0 (A); a–Browder’s theorem, or a − Bt, if σaw (A) = σab (A); a–Weyl’s theorem, or a − W t, if σa (A) \ σaw (A) = π0a (A). An operator A ∈ B(X ) has the single-valued extension property at λ0 ∈ C, SVEP at λ0 , if for every open disc Dλ0 centered at λ0 the only analytic function f : Dλ0 → X which satisfies (A − λ)f (λ) = 0 for all λ ∈ Dλ0 is the function f ≡ 0. Trivially, every operator A has SVEP on the resolvent set ρ(A) = C\σ(A); also A has SVEP at points λ ∈ isoσ(A). Let Ξ(A) denote the set of λ ∈ C where A does not have SVEP: we say that A has SVEP if Ξ(A) = ∅. SVEP plays an important role in determining the relationship between the Browder and Weyl spectra, and the Browder and Weyl theorems. Thus σb (A) = σw (A)∪Ξ(A) = σw (A) ∪ Ξ(A∗ ), and if A∗ has SVEP then σb (A) = σw (A) = σab (A) [1, pp 141142]; A satisfies Bt (resp., a − Bt) if and only if A has SVEP at λ ∈ / σw (A) (resp., λ∈ / σaw (A)) [5, Lemma 2.18]; and if A∗ has SVEP, then A satisfies W t implies A satisfies a − W t [1, Theorem 3.108]. For A, B and C ∈ B(X ), let MC denote the upper triangular operator matrix A C MC = . A study of the spectrum, the Browder and Weyl spectra, and 0 B the Browder and Weyl theorems for the operator MC , and the related diagonal operator M0 = A ⊕ B, has been carried by a number of authors in the recent past (see [2, 3, 4, 8] for further references). Thus, if either Ξ(A∗ ) = ∅ or Ξ(B) = ∅, then σ(MC ) = σ(M0 ) = σ(A) ∪ σ(B); if Ξ(A) ∪ Ξ(B) = ∅, then MC has SVEP, σb (MC ) = σw (MC ) = σb (M0 ) = σw (M0 ), and MC satisfies a − Bt. Browder’s theorem, much less Weyl’s theorem, does not transfer from individual operators to

Vol. 63 (2009)

Operator Matrices

19

direct sums: for example, the forward unilateral shift and the backward unilateral shift on a Hilbert space satisfy Bt, but their direct sum does not. However, if (Ξ(A) ∩ Ξ(B ∗ )) ∪ Ξ(A∗ ) = ∅, then : M0 satisfies Bt (resp., a − Bt) implies MC satisfies Bt (resp., a − Bt); if points λ ∈ iso σ(A) are eigenvalues of A, A satisfies W t, then M0 satisfies W t implies MC satisfies W t [4, Proposition 4.1 and Theorem 4.2]. Our aim in this paper is to fine tune some of the extant results to prove that: σb (M0 ) = σb (MC ) ∪ {Ξ(A∗ ) ∩ Ξ(B)}; σab (MC ) ⊆ σab (M0 ) ⊆ σab (MC ) ∪ Ξ∗+ (A) ∪ Ξ+ (B); and σw (A) ∪ σw (B) ⊆ σw (MC ) ∪ {Ξ)P ) ∪ Ξ(Q)}, where,except for P = A and Q = B ∗ , P = A or A∗ and Q = B or B ∗ , Ξ∗+ (A) = {λ : λ ∈ / Φ+ (A) or A∗ does not have SVEP at λ} and Ξ+ (B) = {λ : λ ∈ / Φ+ (B) or B does not have SVEP at λ}. Let σSF+ (A) (resp., σSF− (A)) denote the upper semi– Fredholm spectrum (resp., lower semi–Fredholm spectrum) of A. It is proved that if points λ ∈ π0 (MC ) are poles of A, M0 satisfies W t, and A and B satisfy either of the hypotheses (i) A has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B has SVEP at points µ ∈ σw (M0 ) \ σSF− (B) or both A and A∗ have SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) or A∗ has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B ∗ has SVEP at points µ ∈ σw (M0 ) \ σSF− (B), then MC satisfies W t. Here the hypothesis that points λ ∈ π0 (MC ) are poles of A is essential. We prove also that if Ξ(A∗ ) ∪ Ξ(B ∗ ) = ∅, points λ ∈ π0a (MC ) are poles of A and points µ ∈ π0a (B) are poles of B, then MC satisfies a − W t. Throughout the following, the operators A, B and C shall be as in the operator matrix MC ; we shall write T ∈ B(Y) for a general Banach space operator.

2. Browder, Weyl Spectra and SVEP We start by recalling some results which will be used in the sequel without further reference. For an operator T ∈ B(Y) such that λ ∈ Φ± (T ), the following statements are equivalent [1, Theorems 3.16 and 3.17]: a) T (resp., T ∗ ) has SVEP at λ; b) asc(T − λ) < ∞ (resp., dsc(T − λ) < ∞). Furthermore, if λ ∈ Φ± (T ), and both T and T ∗ have SVEP at λ, then asc(T −λ) = dsc(T − λ) < ∞, λ ∈ iso σ(T ) and λ is a pole of (the resolvent of) T [1, Corollary 3.21]. For an operator T ∈ B(Y) such that λ ∈ Φ(T ) with ind(T − λ) = 0, T or T ∗ has SVEP at λ if and only if asc(T − λ) = dsc(T − λ) < ∞. Evidently, asc(MC ) < ∞ =⇒ asc(A) < ∞. It is not difficult to verify, [4, Lemma 2.1], that dsc(B) = ∞ =⇒ dsc(MC ) = ∞. In general, σ(MC ) ⊆ σ(A) ∪ σ(B) = σ(M0 ) = σ(MC ) ∪ {Ξ(A∗ ) ∪ Ξ(B)}; σb (MC ) ⊆ σb (A) ∪ σb (B) = σb (M0 ); and σw (MC ) ⊆ σw (M0 ) ⊆ σw (A) ∪ σw (B).

20

B.P. Duggal

IEOT

If λ ∈ / σaw (MC ), then λ ∈ Φ+ (A), and either α(B − λ) < ∞ and ind(A − λ) + ind(B − λ) ≤ 0, or β(A − λ) = α(B − λ) = ∞ and (B − λ)X is closed, or β(A − λ) = ∞ and (B − λ)X is not closed [3, Theorem 4.6]. Since I 0 I C A−λ 0 MC − λ = , 0 B−λ 0 I 0 I λ ∈ Φ(MC ) implies that λ ∈ Φ+ (A) ∩ Φ− (B). Using this it is seen that if λ ∈ / σb (MC ), then λ ∈ Φ+ (A) ∩ Φ− (B), ind(A − λ) + ind(B − λ) = 0, asc(A − λ) < ∞ and dsc(B − λ) < ∞. If in addition λ ∈ / Ξ(P ) ∪ Ξ(Q), where (except for P = A and Q = B ∗ ) P = A or A∗ and Q = B or B ∗ , then it is seen (argue as in the proof of Proposition 2.1 below) that ind(A − λ) = ind(B − λ) = 0. Thus λ ∈ / σb (M0 ), which implies that σb (M0 ) ⊆ σb (MC ) ∪ {Ξ(P ) ∪ Ξ(Q)}, where, except for P = A and Q = B ∗ , P = A or A∗ and Q = B or B ∗ . The following proposition gives more. Proposition 2.1. σb (M0 ) = σb (MC ) ∪ {Ξ(A∗ ) ∩ Ξ(B)}. Proof. If λ ∈ / σb (M0 ), then λ ∈ Φ(A) ∩ Φ(B), asc(A − λ) = dsc(A − λ) < ∞ and asc(B − λ) = dsc(B − λ) < ∞. Since dsc(A − λ) =⇒ λ ∈ / Ξ(A∗ ) and asc(B − λ) =⇒ ∗ λ ∈ / Ξ(B), λ ∈ / Ξ(A ) ∩ Ξ(B). Hence, since σb (MC ) ⊆ σb (M0 ), λ ∈ / σb (MC ) ∪ {Ξ(A∗ ) ∩ Ξ(B)}. Conversely, if λ ∈ / σb (MC ) ∪ {Ξ(A∗ ) ∩ Ξ(B)}, then λ ∈ Φ+ (A) ∩ Φ− (B), asc(A−λ < ∞ (=⇒ ind(A−λ) ≤ 0), dsc(B −λ) < ∞ (=⇒ ind(B −λ) ≥ 0) and ind(A − λ) + ind(B − λ) = 0. If A∗ has SVEP, then ind(A − λ) ≥ 0; hence ind(A − λ) = 0, which implies that ind(B − λ) = 0. But then both A − λ and B − λ have finite (hence, equal) ascent and descent. Thus λ ∈ / σb (M0 ). Arguing similarly in the case in which λ ∈ / Ξ(B) (this time using the fact that λ ∈ Φ− (B), dsc(B − λ) < ∞ and λ ∈ / Ξ(B) imply ind(B − λ) = 0), it is seen (once again) that λ∈ / σb (M0 ). If we let σsb (T ) = {λ ∈ C : either λ ∈ / Φ− (T ) or dsc(T − λ) = ∞}, then σsb (T ) = σab (T ∗ ), σb (T ) = σab (T ) ∪ σsb (T ) and σb (T ) = σab (T ) ∪ Ξ(T ∗ ) = σsb (T ) ∪ Ξ(T ) [1, p 141]. Evidently, σab (M0 ) ∪ {Ξ(A∗ ) ∪ Ξ(B ∗ )} = σb (M0 ). Let Ξ+ (T ) and Ξ∗+ (T ) denote the sets of λ such that λ∈ / Ξ+ (T ) =⇒ λ ∈ Φ+ (T ) and T has SVEP at λ and λ∈ / Ξ∗+ (T ) =⇒ λ ∈ Φ+ (T ) and T ∗ has SVEP at λ. Proposition 2.2. σab (MC ) ⊆ σab (M0 ) ⊆ σab (MC ) ∪ {Ξ∗+ (A) ∪ Ξ+ (B)}. Proof. The inclusion σab (MC ) ⊆ σab (M0 ) being evident, we prove σab (M0 ) ⊆ σab (MC ) ∪ Ξ∗+ (A). Let λ ∈ / σab (MC ) ∪ Ξ∗+ (A). Then λ ∈ Φ+ (A), asc(A − λ) < ∞, ∗ and A has SVEP at λ. Hence ind(A − λ) = 0 and λ ∈ Φ(A) (=⇒ λ ∈ / σab (A)),

Vol. 63 (2009)

Operator Matrices

21

which implies that λ ∈ Φ+ (B) and ind(B − λ) ≤ 0(since asc(MC − λ) < ∞ =⇒ ind(A − λ) + ind(B − λ) ≤ 0). But then the hypothesis λ ∈ / Ξ+ (B) implies that λ∈ / σab (B) =⇒ λ ∈ / σab (M0 ). The following corollary is immediate from Proposition 2.2. Corollary 2.3. If Ξ∗+ (A) ∪ Ξ+ (B) = ∅, then σab (M0 ) = σab (MC ). It is easy to see (from the definitions of σb (T ) and σw (T )) that σb (T ) = σw (T ) ∪ Ξ(T ) = σw (T ) ∪ Ξ(T ∗ ). Hence σb (M0 ) = {σw (A) ∪ σw (B)} ∪ {Ξ(P ) ∪ Ξ(Q)}, where P = A or A∗ and Q = B or B ∗ . Proposition 2.4. σw (A) ∪ σw (B) ⊆ σw (MC ) ∪ {Ξ(P ) ∪ Ξ(Q)}, where P = A and Q = B or P = A∗ and Q = B ∗ . Proof. The proof in both the cases is similar: we consider P = A and Q = B. If λ∈ / σw (MC ), then λ ∈ Φ+ (A) ∩ Φ− (B) and ind(A − λ) + ind(B − λ) = 0. Thus, since λ ∈ / Ξ(A) ∪ Ξ(B) implies ind(A − λ) ≤ 0 and ind(B − λ) ≤ 0, ind(A − λ) = ind(B − λ) = 0 and λ ∈ Φ(A) ∩ Φ(B). Hence λ ∈ / σw (A) ∪ σw (B). Proposition 2.4 implies that if Ξ(P ) ∪ Ξ(Q) = ∅, P and Q as above, then σw (M0 ) = σw (MC ) = σw (A) ∪ σw (B). More is true. Since λ ∈ / σw (MC ) ∪ {Ξ(P ) ∪ Ξ(Q)} implies λ ∈ Φ(A) ∩ Φ(B) and ind(A − λ) = ind(B − λ) = 0, asc(A − λ) = dsc(A − λ) < ∞ and asc(B − λ) = dsc(B − λ) < ∞. Hence σb (MC ) ⊆ σw (MC ) ∪ {Ξ(P ) ∪ Ξ(Q)}. Corollary 2.5. If Ξ(P ) ∪ Ξ(Q) = ∅, P and Q as in Proposition 2.4, then σb (M0 ) = σw (M0 ) = σb (MC ) = σw (MC ) = σw (A) ∪ σw (B). Proof. σw (MC ) ⊆ σb (MC ) ⊆ σb (M0 ).

The following theorem gives a necessary and sufficient condition for σw (M0 ) = σb (M0 ) and σaw (M0 ) = σab (M0 ). Theorem 2.6. (i) σw (M0 ) = σb (M0 ) if and only if A and B have SVEP on {λ : λ ∈ Φ(A) ∩ Φ(B), ind(A − λ) + ind(B − λ) = 0}. (ii) σaw (M0 ) = σab (M0 ) if and only if A and B have SVEP on {λ : λ ∈ Φ+ (A) ∩ Φ+ (B), ind(A − λ) + ind(B − λ) ≤ 0}. Proof. (i) If σw (M0 ) = σb (M0 ), then σw (M0 ) = σw (A) ∪ σw (B) = σb (A) ∪ σb (B). Hence λ ∈ Φ(A)∩Φ(B) with ind(A−λ) = ind(B −λ) = 0 if and only if λ ∈ Φ(A)∩ Φ(B), asc(A − λ) = dsc(A − λ) < ∞ and asc(B − λ) = dsc(B − λ) < ∞. Evidently, A and B have SVEP at points {λ : λ ∈ Φ(A) ∩ Φ(B), ind(A − λ) + ind(B − λ) = 0}. Conversely, if λ ∈ / σw (M0 ), then λ ∈ Φ(A) ∩ Φ(B) and ind(A − λ) + ind(B − λ) = 0. Since A and B have SVEP at λ, ind(A − λ) and ind(B − λ) are both ≤ 0. Hence ind(A − λ) = ind(B − λ) = 0, which (because of SVEP) implies that asc(A − λ) = dsc(A − λ) < ∞ and asc(B − λ) = dsc(B − λ) < ∞.

22

B.P. Duggal

IEOT

Thus λ ∈ / σb (A) ∪ σb (B) =⇒ σb (M0 ) ⊆ σw (M0 ). Since σw (M0 ) ⊆ σb (M0 ) always, σw (M0 ) = σb (M0 ). (ii) Since σaw (M0 ) ⊆ σaw (A) ∪ σaw (B) ⊆ σab (A) ∪ σab (B) = σab (M0 ), σaw (M0 ) = σab (M0 ) implies that σaw (M0 ) = σab (A) ∪ σab (B). Equivalently, {λ : λ ∈ Φ+ (A) ∩ Φ+ (B), ind(A − λ) + ind(B − λ) ≤ 0} ={λ : λ ∈ Φ+ (A) ∩ Φ+ (B), asc(A − λ) < ∞, asc(B − λ) < ∞}. Hence A and B have SVEP on {λ : λ ∈ Φ+ (A) ∩ Φ+ (B), ind(A − λ) + ind(B − λ) ≤ 0}. Conversely, if λ ∈ / σaw (M0 ), then λ ∈ Φ+ (A) ∩ Φ+ (B) and ind(A − λ) + ind(B −λ) ≤ 0. Since A and B have SVEP at λ, both asc(A−λ) and asc(B −λ) are finite. Hence λ ∈ / σab (A) ∪ σab (B), which implies that λ ∈ / σab (M0 ) =⇒ σab (M0 ) ⊆ σaw (M0 ). Since σaw (M0 ) ⊆ σab (M0 ) always, the proof is complete.

3. Browder, Weyl Theorems Translating Theorem 2.6 to the terminology of Browder’s theorem, Bt, and aBrowder’s theorem, a − Bt, we see that a necessary and sufficient condition for M0 to satisfy Bt is that A and B have SVEP at points λ ∈ / σw (M0 ), and that a necessary and sufficient condition for M0 to satisfy a − Bt is that A and B have SVEP at points λ ∈ / σaw (M0 ). The following theorem relates Bt (resp., a − Bt) for M0 to Bt (resp., a − Bt) for MC . Let σSF+ (T ) (resp., σSF− (T )) denote the upper semi–Fredholm spectrum (resp., the lower semi–Fredholm spectrum) of T . Theorem 3.1. (a) If either (i) A has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B has SVEP at points µ ∈ σw (M0 ) \ σSF− (B), or (ii) both A and A∗ have SVEP at points λ ∈ σw (M0 ) \ σSF+ (A), or (iii) A∗ has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B ∗ has SVEP at points µ ∈ σw (M0 ) \ σSF− (B), then M0 satisfies Bt implies MC satisfies Bt. (b) If either (i) A has SVEP at points λ ∈ σaw (M0 ) \ σSF+ (A) and A∗ has SVEP at points µ ∈ σw (M0 ) \ σSF+ (A), or (ii) A∗ has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B ∗ has SVEP at points µ ∈ σw (M0 ) \ σSF+ (B), then M0 satisfies a − Bt implies MC satisfies a − Bt. Proof. (a) Recall that M0 satisfies Bt if and only if M0∗ satisfies Bt, and that σw (M0∗ ) = σw (M0 ). Hence A, B, A∗ and B ∗ have SVEP at points λ ∈ / σw (M0 ). In view of this, hypotheses (i), (ii) and (iii) imply respectively that either (i)’ A has SVEP at λ ∈ Φ+ (A) and B has SVEP at µ ∈ Φ− (B), or (ii)’ A and A∗ have SVEP at λ ∈ Φ+ (A), or (iii)’ A∗ has SVEP at λ ∈ Φ+ (A) and B ∗ has SVEP at µ ∈ Φ− (B). Evidently, σw (MC ) ⊆ σb (MC ): we prove that σb (MC ) ⊆ σw (MC ). For this, let λ ∈ / σw (MC ). Then λ ∈ Φ+ (A) ∩ Φ− (B) and ind(A − λ) + ind(B − λ) = 0. Since both ind(A − λ) and ind(B − λ) are ≤ 0 if (i)’ holds, ind(A − λ) = 0 if (ii)’ holds, and both ind(A − λ) and ind(B − λ) are ≥ 0 if (iii)’ holds, we conclude that ind(A − λ) = ind(B − λ) = 0, λ ∈ Φ(A) ∩ Φ(B). Furthermore, since A and B have

Vol. 63 (2009)

Operator Matrices

23

SVEP at λ, asc(A − λ) = dsc(A − λ) < ∞ and asc(B − λ) = dsc(B − λ) < ∞. Hence λ ∈ / σb (A) ∪ σb (B) =⇒ λ ∈ / σb (MC ). (b) Since a−Bt implies Bt, M0∗ satisfies Bt, which implies that A∗ has SVEP at λ ∈ / σw (M0 ). Hence the hypothesis A∗ has SVEP at λ ∈ σw (M0 ) \ σSF+ (A) implies that A∗ has SVEP at λ ∈ Φ+ (A). Since the hypothesis M0 satisfies a − Bt implies that A and B have SVEP on {λ : λ ∈ Φ+ (A) ∩ Φ+ (B), ind(A − λ) + ind(B − λ) ≤ 0}, it follows from the SVEP hypotheses of the statement that either (i)’ both A and A∗ have SVEP at λ ∈ Φ+ (A) or (ii)’ A∗ has SVEP at λ ∈ Φ+ (A) and B ∗ has SVEP at µ ∈ Φ+ (B). Evidently, σaw (MC ) ⊆ σab (MC ). For the reverse inclusion, let λ ∈ / σaw (MC ). Then λ ∈ Φ+ (A). If (i)’ is satisfied, then A and A∗ have SVEP at λ =⇒ ind(A − λ) = 0, which (because λ ∈ Φ+ (A)) implies that λ ∈ Φ(A) and ind(A − λ) = 0; if (ii)’ holds, then A∗ has SVEP at λ implies ind(A − λ) ≥ 0 =⇒ λ ∈ Φ(A) with ind(A − λ) ≥ 0. In either case it follows that λ ∈ Φ+ (B) and ind(A − λ) + ind(B − λ) ≤ 0. Hence λ ∈ Φ(A) ∩ Φ(B), ind(A − λ) = 0 and ind(B − λ) ≤ 0. Since both A and B have SVEP on {λ : λ ∈ Φ+ (A)∩Φ+ (B), ind(A−λ)+ind(B −λ) ≤ 0}, asc(A−λ) < ∞ and asc(B −λ) < ∞. Thus λ ∈ / σab (M0 ) =⇒ λ ∈ / σab (MC ). Remark 3.2. We note, for future reference, that if M0 satisfies Bt and either of the hypotheses (i) to (iii) of Theorem 3.1(a) is satisfied, then σw (M0 ) = σw (MC ). Furthermore, σ(MC ) = σ(M0 ), as the following argument shows. If M0 satisfies Bt, and one of the hypotheses (i), (ii) and (iii) is satisfied, then either A∗ has SVEP at λ ∈ Φ+ (A) or B has SVEP at µ ∈ Φ− (B). Since λ ∈ / σ(MC ) implies A − λ is left invertible and B − λ is right invertible, A∗ has SVEP at λ ∈ Φ+ (A) if and only if A − λ is onto and B has SVEP at λ ∈ Φ− (B) if and only if B − λ is injective [1, Corollary 2.4]. In either case, both A − λ and B − λ are invertible, which implies that λ ∈ / σ(M0 ) =⇒ σ(M0 ) ⊆ σ(MC ). Since σ(MC ) ⊆ σ(M0 ), the equality of the spectra follows. Corollary 3.3. (a) [4, Proposition 4.1] If {Ξ(A) ∩ Ξ(B ∗ )} ∪ Ξ(A∗ ) = ∅, then M0 satisfies Bt (resp., a − Bt) implies MC satisfies Bt (resp., a − Bt). (b) [2, Theorem 3.2] If either σaw (A) = σSF+ (B) or σSF− (A) ∩ σSF+ (B) = ∅, then M0 satisfies Bt (resp., a − Bt) implies MC satisfies Bt (resp., a − Bt). Proof. Observe that if {Ξ(A) ∩ Ξ(B ∗ )} ∪ Ξ(A∗ ) = ∅, then either A and A∗ have SVEP or A∗ and B ∗ have SVEP. Hence the proof for (a) follows from Theorem 3.1(b). Assume now that σaw (A) = σSF+ (B). If λ ∈ / σaw (MC ), then λ ∈ Φ+ (A) and either α(B − λ) < ∞ and ind(A − λ) + ind(B − λ) ≤ 0, or β(A − λ) = α(B − λ) = ∞ and (B − λ)X is closed, or β(A − λ) = ∞ and (B − λ)X is not closed. Observe that if α(B − λ) = β(A − λ) = ∞ or β(A − λ) = ∞, then the hypothesis λ ∈ Φ+ (A) with ind(A − λ) ≤ 0 ⇐⇒ λ ∈ Φ+ (B) =⇒ α(B − λ) < ∞ – a contradiction. Hence λ ∈ Φ+ (A) ∩ Φ+ (B) and ind(A − λ) + ind(B − λ) ≤ 0. Again, if σSF− (A) ∩ σSF+ (B) = ∅, then Φ− (A) ∪ Φ+ (B) = C. If λ ∈ / σaw (MC ), then λ ∈ Φ+ (A) =⇒ λ ∈ Φ(A), which (see above) implies that λ ∈ Φ+ (A) ∩ Φ+ (B) and

24

B.P. Duggal

IEOT

ind(A − λ) + ind(B − λ) ≤ 0. Hence if either of the hypotheses σaw (A) = σSF+ (B) and σSF− (A)∩σSF+ (B) = ∅ holds, then σaw (MC ) = σaw (M0 ). A similar argument, this time using the fact that λ ∈ / σw (MC ) =⇒ λ ∈ / σaw (MC ) = σaw (M0 ), shows that σw (MC ) = σw (M0 ). (See [2, Corollary 3.2 and Theorem 3.2] for a slightly different argument.) Thus if M0 satisfies Bt (resp., a − Bt), then MC has SVEP at λ ∈ / σw (MC ) (resp., λ ∈ / σaw (MC )), which implies that MC satisfies Bt (resp., a − Bt). Remark 3.4. If Ξ(A∗ )∪Ξ(B ∗ ) = ∅, then MC∗ has SVEP: this follows from a straightforward application of the definition of SVEP (applied to (MC∗ − λI ∗ )(f1 (λ) ⊕ f2 (λ)) = 0). Hence σ(M0 ) = σ(MC ) = σa (MC ), σaw (MC ) = σw (MC ) = σw (M0 ) and p0 (MC ) = pa0 (MC ). Evidently, both M0 and MC satisfy a − Bt. We call an operator T ∈ B(Y) polaroid [7] (resp., isoloid) at λ ∈ iso σ(T ) if asc(T − λ) = dsc(T − λ) < ∞ (resp., λ is an eigenvalue of T ). Trivially, T polaroid at λ implies T isoloid at λ. Since π0 (M0 ) = {π0 (A) ∩ ρ(B)} ∪ {ρ(A) ∩ π0 (B)} ∪ {π0 (A) ∩ π0 (B}, if M0 is polaroid at λ ∈ π0 (M0 ), then either A or B is polaroid at λ; in particular, A and B are polaroid at λ ∈ π0 (A) ∩ π0 (B). Conversely, if A is polaroid at λ ∈ π0 (A) and B is polaroid at µ ∈ π0 (B), then M0 is polaroid at ν ∈ π0 (M0 ). We say that T is a-polaroid if T is polaroid at λ ∈ iso σa (T ). Proposition 3.5. (i) M0 satisfies W t if and only if M0 has SVEP at λ ∈ / σw (M0 ) and M0 is polaroid at µ ∈ π0 (M0 ). (ii) M0 satisfies a − W t if and only if M0 has SVEP at λ ∈ / σaw (M0 ) and M0 is polaroid at µ ∈ π0a (M0 ). Proof. (i) is proved in [6, Theorem 2.2(i) and (ii)]. To prove (ii) we start by observing that if M0 has SVEP at λ ∈ / σaw (M0 ), then (M0 satisfies a − Bt =⇒) σa (M0 ) \ σaw (M0 ) = pa0 (M0 ) ⊆ π0a (M0 ), which if points in π0a (M0 ) are poles implies that π0a (M0 ) ⊆ pa0 (M0 ). Conversely, M0 satisfies a − W t implies M0 satisfies a − Bt, which in turn implies that M0 has SVEP at λ ∈ / σaw (M0 ). Again, since M0 (satisfies a − Bt and) a − W t, π0a (M0 ) = pa0 (M0 ). A similar argument proves the following: Proposition 3.6. (i) MC satisfies W t if and only if MC has SVEP at λ ∈ / σw (MC ) and MC is polaroid at µ ∈ π0 (MC ). (ii) MC satisfies a − W t if and only if MC has SVEP at λ ∈ / σaw (MC ) and MC is polaroid at µ ∈ π0a (MC ). The following theorem gives a necessary and sufficient condition for MC to satisfy W t in the case in which either of the hypotheses (i), (ii) and (iii) of Theorem 3.1 is satisfied. Theorem 3.7. If either of the SVEP hypotheses (i), (ii) and (iii) of Theorem 3.1(a) is satisfied, then MC satisfies W t for every C ∈ B(X ) if and only if M0 satisfies W t and A is polaroid at λ ∈ π0 (MC ).

Vol. 63 (2009)

Operator Matrices

25

Proof. Sufficiency. If M0 satisfies W t (hence, Bt) and either of the hypotheses (i), (ii) and (iii) of Theorem 3.1 is satisfied, then A satisfies Bt, σ(MC ) = σ(M0 ), σw (MC ) = σw (M0 ) and MC satisfies Bt (see Remark 3.2 and Theorem 3.1(a)). Hence σ(MC ) \ σw (MC ) = σ(M0 ) \ σw (M0 ) = p0 (M0 ) = π0 (M0 ) ⊆ π0 (MC ), where the final inclusion follows from the fact that σ(MC ) \ σw (MC ) = p0 (MC ) ⊆ π0 (MC ). Hence to prove sufficiency, we have to prove the reverse inclusion. Let λ ∈ π0 (MC ). Then λ ∈ iso σ(M0 ). Start by observing that (MC − λ)−1 (0) 6= ∅ =⇒ (M0 − λ)−1 (0) 6= ∅; also, dim(MC − λ)−1 (0) < ∞ =⇒ dim(A − λ)−1 (0) < ∞. We claim that dim(B−λ)−1 (0) < ∞. For suppose to the contrary that dim(B−λ)−1 (0) is infinite. Since (MC − λ)(x ⊕ y) = {(A − λ)x + Cy} ⊕ (B − λ)y, either dim(C(B − λ)−1 (0)) < ∞ or dim(C(B − λ)−1 (0)) = ∞. If dim(C(B − λ)−1 (0)) < ∞, then (B − λ)−1 (0) contains an orthonormal sequence {yj } such that (MC − λ)(0 ⊕ yj ) = 0 for all j = 1, 2, . . .. But then dim(MC − λ) = ∞, a contradiction. Assume now that dim(C(B − λ)−1 (0)) = ∞. Since λ ∈ ρ(A) ∪ iso σ(A), A satisfies Bt, A is polaroid at λ ∈ π0 (MC ) and α(A − λ) < ∞, β(A − λ) < ∞. Hence dim{C(B − λ)−1 (0) ∩ (A − λ)X } = ∞ implies the existence of a sequence {xj } such that (A − λ)xj = Cyj for all j = 1, 2, . . . . But then (MC − λ)(xj ⊕ −yj ) = 0 for all j = 1, 2, . . . . Thus dim(MC − λ)−1 (0) = ∞, again a contradiction. Our claim having been proved, we conclude that λ ∈ π0 (M0 ). Thus π0 (MC ) ⊆ π0 (M0 ). Necessity. Evidently, MC satisfies W t for all C implies M0 satisfies W t. Hence p0 (MC ) = π0 (MC ) = p0 (M0 ) = π0 (M0 ), which implies that M0 is polaroid at points λ ∈ π0 (MC ). Since π0 (MC ) = p0 (M0 ), and since λ ∈ p0 (M0 ) implies λ ∈ p0 (A) ∪ ρ(A), A is polaroid at λ ∈ π0 (MC ). Remark 3.8. An examination of the proof of the sufficiency part of the theorem above shows that if either of the SVEP hypotheses (i), (ii) and (iii) of Theorem 3.1(a) is satisfied and M0 satisfies W t, then either of the hypotheses that A is polaroid or A is isoloid and satisfies W t is sufficient for MC to satisfy W t. Corollary 3.9. (a) [4, Theorem 4.2] If {Ξ(A) ∩ Ξ(B ∗ )) ∪ Ξ(A∗ ) = ∅, A is polaroid at λ ∈ π0 (MC ) (or A is isoloid and satisfies W t) and M0 satisfies W t, then MC satisfies W t. (b) [2, Theorem 3.3] If σaw (A) = σSF+ (B) or σSF− (A) ∩ σSF+ (B) = ∅, A is polaroid at λ ∈ π0 (MC ) (or A is isoloid and satisfies W t) and M0 satisfies W t, then MC satisfies W t. Proof. (a) Theorem 3.7, and Remark 3.8, apply. (b) Recall from Corollary 2.5(b) that MC satisfies Bt ⇐⇒ σ(MC ) \ σw (MC ) = p0 (MC ). Hence σ(MC ) \ σw (MC ) ⊆ π0 (MC ). For the reverse inequality, start by recalling from the proof of Corollary 2.5(b) that σw (MC ) = σw (M0 ). If λ0 ∈ π0 (MC ), then there exists an – neighbourhood N of λ0 such that MC − λ is

26

B.P. Duggal

IEOT

invertible (implies A − λ is left invertible and B − λ is right invertible), hence Weyl, for all λ ∈ N not equal to λ0 . Thus M0 − λ is Weyl for all λ ∈ N not equal to λ0 . Since M0 satisfies Bt, M0 − λ is Browder for all λ ∈ N not equal to λ0 , which implies that both A − λ and B − λ are invertible. Hence λ ∈ iso σ(M0 ). Now argue as in the sufficiency part of the proof of Theorem 3.7. The following examples, [8] and [4], show that MC in theorem above may fail to satisfy W t if one assumes only that A is isoloid but not polaroid at λ ∈ π0 (MC ), or only that A is polaroid at λ ∈ π0 (A). Example. Let A, B and C ∈ B(`2 ) be the operators 1 1 A(x1 , x2 , x3 , . . .) = (0, x1 , 0, x2 , 0, x3 , . . .), 2 3 B(x1 , x2 , x3 , . . .) = (0, x2 , 0, x4 , 0, . . .), and C(x1 , x2 , x3 , . . .) = (0, 0, x2 , 0, x3 , . . .). ∗

Then A,A , B and B ∗ have SVEP, σ(A) = σw (A) = {0}, π0 (A) = p0 (A) = ∅, and A satisfies Weyl’s theorem. Since σ(M0 ) = σw (M0 ) = {0, 1} and π0 (M0 ) = p0 (M0 ) = ∅, M0 satisfies W t. However, since σ(MC ) = σw (MC ) = {0, 1} and π0 (MC ) = {0}, MC does not satisfy W t. Observe that A is not polaroid on π0 (MC ). Again, let A, B and C ∈ B(`2 ) be the operators 1 1 A(x1 , x2 , x3 , . . .) = (0, 0, 0, x2 , 0, x3 , . . .), 2 3 B(x1 , x2 , x3 , . . .) = (0, x2 , 0, x4 , 0, . . .), and C(x1 , x2 , x3 , . . .) = (x1 , 0, x2 , 0, x3 , . . .). Then A, B (and C) have SVEP, σ(A) = σw (A) = π0 (A) = {0}, and σ(B) = σw (B) = {0, 1}, π0 (B) = p0 (B) = ∅. Since σ(M0 ) = σw (M0 ) = {0} and π0 (M0 ) = p0 (M0 ) = ∅, M0 satisfies W t. However, since σ(MC ) = σw (MC ) = {0, 1} and π0 (MC ) = {0}, MC does not satisfy W t. Observe that 0 ∈ / p0 (A); A satisfies Bt, but does not satisfy W t. More can be said in the case in which Ξ(A∗ ) ∪ Ξ(B ∗ ) = ∅. Recall from Remark 3.4 that if Ξ(A∗ ) ∪ Ξ(B ∗ ) = ∅, then MC∗ has SVEP and MC satisfies a − Bt. Theorem 3.10. If Ξ(A∗ ) ∪ Ξ(B ∗ ) = ∅, A is polaroid at λ ∈ π0a (MC ) (or, A is isoloid and satisfies W t) and B is polaroid at µ ∈ π0a (B), then MC satisfies a − W t.

Vol. 63 (2009)

Operator Matrices

27

Proof. Since A∗ and B ∗ have SVEP, both M0∗ and MC∗ have SVEP. Hence MC (also, M0 ) satisfies Bt, which implies that σ(MC ) \ σw (MC ) = p0 (MC ) ⊆ π0 (MC ). Apparently, σ(M0 ) = σ(MC ) = σa (MC ), σw (M0 ) = σw (MC ) = σaw (MC ), π0 (MC ) = π0a (MC ) and iso σ(MC ) = iso σ(M0 ). Following (part of) the argument of the proof of the sufficiency part of Theorem 3.7, it follows that if λ ∈ π0 (MC ), then λ ∈ π0 (A) ∩ π0 (B). By assumption, both A and B are polaroid at λ. Hence M0 is polaroid at λ, which implies that λ ∈ p0 (M0 ). Since M0 satisfies Bt, λ ∈ / σw (M0 ) = σw (MC ), which in view of the fact that MC satisfies Bt implies that λ ∈ p0 (MC ). Hence σ(MC ) \ σw (MC ) = π0 (MC ) =⇒ σa (MC ) \ σaw (MC ) = π0a (MC ), i.e., MC satisfies a − W t. Theorem 3.10 holds for polaroid operators A and B: for the polaroid hypothesis implies that M0 is polaroid, hence satisfies W t, which by Theorem 3.7 implies that MC satisfies W t. If the operators A and B have SVEP, then M0 and MC have SVEP, σ(M0 ) = σ(MC ) = σ(MC∗ ) = σa (MC∗ ), iso σ(M0∗ ) = iso σ(MC∗ ) = iso σa (MC∗ ), π0 (MC∗ ) = π0a (MC∗ ) and σw (M0 ) = σw (MC ) = σw (MC∗ ) = σaw (MC∗ ). Evidently, A∗ , B ∗ , M0∗ and MC∗ satisfy Bt; in particular, p0 (M0∗ ) = p0 (MC∗ ) ⊆ π0 (MC∗ ). Corollary 3.11. If the polaroid operators A and B have SVEP, then M0 satisfies W t and MC∗ satisfies a − W t. Proof. Apparently, MC satisfies W t. Since the polaroid hypothesis on A and B implies that A∗ and B ∗ are polaroid, an argument similar to that in the theorem above applied to MC∗ implies that if λ ∈ π0 (MC∗ ), then λ ∈ π0 (B ∗ ) ∩ π0 (A∗ ) =⇒ λ ∈ p0 (B ∗ ) ∩ p0 (A∗ ) =⇒ λ ∈ / σw (M0∗ ) = σw (MC∗ ) =⇒ MC∗ satisfies W t. Hence MC∗ satisfies a − W t.

References [1] P. Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer, 2004. [2] X. Cao and B. Meng, Essential approximate point spectra and Weyl’s theorem for operator matrices, J. Math. Anal. Appl. 304 (2005), 759-771. [3] D. Djordjevi´ c, Perturbation of spectra of operator matrices, J. Oper. Th. 48 (2002), 467-486. [4] S. V. Djordjevi´ c and Hassan Zguitti, Essential point spectra of operator matrices through local spectral theory, J. Math. Anal. Appl. 338 (2008), 285-291. [5] B.P. Duggal, Hereditarily normaloid operators, Extracta Math. 20 (2005), 203-217. [6] B. P. Duggal, Polaroid operators satisfying Weyl’s theorem, Lin. Alg. Appl. 414 (2006), 271-277. [7] B. P. Duggal, R. E. Harte and I. H. Jeon, Polaroid operators and Weyl’s theorem, Proc. Amer. Math. Soc. 132 (2004), 1345-1349. [8] W. Y. Lee, Weyl’s theorem for operator matrices, Integr. Equat. Op. Th. 32 (1998), 319-331

28 B.P. Duggal 8 Redwood Grove Ealing London W5 4SZ United Kingdom e-mail: [email protected] Submitted: August 12, 2007. Revised: August 17, 2008.

B.P. Duggal

IEOT

Integr. equ. oper. theory 63 (2009), 29–46 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010029-18, published online December 22, 2008 DOI 10.1007/s00020-008-1647-9

Integral Equations and Operator Theory

On Toeplitz-type Operators Related to Wavelets Ondrej Hutn´ık Abstract. Let G be the “ax + b”-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on L2 (R). The structure of the space of Calder´ on (wavelet) transforms Wψ (L2 (R)) inside L2 (G, dν) is described. Using this result some representations, properties and the Wick calculus of the Calder´ on-Toeplitz operators Ta acting on Wψ (L2 (R)) whose symbols a = a(ζ) depend on v = =ζ for ζ ∈ G are investigated. Mathematics Subject Classification (2000). Primary 46E22, 47B35; Secondary 42C40. Keywords. Calder´ on reproducing formula, admissible wavelet, Toeplitz operator, Wick calculus, localization operator.

1. Introduction Let G be the noncompact group of shifts and dilations acting on L2 (R) with the left invariant Haar measure dν, the so called “ax+b”-group. In the first part of this paper we study the associated space of Calder´on (or wavelet) transforms Wψ (L2 (R)) and show its structure inside the space L2 (G, dν) after identifying the group G with the upper half-plane Π in the complex plane C. The main idea is based on Vasilevski’s paper [24], where the structure of Bergman and poly-Bergman spaces in L2 (Π) was obtained (and its complete decomposition onto them). The representation of the space of Calder´on transforms Wψ (L2 (R)) is especially important in the study of the Calder´on-Toeplitz operators with symbols depending only on v, the imaginary part of a variable ζ = u + iv (as in the case of Bergman spaces and Toeplitz operators acting on them, cf. [9]). For a given bounded function a on G and an admissible wavelet ψ on L2 (R), the Calder´ onToeplitz operator Ta with symbol a is defined to be the map of L2 (R) to L2 (R) This paper was supported by Grant VEGA 2/0097/08.

30

Ondrej Hutn´ık

IEOT

given by Z a(ζ)hf, ψζ iψζ dν(ζ),

Ta f =

f ∈ L2 (R).

G

Taking the inner product with a function g ∈ L2 (R), the definition of Ta may be written in a weak sense, namely, Z hTa f, gi = a(ζ)hf, ψζ ihψζ , gi dν(ζ), f, g ∈ L2 (R), (1.1) G

according to the Calder´ on reproducing formula (a particular case of reproducing formulas related to square integrable representations defining a class of Hilbert spaces with reproducing kernels, see Section 2). Alternatively, the Calder´ on-Toeplitz operator Ta may be viewed as acting on L2 (G, dν) and given as Pψ Ma Pψ , where Pψ is the orthogonal projection from L2 (G, dν) onto Wψ (L2 (R)) and Ma is the operator of pointwise multiplication by a on L2 (G, dν). Thus, Pψ Ma Pψ is a Toeplitz operator acting on the Hilbert space Wψ (L2 (R)) and it has the matrix representation Wψ Ta Wψ∗ 0 0 0 with respect to the decomposition L2 (G, dν) = Wψ (L2 (R)) ⊕ Wψ (L2 (R))⊥ , where Wψ , resp. Wψ∗ , is the continuous wavelet transform operator, resp. its adjoint (see Section 2). Hence the name Calder´on-Toeplitz operator. The realization on L2 (R) is of intrinsic interest as an alternative quantization to classical pseudodifferential calculus. The realization using Wψ (L2 (R)) is of our present interest. Clearly, it is also possible to define a more general class of Toeplitz-type operators including these of Calder´on-Toeplitz. Such operators are based on the group theoretical approach (the group representations): let U be a representation of a group G acting on functions defined on Rn , a be a function defined on G and φ be a function defined on Rn . The operator Ta,φ acting on functions on Rn by Z Ta,φ f = a(g)hf, Ug φiUg φ dg, G

where h·, ·i is the inner product of L2 (Rn ) and dg is the left invariant measure on G, is called the Toeplitz operator based on the representation U . Note that such operators are, in fact, examples of a more general A-Toeplitz operator defined in [7]. The particular case of Toeplitz-type operators with respect to specific representations (the Schr¨ odinger representation of the reduced Heisenberg group and the natural representation of the “ax + b”-group) was investigated in [16]. The study of Toeplitz-type operators based on the Calder´on reproducing formula is a relatively new area of operator research which began in the early 1990’s. These operators were introduced by R. Rochberg in [21] as a wavelet counterpart of Toeplitz operators defined on Hilbert spaces of holomorphic functions. They are the model operators that fit nicely in the context of wavelet decomposition

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

31

of function spaces and almost diagonalization of operators. This class of operators includes as Toeplitz operators on classical Fock spaces and weighted Bergman spaces of the unit disk as many interesting classes of Fourier multiplier operators, singular integral operators, paracommutators and paraproducts. Also, these operators are an effective time-frequency localization tool, cf. [6], which provides ways of analyzing signals by describing their frequency content as it varies over time, cf. [14]. As far as we know, there is only a few papers of R. Rochberg and K. Nowak on this topic investigating the Calder´on-Toeplitz operators mainly with respect to their eigenvalues estimates and Schatten ideal criteria, cf. [15], [16], [21], [22]. However, the deeper results on the properties of Calder´on-Toeplitz operators (e.g. compactness, boundedness, spectra, etc.) are not known in general. We start with the description of the transform, the unitary operator which maps the space of Calder´ on transforms Wψ (L2 (R)) onto L2 -type spaces. Besides of immediate necessity of this transform, it provides the unitary equivalence of Calder´ on-Toeplitz operators acting on Wψ (L2 (R)) whose symbols depend only on v = =ζ, ζ ∈ G, with the multiplication operators acting on L2 (R), this type of transform is also of great importance itself in wavelet analysis. The key result, which gives an access to the properties of Calder´on-Toeplitz operators studied in the second part of this paper, is established in Section 4. Namely, we prove that the Calder´on-Toeplitz operator Ta with symbol a(v) depending only on v = =ζ, ζ ∈ G, is unitarily equivalent to the multiplication operator γa I acting on L2 (R), where the function γa is given by Z γa (ξ) = R+

2 ˆ a(v)|ψ(vξ)|

dv , v

ξ ∈ R.

In this context we also mention the Wick (or covariant, or Berezin) symbol e a(ζ) of the Calder´ on-Toeplitz operator Ta which together with the star product carries as well many essential properties of the corresponding operator. The star product defines the composition of two Wick symbols e aA and e aB of the operators A and B, respectively, as the Wick symbol of the composition AB, i.e., e aA ∗ e aB = e aAB . In Section 4 we also state the formulas for the Wick symbols of Calder´on-Toeplitz operators Ta whose symbols depend only on v = =ζ, as well as the formulas for the star product in terms of our function γa . The obtained results shed new light upon the properties of Calder´on-Toeplitz operator Ta , e.g. boundedness (Corollary 4.2), spectral-type representation (Theorem 4.3), and serve as useful tools for further investigating of spectra, compactness and other properties of these wavelet-based Toeplitz operators. Also, using these results in connection with harmonic analysis and time-frequency analysis may bring new insight on problems in these areas.

32

Ondrej Hutn´ık

IEOT

2. Preliminaries Here we use the obvious notations: R is the set of all real numbers, R+ (R− ) is the positive (negative) half-line and χ+ (χ− ) is the characteristic function of R+ (R− ). The Calder´ on reproducing formula is the following resolution of unity on L2 (R), Z hf, gi = hf, ψζ ihψζ , gi dν(ζ), f, g ∈ L2 (R), (2.1) G

where G = {ζ = (u, v); u ∈ R, v > 0} is the “ax + b”-group with the left invariant Haar measure dν(ζ) = v −2 du dv. The Calder´on reproducing formula is used as a starting point for the construction of time-frequency localization or filter operators and is used in many fields of science and technology. Historically, it is usually connected with the paper [5], but its basic idea was known before. The group G acts on L2 (R) via translations and dilations, i.e. for ζ = (u, v) ∈ G, the unitary representation ρ of G is given by x−u 1 , (ρζ ψ)(x) = ψu,v (x) = √ ψ v v where ψ ∈ L2 (R) is an admissible wavelet satisfying Z 2 dξ ˆ = 1, |ψ(xξ)| ξ R+

(2.2)

for almost every x ∈ R, and ψˆ stands for the Fourier transform F : L2 (R) → L2 (R) given by Z F{g}(ξ) = gˆ(ξ) = g(x)e−2πixξ dx. (2.3) R

Thus, ψu,v is the function ψ translated to be centered at u, scaled to a width ˆ of v, and renormalized to be a unit vector. Recall that (ˆ ρζ ψ)(ξ) = ψˆu,v (ξ) = √ ˆ −2πiuξ v ψ(vξ)e on Fourier transform side. The integral (2.1) is understood in a weak sense. It is not hard to check that the admissibility condition (2.2) is not only sufficient, but also necessary for the Calder´on reproducing formula to hold. In what follows we identify the group G with the upper half-plane Π = {ζ = u + iv; u ∈ R, v > 0} in the complex plane C and consider an arbitrary realvalued admissible wavelet ψ ∈ L2 (R). Also, h·, ·i always means the inner product on L2 (R), whereas h·, ·iG denotes the inner product on L2 (G, dν). For a fixed ψ ∈ L2 (R), the functions Wψ f on G of the form (Wψ f )(η) = hf, ψη i,

f ∈ L2 (R),

(2.4)

form a reproducing kernel Hilbert space Wψ (L2 (R)) called the space of Calder´ on (wavelet) transforms (since Wψ f is called the continuous wavelet transform of a function f with respect to the analyzing wavelet ψ). The space Wψ (L2 (R)) is a

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

33

closed subspace of L2 (G, dν). Now, for an admissible ψ, the Calder´on reproducing formula (2.1) reads as follows: hf, gi = hWψ f, Wψ giG ,

f, g ∈ L2 (R),

which means that the continuous wavelet transform operator Wψ : L2 (R) → L2 (G, dν) given by (2.4) is an isometry. Consequently, for an admissible wavelet ψ ∈ L2 (R) and for all f ∈ L2 (R) holds Z |(Wψ f )(η)|2 dν(η) = kf k2 , (2.5) G

and the integral operator Pψ : L2 (G, dν) → L2 (G, dν) given by Z (Pψ F )(η) = F (ζ)Kζ (η) dν(ζ), F ∈ L2 (G, dν),

(2.6)

G

is the orthogonal projection onto Wψ (L2 (R)), where Kζ (η) = hψζ , ψη i is the reproducing kernel in Wψ (L2 (R)). Since ψ is a real-valued admissible wavelet, then Kζ (η) is a real symmetric (reproducing) kernel. Obviously, hKζ , Kη iG = Kζ (η), and thus |hKζ , Kη iG | = |hψζ , ψη i|. Easily we have Lemma 2.1. If F ∈ Wψ (L2 (R)), then Pψ F = F , i.e. for all ζ ∈ G, Z F (ζ) = F (η)Kζ (η) dν(η).

(2.7)

G

This means that a function f ∈ L2 (G, dν) is the wavelet transform of a certain signal if and only if it satisfies the reproducing property (2.7) with (Wψ f )(·) = F (·). A special case of the reproducing formula above is the following identity: Z 1= |Kζ (η)|2 dν(η), ζ ∈ G. G

As it was stated in introduction we may identify operators Ta and Pψ Ma Pψ on the Hilbert space Wψ (L2 (R)). Thus for F ∈ Wψ (L2 (R)) we get (Ta F )(ζ) = haPψ F, Kζ iG = haF, Kζ iG .

(2.8)

The following easy result implies that the Calder´on-Toeplitz operator acting on Wψ (L2 (R)) is the integral operator with kernel (Ta Kη )(ζ). Theorem 2.2. Let a be a real bounded integrable function on G and F ∈ Wψ (L2 (R)). Then the Calder´ on-Toeplitz operator acting on Wψ (L2 (R)) has the form Z (Ta F )(ζ) = F (η)(Ta Kη )(ζ) dν(η). (2.9) G

Moreover, if Ta is bounded on Wψ (L2 (R)), then Ta is self-adjoint.

34

Ondrej Hutn´ık

IEOT

Proof. According to (2.8) and (2.7) we may write Z (Ta F )(ζ) = a(θ)F (θ)Kζ (θ) dν(θ) G Z Z = a(θ) F (η)Kθ (η) dν(η) Kζ (θ) dν(θ) G ZG Z = F (η) a(θ)Kη (θ)Kζ (θ) dν(θ) dν(η) G ZG = F (η)(Ta Kη )(ζ) dν(η). G

If Ta is bounded on Wψ (L2 (R)), then Ta∗ is bounded on Wψ (L2 (R)) as well. Thus, for F ∈ Wψ (L2 (R)) we have (Ta∗ F )(ζ) = hTa∗ F, Kζ iG = hF, Ta Kζ iG = hF, aKζ iG = (Ta F )(ζ).

3. Representation of Wψ (L2 (R)) It is well known that the one-dimensional wavelet analysis is an intermediate between the function theory on the upper half-plane of one complex variable and the harmonic analysis on the real line, cf. [19]. As a motivation, let us recall the following well known results describing the relation between weighted Bergman spaces on the upper half-plane and the space of Calder´on transforms of Hardy space functions with respect to Bergman wavelets, cf. [8], [19]: Let H2 (R) be the Hardy space of all square integrable functions whose Fourier transform is supported on R+ , i.e. H2 (R) = {f ∈ L2 (R); fˆ(ξ) = 0 a.e. ξ ≤ 0}, and take the specific (Bergman) wavelet ψ α , i.e. for α > 0, cα ξ α e−2πξ for ξ > 0, α ˆ ψ (ξ) = 0 for ξ ≤ 0, where Z cα =

dξ |ξ α e−2πξ |2 ξ R+

!− 21

(4π)α =p Γ(2α)

is the normalization factor (Γ(z) is the Euler gamma function). Let us denote by Wψα (H2 (R)) the space of wavelet transforms of functions in the Hardy space H2 (R) with respect to Bergman wavelet ψ α . Clearly, Wψα (H2 (R)) is a reproducing kernel Hilbert space with the following kernel 2α+1 α 2i α α α+1/2 hψη , ψζ i = (tv) , 2π ζ −η

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

35

where ζ = (u, v), η = (s, t) ∈ G. Let Aβ (G), β > −1, stand for the weighted Bergman space of holomorphic functions on G satisfying Z Z kF k2Aβ = |F (u + iv)|2 v β dudv < ∞. R+

R

The identity (2.5) motivates the definition of the (Bergman) transform 1

(B α F )(u, v) = v −α− 2 F (u, v), which leads to the following Theorem 3.1. The unitary operator B α gives an isometrical isomorphism of the space L2 (G, dν) onto L2 (G, v 2α−1 dudv) under which the space of Calder´ on transforms Wψα (H2 (R)) is mapped onto the weighted Bergman space A2α−1 (G). The unitary map B α from Cψα to A2α−1 provides a unitary equivalence between commutators defined by the wavelet ψ α and their Bergman space analogues, cf. [17]. Also, this result provides a good tool for studying Toeplitz operators on weighted Bergman spaces on the upper half-plane which (in this case) become unitarily equivalent to Calder´ on-Toeplitz operators, see [15]. A direct Fourier transform calculation yields the following useful lemma. Its proof is easy and therefore omitted. Lemma 3.2. Let ζ = (u, v) ∈ G and η = (s, t) ∈ G. Then the Fourier transform (with respect to s = <η) of the kernel function Kη (ζ) is equal to √ −2πiuξ ˆ ˆ tv ψ(vξ) ψ(tξ)e . Our first aim is to give some more general results (analogous to Theorem 3.1 in some sense) involving an arbitrary real-valued admissible wavelet ψ ∈ L2 (R) and to obtain new representation of the space of wavelet transforms Wψ (L2 (R)), which will be useful later for investigating the Calder´on-Toeplitz operators. For this purpose, let us introduce the unitary operator U1 = (F ⊗ I) : L2 (G, dν(ζ)) = L2 (R, du) ⊗ L2 (R+ , v −2 dv) → L2 (R, du) ⊗ L2 (R+ , v −2 dv), where ζ = (u, v) ∈ G, and denote by ∆1 the image of the space Wψ (L2 (R)) in the mapping U1 . The space ∆1 consists of all functions √ ˆ F (u, v) = v f (u)ψ(uv), (3.1) where f ∈ L2 (R) and ψ ∈ L2 (R) is a real admissible wavelet. Moreover, we have kF (u, v)k∆1 = kf (u)kL2 (R, du) . The orthogonal projection Λ1 : L2 (G, dν) → ∆1 has obviously the form Λ1 = U1 Pψ U1−1 = (F ⊗ I)Pψ (F −1 ⊗ I), and therefore (F −1 ⊗ I)Λ1 F = Pψ (F −1 ⊗ I)F = h(F −1 ⊗ I)F, Kη iG = hF, (F ⊗ I)Kη iG .

36

Ondrej Hutn´ık

IEOT

According to Lemma 3.2 we get (F −1 ⊗ I)Λ1 F = hF, (F ⊗ I)Kη iG Z Z √ dt 2πiuξ ˆ ˆ dξ tv F (ξ, t) ψ(tξ) ψ(vξ)e = t2 R+ R ! Z Z √ dt ˆ ˆ = v e2πiuξ dξ ψ(vξ) F (ξ, t) ψ(tξ) 3/2 t R R+ ! Z √ dt −1 ˆ ˆ = (F ⊗ I) , v ψ(uv) F (u, t) ψ(ut) t3/2 R+ or equivalently, (Λ1 F )(u, v) =

√

ˆ v ψ(uv)

Z

dt ˆ F (u, t) ψ(ut) . 3/2 t R+

Thus for each function F ∈ L2 (G, dν) its image (Λ1 F )(u, v) has the form (3.1) with Z f (u) =

dt ˆ F (u, t)ψ(ut) , 3/2 t R+

and therefore for each function F0 of the form (3.1) we have (Λ1 F0 )(u, v) = F0 (u, v). Indeed √

Z

dt 3/2 t R+ Z √ 2 dt ˆ ˆ = v ψ(uv)f (u) |ψ(ut)| t R+ √ ˆ = v ψ(uv)f (u) = F0 (u, v).

(Λ1 F0 )(u, v) =

ˆ v ψ(uv)

ˆ F0 (u, t) ψ(ut)

Thus ∆1 coincides with the space of all functions of the form (3.1). Introduce now the unitary operator U2 : L2 (R, du) ⊗ L2 (R+ , v −2 dv) → L2 (R, dx) ⊗ L2 (R+ , dy) by the rule p U2 : F (u, v) 7→

|x| F y

y x, |x|

.

Then the inverse operator U2−1 = U2∗ : L2 (R, dx) ⊗ L2 (R+ , dy) → L2 (R, du) ⊗ L2 (R+ , v −2 dv) is given by U2−1 : F (x, y) 7→

p |u|v F (u, |u|v).

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

37

Denote by ∆2 = U2 (∆1 ). Then the operator Λ2 = U2 Λ1 U2−1 is obviously the orthogonal projection of L2 (G, dν) onto ∆2 , and (Λ2 F )(x, y) = (U2 Λ1 U2−1 F )(x, y) Z p ˆ |u|v ψ(uv) = U2

dθ ˆ √ F (u, |u|θ) ψ(uθ) θ R+

1 ˆ = √ ψ(sgn(x)y) y

Z R+

!

dτ ˆ F (x, τ ) ψ(sgn(x)τ )√ . τ

Now with F ∈ ∆1 the space ∆2 ⊂ L2 (R, dx) ⊗ L2 (R+ , dy) consists of all functions of the form p |x| y 1 ˆ (U2 F )(x, y) = F x, = √ ψ(sgn(x)y) f (x). y |x| y Introducing 1 ˆ `± (y) = √ ψ(±y), y we obviously have that `± (y) ∈ L2 (R+ , dy) and from the admissibility condition (2.2) k`± (y)k = 1. For each f ∈ L2 (R), let f± (x) = χ± (x)f (x) be its restriction onto positive and negative half-lines. Then (U2 F )(x, y) = f+ (x)`+ (y) + f− (x)`− (y). Denote by L± the one-dimensional subspaces of L2 (R+ , dy) generated by functions `± (y). Then ∆2 = L2 (R+ ) ⊗ L+ ⊕ L2 (R− ) ⊗ L− , and the one-dimensional projections P± of L2 (R+ , dy) onto L± have the form Z dτ 1 ˆ ˆ (P± H)(y) = hH, `± i`± = √ ψ(±y) H(τ )ψ(±τ )√ . (3.2) y τ R+ Thus, Λ2 = χ+ I ⊗P+ ⊕χ− I ⊗P− . This leads immediately to the following theorem which describes the structure of the space of wavelet transforms Wψ (L2 (R)) inside L2 (G, dν). Theorem 3.3. The unitary operator U = U2 U1 gives an isometrical isomorphism of the space L2 (G, dν) = L2 (R, du) ⊗ L2 (R+ , v −2 dv) onto L2 (R, dx) ⊗ L2 (R+ , dy) under which (i) the space of Calder´ on transforms Wψ (L2 (R)) is mapped onto L2 (R+ ) ⊗ L+ ⊕ L2 (R− ) ⊗ L− with U : Wψ (L2 (R)) 7→ L2 (R+ ) ⊗ L+ ⊕ L2 (R− ) ⊗ L− , where L± are the one-dimensional subspaces of L2 (R+ , dy) generated by functions 1 ˆ `± (y) = √ ψ(±y); y

38

Ondrej Hutn´ık

IEOT

(ii) the projection Pψ is unitarily equivalent to U Pψ U −1 = χ+ I ⊗ P+ ⊕ χ− I ⊗ P− , where P± are the one-dimensional projections of L2 (R+ , dy) onto L± given in (3.2). Remark 3.4. Let us mention that by taking a system of admissible wavelets ψn forming all together an orthonormal basis of L2 (R) one gets a full decomposition of the space L2 (G, dν). We also mention the paper [10] where the above construction is applied for the specific wavelets on R and wavelet subspaces are studied in more detail therein. An orthogonal decomposition of admissible wavelets is also constructed e.g. in [11], and [13]. The result of Theorem 3.3 may be viewed as a “wavelet version” of the result of Vasilevski [24] obtained for the Bergman spaces. In what follows we continue in this direction and construct operators which provides a decomposition of the projection Pψ and of the identity operator on L2 (R). Introduce the operator R0 : L2 (R) → L2 (R+ ) ⊗ L2 (R+ ) ⊕ L2 (R− ) ⊗ L2 (R+ ) by the rule (R0 f )(x, y) = f+ (x)`+ (y) + f− (x)`− (y). Obviously, the image of R0 coincides with the space ∆2 . The adjoint operator R0∗ : L2 (R+ ) ⊗ L2 (R+ ) ⊕ L2 (R− ) ⊗ L2 (R+ ) → L2 (R) is given by (R0∗ F )(x)

Z

Z

F (x, τ )`− (τ ) dτ,

F (x, τ )`+ (τ ) dτ + χ− (x)

= χ+ (x) R+

R+

and R0∗ R0 = I : L2 (R) → L2 (R), R0 R0∗ = Λ2 : L2 (R+ ) ⊗ L2 (R+ ) ⊕ L2 (R− ) ⊗ L2 (R+ ) → ∆2 . Summarizing the above construction we have Theorem 3.5. The operator R = R0∗ U maps the space L2 (G, dν) onto L2 (R), and the restriction R|Wψ (L2 (R)) : Wψ (L2 (R)) → L2 (R) is an isometrical isomorphism. The adjoint operator R∗ = U ∗ R0 : L2 (R) → Wψ (L2 (R)) ⊂ L2 (G, dν) is an isometrical isomorphism of the space L2 (R) onto the subspace Wψ (L2 (R)) of the space L2 (G, dν).

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

39

Corollary 3.6. According to the construction of operators R and R∗ we have: RR∗ = I : L2 (R) → L2 (R), R∗ R = Pψ : L2 (G, dν) → Wψ (L2 (R)). Theorem 3.7. The isometrical isomorphism R∗ = U ∗ R0 : L2 (R) → Wψ (L2 (R)) is given by (R∗ f )(z) =

√

Z y

2πixξ ˆ f (ξ)ψ(yξ)e dξ,

z = (x, y) ∈ G.

(3.3)

R

Proof. The direct calculation yields (R∗ f )(z) = (U ∗ R0 f )(z) = (U1∗ U2∗ R0 f )(z) p = (F −1 ⊗ I) |ξ|y (f+ (ξ)`+ (y|ξ|) + f− (ξ)`− (y|ξ|)) √ ˆ ˆ = (F −1 ⊗ I) y f+ (ξ)ψ(y|ξ|) + f− (ξ)ψ(−y|ξ|) √ ˆ = (F −1 ⊗ I) y f (ξ)ψ(yξ) Z √ 2πixξ ˆ = y f (ξ)ψ(yξ)e dξ.

R

Corollary 3.8. The inverse isomorphism R = R0∗ U : Wψ (L2 (R)) → L2 (R) is given by Z (RF )(ξ) =

−2πiuξ dudv ˆ F (ζ)ψ(vξ)e , v 3/2 G

ζ = (u, v) ∈ G.

(3.4)

Remark 3.9. The results of this section may be viewed as a special case of constructing of operators which is based on a general scheme recently presented in [20] and applied in many different settings (e.g., Bergman-type spaces, Fock-type spaces, etc.).

4. Calder´on-Toeplitz operators with symbols depending on v = =ζ Now, using the representation of Wψ (L2 (R)) and operators from the previous section we may state the following theorem describing the important tool for studying the properties of Calder´ on-Toeplitz operator Ta with symbol a depending only on v = =ζ. Theorem 4.1. Let a = a(v) be a measurable function on L2 (R+ ). Then the Calder´ on-Toeplitz operator Ta acting on Wψ (L2 (R)) is unitarily equivalent to the

40

Ondrej Hutn´ık

IEOT

multiplication operator γa I = RTa R∗ acting on L2 (R), where R and R∗ are given by (3.4) and (3.3), respectively, and γa (u) = χ+ (u)γa+ (u) + χ− (u)γa− (u), where Z τ 2 ± |`± (τ )| dτ, u ∈ R. γa (u) = χ± (u) a |u| R+ Proof. The operator Ta is obviously unitarily equivalent to the operator RTa R∗ = RPψ aPψ R∗ = R(R∗ R)a(R∗ R)R∗ = (RR∗ )RaR∗ (RR∗ ) = RaR∗ = R0∗ U2 (F ⊗ I)a(v)(F −1 ⊗ I)U2−1 R0 = R0∗ U2 a(v)U2−1 R0 y ∗ R0 . = R0 a |x| Finally,

R0∗ a

y |x|

R0 f

Z (x) = f+ (u)χ+ (u)

τ |u|

τ |u|

a R+

Z + f− (u)χ− (u)

a R+

|`+ (τ )|2 dτ |`− (τ )|2 dτ

= f (u) χ+ (u)γa+ (u) + χ− (u)γa− (u) = f (u)γa (u), where for u ∈ R γa± (u) = χ± (u)

Z a R+

and γa (u) = χ+ (u)γa+ (u) + χ− (u)γa− (u).

τ |u|

2

|`± (τ )| dτ,

Corollary 4.2. The Calder´ on-Toeplitz operator Ta with symbol a = a(v) is bounded on Wψ (L2 (R)) if and only if the corresponding function γa (u) is bounded. Reverting the statement of Theorem 4.1 we come to the following spectraltype representation of a Calder´on-Toeplitz operator (which is much easier than the representation (2.9)). Its proof goes directly from Theorem 4.1, Theorem 3.7 and Corollary 3.8. Theorem 4.3. Let ζ = (u, v) ∈ G. If a(ζ) = a(v) is a measurable function on L2 (R+ ), then the Calder´ on-Toeplitz operator Ta acting on Wψ (L2 (R)) has the following representation Z √ ˆ (Ta F )(ζ) = v γa (ξ)ψ(vξ)f (ξ)e2πiuξ dξ, (4.1) R

where f (ξ) = (RF )(ξ). At the same time it is instructive to give a direct proof of the theorem which does not use the results of the previous section.

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

41

Proof. Indeed, for a symbol a = a(v) depending only on v = =ζ consider the Calder´ on-Toeplitz operator, see (2.8), Z (Ta F )(ζ) = a(t)F (η)Kζ (η) dν(η), G

where η = (s, t) ∈ G. Represent the function F (η) as follows √ Z 2πisξ ˆ dξ, F (η) = t f (ξ)ψ(tξ)e R

where f ∈ L2 (R). Thus, Z Z (Ta F )(ζ) =

! dt 2πisξ ˆ t a(t) f (ξ)ψ(tξ)e dξ Kζ (η) 2 ds t R R R+ Z ! Z √ Z dt ˆ t a(t)ψ(tξ) Kζ (η)e2πisξ ds = f (ξ) dξ. t2 R+ R R √

Z

Using Lemma 3.2 we have Z √ 2πiuξ ˆ ˆ ψ(vξ)e , Kζ (η)e2πisξ ds = F(Kη (ζ))(ξ) = tv ψ(tξ) R

and therefore (Ta F )(ζ) =

√

Z v

2πiuξ ˆ f (ξ)ψ(vξ)e

R

It remains to prove that

R R+

2 ˆ a(t)|ψ(tξ)|

dt t

Z

2 dt ˆ a(t)|ψ(tξ)| t R+

! dξ.

= γa (ξ). Indeed,

γa (ξ) = χ+ (ξ)γa+ (ξ) + χ− (ξ)γa− (ξ) Z Z τ τ 2 = χ+ (ξ) a |`+ (τ )| dτ + χ− (ξ) a |`− (τ )|2 dτ |ξ| |ξ| R+ R+ Z Z τ τ dτ ˆ )|2 dτ + χ− (ξ) ˆ |ψ(τ a |ψ(−τ )|2 . = χ+ (ξ) a |ξ| τ |ξ| τ R+ R+ Using the substitution τ = t|ξ|, we get Z Z 2 dt 2 dt ˆ ˆ + χ− (ξ) a(t)|ψ(−t|ξ|)| γa (ξ) = χ+ (ξ) a(t)|ψ(t|ξ|)| t t R R Z + Z + 2 dt 2 dt ˆ ˆ = χ+ (ξ) a(t)|ψ(tξ)| + χ− (ξ) a(t)|ψ(tξ)| t t R+ R+ Z 2 dt ˆ = a(t)|ψ(tξ)| . t R+

Remark 4.4. Observe that the function γa is constructed by putting a multiplier in admissibility condition (2.2).

42

Ondrej Hutn´ık

IEOT

It is straightforward to check that if the symbol a of the Calder´on-Toeplitz operator Ta depends on u, i.e. a(ζ) = a(u) with ζ = (u, v) ∈ G, then the operator Ta is closely related to the operator of multiplication by a(u). For example, kTa k ∼ kak∞ , and such a Ta will never be compact. Also the case a(ζ) = a(v) was studied in [16] and the following statement was given therein without the proof. It may be seen that Ta is given by a convolution with the inverse Fourier transform of γa . For the sake of completeness we state it in our notation and give its short proof. Lemma 4.5. (cf. [16]) If a(ζ) = a(v) with ζ = (u, v) ∈ G, then the Calder´ onToeplitz operator Ta has the form Z g (ξ) dξ. hTa f, gi = γa (ξ)fˆ(ξ)ˆ R

Proof. By Fourier representation of Wψ we have Z √ 2πiuξ ˆ Wψ f (u, v) = v fˆ(ξ)ψ(vξ)e dξ. R

Then according to (1.1) we may write Z dv hTa f, gi = haWψ f, Wψ giG = a(v)Wψ f (u, v)Wψ g(u, v) du 2 v R×R+ Z Z Z dudv 2πiuξ 2πiuξ ˆ ˆ ˆ = a(v) f (ξ)ψ(vξ)e dξ · gˆ(ξ)ψ(vξ)e dξ v R×R+ R R Z Z Z dudv −2πiuξ −2πiuξ dξ ˆ ˆ a(v) gˆ(ξ)ψ(vξ)e dξ · fˆ(ξ)ψ(vξ)e = v R R×R+ R Z Z dv 2 ˆ = a(v) fˆ(ξ)ˆ . g (ξ) |ψ(vξ)| dξ v R+ R Applying Fubini’s theorem yields ! Z Z dv 2 ˆ a(v)|ψ(vξ)| hTa f, gi = g (ξ) dξ. fˆ(ξ)ˆ v R R+ Hence the result.

Remark 4.6. In the other words, the Fourier transform computation shows that Ta is a (Fourier) multiplier with γa . This is in line with the informal analysis. The 2 ˆ normalization of ψ ensures that |ψ(vξ)| is a probability density with respect to dv the measure v . Also, if we select ψ ∈ L2 (R) with ψˆ concentrated near the unit 2 ˆ sphere, then |ψ(vξ)| will be concentrated near the sphere of radius v −1 . Remark 4.7. If the symbol a depends only on =ζ, then roughly we are multiplying the part of the transform that contains information about scale t (and hence frequency t−1 ) by a(t).

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

43

Remark 4.8. The results involving Calder´on-Toeplitz operators with symbols depending on the individual coordinate functions describe an analogy between the Calder´ on-Toeplitz operators and the calculus of pseudodifferential operators, cf. [16]. Since the related operators show up in physics when working with coherent states, see e.g. [1], and [12], now we recall the necessary ingredients of the Berezin theory, cf. [2], [4] (for a connection with Toeplitz operators on Bergman spaces see [25]). Let {kg }g∈S be a system of coherent states parametrized by elements g of some set S carrying a measure dµ in a Hilbert space H. If V : H → L2 (S) is the isomorphic inclusion V : f ∈ H 7→ hf, kg iH ∈ L2 (S), and the integral operator Z f (h)hkh , kg iH dµ(h)

(P f )(g) = S

is the orthogonal projection from L2 (S) onto V (H), then the function a(g), g ∈ S is called the anti-Wick (or contravariant) symbol of an operator A : H → H, if V AV −1 |V (H) = P a(g)P = P a(g)I|V (H) : V (H) → V (H), or, in the other terminology, the operator V AV −1 |V (H) is the Toeplitz operator, Aa(g) = P a(g)I|V (H) : V (H) → V (H), with the symbol a(g). For a given bounded linear operator A : H → H, introduce the (Wick) function e aA (g, h) =

hAkh , kg iH , hkh , kg iH

g, h ∈ S.

The restriction of the function e aA (g, h) onto the diagonal e aA (g) = e aA (g, g) =

hAkg , kg iH , hkg , kg iH

g ∈ S,

is called the Wick (or covariant) symbol of the operator A. Note that the Wick and anti-Wick symbols of an operator A : H → H are connected by the Berezin transform, Z hkg , kh iH hkh , kg iH e a(g) = a(h) dµ. hkg , kg iH S In our particular case A = Ta , H = L2 (R), and S = G = Π equipped with the measure dµ = dν(ζ) = v −2 dudv. Note that admissible wavelets in L2 (R) forming a dense vector subspace of L2 (R) are, in fact, nothing but coherent states related to “ax + b”-group G, and therefore kg = ψζ (x) = v −1/2 ψ( x−u v ). The following result gives the form of the Wick symbol of Calder´on-Toeplitz operator Ta depending on v = =ζ.

44

Ondrej Hutn´ık

IEOT

Theorem 4.9. Let a = a(v) be a measurable function on L2 (R+ ). Then the Wick symbol e a(ζ) of the Calder´ on-Toeplitz operator Ta depends only on v as well, and has the form Z hTa ψζ , ψζ i 2 ˆ e a(v) = e a(ζ, ζ) = = v γa (ξ)|ψ(vξ)| dξ. (4.2) hψζ , ψζ i R The corresponding Wick function is given by the formula √ Z tv hTa ψζ , ψη i −2πiξ(u−s) ˆ ˆ = γa (ξ)ψ(vξ) ψ(tξ)e dξ, e a(ζ, η) = hψζ , ψη i Kζ (η) R

(4.3)

with ζ = (u, v), η = (s, t) ∈ G. Proof. Consider the kernel Kζ (η) = hψζ , ψη i with ζ = (u, v), η = (s, t) ∈ G. By Lemma 3.2, √ −2πisξ ˆ ˆ ψ(vξ)e , U1 Kζ (η) = tv ψ(tξ) and therefore from representation (1.1) we get hTa ψζ , ψζ i = haKζ , Kζ iG = hU1 (aKζ ), U1 Kζ iG = haU1 Kζ , U1 Kζ iG Z dξdt 2 ˆ ˆ a(t)|ψ(vξ)| |ψ(tξ)|2 =v t R×R+ ! Z Z 2 dt 2 ˆ ˆ =v a(t)|ψ(tξ)| |ψ(vξ)| dξ t R R+ Z 2 ˆ = v γa (ξ)|ψ(vξ)| dξ. R

Thus, we have (4.2). The equality (4.3) may be verified by direct calculations, or using Lemma 4.5. Remark 4.10. Let ζ = (u, v), η = (s, t) ∈ G, a(ζ) = a(v), and F ∈ Wψ (L2 (R)). Applying the general approach to coherent states, cf. [4], to our particular case it is immediate that hTa ψζ , ψη i haKζ , Kη iG e a(ζ, η) = = , hψζ , ψη i Kζ (η) and therefore by using (2.7) we have Z (Ta F )(η) = a(θ)F (θ)Kη (θ) dν(θ) Z ZG = a(θ) F (ζ)Kθ (ζ) dν(ζ) Kη (θ) dν(θ) G ZG Z = F (ζ) a(θ)Kθ (ζ)Kη (θ) dν(θ) dν(ζ) G

G

Vol. 63 (2009)

On Toeplitz-type Operators Related to Wavelets

Z =

F (ζ)Kζ (η) ZG

1 Kζ (η)

Z

45

a(θ)Kζ (θ)Kη (θ) dν(θ) dν(ζ)

G

=

e a(ζ, η)F (ζ)Kζ (η) dν(ζ) Z √ Z dudv −2πiξ(u−s) ˆ ˆ = t ψ(tξ)e dξ F (ζ) γa (ξ)ψ(vξ) v 3/2 R×R+ R ! Z √ Z dudv 2πisξ −2πiuξ ˆ ˆ = t γa (ξ)ψ(tξ)e dξ F (ζ)ψ(vξ)e v 3/2 R R×R+ √ Z ˆ = t γa (ξ)ψ(tξ)(RF )(ξ)e2πisξ dξ, G

R

i.e. writing the Calder´ on-Toeplitz operator Ta in terms of its Wick symbol yields the representation (4.1). Corollary 4.11. Let Ta and Tb be two Calder´ on-Toeplitz operators with symbols a(v) and b(v), where a(v), b(v) are measurable functions on L2 (R+ ), and let e a(v) and eb(v) be their Wick symbols, respectively. Then the Wick symbol e c(v) of the composition Ta Tb is given by Z 2 e ˆ e c(v) = (e a ? b)(v) = v γa (ξ)γb (ξ)|ψ(vξ)| dξ. R

Proof. The result follows immediately from Theorem 4.1 and Theorem 4.9.

References [1] S. T. Ali, J-P. Antoine and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 2000. [2] F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR Izvestija 6(1972), 1117–1151. [3] F. A. Berezin, General concept of quantization, Comm. Math. Phys. 40(1975), 135174. [4] F. A. Berezin, Method of Second Quantisation, Nauka, Moscow, 1988. [5] A. Calder´ on, Intermediate spaces and interpolation, the complex method, Studia Math. 24(1964), 113–190. [6] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. [7] M. Engliˇs, Toeplitz operators and group representations, J. Fourier Anal. Appl. 13(2007), 243–265. [8] A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations II: Examples, Ann. Inst. Henri Poincar´e 45(1986), 293–309. [9] S. Grudsky, A. Karapetyants and N. Vasilevski, Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case, J. Operator Theory 52(2004), 185– 204.

46

Ondrej Hutn´ık

IEOT

[10] O. Hutn´ık, A note on wavelet subspaces (submitted). [11] Q. Jiang and L. Z. Peng, Toeplitz and Hankel type operators on the upper half-plane, Integral Equations Operator Theory 15(1992), 744–767. [12] J. R. Klauder and B. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics, World Scientific, Singapore, 1985. [13] H. P. Liu and L. Z. Peng, Admissible wavelets associated with the Heisenberg group, Pacific J. Math. 180(1997), 101–123. [14] F. de Mari, H. G. Feichtinger and K. Nowak, Uniform eigenvalue estimates for timefrequency localization operators, J. London Math. Soc. 65(2002), 720–732. [15] K. Nowak, Weak type estimates for singular values of commutators on weighted Bergman spaces, Indiana Univ. Math. J. 40(1991), 1315–1331. [16] K. Nowak, On Calder´ on-Toeplitz operators, Mh. Math. 116(1993), 49–72. [17] K. Nowak, Commutators based on the Calder´ on reproducing formula, Studia Math. 104(1993), 285–306. [18] K. Nowak, Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets, Studia Math. 119(1996), 37–64. [19] T. Paul, Estats quantiques realis´es par des fonctions analitiques dans le demi-plan, Th`ese (1984), L’Universit´e Pierre et Marie Curie, Paris. [20] R. Quiroga-Barranco and N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball, I. Bargman-type transforms and spectral representations of Toeplitz operators, Integral Equations Operator Theory 59(2007), 379–419. [21] R. Rochberg, Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators, part I., Proc. Symp. Pure Math. 51(1990), 425–444. [22] R. Rochberg, A correspondence principle for Toeplitz and Calder´ on-Toeplitz operators, M. Cwikel (ed.) etAAsal. (ed.), Israel Math. Conf. Proc. 5(1992), pp. 229-243. [23] R. Rochberg, Eigenvalue estimates for Calder´ on-Toeplitz operators, K. Jarosz (ed.), Lecture Notes in Pure and Appl. Math. 136(1992), pp. 345-357. [24] N. Vasilevski, On the structure of Bergman and poly-Bergman spaces, Integral Equations Operator Theory 33(1999), 471–488. [25] N. Vasilevski, Toeplitz Operators on the Bergman Spaces: Inside-the-Domain Effects, Contemporary Mathematics 289(2001), 79–146. Ondrej Hutn´ık Institute of Mathematics Faculty of Science ˇ arik University Pavol Jozef Saf´ Jesenn´ a5 041 54 Koˇsice Slovakia e-mail: [email protected] Submitted: February 8, 2008.

Integr. equ. oper. theory 63 (2009), 47–93 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010047-47, published online December 22, 2008 DOI 10.1007/s00020-008-1644-z

Integral Equations and Operator Theory

Invariant Subspaces of Subgraded Lie Algebras of Compact Operators Matthew Kennedy, Victor S. Shulman and Yuri V. Turovskii Abstract. We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the structure of such algebras. As an application, we prove a number of results on the existence of invariant subspaces for algebraic structures of compact operators, in particular for Jordan algebras and Lie triple systems of Volterra operators. Along the way we obtain new criteria for the triangularizability of a Lie algebra of compact operators. Mathematics Subject Classification (2000). Primary 47L70, 47A15; Secondary 47L10, 16W25. Keywords. Graded Lie algebra, Engel Lie algebra, Lie triple system, Jordan algebra, compact Banach algebra, compact operator, invariant subspace, triangularization, Cartan criterion.

1. Introduction 1.1. Aim of the work and remarks on the history of the topic The classical Invariant Subspace Problem, the problem of the existence of a nontrivial closed subspace invariant under an operator from a given class, extends in various forms to families of operators. Even partial positive results on this subject can be of considerable importance for the analysis of various algebraic systems (groups, semigroups, and both associative and non-associative algebras) of operators, and for the structure and representation theory of abstract topologicalalgebraic systems. Typically, the application of these results involves the construction of a maximal chain of invariant subspaces for a system of operators. This process is often referred to as triangularization, and in fact, it provides many The support received from INTAS project No 06-1000017-8609 is gratefully acknowledged by the third author.

48

Kennedy, Shulman and Turovskii

IEOT

of the same advantages that triangular forms give for the analysis of systems of matrices. There are two popular versions of the invariant subspace problem. The first concerns commutative families of operators, and the second general operator algebras (i.e. problems of Burnside type). The solution of both versions for the class of compact operators was given in 1973 by Victor Lomonosov [L], who proved that every algebra of compact operators on a Banach space X possessing no invariant subspaces (i.e., which is irreducible) is dense in the algebra B(X ) of all bounded operators on X , with respect to the weak operator topology. (Notice this implies that any commutative family of compact operators has an invariant subspace.) An important consequence is that an algebra of compact quasinilpotent (Volterra, for brevity) operators has invariant subspace (i.e., is reducible). This is the core of the results of [L]; the general case can be easily deduced from it by using spectral projections. For the investigation and classification of general algebraic systems of compact operators, the case when the system consists of Volterra operators is a natural first step. It can be said that such systems play the central role in the so called “radical” part of the more general structural theory. Indeed, such an approach was very successful in the theory of finite-dimensional Lie algebras. We refer, for instance, to the classical theorems of Engel and Lie. The first attempts to extend the results of Lomonosov to Lie algebras of compact operators [W], and to semigroups of compact operators [R], used techniques based on the trace, so positive results were obtained under the condition that the Lie algebra or semigroup in question contain a nonzero trace class operator (or more generally, an operator from a Schatten class). With the introduction of more general techniques based on the joint spectral radius [S, T84, T85], more general results, free from the above restriction were obtained. It was proved in [T98] that a semigroup of Volterra operators has an invariant subspace, and the reducibility of Lie algebras of Volterra operators was proved in [ST0]. The latter result allowed for the development of a detailed theory of the structure of Lie algebras of compact operators in [ST5]. More recently, in [K], the existence of invariant subspaces was established for Jordan algebras of Volterra operators which contain nonzero trace class operators. The initial aim of the present work was to extend this result by avoiding the hypothesis of the existence of nonzero trace class operators, that is, to prove that every Jordan algebra of Volterra operators has an invariant subspace. Upon further examination, we realized that the needed result could be obtained within the much more general framework of the theory of subgraded Lie algebras of compact operators. Definition 1. Let Γ be an abelian group. A Lie algebra L is said to be Γ-subgraded if it can be written as a sum of subspaces indexed by the elements of Γ, X L= Lγ , γ∈Γ

Vol. 63 (2009)

Invariant Subspaces

49

which satisfy the condition [Lγ , Lδ ] ⊆ Lγ+δ . If L = ⊕γ∈Γ Lγ , i.e. if the sum is direct, then L is said to be Γ-graded. Elements of ∪γ∈Γ L are called homogeneous, and elements of Lγ are called γ-homogeneous. Unless otherwise specified, the index group Γ is assumed to be finite, and in this case we say that L is finitely subgraded. There are two types of quasinilpotence conditions that we will impose on an element a of a Lie algebra L of compact operators. The first condition, which is purely operator-theoretic, is that a is a Volterra (i.e. quasinilpotent) operator. The second condition, which is less restrictive, is that the operator ad a : x → [a, x], defined on L, is quasinilpotent. Notice that the second condition makes sense in the context of any abstract Banach Lie algebra, since ad a is the image of a in the adjoint representation of L. When a satisfies the second condition, we say that a is an Engel element of L. In dealing with subgraded Lie algebras of compact operators, the task we set for ourselves is to prove their reducibility or triangularizability by imposing a minimal number of quasinilpotence conditions. In particular, we will consider two very natural conditions – that all their homogeneous elements are Engel, or that all their 0-homogeneous elements are Engel. There is a special motivation for considering the case of Z2 -subgraded Lie algebras with the condition that the 1-homogeneous elements are Engel. For, if J is a Jordan algebra of operators then it can be checked that L = [J, J] + J is a Z2 -subgraded Lie algebra, with L0 = [J, J] and L1 = J. One should note that the intersection of L0 and L1 may be nonzero. During our work, we obtained a number of conditions, interesting in their own right, which imply the reducibility and triangularizability of (not necessarily subgraded) Lie algebras of compact operators. These conditions proved valuable in our investigation of subgraded Lie algebras and Jordan algebras of compact operators. 1.2. Structure of paper and outline of results This paper is naturally partitioned into three interrelated parts, with each part building up to “main results.” 1.2.1. Triangulation criteria for Lie algebras of compact operators. In Part 1 (Sections 2–4), we develop a list of criteria for the reducibility and triangularizability of Lie algebras of compact operators. Section 2, being a background for the rest of the paper, contains a review of the known criterion for reducibility and triangularizability. Among the most important is an infinite dimensional analogue of Cartan’s Solvability Lemma (see Lemma 2.3), which is a variation of a result in [K].

50

Kennedy, Shulman and Turovskii

IEOT

There is a well-known condition [M1] for the triangularizability of an associative algebra of compact operators. Namely, a closed associative algebra of compact operators is triangularizable if and only if it is commutative modulo its Jacobson radical. For Lie algebras, a similar result, formulated in ”radical-like” terms, and without referring to any underlying space, was obtained in [ST5]. Let us call a Banach Lie algebra L Engel if all its elements are Engel, and E-solvable if every non-zero quotient of L by a closed ideal of L has a nonzero closed Engel ideal. (When L is finite dimensional, this condition is equivalent to the solvability of L.) The above-mentioned result of [ST5] states that a Lie algebra of compact operators is triangularizable if and only if it is E-solvable. Furthermore, if L has a non-scalar E-solvable ideal, then it is reducible. The main result of Section 3 is that for a Lie algebra L of compact operators, the Engel elements of L which belong to an E-solvable subalgebra W of L form an ideal of W . Under certain additional conditions, this allows us to obtain an Engel ideal of L itself, and as a consequence, to prove the reducibility of L. In the proof we make use of techniques from the theory of compact Banach algebras. In Section 4, we further develop the list of criteria which imply the reducibility of a Lie algebra of compact operators. Among the most effective are those that impose restrictions on finite rank or ad-nilpotent elements. For example, when L is closed, we show that if the commutator [a, b] of arbitrary finite rank operators a, b ∈ L is nilpotent, then L is triangularizable. This greatly extends the result of Katavolos and Radjavi [KaR] that an associative algebra of compact operators is triangularizable if all its commutators are quasinilpotent. In general, the first step in establishing the triangularizability of a Lie algebra L of compact operators is to show that it is reducible. Unfortunately however, conditions imposed only on finite rank operators are not necessarily preserved in passing to the (closure of the) Lie algebras induced by quotients of invariant subspaces, and as a result, the argument cannot be repeated. This is because it is possible for extra finite rank operators to appear. To circumvent this obstacle, we first prove that the closed associative algebra A(L), generated by L, is commutative modulo its radical rad A(L), which establishes the triangularizability of L by the above-mentioned criterion of [M1]. The quotient A(L)/ rad A(L) is not an algebra of compact operators, but it is a compact Banach algebra. This is why the results on compact Banach algebras obtained in Section 3 are important for our work. For (not necessarily closed) Lie algebras of compact operators, another particularly important criterion for triangularizability is the condition that all commutators are Engel elements (or Volterra operators). In Part 2, we extend this result to subgraded Lie algebras of compact operators, requiring only that homogeneous commutators are Engel elements (or Volterra operators). 1.2.2. Subgraded Lie algebras of compact operators. In Part 2 (Sections 5-8), we consider general finitely subgraded Lie algebras of compact operators. Results

Vol. 63 (2009)

Invariant Subspaces

51

establishing the existence of invariant subspaces for these algebras are very interesting, in the sense that hypotheses of a highly algebraic nature are able to imply the existence of intrinsically geometric objects. In Section 5 a special tensor product construction is used to reduce the consideration of Γ-subgraded Lie algebras of compact operators to Γ-graded ones. More concretely, let π be a faithful representation of Γ on a finite-dimensional space Y. To every Γ-subgraded Lie algebra L of operators on a Banach space X one can associate a Γ-graded Lie algebra Lπ of operators on X ⊗ Y, with components Lγ ⊗ π(γ) for γ ∈ Γ. We say that Lπ is a graded ampliation of L. If L consists of compact operators, then it turns out that L is Engel or Esolvable whenever Lπ is. This allows us to establish the result that L is Engel if the closure of each component of L has no nonzero finite rank operators. For cyclic Γ, we prove that L is E-solvable if the closure of every component contains no nonzero nilpotent finite rank operators. This is a direct extension of the above-mentioned result from [ST5]. Thus, in considering a subgraded Lie algebra of compact operators, we may restrict ourselves to the case when Lie algebra contains nonzero homogeneous finite rank operators. Such operators generate an ideal which also a subgraded Lie algebra. In Section 7 we reduce the study of subgraded Lie algebras of finite rank operators to the case when the underlying space is finite-dimensional. In Section 6, we consider this case in great detail. Results on graded Lie algebras of operators on a finite-dimensional space can be considered as an aspect of the purely algebraic theory of graded groups and Lie rings; for a treatment of the theory in this context, see [Hig, Th, KrK, Kr, STW, KhM, MKh]. It was shown in [KrK, Kr] that every Zn -graded Lie ring L with L0 = 0 is solvable, with derived length less than 2n−1 − 1, and with L being nilpotent whenever n is prime. These estimates were improved in [STW, Theorems A, B and C]. Here we do not require estimates on the derived length; for the convenience of readers we provide, in Lemma 6.2, a self-contained and comparatively simple proof of the solvability of L when L consists of operators on a finite-dimensional space with scalar L0 . Another result, Lemma 6.4, is that a Γ-subgraded Lie algebra L of operators on a finite-dimensional space is solvable if all its homogeneous elements are Engel elements of L; in the case when Γ = Zn , it suffices to assume that only L0 consists of Engel elements of L. For applications in Part 3, we require Lemma 6.5, that if L0 is solvable and noncommutative, then L has a non-scalar solvable ideal, and as a consequence, is reducible. The main result of Part 2 is Theorem 7.3; it states that if L is a Γ-subgraded Lie algebra of compact operators, and if the homogeneous elements of L are Volterra operators, or more generally, are Engel elements of L, then L is triangularizable. For Γ = Zn , it is sufficient to require only that the component L0 consists of Engel elements. In Section 8 we discuss some of the consequences of this result. It is well known to algebraists that for any Zn -grading of a Lie algebra L, one can find an

52

Kennedy, Shulman and Turovskii

IEOT

automorphism φ of L satisfying the condition φn = 1 such that the components of L are the spectral subspaces of φ. Conversely, each φ defines a Zn -grading of L. Thus when L is Zn -graded, the statement of Theorem 7.3 can be reformulated in the following way (see Corollary 8.7(i)): if φ is an automorphism of finite order of a closed Lie algebra L of compact operators, and if every element invariant under φ is a Volterra operator, then L is triangularizable. In Corollary 8.7(ii) we extend this result to automorphisms of possibly infinite order, when the spectrum of φ be contained in a finite subgroup of the unit circle. Finally, Corollary 8.4 combines both statements of the main result in the following way: if Γ has a subgroup H such that Γ/H is cyclic, and such that every elements in Lγ is Engel, for γ ∈ H, then L is triangularizable. It should be noted that one can obtain further results by imposing quasinilpotence conditions on certain other “valuable” Lie multiplicative sets of homogeneous elements. The first result of this kind is Theorem 5.4, which states that for the triangularizability of a Γ-subgraded Lie algebra L of compact operators, it suffices to require that all the homogeneous commutators are Engel elements of L. We next prove a stronger assertion (see Corollary 8.1) in which the quasinilpotence condition is imposed only for homogeneous commutators in certain components (depending on Γ). For instance, if Γ = Zn , it suffices to require that every commutator [a, b] is an Engel element of L, for a ∈ Lk , b ∈ L−k , and k ∈ Zn . Of course, these conditions are also necessary for the triangularizability of L. An easy consequence of these results on homogeneous commutators is that for a Γ-subgraded Lie algebra L of compact operators, homogeneous local triangularizability implies global triangularizability. The simplest form of this statement, when every pair of homogeneous elements is triangularizable, is shown in Corollary 5.5. A stronger result, requiring only the homogeneous local triangularizability of certain components (again, depending on Γ), is given in Corollary 7.8. At the end of Part 2 we discuss an extension of our results to the case when Γ is an infinite commutative group (see Theorem 8.12). 1.2.3. Z2 -subgraded Lie algebras, Lie triple systems, and Jordan algebras. In Part 3, Z2 -subgraded Lie algebras, Lie triple systems and Jordan algebras are treated. As was mentioned above, in the case of Z2 -subgraded Lie algebra L = L0 + L1 , we consider the case when the condition of quasinilpotence is imposed on the component L1 , and not on L0 . (The case when the condition of quasinilpotence is imposed on L0 was treated in Part 2.) Note that [L1 , [L1 , L1 ]] ⊂ L1 , which means that L1 is a Lie triple system. The main result of Part 3, Theorem 9.4, states that if L is a Z2 -subgraded Lie algebra of compact operators, and if L1 is non-scalar and consists of Engel elements of L (in particular, if L1 consists of Volterra operators), then L is reducible. Of course, if in addition L1 generates L as a Lie algebra then L is triangularizable. Example 10.3 shows that this is not valid for n = 4, i.e. there exists an irreducible Z4 -subgraded Lie algebra of finite rank operators such that every element in L1

Vol. 63 (2009)

Invariant Subspaces

53

is nilpotent, and L1 generates L as a Lie algebra. It seems that the corresponding problem for n = 3 is still open. Applying the results of Part 2 to the case when L is a Z2 -subgraded Lie algebra of compact operators, we get in Theorem 9.5(ii) that if [L0 , L0 ] and [L1 , L1 ] consist of Volterra operators, then L is triangularizable. Every Lie algebra or Jordan algebra of operators forms a Lie triple system. We investigate operator Lie triple systems, and prove the result that a Lie triple system of Volterra operators is triangularizable (see Theorem 10.2). This is an infinite dimensional extension of a result in [J8]. The proof of this result for operator Lie triple systems illustrates the benefits of working with subgraded, and hence not necessarily graded, Lie algebras. In fact, if L1 is an operator Lie triple system then L1 + [L1 , L1 ] is a Z2 -subgraded Lie algebra when L0 = [L1 , L1 ] (see Lemma 10.1), but it is possible that L0 ∩ L1 6= 0. As an application, we establish that a Jordan algebra of Volterra operators is triangularizable (see Theorem 11.1), and moreover, that a Jordan algebra of compact operators which has a nonzero Jordan ideal of Volterra operators is reducible (see Theorem 11.3). These results are the Jordan analogues of results for Lie algebras which were established in [ST0] and [ST5] respectively. The authors would like to express their gratitude to Matej Breˇsar and Heydar Radjavi, who provided them with helpful discussions and moral support. Part 1. Triangularization Criteria for Lie Algebras of Compact Operators In Theorem 2.5 we collect the known necessary and sufficient conditions for the reducibility and triangularizability of Lie algebras of compact operators. Our goal here is to add to these conditions in order to prove our main results. This is done in Theorem 4.7 and Remark 4.8. Many other results of this part can be treated as preliminary ones for consideration of subgraded Lie algebras of compact operators.

2. Preliminaries For a complex Banach space X , we denote by B(X ) the algebra of all bounded linear operators on X , and by K(X ) (respectively, F(X )) the ideal of all compact (respectively, finite rank) operators on X . Recall that an operator is called Volterra if it is compact and quasinilpotent. A set of operators is called Volterra if all its elements are Volterra. We will implicitly use the well known fact (see, for instance, [M1]) that the spectrum is continuous on K(X ), and more generally, on the set of operators with countable spectra. In particular, the limit of a convergent sequence of Volterra operators is Volterra. A set M of bounded linear operators on a Banach space X is said to be triangularizable if there exists a maximal subspace chain consisting of closed subspaces

54

Kennedy, Shulman and Turovskii

IEOT

that are invariant under all operators in M . The usage of the word triangularizable stems from the fact that if X is finite-dimensional, the existence of such a subspace chain is equivalent to the existence of basis of X under which all operators in M can be represented by upper-triangular matrices. As with matrices, it is often much easier to work with triangularizable families of operators, and in fact, it could be said that one of the aims of Invariant Subspace Theory is to establish triangularizability results. It is easy to see that if a property of a set M of operators implies reducibility of M and is preserved for the related set M |V of induced operators on the quotient V = Y/Z of arbitrary invariant closed subspaces Y and Z for M , with Z ⊂ Y, then this property implies triangularizability of M (this is the Triangularization Lemma in [RR0]). That is why to establish the triangularizability of a family of operators in many cases it actually suffices to establish its reducibility. For a set M of compact operators, the triangularizability of M is equivalent to the commutativity of A(M ) modulo the Jacobson radical rad A(M ) (see [M2]), where A(M ) is the closed algebra generated by M . This implies important spectral properties of M , for instance that the spectral radius (and also the spectrum itself) is subadditive and submultiplicative on A(M ), that commutators of elements of A(M ) are Volterra, etc. In what follows, we will frequently make use of these properties. A linear subspace L of B(X ) is called an operator Lie algebra if it is closed under the Lie product [a, b] = ab − ba. For each a ∈ L, we denote by ad a, or adL a, the operator on L which maps x to [a, x], for every x ∈ L. For brevity, we will occasionally make use of shorthand notation. For example, we will write U V to denote the (algebraic) associative product of linear subspaces U and V of B(X ), i.e. U V = span{ab : a ∈ U, b ∈ V }, and similarly, we will write [U, V ] to denote the (algebraic) Lie product, i.e. [U, V ] = span {[a, b] : a ∈ U, b ∈ V } . Recall that a finite-dimensional Lie algebra L is called semisimple if it does not have non-zero abelian ideals. Cartan’s well-known Semisimplicity Criterion (see for example [J2, Section 3.4]) states that L is semisimple if and only if the Killing form hx, yi := tr((ad x)(ad y)) of L is non-degenerate on L; here tr(a) means the trace of operator a. For a Lie algebra L, the lower central (respectively, derived ) series of ideals L[k] (respectively, L(k) ) is defined [Hum, Section 1.3] by the rules L[0] = L and L[k+1] = [L, L[k] ] (respectively, L(0) = L and L(k+1) = [L(k) , L(k) ])

Vol. 63 (2009)

Invariant Subspaces

55

for k = 0, 1, . . .. Note that we use the notation L[k] instead of the standard Lk to differentiate ideals of the lower central series from corresponding associative structures. If L[k] = 0 (respectively, L(k) = 0) for some k, then L is called nilpotent (respectively, solvable). One of the forms of Lie’s classical Theorem is that a Lie algebra of operators on a finite-dimensional space is solvable if and only if it is triangularizable (or more precisely, is representable by upper-triangular matrices in some basis of the space). The following lemma of Cartan, which is also well-known, provides a criterion for the solvability of L (see [B, Theorem 1.5.2] and [Hum, Section 4.3]). Lemma 2.1 (Cartan’s Solvability Criterion). Let L be a Lie algebra of operators on a finite-dimensional vector space. Then the following conditions are equivalent. • L is solvable. • ha, bi = 0 for all a ∈ [L, L] and b ∈ L. • tr(ab) = 0 for all a ∈ [L, L] and b ∈ L. In particular, it follows from Cartan’s Solvability Criterion that if tr(ab) = 0 for all a,b ∈ L then [L, L] consists of nilpotent operators. This was extended in [K] to infinite dimensions as follows. Theorem 2.2. [K, Theorem 4.6] Let L be a Lie algebra of trace class operators (on a Hilbert space) satisfying tr(ab) = 0 for all a, b ∈ L. Then every finite rank operator in [L, L] is nilpotent. For subspaces N and M of B(X ), we will write tr(N M ) = 0

(2.1)

to indicate that tr(ab) = 0 holds for every a ∈ N and b ∈ M , requiring of course that each ab is an operator with trace. Note that in virtue of linearity of trace, condition (2.1) is equivalent to the condition tr(c) = 0 for all c ∈ N M . Similarly, we will write hN, M i = 0 to indicate that ha, bi = 0 for every a ∈ N and b ∈ M . Let us give a somewhat more general formulation of Theorem 2.2. It should be noted that its proof doesn’t need any modification for operators in a normed operator ideal (see [P]) with spectral trace (recall that the trace is spectral if it coincides with the sum of eigenvalues, taken with multiplicity). Lemma 2.3. Let L be a Lie algebra of compact operators. If there is a non-zero finite rank operator a in L such that tr(aL) = 0, then L contains either a nonscalar commutative ideal, or a non-zero ideal consisting of nilpotent operators. In both cases, L is reducible. Proof. Let I = {x ∈ L ∩ F(X ) : tr(xL) = 0}.

56

Kennedy, Shulman and Turovskii

IEOT

Then I is an ideal of L. Indeed, for arbitrary x ∈ I and y ∈ L, we have that tr([x, y]L) = tr(x[y, L]) = 0. Therefore, we obtain that tr(II) = 0. It follows from Theorem 2.2 that [I, I] consists of nilpotent operators. If [I, I] 6= 0, then it is a non-zero Volterra ideal of L. Otherwise, [I, I] = 0, and we have that I is a commutative ideal of L containing a non-scalar operator a (since tr(a2 ) = 0). In both cases, L is reducible by [ST5, Theorem 4.26] (see Theorem 2.6 below). For an operator a ∈ B(X ) and λ ∈ C, let 1/n

n

Eλ (a) = {x ∈ X : lim k(a − λ) xk

= 0}.

This linear subspace of X is an elementary spectral manifold of a [ST5, Section 3]. It is a useful fact (see for instance [ST5, Proposition 3.3]) that Eλ (a) is always closed if a has countable spectrum (or, more generally, if a is decomposable). Thus, if a decomposable operator a is locally quasinilpotent on a subset G of X (this means that 1/n lim kan xk =0 for every x ∈ G), a is locally quasinilpotent, and hence is quasinilpotent, on the closed linear span of G in virtue of well-known fact that if a bounded operator is locally quasinilpotent on a Banach space then it is quasinilpotent. We use in the paper a stronger version of this result (see for instance [M, Corollary 14.19]), namely the spectral radius ρ(a) of an operator a ∈ B(X ) satisfies to the equality ρ(a) = sup lim sup kan xk

1/n

.

(2.2)

x∈X

Recall that an element a of a normed Lie algebra L is called an Engel element of L if the operator ad a is quasinilpotent on L, i.e. if lim k(ad a)n k

1/n

= 0.

It is clear that if L is an operator Lie algebra, and if a is quasinilpotent, then a is an Engel element of L, but the converse is not true in general. We say that L is Engel if all its elements are Engel. If a is an element of a normed Lie algebra L, the following useful property [Eλ (ad a), Eµ (ad a)] ⊂ Eλ+µ (ad a) holds for every λ, µ ∈ C; in the case when L is a normed algebra we have also that Eλ (ad a)Eµ (ad a) ⊂ Eλ+µ (ad a)

(2.3)

(see a more general fact in [ST5, Lemma 3.5]). In what follows we will also need the fact that if a is a compact operator, and λ = 6 0, then Eλ (ad a) consists of nilpotent finite rank operators; the inclusion Eλ (ad a) ⊂ F(X )

Vol. 63 (2009)

Invariant Subspaces

57

for λ 6= 0 was proved in [W, Theorem 3], and we refer the nilpotency of elements of Eλ (ad a) for λ 6= 0 to mathematical folklore. For instance, it is an easy consequence of (2.3) and boundedness of σ(ad a). It is convenient to reformulate this fact as follows (see, for example, [ST9, Corollary 2]). Theorem 2.4. Let L be a closed Lie subalgebra of compact operators on an infinite dimensional space X . If L doesn’t contain non-zero nilpotent finite rank operators, then L is Engel. Recall that a normed Lie algebra is called Engel-solvable, or E-solvable, if, for each proper closed ideal I of L, the quotient Lie algebra L/I contains a non-zero closed Engel ideal; an ideal of a normed Lie algebra is called Engel (respectively, E-solvable) if it is an Engel (respectively, E-solvable) Lie algebra. It was proved in [ST5, Corollary 4.25] that, for a Lie algebra L of compact operators, the condition of E-solvability is equivalent to the geometric condition of triangularizability. As we will use this result frequently in what follows, it is convenient to collect in one statement several known conditions in [ST5] that are equivalent to L being E-solvable; note that these conditions and proofs of their implications are distributed in [ST5, Corollaries 4.20, 4.25, 5.13 and 5.16, Theorem 5.19, etc]. To aid the reader, we spell out exactly the nontrivial implications contained in the following theorem: (1) ⇐⇒ (2) is a part of [ST5, Corollary 4.25], (3) =⇒ (2) follows immediately from [ST5, Corollary 5.16 and Theorem 5.19], (5) ⇐⇒ (2) ⇐⇒ (6) is an easy consequence of [ST5, Theorems 5.2 and 5.9, Corollaries 5.13 and 5.16]. All other implications required, namely (2) =⇒ (4) =⇒ (3), are trivial. Theorem 2.5. For a Lie subalgebra L of K(X ) the following conditions are equivalent. 1. L is E-solvable. 2. L is triangularizable. 3. [L, L] is Engel. 4. [L, L] is Volterra. 5. The spectral radius is subadditive on ad L. 6. The spectral radius is subadditive on L (that is, ρ (a + b) ≤ ρ (a) + ρ (b) for every a, b ∈ L). An immediate consequence of the above results is that an Engel Lie algebra of compact operators is triangularizable. The following result will be our main tool for establishing reducibility results. Theorem 2.6. [ST5, Theorem 4.26] Let L be a Lie subalgebra of K(X ). If there is a non-scalar E-solvable ideal of L then L is reducible. Here, a family of operators is non-scalar if it contains a non-scalar operator (recall that operator is scalar if it is a scalar multiple of the identity operator). Trivially, a non-zero compact operator on X is non-scalar if X is infinite-dimensional. We will also need the following structural result.

58

Kennedy, Shulman and Turovskii

IEOT

Theorem 2.7. [ST5, in Theorem 4.32] Let L be a Lie subalgebra of K(X ). If L has two non-scalar ideals with scalar intersection then L is reducible. For finite-dimensional Lie algebras, one can easily check that L is Engel (respectively, E-solvable) if and only if it is nilpotent (respectively, solvable). It is well known (see for example [J2]) that any finite-dimensional Lie algebra L has a largest solvable ideal R (the radical of L), and that the quotient Lie algebra L/R is semisimple. A similar result for existence of the largest E-solvable ideal was proved in [ST5, Corollary 5.15] for a Lie algebra of compact operators. We note the following important property of this ideal. Theorem 2.8. [ST5, Corollary 5.15] Let L be a Lie algebra of compact operators, and let R be the largest E-solvable ideal of L. Then R is inner-characteristic, i.e. [a, R] ⊂ R for every a ∈ B(X ) such that [a, L] ⊂ L.

3. Engel Elements of a Lie Algebra in its Subalgebra In this section, we will consider a Lie algebra L of compact operators having a subalgebra L0 . We will be particularly interested in Engel elements of L that also belong to L0 . In applications L will be subgraded and L0 will be its 0-component. Lemma 3.1. Let L be a closed Lie algebra of compact operators, L0 a closed subalgebra of L, N the set of all elements of L0 that are Engel elements of L (or the set of all Volterra operators in L0 ), Nf the set of all finite-rank operators in N . If N (respectively Nf ) is a subspace of L0 then it is an ideal of L0 . Proof. Let b ∈ L0 . Then exp (λ ad b) is a bounded automorphism of L for every λ ∈ C. If a is in N then (exp (λ ad b)) (a) = exp(λb)a exp(−λb) ∈ N for every λ ∈ C, whence we obtain by hypothesis that υ(λ) := λ−1 [(exp (λ ad b)) (a) − a] is in N for every λ 6= 0, and that [b, a] = lim υ(λ) λ→0

is in N . Note that if a ∈ Nf then so is [b, a].

We will need the following three lemmas about Banach algebras. Lemma 3.2. Let A, B be Banach algebras, φ : A −→ B a bounded homomorphism. Suppose that A consists of elements with countable spectra, and that A/ rad A is commutative. If φ(x) and φ(y) are quasinilpotent elements of B for x, y ∈ A then φ(x + y) is quasinilpotent, too.

Vol. 63 (2009)

Invariant Subspaces

59

Proof. One may assume that A and B are unital and φ(1) = 1. It is clear that A/ ker φ is commutative modulo the Jacobson radical and consists of elements with countable spectra. So we can assume that φ is injective. We claim that x, y are quasinilpotent elements of A. Indeed, if x is not quasinilpotent then there is a non-zero isolated point λ ∈ σ(x). Let p be the Riesz idempotent of x corresponding to λ. Then p 6= 0 and (x − λ)p is quasinilpotent, whence so is (φ(x) − λ) φ(p). If φ(p) 6= 0 then λ ∈ σ(φ(x)), a contradiction. Therefore φ(p) = 0. But, since φ is injective, p = 0, a contradiction. This proves our claim. As A is commutative modulo the Jacobson radical, x + y is a quasinilpotent element of A. Then φ(x + y) is quasinilpotent, too. Recall [A] that a Banach algebra A is called compact if the map Lx Rx : y 7−→ xyx is compact on A, for every x ∈ A, where multiplication operators Lx and Rx in B(A) are defined by Lx y = xy

and

Rx y = yx

for every y ∈ A. It should be noted [A, Theorem 4.4] that each element of a compact Banach algebra has countable spectrum, and every closed subalgebra of K(X ) is a compact Banach algebra. The following result is contained in [ST1, Theorems 2 and 7]. Since the proofs of these statements are still not published, we include the proof adapted for this special case. Lemma 3.3. Let A and B be compact Banach algebras, and let A1 = A ⊕ C and B 1 = B ⊕ C be the algebras obtained by adjoining the identity element. If A and B are commutative modulo the Jacobson radical, then so is the projective tensor b 1 , and all elements of W have countable spectra. product W = A1 ⊗B b Proof. Let I = A⊗B. As is known, under the natural embedding of I into W , the algebra I is topologically isomorphic to a closed subspace of W which is an ideal of W . So we will identify I with a closed ideal of W . Let τ be a strictly irreducible representation of W . If τ (I) = 0 then τ is also a strictly irreducible representa1 tion of the algebra (A ⊕ B) (with coordinate-wise multiplication). This algebra is clearly commutative modulo the Jacobson radical, whence τ is one-dimensional. Thus, one can assume that τ (I) 6= 0. Then τ is also a strictly irreducible representation of I. Using the fact that the tensor product of compact operators is a compact operator (on the projective tensor product of corresponding spaces), we obtain that I is a compact Banach algebra. By [A, Theorem 5.1], τ (I) consists of compact operators. Clearly, the set G = {x ⊗ y : x ∈ rad A, y ∈ B} ∪ {z ⊗ w : z ∈ A, w ∈ rad B}

60

Kennedy, Shulman and Turovskii

IEOT

is a multiplicative semigroup of quasinilpotent elements of I. Therefore τ (G) is a Volterra semigroup. Let N be the closure of span τ (G) in τ (I). By [T98, Theorem 4], N is Volterra. But N is an ideal of τ (I), whence N = 0. Since it is easy to check that, for every x, z ∈ A and y, w ∈ B, [x ⊗ y, z ⊗ w] = xz ⊗ yw − zx ⊗ wy = [x, z] ⊗ yw + zx ⊗ [y, w] with [x, z] ∈ rad A and [y, w] ∈ rad B, we obtain that τ ([I, I]) ⊂ N. Then τ (I) is commutative and τ is one-dimensional. In any case, we obtain that W is commutative modulo the Jacobson radical. Now we have to prove that W consists of elements with countable spectra. For every g ∈ W there are x ∈ A, y ∈ B, λ ∈ C, and f ∈ I such that g = λ(1 ⊗ 1) + x ⊗ 1 + 1 ⊗ y + f. Then, by commutativity of W modulo the Jacobson radical, we obtain that σ(g) ⊂ λ + σ(x ⊗ 1) + σ(1 ⊗ y) + σ(f ) ⊂ λ + σ(x) + σ(y) + σ(f ) As all last spectra are countable (in particular, f has countable spectrum by [A, Theorem 4.4]), g has countable spectrum. Recall that an element b of a Banach algebra B is of finite rank if the map Lb Rb : x 7→ bxb is of finite rank on B. By [A, Theorem 7.2], for a semisimple compact Banach algebra B, an element b of B is of finite rank if and only if b is in the socle soc B of B. Of course, every quasinilpotent finite rank element of B is in fact nilpotent. The following lemma strongly generalizes [W, Theorem 3]. Lemma 3.4. Let B be a semisimple compact Banach algebra B, and let a ∈ B. Then E1 (adB a) consists of nilpotent elements of soc B. Proof. Recall that the spectrum of a is considered with respect to B 1 . By [A, Theorem 4.4], there is a finite number of points in σ(a) ∩ Cε for every ε > 0, where Cε = {µ ∈ C : |µ| > ε}, and by [ST2, Lemma 3], the Riesz idempotent of a corresponding to a non-zero number λ in σ(a) is of finite rank in B. Thus, if pε is the Riesz idempotent of a corresponding to σ(a) ∩ Cε , then pε is of finite rank in B. Let qε = 1 − pε . Since a commutes with qε and Lqε Rqε is a projection, it is easy to verify that Lqε Rqε E1 (ad a) ⊂ E1 (ad a) ∩ E1 (Lqε Rqε ad a) , and since ρ(aqε ) ≤ ε, that ρ (Lqε Rqε ad a) = ρ (Laqε Rqε − Lqε Raqε ) ≤ ρ (aqε ) ρ (qε ) + ρ (qε ) ρ (aqε ) ≤ 2ρ (aqε ) ≤ 2ε.

(3.1)

Vol. 63 (2009)

Invariant Subspaces

61

If ε < 1/2, then ρ (Lqε Rqε ad a) < 1, and then E1 (Lqε Rqε ad a) = 0, whence, by (3.1), (1 − pε ) b (1 − pε ) = 0 for every b ∈ E1 (ad a), that completes the proof that E1 (ad a) is in soc B. Since n

(E1 (ad a)) ⊂ En (ad a) for every integer n > 0 by [ST5, Lemma 3.5], and since σ (ad a) is bounded, we may take an integer n outside of σ (ad a) and obtain that En (ad a) = 0, whence every element of E1 (ad a) is nilpotent.

For an algebra B, we denote by B op the algebra that is opposite to B; this algebra coincides with B as a linear space, but has opposite multiplication, namely a · b = ba op

for every a, b ∈ B . It is clear from this that if B is a compact Banach algebra then so is B op . Theorem 3.5. Let L be a Lie algebra of compact operators, L0 an E-solvable subalgebra of L. If a, b ∈ L0 are Engel elements of L then so is a + b. b op . As L0 is triangularizable by Proof. Let D = A(L0 ∪ {1}) and A = D⊗D op Theorem 2.5, then the algebras D and D are commutative modulo the Jacobson radical by [M2]. Since A(L0 ) is a compact Banach algebra by [V] and [A, Lemma 3.4], then A is commutative modulo the Jacobson radical, and elements of A have countable spectra, by Lemma 3.3. Let C = A(L) and B = B(C). For every x ∈ C, it is convenient to work with left and right multiplication operators Lx and Rx on C. Define a map φ : A −→ B by setting φ(x ⊗ y) = Lx Ry for every x ⊗ y ∈ A, and by extending the map to the whole A by linearity and continuity. Then φ is a bounded homomorphism of Banach algebras and φ(1 ⊗ 1) is the identity operator on C. We claim that if a is an Engel element of L then a is an Engel element of A(L). Indeed, since E0 (adC a) is a closed algebra (see, for instance, [ST5, Proposition 3.3 and Lemma 3.5]) and L ⊂ E0 (adL a) ⊂ E0 (adC a) , E0 (adC a) contains the closed algebra generated by L. This means that a is an Engel element of A(L). Now let a, b ∈ L0 be Engel elements of L. As C = A(L), they are Engel elements of C by above. Take elements x = a ⊗ 1 − 1 ⊗ a and y = b ⊗ 1 − 1 ⊗ b in A. Then it is clear that φ(x) = La − Ra = adC a and φ(y) = Lb − Rb = adC b.

62

Kennedy, Shulman and Turovskii

IEOT

As a and b are Engel elements of C, operators φ(x) and φ(y) are quasinilpotent elements of B. On the other hand, it follows from Lemma 3.3 that A satisfies conditions of Lemma 3.2, whence, by Lemma 3.2, φ(x + y) is a quasinilpotent element of B. Since φ(x + y) = adC (a + b) , this means that a + b is an Engel element of C. But then a + b is an Engel element of L, too. For a subspace N ⊂ B(X ), we will denote by N the closure of N in B(X ). Corollary 3.6. Let L be a Lie algebra of compact operators, L0 an E-solvable subalgebra of L, and let N be the set of all elements of L0 that are Engel elements of L. Then N is a closed ideal of L0 . Proof. Let M be the set of all elements of L0 that are Engel elements of L. It follows from Theorem 3.5 that M is a subspace of L0 , whence M is a closed ideal of L0 by Lemma 3.1. If N is the set of all elements of L0 that are Engel elements of L, then it is clear that N = L0 ∩ M, and that L0 ∩ M is a closed ideal of L0 . Remark 3.7. Note that for a finite rank operator a in a Lie algebra L ⊂ B (X ) the condition of being an Engel element of L is equivalent to the condition of being an ad-nilpotent element of L. Indeed, since a is an algebraic operator, the operators La and Ra are also algebraic. As they commute, their difference La −Ra is algebraic operator whence ad a, being the restriction of La − Ra to L, is algebraic. But it is evident that a quasinilpotent algebraic operator is nilpotent.

4. Further Conditions Equivalent to the Triangularizability of Lie Algebras of Compact Operators In this section we will derive some new criteria for the triangularizability of Lie algebras of compact operators, which, while relevant to the discussion of graded Lie algebras, are also of independent interest. For a subset M of a Lie algebra L, we define C(M ) = {[a, b] ∈ L : a, b ∈ M }. This set is not additive in general, but is a Lie multiplicative set, i.e. it is closed under taking commutators. It is clear that if M is a Lie multiplicative set in a Lie algebra L then C(M ) is also Lie multiplicative and C(M ) ⊂ M . Also, since the map ad : a 7→ ad a is a representation of L on L, i.e. [ad a, ad b] = ad [a, b] for every a, b ∈ L, we have that if M is a Lie multiplicative set in L, then adL M is also a Lie multiplicative set in ad L.

Vol. 63 (2009)

Invariant Subspaces

63

It is well known [J2, Theorem 2.2.1] that the span of a Lie multiplicative set of nilpotent operators on a finite-dimensional space consists of nilpotent operators. The following lemma uses and extends this result. Lemma 4.1. Let L be a Lie algebra of compact operators, M a Lie multiplicative set of finite rank operators that are Engel elements of L, and let L0 = span M . Then L0 is a subalgebra of L consisting of ad-nilpotent elements of L, and [L0 , L0 ] consists of nilpotent finite rank operators. Moreover, if M consists of nilpotent finite rank operators then so does L0 . Proof. It is clear that L0 is a subalgebra of L. Let F be a finite subset of M , and let MF be a Lie multiplicative set generated by F . Then MF ⊂ M, and there is a finite-dimensional subspace Y of X such that span MF ⊂ {a ∈ L0 : aX ⊂ Y}. Suppose, to the contrary, that span C(MF ) does not consist of nilpotent operators. Then there are a ∈ span C(MF ) and x ∈ X such that ax = x. It is obvious that x ∈ Y. Clearly Y is an invariant subspace for span MF , and that span C(MF )|Y contains a finite rank operator having eigenvalue 1. Now let N = span MF |Y. Then N is a finite-dimensional Lie algebra of operators on Y, because Y is finite-dimensional. Since MF consists of Engel elements of L, adN MF |Y is a Lie multiplicative set of nilpotent operators on the finite-dimensional space N , whence so is span adN MF |Y by [J2, Theorem 2.2.1]. Since span adN MF |Y = adN span MF |Y, it follows that the Lie algebra span MF |Y is Engel, and hence is triangularizable by Theorem 2.5. In particular, span C(MF )|Y consists of nilpotent operators, which gives a contradiction. Thus span C(MF ) consists of nilpotent operators. Since F is arbitrary, we obtain in fact that span C(M ) consists of nilpotent finite rank operators. Since span C(M ) = [L0 , L0 ] , L0 is E-solvable and triangularizable by Theorem 2.5. By Theorem 3.5, span M consists of Engel elements of L, but span M = L0 . By Remark 3.7, L0 consists of ad-nilpotent elements of L. This completes the proof of the basic part of the lemma. If M consists of nilpotent finite rank operators then, applying the subadditivity of the spectral radius on L0 (see Theorem 2.5), we obtain that L0 is Volterra, and hence it consists of nilpotent finite rank operators. We need the following technical lemmas.

64

Kennedy, Shulman and Turovskii

IEOT

Lemma 4.2. Let ω : L −→ M be a bounded homomorphism with dense image from a normed Lie algebra L into a Lie subalgebra M of a compact Banach algebra B. Then ρ((ad ω(a)) (ad ω (b))) ≤ ρ((ad a) (ad b)) for every a, b ∈ L. Proof. One can assume that M is closed in B. Let T = (ad ω (a)) (ad ω (b)) and S = Lω(a) − Rω(a) Lω(b) − Rω(b) . Note that S = S1 − S2 = Lω(a)ω(b) + Rω(b)ω(a) − Lω(a) Rω(b) + Lω(b) Rω(a) , where the operator S1 = Lω(a)ω(b) + Rω(b)ω(a) has countable spectrum on B since ω (a) ω (b) and ω (b) ω (a) have countable spectra as compact elements of B, and σ(S1 ) ⊂ σ (ω (a) ω (b)) + σ (ω (b) ω (a)), and the operator S2 = Lω(a) Rω(b) + Lω(b) Rω(a) = Lω(a)+ω(b) Rω(a)+ω(b) − Lω(a) Rω(a) − Lω(b) Rω(b) is compact on B by definition of compact Banach algebras. Therefore S has countable spectrum as a compact perturbation of an operator with countable spectrum (for instance see [ST5, Proposition 3.23]). Then T has countable spectrum as the restriction of S to M . For every c ∈ L, we have that n

k((ad ω (a)) (ad ω (b))) ω (c)k

1/n

n

1/n

= kω (((ad a) (ad b)) c)k 1/n

≤ (kωk kck)

n 1/n

k((ad a) (ad b)) k

for each integer n, whence lim sup kT n xk

1/n

≤ ρ((ad a) (ad b))

for every x in the image of ω. Recall, as T has countable spectrum, that the 1/n subspace {x ∈ M : lim sup kT n xk ≤ r} is in fact closed (see, for instance, [ST5, Proposition 3.3]) for every r ≥ 0. Since one can consider M as a Banach space and ω (L) is dense in M , we obtain by (2.2) that 1/n

ρ(T ) = sup lim sup kT n xk

≤ ρ((ad a) (ad b)).

x∈M

Remark 4.3. The same argument as in Lemma 4.2 proves that if ω : L −→ M is a bounded homomorphism with dense image from a normed Lie algebra L into a Banach Lie algebra M then ρ(ad ω(a)) ≤ ρ(ad a) for every a ∈ L such that ad ω(a) is an operator with countable spectrum. This shows for instance that the property of being Engel is preserved for compact operators induced on a quotient V of invariant closed subspaces; for this, it suffices to take L ⊂ K(X ), M = L|V and ω(a) = a|V for every a ∈ L.

Vol. 63 (2009)

Invariant Subspaces

65

Lemma 4.4. Let L be a Lie algebra of compact operators on X , and let y ∈ L∩F(X ) be such that ρ((ad y) (ad b)) = 0 for every b ∈ L. Then {y, z} is triangularizable for every z ∈ L ∩ F(X ). Proof. If not then there are closed subspaces Z ⊂ Y invariant for both y and z, such that the restriction of {y, z} to the quotient V = Y /Z is irreducible and dim V > 1. It is clear that y|V and z|V are non-scalar. Let N be the Lie algebra generated by y and z. As y|V and z|V are of finite rank, it is clear that V is finite-dimensional. It follows from Lemma 4.2 that ρ((ad y|V) (ad b|V)) = 0 for every b ∈ N , whence in particular we have that hy|V, N |Vi = 0, where the Killing form is taken relative to the Lie algebra N |V which is finitedimensional. Let I = {a ∈ N |V : ha, N |Vi = 0} . Since h[a, b] , N |Vi = ha, [b, N |V]i = 0 for every a ∈ I and b ∈ N |V, the subspace I is in fact an ideal of N |V. Since also hI, [I, I]i = 0, I is solvable by Lemma 2.1. Since I contains a non-scalar operator, N |V is reducible by Theorem 2.6, which is a contradiction. Lemma 4.5. Let L be a Lie algebra of compact operators, B = A(L)/ rad A(L), θ : A(L) −→ B a canonical map, and let M be the closure of θ(L) in B. If M is Engel, then L is E-solvable. Proof. Let us show that B is commutative. Since B is semisimple, we have only to show that each strictly irreducible representation τ of B acts on one-dimensional space. Since τ (E0 (ad z)) ⊂ E0 (ad τ (z)) for every z ∈ M , we obtain from our hypothesis that ad τ (z) is locally quasinilpotent on τ (M ), and hence that it is quasinilpotent. By [A, Theorem 5.1], τ (M ) is an Engel Lie algebra of compact operators, whence τ (B) is triangularizable by Theorem 2.5. As τ (B) is irreducible, τ is one-dimensional. Thus L is triangularizable by [M2], and is E-solvable by Theorem 2.5. Recall [ST5, Theorems 5.19 and 5.5] that there exists a largest Engel (respectively, Volterra) ideal in any Lie algebra of compact operators. Lemma 4.6. Let L be a closed Lie algebra of compact operators on X , and let E be the largest Engel ideal of L. If all nilpotent finite rank operators in L belong to E, then L/E is Engel, L is E-solvable and therefore triangularizable.

66

Kennedy, Shulman and Turovskii

IEOT

Proof. For a bounded operator x on a Banach space, let pλ,r (x) be the Riesz projection of x corresponding to the part of σ(x) enclosed by the circle Ωλ,r of radius r centered at λ ∈ C whenever this circle lies into the resolvent set of x. Recall that Z 1 (µ − x)−1 dµ. (4.1) pλ,r (x) = 2πi Ωλ,r In the following claims we will deal with operators (of adjoint representation) with countable spectra. Let ϕ : L −→ L/E be the canonical map. Claim 1. L/E is an Engel Lie algebra. If not, there are an element a ∈ L and a non-zero λ ∈ C such that λ is an isolated point of σ(ad ϕ(a)). Note that ad ϕ(a) = (ad a) |ϕ(L). It is clear that there are pλ,r (ad a) and pλ,r (ad ϕ(a)), where the circle Ωλ,r encloses only {λ} in σ(ad ϕ(a)) and r < |λ|. Since λ ∈ σ(ad ϕ(a)), we have that pλ,r (ad ϕ(a)) 6= 0. By using the fact that the resolvent set of an operator with countable spectrum is connected, we obtain that (µ − ad a)−1 |ϕ(L) = (µ − (ad a) |ϕ(L))−1

(4.2)

for every µ in the intersection of resolvent sets of a and a|ϕ(L). It follows from (4.1) (under x = ad a) and (4.2) that pλ,r (ad ϕ(a)) = pλ,r ((ad a) |ϕ(L)) = pλ,r (ad a) |ϕ(L). Note that n o 1/n n pλ,r (ad a) L = Eλ,r (ad a) := x ∈ L : lim sup k(ad a − λ) xk ≤r . As r < |λ|, it follows from [ST5, Corollary 3.12] that Eλ,r (ad a) consists of nilpotent finite rank operators. Then we have that Eλ,r (ad a) ⊂ E. This shows that pλ,r (ad a) L ⊂ E and pλ,r (ad ϕ(a)) = pλ,r (ad a) |ϕ(L) = 0, a contradiction. Claim 2. L is E-solvable and triangularizable. Let I be a proper closed ideal of L, and let ω : L −→ L/I be a canonical map. It is clear that ad ω (a) is an operator with countable spectrum for every a ∈ L. If I doesn’t contain E then it follows from Remark 4.3 that ω (E) is a non-zero Engel ideal of L/I. Its closure is also Engel in virtue of continuity of the spectral radius on operators with countable spectra. Suppose now that E ⊂ I. Then there is a natural bounded epimorphism L/E −→ L/I, whence L/I is clearly Engel. Therefore L is E-solvable. Then L is triangularizable by Theorem 2.5.

Vol. 63 (2009)

Invariant Subspaces

67

Now we are in a position to obtain the following result which complements Theorem 2.5. Theorem 4.7. The following conditions can be added to the list of conditions equivalent to triangularizability in Theorem 2.5, for a Lie algebra L of compact operators. (7) The set C(L) of all commutators of L consists of Engel elements of L. (8) C(L) is Volterra. (9) ρ((ad a) (ad b)) ≤ ρ(ad a)ρ(ad b) for every a, b ∈ L. (10) ρ(ab) ≤ ρ(a)ρ(b) for every a, b ∈ L. (11) a + b is an Engel element of L for every ad-nilpotent elements a, b of L. (12) a + b is nilpotent for every nilpotent finite rank operators a, b in L. (13) [a, b] is an Engel element of L for every ad-nilpotent elements a, b of L. (14) The set of all nilpotent finite rank operators in L is Lie multiplicative. Proof. Let B = A(L)/ rad A(L), θ : A(L) −→ B a canonical map, and let M be the closure of θ(L) in B. Recall that (2) of Theorem 2.5 means that L is triangularizable, and we refer to it simply as (2). We start with an easy part of the implications. • (2) implies either of (7), (8), (9), (10), (11), (12), (13), and (14). Indeed, the triangularizability of L implies (9) by [ST5, Theorem 5.12 and Corrolary 5.13], (11) by [ST5, Corollary 5.13 and Theorem 5.19], and (13) by [ST5, Theorem 5.19]. The other implications are obvious, because if L is triangularizable, then A(L) is commutative modulo the Jacobson radical, and hence has the spectral properties required for these implications. Now we must show the reverse implications. • (7) =⇒ (2). Let I = C(L) ∩ F(X ). Then I is a Lie multiplicative set in L. By Lemma 4.2, the property being Engel is preserved for compact operators induced on a quotient of invariant subspaces, whence for the implication it is sufficient only to show that L is reducible. Suppose, to the contrary, that L is irreducible. Then L is not Engel, and hence there is a non-zero nilpotent finite rank operator a of L by Theorem 2.4. If [a, L] = 0, then L0 is reducible by Lomonosov’s Theorem [L], which gives a contradiction. Therefore [a, L] 6= 0 , whence I 6= 0, and moreover, I is non-scalar. Since, by hypothesis, C(L) consists of Engel elements of L so does the set I. By Lemma 4.1, span I consists of Engel elements of L. In particular, span I is Engel. But span I is an ideal of L, whence L is reducible by Theorem 2.6, a contradiction. • (8) =⇒ (2). It is clear that (8) =⇒ (7) is true, whence (8) =⇒ (2) holds in virtue of (7) =⇒ (2).

68

Kennedy, Shulman and Turovskii

IEOT

• (9) =⇒ (2). Assume that (9) holds. It follows from Lemma 4.5 that it is sufficient to show that M is Engel. Claim 1. Let y be a quasinilpotent element in M . Then ρ((ad y) (ad z)) = 0 for every z ∈ M . Suppose first that z = θ(b) for some b ∈ L. Let (ak ) be a sequence of elements of L such that θ(ak ) → y as k → ∞. Then we have that ρ (ak ) = ρ (θ(ak )) → 0, and hence that ρ (ad ak ) ≤ 2ρ (ak ) → 0, as k → ∞. It follows from Lemma 4.2 and (9) that ρ((ad θ(ak )) (ad θ(b))) ≤ ρ((ad ak ) (ad b)) ≤ ρ (ad ak ) ρ (ad b) → 0, and since (ad θ(ak )) (ad θ(b)) → (ad y) (ad z) as k → ∞, we obtain that ρ((ad y) (ad z)) = 0 for every z in θ(L). By continuity of ρ on elements with countable spectra, this equality holds for every z ∈ M . Claim 2. Let y ∈ M ∩soc B be nilpotent. Then [y, z] is a nilpotent element of soc B for every z ∈ M ∩ soc B. Since the spectrum of [y, z] is equal to the union of σ (τ ([y, z])), when τ runs over all strictly irreducible representations of B, it is sufficient to show that σ ([τ (y) , τ (z)]) = 0 for an arbitrary τ . As τ (B) consists of compact operators by [A, Theorem 5.1], τ (M ) is a Lie algebra of compact operators, and also τ (y) and τ (z) are finite rank operators. It follows by Claim 1 and Lemma 4.2 that ρ((ad τ (y)) (ad τ (b))) = 0 for every b ∈ M . By Lemma 4.4, {τ (y) , τ (z)} is triangularizable, whence ρ ([τ (y) , τ (z)]) = 0. This proves the claim. Claim 3. Let N be the set of all nilpotent elements in M ∩ soc B. Then N is an ideal of M ∩ soc B. It is sufficient to show that τ (span N ) consists of nilpotent operators for an arbitrary strictly irreducible representation τ of B. As N is Lie multiplicative in M by Claim 2, τ (N ) is a Lie multiplicative set of nilpotent finite rank operators in τ (M ), whence so is span τ (N ) by Lemma 4.1. This proves the claim.

Vol. 63 (2009)

Invariant Subspaces

69

Claim 4. N = 0. It is sufficient to show that τ (N ) = 0 for an arbitrary strictly irreducible representation τ of B. Suppose, to the contrary, that τ (N ) 6= 0 for some τ . Since τ (N ) is a Volterra ideal of the Lie algebra τ (M ∩ soc B), τ (N ) ⊂ R, where R is the largest E-solvable ideal of τ (M ∩ soc B). Since R is innercharacteristic by Theorem 2.8 and τ (M ∩ soc B) is an ideal of the Lie algebra τ (M ), R is an E-solvable ideal of τ (M ), whence τ (M ) is reducible by Theorem 2.6, which is a contradiction. Claim 5. M is an Engel Lie algebra. Indeed, if not, there are non-zero x, y ∈ M , and a non-zero isolated point λ in the spectrum σ(ad x), such that y ∈ Eλ (ad x). It follows that y is a nilpotent element of M ∩ soc B by Lemma 3.4, whence y ∈ N. But N = 0 by Claim 4, a contradiction. This proves the claim. Now L is triangularizable by Lemma 4.5. • (10) =⇒ (2). Assume that (10) holds. First of all, note that (10) holds for every a, b ∈ M . Indeed, since ρ(θ(a)) = ρ(a) for every a ∈ L, and since the spectral radius is continuous on elements with countable spectra, (10) holds for elements of M . Further, we claim that M is an Engel Lie algebra. If not, there are non-zero x, y ∈ M such that y ∈ E1 (ad x). Since y is nilpotent by Lemma 3.4, we have by (10) (reformulated for elements of M ) that ρ(yz) = 0 for every z ∈ M . Let τ be a strictly irreducible representation of B. By [A, Theorem 5.1], τ (B) consists of compact operators. Since τ (y) ∈ E1 (ad τ (x)),we have that τ (y) is a finite rank operator by [W, Theorem 3], and since ρ(τ (z)) ≤ ρ(z) for every z ∈ B, we have that τ (yM ) consists of nilpotent finite rank operators, whence tr(τ (y)τ (M )) = 0. If τ (y) 6= 0, it follows from Lemma 2.3 that τ (M ) is reducible, which would be a contradiction. Then τ (y) = 0 what implies, since τ is arbitrary, that y = 0, a contradiction. This proves that M is Engel. Now L is triangularizable by Lemma 4.5.

70

Kennedy, Shulman and Turovskii

IEOT

• (11) =⇒ (2). Let E be the set of all Engel elements of L. By Remark 3.7, E ∩ F(X ) consists of ad-nilpotent elements of L, whence E ∩ F(X ) is additive by hypothesis. By Lemma 3.1, E ∩ F(X ) is an ideal of L, whence E ∩ F(X ) is an Engel ideal of L. Then the set of all nilpotent finite rank operators in L is contained in the largest Engel ideal of L. By Lemma 4.6, L is triangularizable. • (12) =⇒ (2). Let F be the set of all nilpotent finite rank operators in L. Since F is a subspace of L, we have by Lemma 3.1 that F is an ideal of L, whence it is contained in the largest Engel ideal of L. By Lemma 4.6, L is triangularizable. • (13) =⇒ (2). Let E be the set of all Engel elements of L. By Remark 3.7, E ∩F(X ) consists of ad-nilpotent elements of L, and is therefore Lie multiplicative by hypothesis. By Lemma 4.1, span E ∩ F(X ) ⊂ E ∩ F(X ), whence E ∩ F(X ) is an ideal of L by Lemma 3.1. Then E ∩ F(X ) is an Engel ideal of L, and the set of all nilpotent finite rank operators in L is contained in the largest Engel ideal of L. By Lemma 4.6, L is triangularizable. • (14) =⇒ (2). Let F be the set of all nilpotent finite rank operators in L. By hypothesis, F is Lie multiplicative. By Lemma 4.1, span F ⊂ F. By Lemma 3.1, F is an ideal of L, whence it is contained in the largest Engel ideal of L. By Lemma 4.6, L is triangularizable. Remark 4.8. Using the continuity of the spectral radius on operators with countable spectra, it is easy to see that (5) and (6) of Theorem 2.5 imply (11) and (12) of Theorem 4.7, respectively. So, for applications to triangularizability, conditions (11) and (12) are more effective than (5) and (6). Similarly, the following conditions (15) The spectral radius is Lie submultiplicative on ad L. (16) The spectral radius is Lie submultiplicative on L (i.e. ρ ([a, b]) ≤ 2ρ (a) ρ (b) for every a, b ∈ L). are stronger than (13) and (14), respectively, and clearly follow from (2) of Theorem 2.5; as a result they are equivalent to the triangularizability of a Lie algebra L of compact operators. Clearly (13) and (14) are more effective tools for establishing triangularizability than (15) and (16). Note that the conditions in Theorems 2.5 and 4.7, and in Remark 4.8, are arranged in pairs; odd conditions are expressed in terms of normed Lie algebras, while even conditions are expressed in terms of operator algebras.

Vol. 63 (2009)

Invariant Subspaces

71

Part 2. Subgraded Lie Algebras of Compact Operators Our aim in this part is to determine which of quasinilpotence conditions on homogeneous elements of a subgraded Lie algebra of compact operators are equivalent to triangularizability. The main result is Theorem 7.3.

5. Graded Ampliations of Subgraded Lie Algebras Let Γ be a finite commutative group, and let π be a faithful representation of Γ on a finite-dimensional space Y (i.e. an injective homomorphism of Γ into the group of invertible operators on Y), and let L be a Γ-subgraded Lie algebra of operators in K(X ). Instead of working directly with L, it is convenient to consider a related Lie algebra M := Lπ of operators in K(Z) where Z = X ⊗ Y, and M is the linear space generated by subspaces Mγ = Lγ ⊗ π(γ), γ ∈ Γ. P It is clear that M = γ∈Γ Mγ where the sum is direct, and that [Mα , Mβ ] ⊂ Mα+β for every α, β ∈ Γ, so M is a Γ-graded Lie algebra. Setting fπ (a ⊗ π(γ)) = a for all γ ∈ Γ, a ∈ Lγ and extending it linearly to M , we obtain a continuous homomorphism fπ of M onto L. Clearly fπ is injective if and only if L is graded. We will refer to Lπ constructed in this way as a graded ampliation of L. For a Zn -subgraded Lie algebra L, it is not difficult to realize Lπ by concrete n × n-matrices with entries in L. For n = 2, this is shown in the following example. Example 5.1. For a Z2 -subgraded Lie algebra L = L0 + L1 , define π by 1 0 0 1 π(0) = , π(1) = . 0 1 1 0 Then Lπ is equal to M = M0 + M1 up to an isomorphism, where a 0 0 b M0 = : a ∈ L0 , M1 = : b ∈ L1 . 0 a b 0 In this case, fπ

a b = a + b. b a

In what follows, Γ is assumed to be finite, unless specifically indicated otherwise. Lemma 5.2. Let L be a Γ-subgraded Lie algebra of compact operators. If Lπ is Engel (or E-solvable) then so is L.

72

Kennedy, Shulman and Turovskii

IEOT

Proof. For an arbitrary homomorphism of Lie algebras, we have (ad fπ (a))fπ = fπ ad a for every a ∈ Lπ , so it follows that (ad fπ (a))n fπ = fπ (ad a)n for all n. If Lπ is Engel then k(ad a)n k1/n → 0 as n → ∞, for every a ∈ Lπ , whence k(ad fπ (a))n fπ k1/n → 0. Since fπ (Lπ ) = L, this shows that k(ad x)n yk1/n → 0 as n → ∞, for every x, y ∈ L. As ad x has countable spectrum, ad x is decomposable. So the local quasinilpotence of ad x on L implies that ad x is quasinilpotent, and hence L is Engel. Now if Lπ is E-solvable then [Lπ , Lπ ] is Volterra, whence [Lπ , Lπ ] is Engel. It follows from above that fπ ([Lπ , Lπ ]) is Engel. Since [L, L] = fπ ([Lπ , Lπ ]), L is E-solvable by Theorem 2.5.

Theorem 5.3. Let L be a Γ-subgraded Lie algebra of compact operators. If the union of the closures of all components Lγ doesn’t contain non-zero finite rank operators, then L is Engel. P Proof. Replacing L by γ∈Γ Lγ if necessary, we may assume that each Lγ is closed, whereupon it follows that the algebra Lπ is closed. If Lπ contains a non-zero finite P rank operator γ aγ ⊗ π(γ) then since each aγ must be an operator of finite rank, the hypothesis implies that each aγ = 0, and hence that a = 0. It follows that Lπ does not contain any nonzero finite rank operators, and hence that it is Engel by Theorem 2.4. The result now follows by applying Lemma 5.2. For a Γ-subgraded Lie algebra L, define CΓ (L) = {[a, b] : a, b ∈ ∪γ∈Γ Lγ }. Then CΓ (L) is the set of all homogeneous commutators in L. It is obvious that CΓ (L) is a Lie submultiplicative set in L. Now we may apply Theorem 5.3 to obtain the following result. Theorem 5.4. If L is a Γ-subgraded Lie algebra of compact operators on X , and if CΓ (L) consists of Engel elements of L, then L is E-solvable and is therefore triangularizable. Proof. It is sufficient to show that L is reducible. One may assume that every component of L is a closed subspace in B(X ). Let I = CΓ (L) ∩ F(X ). Then I is a Lie multiplicative set in L. Suppose, to the contrary, that L is irreducible. Then L is not Engel, and then there is a component Lγ containing a

Vol. 63 (2009)

Invariant Subspaces

73

non-zero finite rank operator, say a by Theorem 5.3. If a is scalar then X is finitedimensional and span I consists of Engel elements of L by Lemma 4.1. But in this case I = CΓ (L), whence span I = [L, L], and L is triangularizable by Theorem 2.5, a contradiction. So one can assume that a is not scalar. If [a, Lβ ] = 0 for every β ∈ Γ then [a, L] = 0, whence L0 is reducible by Lomonosov’s Theorem [L], which gives a contradiction. Therefore [a, Lβ ] 6= 0 for some β ∈ Γ, whence I 6= 0 and, moreover, I is non-scalar. Since CΓ (L) consists of Engel elements of L by hypothesis, so is the set I. By Lemma 4.1, span I consists of Engel elements of L. In particular, span I is Engel. But span I is an ideal of L, whence L is reducible by Theorem 2.6, a contradiction. One consequence of Theorem 5.4 is the following homogeneous version of the local triangularizability result from [ST0, Corollary 5.17]. P Lγ be a Γ-subgraded Lie algebra of compact operators. Corollary 5.5. Let L = If for any homogeneous a, b the set {a, b} is triangularizable, then L is triangularizable, and hence is E-solvable. Proof. It follows from homogeneous local triangularizability of L that CΓ (L) is Volterra, whence L is triangularizable by Theorem 5.4. A stronger result is contained in Corollary 7.8.

6. The Case that the Underlying Space is Finite-Dimensional Theorem 5.3 shows that one should analyze graded Lie algebras with non-zero finite rank operators. In this section, we consider the case when L is a Lie algebra of operators on a finite-dimensional vector space. We begin with a graded version of Cartan’s criterion. Lemma 6.1. Let L be a Γ-graded Lie subalgebra of operators on a finite-dimensional space. If L0 is scalar, and some component Lα contains an Engel element a of L, then a belongs to the radical R of L. If in particular, a is non-scalar, then L is reducible. Proof. Using the non-graded version of Cartan’s Criterion, it suffices to show that hx, ai = 0 for every x in L. For arbitrary x in Lγ , the operator (ad x)(ad a) maps each Lδ to Lα+γ+δ . If α + γ 6= 0, then (ad x)(ad a) doesn’t fix any component of L. Thus, an easy calculation shows that in some basis of L, the operator (ad x)(ad a) is represented by a matrix with a zero main diagonal, whence hx, ai = 0. On the other hand, if α + γ = 0, then [x, a] is in L0 . Since L0 is scalar, this implies that x and a commute, whence ad x and ad a commute. Since a is an Engel element,

74

Kennedy, Shulman and Turovskii

IEOT

ad a is nilpotent, and it follows that (ad x)(ad a) is nilpotent, whence hx, ai = 0. Hence hx, ai = 0 for all x in L. Let I = {u ∈ L : hu, Li = 0}. Since [v, L] ⊂ L for every v ∈ L, it follows from h[u, v], Li = hu, [v, L]i = 0 for every u ∈ I that in fact the subspace I is an ideal of L. By Lemma 2.1, I is solvable, and therefore a ∈ I ⊂ R. If a is not scalar, then L is reducible by Theorem 2.6. One can readily deduce the following lemma from algebraic results of [Kr, KrK], but we prefer to give a self-contained and comparatively simple proof for the convenience of readers. Lemma 6.2. Let L be a Zn -graded Lie subalgebra of operators on a finitedimensional space X . If L0 is scalar, then L is solvable, and therefore is triangularizable. Proof. One may assume that dim X > 1. Then, proceeding inductively, it suffices to prove that L is reducible. Suppose, to the contrary, that L is irreducible. It is clear that L is not scalar. Claim 1. If [Lm , [Lm , Lk ]] is scalar then [Lm , Lk ] = 0. Suppose that [a, b] 6= 0 for some a ∈ Lm and b ∈ Lk . By condition, [a, [a, b]] is scalar. As tr([a, [a, b]]) = 0, we obtain that [a, [a, b]] = 0. Then by the Kleinecke-Shirokov Theorem [Kl, Sh], [a, b] is nilpotent, meaning Lm+k contains a non-scalar Engel element of L, whence L is reducible by Lemma 6.1, which gives a contradiction. Therefore we have that [a, b] = 0, and this proves Claim 1. Claim 2. If Lm is not scalar then there is a positive integer k < n such that [Lm , Lk ] and [Lm , [Lm , Lk ]] are not scalar. If Lm is in the center of L then L is reducible by [L], a contradiction. Thus there is an integer k > 0 such that k < n, and [Lm , Lk ] 6= 0. It follows from Claim 1 that [Lm , [Lm , Lk ]] and [Lm , Lk ] are non-scalar. Claim 3. If Lm is not scalar and an integer k is as in Claim 2 then Lk+pm is not scalar for every integer p > 0.

Vol. 63 (2009)

Invariant Subspaces

75

Inductively applying Claims 1 and 2, we obtain that p times

Lk+pm

z }| { ⊃ [Lm , . . . , [Lm , [Lm , [Lm , Lk ]]] · · · ]

is non-scalar for every p. Let gcd(m, n) be the greatest common divisor of integers m and n. Claim 4. If gcd(m, n) = 1 for some integer m > 0, then Lm is scalar. Suppose, to the contrary, that Lm is non-scalar. Then it follows from Claim 3 that there is an integer k such that Lk+pm is non-scalar for every integer p > 0. On the other hand, since gcd(m, n) = 1, one can find a positive integer p such that n divides k + pm. As Lk+pm = L0 , Lk+pm is scalar, a contradiction. This proves Claim 4. Claim 5. Ln−1 , Ln and Ln+1 are scalar. Indeed, it follows from Claim 4 that Ln−1 and Ln+1 (= L1 ) consist of scalar operators. Recall that Ln (= L0 ) is also scalar, by hypothesis. Claim 6. Lm is scalar for every positive integer m < n. If not, let m be the least positive integer such that L0 , L1 , ..., Lm−1 consist of scalar operators, but Lm is non-scalar. Then there is an integer k satisfying Claim 2. On the other hand, there is an integer p > 0 such that k + pm ∈ {n − 1, n, n + 1, ..., n + m − 1}. Taking into account Claim 5, we conclude that for any integer q in this set, Lq must be scalar. But Lk+pm is non-scalar by Claim 3, a contradiction. This proves Claim 6. Now it readily follows by Claim 6 that L consists of scalar operators, a contradiction. We have therefore proved that L is triangularizable, and hence is solvable. The following example shows that the restriction to cyclic groups in the above lemma cannot be removed. Example 6.3. Define 0 a= −1

1 , 0

b=

0 −i , −i 0

c=

−i 0 . 0 i

These matrices form a basis of L := sl(2, C), the Lie algebra of all operators on C2 with zero trace, and satisfy the identities [a, b] = 2c,

[b, c] = 2a,

[c, a] = 2b.

Let Γ = Z2 × Z2 ; then we may define a Γ-grading on L by setting L(0,0) = 0,

L(0,1) = Ca,

It is clear that L is irreducible.

L(1,0) = Cb,

L(1,1) = Cc.

76

Kennedy, Shulman and Turovskii

IEOT

Note that the spectral radii of a, b, c are equal to 1. It is easy to calculate that

1/2 ≤ |λ| + |µ| = ρ(λx) + ρ(µy), ρ(λx + µy) = λ2 + µ2

and ρ(λxµy) = |λ| |µ| ρ(xy) = |λ| |µ| = ρ(λx)ρ(µy), for every λ, µ ∈ C, and x, y ∈ {a, b, c} with x 6= y. This shows that the following conditions (even taken together) • ρ(u + v) ≤ ρ(u) + ρ(v) for every u, v ∈ ∪γ∈Γ Lγ . • ρ(uv) ≤ ρ(u)ρ(v) for every u, v ∈ ∪γ∈Γ Lγ . are not sufficient to imply the reducibility of a Γ-graded Lie algebra of compact operators with L0 = 0. Lemma 6.4. Let L be a Γ-subgraded Lie algebra of operators on a finite-dimensional space. Then L is solvable if one of the following conditions holds. (i) Lα consists of Engel elements of L for every α ∈ Γ. (ii) Γ = Zn and L0 consists of Engel elements of L. Proof. First, assume that L is graded. By Theorem 2.5, we must show that L is triangularizable. For this, it suffices to prove that L is reducible. Case 1. L0 is not scalar. For α 6= 0, as in Lemma 6.1, we obtain that hL0 , Lα i = 0. Since ad L0 consists of nilpotent elements, we also have hL0 , L0 i = 0. Since the Killing form is linear, we therefore obtain that hL0 , Li = 0. Let I = {a ∈ L : ha, Li = 0}. Then I is a solvable ideal by Lemma 2.1. Since L0 ⊂ I, I is non-scalar, whence L is reducible by Theorem 2.6. This proves both (i) and (ii). Case 2. L0 is scalar. In this case, (i) follows from Lemma 6.1, and (ii) follows from Lemma 6.2. Case 3. L is not necessarily graded (but of course is Γ-subgraded). Let M = Lπ (see the beginning of Section 5). Then M is a Γ-graded Lie algebra of compact operators on a finite-dimensional space. It is easy to see that if some Lγ consists of Engel elements of L then Mγ consists of Engel elements of M . Indeed, one can check that adM (a ⊗ π(γ)) = (adL a) ⊗ π + (γ)

Vol. 63 (2009)

Invariant Subspaces

77

where π + is the representation of Γ on the image of π defined by π + (γ)π(α) = π(γ + α) for every α ∈ Γ. Now if adL a is nilpotent, then so is (adL a) ⊗ π + (γ), and we conclude that adM (a ⊗ π(γ)) must also be nilpotent. It now follows from above that M is solvable; the solvability of L follows from Lemma 5.2. In the section on Z2 -graded algebras we will make use of the following result which is reminiscent of Cartan’s Criterion. Lemma 6.5. Let L be a Γ-graded Lie algebra of operators on a finite-dimensional space. If L0 is solvable and non-commutative, then L has a non-scalar solvable ideal, and is therefore reducible. Proof. By Lemma 2.1, we have that h[L0 , L0 ], L0 i = 0. Since, as in Lemma 6.1, hL0 , Lα i = 0 for all α 6= 0, and [L0 , L0 ] ⊂ L0 , we obtain that h[L0 , L0 ], Lα i = 0 for all α 6= 0. This implies that h[L0 , L0 ], Li = 0. Let I = {a ∈ L : ha, Li = 0}. As we have seen, I is a solvable ideal of L by Lemma 2.1. Note that it is non-scalar because [L0 , L0 ] ⊂ I, and [L0 , L0 ] contains a non-zero nilpotent operator. Therefore, L is reducible by Theorem 2.6.

7. Triangularization Theorems Taking Theorem 5.3 into account, we have to consider those subgraded Lie algebras of compact operators on infinite-dimensional spaces whose components contain finite rank operators. Recall (Remark 3.7) that a finite rank operator a in an operator Lie algebra L is an Engel element of L if and only if the operator ad a is nilpotent on L. Lemma 7.1. Suppose that a Γ-subgraded Lie algebra L consists of finite rank operators and that all components Lγ consist of Engel elements of L (if Γ = Zn , then we impose the last condition only on operators in L0 ). Then L is E-solvable and is therefore triangularizable.

78

Kennedy, Shulman and Turovskii

IEOT

Proof. Given a finite-dimensional subspace Y of X , let Y LY γ = {a ∈ Lγ : aX ⊂ Y} and L =

X

LY γ.

γ∈Γ

Then LY is a Γ-subgraded Lie algebra of operators on X . It is evident that Y is invariant for LY , and the restriction LY |Y of LY to Y is a Γ-subgraded Lie algebra of operators on Y, namely X LY |Y = LY γ |Y. γ∈Γ Y

As Y is finite-dimensional, L |Y is also finite-dimensional. Suppose that some Lγ consists of Engel elements of L, and let a be an arbitrary element of LY γ . Since a ∈ Lγ , we obtain for every b ∈ LY that n

k(ad a|Y) (b|Y)k

1/n

n

= k((ad a) b)|Yk

1/n

n

≤ k(ad a) bk

1/n

→0

as n → ∞, whence ad a|Y is nilpotent on LY |Y. Therefore, the component LY γ |Y of LY |Y consists of Engel elements of LY |Y. Thus LY |Y satisfies conditions of Lemma 6.4. Then, by Lemma 6.4, LY |Y is solvable and is triangularizable, whence any sum of commutators in P LY |Y is nilpotent.PLet ai , bi (1 ≤ i ≤ n) be arbitrary operators in L. Then ai = γ ai,γ and bi = γ bi,γ , where the sums are finite, in accordance of grading on L. This implies the existence of a finite-dimensional subspace Y that contains all ai,γ X and bi,γ X . It follows that ai , bi ∈ LY for every i. Let X c= [ai , bi ]. i

If c is not nilpotent then there is a non-zero vector x ∈ X such that cx = λx for some non-zero λ ∈ C; but x ∈ Y and c|Y is not nilpotent, which gives a contradiction. This proves that [L, L] consists of nilpotent operators, and hence that L is E-solvable and triangularizable by Theorem 2.5. P Lemma 7.2. Let L = Lγ be a Γ-subgraded Lie algebra of compact operators. Suppose that ∪γ∈Γ Lγ contains a non-scalar finite rank operator and each Lγ consists of Engel elements of L (if Γ = Zn , then we again impose this condition only on L0 ). Then L contains a non-scalar E-solvable ideal. P Proof. Let Iγ = Lγ ∩ F(X ) and I = γ∈Γ Iγ . Then I is a non-scalar ideal of L, so it suffices to show that I is triangularizable. But this follows from Lemma 7.1, because I is a Γ-subgraded Lie algebra of finite rank operators and each Iγ consists of Engel elements of I (if Γ = Zn , then this need be true only of I0 ). Now we obtain our main result on triangularization of Γ-subgraded Lie algebras.

Vol. 63 (2009)

Invariant Subspaces

79

Theorem 7.3. Let L be a Γ-subgraded (respectively, Zn -subgraded) Lie algebra of compact operators. If each Lγ (respectively, only L0 ) consists of Engel elements of L, then L is E-solvable, and is therefore triangularizable. Proof. Follows from Lemma 7.2 and Theorem 5.3.

Corollary 7.4. If the components of a Γ-subgraded Lie algebra L of compact operators consist of Engel elements of L (respectively, of Volterra operators) then L is Engel (respectively, Volterra). Proof. Indeed, L is triangularizable by Theorem 7.3. Since the spectral radius is subadditive on ad L (respectively, on L) by Theorem 2.5, and since every operator in L is a sum of elements of components of L, every operator in L is an Engel element of L (respectively, is a Volterra operator). The following example shows that one cannot weaken the assumptions of Theorem 7.3 to the case that the components are triangularizable. Example 7.5. Let X be two-dimensional and L = sl(2, C). Let e, f, g be operators in L such that [e, f ] = g, [g, e] = e, [g, f ] = −f. Then L has a Z3 -grading, defined by setting L0 = Cg,

L1 = Ce,

L2 = Cf.

The components L1 and L2 are Volterra, L0 is triangularizable (more precisely, is commutative), but L is clearly irreducible. Note that L0 doesn’t consist of Engel elements of L, and that [L1 , L2 ] doesn’t consist of Volterra operators. The commutativity of L0 is a necessary condition for such examples in virtue of Lemma 6.5. It follows from Remark 8.3 below that the condition [L1 , L2 ] is Volterra is essential for reducibility of Z3 -subgraded Lie algebras of compact operators with triangularizable L0 . For a Γ-subgraded Lie algebra L, let X L00 = [Lγ , L−γ ] and L000 = γ∈Γ

X

[Lγ , L−γ ].

γ∈Γ\{0}

It is clear that L000 ⊂ L00 ⊂ L0 . We also set L0 = L00 +

X

Lγ

and L00 = L000 +

γ∈Γ\{0}

X

Lγ .

γ∈Γ\{0}

It is easy to see that L0 and L00 are Γ-subgraded Lie algebras whose zero components are L00 and L000 , respectively, and that L0γ = L00γ = Lγ for every γ ∈ Γ\{0}. The following lemma motivates the definition of L0 and L00 .

80

Kennedy, Shulman and Turovskii

IEOT

Lemma 7.6. For a Γ-subgraded Lie algebra L of compact operators, the Lie algebras L0 and L00 are ideals of L. Moreover, L is E-solvable if and only if is L0 . Proof. It is easy to check that [L0 , L] ⊂ L0 and [L00 , L] ⊂ L00 . If L is E-solvable, then L is triangularizable, whence L0 is triangularizable, and is therefore E-solvable. If L0 is E-solvable, then taking into account that [L, L] ⊂ L0 , we have that [L, L] is triangularizable, whence L is triangularizable, and is therefore E-solvable. Define the relation Γ] in Γ × Γ by setting Γ] = {(α, β) ∈ Γ × Γ : no cyclic subgroup of Γ contains α and β}. It is clear that Γ] is an empty relation if Γ is cyclic. Now we are in a position to obtain a stronger version of Theorem 5.4. Theorem 7.7. Let L be a Γ-subgraded Lie algebra of compact operators. If every commutator [a, b] is an Engel element of L for a ∈ Lγ and b ∈ Lδ such that γ + δ = 0 or (γ, δ) ∈ Γ] , then L is E-solvable and is triangularizable. Proof. Since C(L0 ) consists of Engel elements of L, it follows from Theorem 4.7 that L0 is triangularizable. Claim 1. L00 consists of Engel elements of L0 . Indeed, since L00 ⊂ L0 , we have that is triangularizable. By hypothesis, every commutator [a, b], with a ∈ Lγ and b ∈ L−γ , is an Engel element of L for every γ ∈ Γ. Since a finite sum of Engel elements of L in the triangularizable Lie algebra L0 is again an Engel element of L by Theorem 3.5, it follows that L00 consists of Engel elements of L. Since L0 ⊂ L, L00 consists of Engel elements of L0 . L00

Claim 2. The set CΓ (L0 ) of homogeneous commutators in L0 consists of Engel elements of L0 . Let a ∈ L0γ and b ∈ L0δ . Assume first that there is a cyclic subgroup Λ ⊂ Γ containing γ and δ. Then X a, b ∈ N := L0β , β∈Λ

where N is a Λ-subgraded Lie algebra of compact operators. Since the zero component of N consists of Engel elements of L0 by Claim 1, this component consists of Engel elements of N . Then N is triangularizable by Theorem 7.3, whence [a, b] is a Volterra operator. In particular, [a, b] is an Engel element of L0 . If (γ, δ) ∈ Γ] then [a, b] is an Engel element of L, and hence of L0 , by hypothesis.

Vol. 63 (2009)

Invariant Subspaces

81

Claim 3. L is E-solvable and triangularizable. Indeed, by Claim 2 and Theorem 5.4, L0 is triangularizable. By Theorem 2.5, L is E-solvable. Now the claim follows from Lemma 7.6. 0

Theorem 7.7 allows us to obtain the following result which extends Corollary 5.5. Corollary 7.8. Let L be a Γ-subgraded Lie algebra of compact operators. If every pair {a, b} is triangularizable for a ∈ Lγ and b ∈ Lδ such that γ + δ = 0 or (γ, δ) ∈ Γ] , then L is triangularizable. Proof. Indeed, every commutator pointed out in Theorem 7.7 is Volterra in L, whence L is E-solvable and, by Theorem 2.5, is triangularizable.

8. Consequences Corollary 8.1. Let L be a Γ-subgraded Lie algebra of compact operators. Then L is E-solvable if one of the following conditions holds. (i) L0 consists of Engel elements of L, and [Lγ , Lδ ] consists of Engel elements of L for every non-zero γ, δ ∈ Γ with (γ, δ) ∈ Γ] . (ii) L0 is E-solvable, and [Lγ , Lδ ] consists of Engel elements of L for every nonzero γ, δ ∈ Γ such that γ + δ = 0 or (γ, δ) ∈ Γ] . Proof. First, we check that the conditions of Theorem 7.7 hold in each case. For (ii), it is sufficient to show that [L0 , L0 ] is Volterra. As L0 is E-solvable, this is obvious by Theorem 2.5. For (i), it is sufficient to show that [Lγ , L−γ ] is Volterra for every γ ∈ Γ. For γ = 0, we have that [L0 , L0 ] is Volterra, because L0 is triangularizable. Now take a non-zero γ ∈ Γ, and let Λ be a subgroup of Γ generated by γ. Then Λ is cyclic. Let X N= Lα . α∈Λ

Then N is a Λ-subgraded Lie algebra of compact operators. Since Λ is cyclic, and since the component L0 consists of Engel elements of L, and hence of N , this Lie algebra N is triangularizable by Theorem 7.3. Then [N, N ] is Volterra by Theorem 2.5, and since [Lγ , L−γ ] ⊂ [N, N ] , we have that [Lγ , L−γ ] is Volterra. Therefore, L is E-solvable in both cases by Theorem 7.7. In particular, Corollary 8.1 implies the following result. Corollary 8.2. Let L be a Zn -subgraded Lie algebra of compact operators. If L0 is E-solvable, and [Lk , L−k ] consists of Engel elements of L for every integer k > 0 such that k ≤ n/2, then L is E-solvable and is therefore triangularizable. Proof. Indeed, L satisfies conditions of Corollary 8.1(ii).

82

Kennedy, Shulman and Turovskii

IEOT

Remark 8.3. The special case of Corollary 8.2 for Z3 -subgraded Lie algebras gives us the following result: Let L = L0 + L1 + L2 be a Z3 -subgraded Lie algebra of compact operators. If L0 is E-solvable and [L1 , L2 ] consists of Volterra operators then L is E-solvable. The condition that [L1 , L2 ] is Volterra is essential in virtue of Example 7.5. Both of the assertions of Theorem 7.3 are contained in the following result. Corollary 8.4. Let L be a Γ-subgraded Lie algebra of compact operators. Suppose that Γ has a subgroup Λ, such that Γ/Λ is cyclic. If Lδ consists of Engel elements of L for every δ ∈ Λ, then L is E-solvable. P P Proof. For α ∈ Γ/Λ, let Mα = γ∈α Lγ . Then M := α∈Γ/Λ Mα is a (Γ/Λ)subgraded Lie algebra of compact operators and M = L. Note that X M0 = Lδ . δ∈Λ

By Corollary 7.4, M0 is an Engel Lie algebra. As M0 is triangularizable by Theorem 2.5, M0 consists of Engel elements of M by Theorem 3.5. So M is E-solvable by Theorem 7.3. Theorem 7.3 allows us, with certain hypotheses in place, to extend the triangularizability of the zero component of a Zn -subgraded Lie algebra of compact operators to the entire algebra. It is natural to ask for the cases in which one can repeat this procedure of inflation. More specifically, let M be a Zm -graded Lie algebra, and let N := M0 be a Zn -graded Lie algebra such that N0 is Volterra. Is it true that M is E-solvable? The following example shows that the answer to this question is negative in general. Example 8.5. Let L be as in Example 6.3, and define M0 = L(0,0) + L(0,1) ,

M1 = L(1,0) + L(1,1) ,

and N0 = L(0,0) ,

N1 = L(0,1) .

Then M := M0 + M1 and N := N0 + N1 are Z2 -graded Lie algebras, N = M0 , and N0 = 0, but it is clear that M = L is irreducible. Lemma 8.6. Let L be a normed Lie algebra (respectively, a normed algebra), and let ϕ be a bounded endomorphism of L. Then [Eλ (ϕ), Eµ (ϕ)] ⊂ Eλµ (ϕ) (respectively, Eλ (ϕ)Eµ (ϕ) ⊂ Eλµ (ϕ)). Proof. Put ϕλ = ϕ − λ for every λ ∈ C. Let x, y be arbitrary elements of L. Claim 1. ϕλµ ([x, y]) = [ϕλ (x), ϕµ (y)] + [λx, ϕµ (y)] + [ϕλ (x), µy]. It is an easy calculation.

Vol. 63 (2009)

Invariant Subspaces

83

Claim 2. ϕnλµ ([x, y]) is the sum of 3n summands of type λn−i ϕiλ (x), µn−j ϕjµ (y) for integers i and j such that 0 ≤ i ≤ n, 0 ≤ j ≤ n and i + j ≥ n. The claim follows easily by induction. Indeed, let zi = λn−i ϕiλ (x) and wj = µ ϕjµ (y) for integers i, j such that 0 ≤ i ≤ n, 0 ≤ j ≤ n and i + j ≥ n. It is sufficient to show that ϕλµ ([zi , wj ]) may be written as the sum of three terms satisfying the condition of Claim 2 for n + 1, and this follows by Claim 1. n−j

Claim 3. [Eλ (ϕ), Eµ (ϕ)] ⊂ Eλµ (ϕ). Let x ∈ Eλ (ϕ) and y ∈ Eµ (ϕ). Then for every ε > 0 there is a constant C > 0 such that

m m

kϕm and ϕm µ (y) ≤ Cε λ (x)k ≤ Cε for every integer m ≥ 0. Let t = max {|λ| , |µ| , 1}. Assume that ε ≤ 1. Then, by Claim 2, we obtain that

n

ϕλµ ([x, y]) ≤ 3n max λn−i ϕiλ (x), µn−j ϕjµ (y) : i + j ≥ n ≤ 3n t2n−i−j 2C 2 εi+j ≤ 2C 2 (3tε)n for every integer n ≥ 0. Hence

1/n

≤ 3tε. lim sup ϕnλµ ([x, y]) Since ε is arbitrary, we have that [x, y] ∈ Eλµ (ϕ). This proves Claim 3. The case of an endomorphism of a normed algebra is similar. Corollary 8.7. Let L be a Lie algebra of compact operators, ϕ an automorphism of L. Then L is E-solvable if one of the following conditions holds. (i) ϕ has a finite order, say ϕn = 1, and the set of fixed points of ϕ consists of Engel elements of L. (ii) L is closed, ϕ is bounded, there is an integer n > 0 such that ϕn − 1 is a quasinilpotent operator on L, and E1 (ϕ) consists of Engel elements of L. Proof. Let θ = e2πi/n . (i) For each k ∈ {0, 1, . . . , n − 1}, let Lk = {x ∈ L : ϕ(x) = θk x}. Then it is not difficult to check that L = L0 + . . . + Ln−1 , and that L0 is the Lie algebra of fixed points of ϕ. It is also clear that [Lk , Lj ] ⊂ Lk+j

(8.1)

where the addition is modulo n. In other words, L is Zn -graded. As L0 consists of Engel elements of L, L is E-solvable by Theorem 7.3.

84

Kennedy, Shulman and Turovskii

IEOT

(ii) For each k ∈ {0, 1, . . . , n − 1}, let Lk = Eθk (ϕ).

n−1

Since σ(ϕ) = 1, θ, . . . , θ and Lk is the image of the Riesz projection of ϕ corresponding to θk , we have that L = L0 + . . . + Ln−1 , and L0 = E1 (ϕ). It follows from Lemma 8.6 that (8.1) holds. Therefore L is a Zn -graded Lie algebra and L is E-solvable. Remark 8.8. Recall that if L is Zn -graded, then there is an automorphism ϕ of L such that ϕn = 1 and L0 = {a ∈ L : ϕ(a) = a}. Indeed, it suffices to set ϕ(a) = θk a for a ∈ Lk and extend ϕ to L by linearity. Thus Corollary 8.7(i) is in fact a reformulation of the part of Theorem 7.3, which tells about Zn -graded Lie algebras. Corollary 8.9. Let L be a Zn -subgraded Lie algebra of compact operators, and let I be L0 or L00 . If at least one of the components of I is non-scalar, and I0 consists of Engel elements of I, then L is reducible. In particular, if I = L0 then L is E-solvable. Proof. As I is an ideal of L, it follows from Theorem 7.3 that I is an E-solvable Lie algebra, and hence L is reducible by Theorem 2.6. If I = L0 , then it follows from Lemma 7.6 that L is E-solvable. Corollary 8.10. Let L be a Zn -subgraded Lie algebra of compact operators. If Lk commutes with L−k for every k 6= 0, and at least one of components of L00 is not scalar, then L is reducible. Proof. By hypothesis, L000 = 0, whence L is reducible by Corollary 8.9.

The following statement is a variation of Theorem 5.3. Corollary 8.11. Let L be a Zn -subgraded Lie algebra of compact operators, and let I be one of L00 , L0 and L. If at least one of the components of I is non-scalar, and if there are no non-zero nilpotent finite rank operators in the the closure of any Ik , then I0 consists of Engel elements of I, and L is reducible. In particular, if I = L or I = L0 , then L is E-solvable. Proof. Let a be an arbitrary element of L0 . If (ad a)|Ik has a non-zero isolated point λ in its spectrum for some k then there is a non-zero x ∈ Ik such that x ∈ Eλ (ad a). By [W, Theorem 3], x is a nilpotent finite rank operator, which would be a contradiction. So ad a is locally quasinilpotent on every Ik and therefore on their sum, that is, I ⊂ E0 (ad a).

Vol. 63 (2009)

Invariant Subspaces

85

The last inclusion clearly holds if we consider ad a as an operator on I. Since ad a has countable spectrum, E0 (ad a) is closed. So one can consider E0 (ad a) as a Banach space, whence ad a is quasinilpotent on I. In any of the given cases, I0 consists of Engel elements of I, so by Theorem 7.3, I is E-solvable. Thus, if I = L00 , L is reducible by Corollary 8.9, and if I = L0 , L is E-solvable by Lemma 7.6. Theorem 7.3 can be partially extended to the case of a Lie algebra graded by an infinite group. We present below an appropriate version of such an extension. Let P Γ be an arbitrary (not necessarily finite) commutative group, and let L := a Γ-subgraded Lie algebra. Let Λ be another commutative γ∈Γ Lγ be P group and M := λ∈Λ Mλ be a Λ-subgraded Lie algebra. We say that M is a subgraded subalgebra of L if for every δ ∈ Λ there is γ ∈ Γ such that Mδ ⊂ Lγ . We say that L is locally finitely subgraded if every finite subset of homogeneous elements of L is contained in a Λ-subgraded subalgebra M for some finite group Λ. Examples of locally finitely subgraded Lie algebras include those which are graded with a locally finite group Γ. Theorem 8.12. Let Γ be a not necessarily finite, commutative group, and let a Γsubgraded Lie algebra L ⊂ K(X ) be locally finitely subgraded. If every component Lγ consist of Engel (respectively, Volterra) elements of L then L is E-solvable (respectively, is Volterra). Pn Pm Proof. Let a, b ∈ L be arbitrary. Then we may write a = i=1 ai , and b = j=1 bj , where all ai , bj are homogeneous elements of L. The set {ai : 1 ≤ i ≤ n} ∪ {bj : 1 ≤ j ≤ m} is contained in a finitely subgraded Lie subalgebra M of L, and it is clear that every component of M consists of Engel elements of M , whence M is triangularizable. In particular, [a, b] is Volterra, whence by Theorem 2.5, L is E-solvable. The case with Volterra components is proved similarly. Indeed, since, by Corollary 7.4, every finitely subgraded Lie algebra of compact operators with Volterra components is Volterra, a is a Volterra operator. Part 3. Z2 -Subgraded Lie Algebras, Lie Triple Systems and Jordan Algebras The situation we treat in this part differs from one in Part 2. For a Z2 -subgraded Lie algebra of compact operators L = L0 + L1 , we impose a quasinilpotence condition to L1 instead of L0 . The main result is Theorem 9.4. As an application, we establish that every Lie triple system or Jordan algebra of Volterra operators is triangularizable, and also that every Jordan algebra of compact operators containing a non-zero Volterra ideal is reducible.

9. Lie Algebras Graded by Z2 Z2 -subgraded Lie algebras are of special interest. We have already seen that if L = L0 + L1 is a Z2 -subgraded Lie algebra of compact operators, and L0 consists

86

Kennedy, Shulman and Turovskii

IEOT

of Engel elements of L, then L is triangularizable. It turns out that if L1 generates L and consists of Engel elements of L then L is triangularizable. As the example 10.3 below shows, this does not extend to Zn -subgraded Lie algebras for arbitrary n. Note that if L = L0 + L1 is a Z2 -subgraded Lie algebra of operators then L−1 = L1 00

and so L is equal to L1 + [L1 , L1 ]. Recall that L00 is an ideal of L. Lemma 9.1. Let L = L0 +L1 be a Z2 -subgraded Lie algebra of operators on a finitedimensional space X . If L1 consists of Engel elements of L, then L00 is solvable. In particular, if L1 is non-scalar, then L is reducible. Proof. Let I = L00 . As I is an ideal of L, it suffices to show that I is triangularizable. For this, it suffices to prove that I is reducible. Suppose, to the contrary, that I is irreducible. Then setting I0 = [L1 , L1 ] and I1 = L1 , we have that I = I0 + I1 is a Z2 -subgraded Lie algebra. Consider the following two cases. Case 1. I1 consists of nilpotent operators. Note that I0 ∩ I1 is an ideal of I, and that moreover, since it consists of nilpotent operators, it is a Volterra ideal of L. If it is non-zero then I is reducible (by Theorem 2.6), which would be a contradiction. So we have that I0 ∩ I1 = 0. In other words, I is a Z2 -graded Lie algebra. Since L1 consists of nilpotent operators, we have that tr(a2 ) = 0 for each a ∈ L1 . Since tr(ab) =

1 2

tr((a + b)2 − a2 − b2 ), we see that tr(ab) = 0

for all a, b ∈ L1 . In other words, we have that tr(L1 L1 ) = 0. Using this and [L1 , [L1 , L1 ]] ⊂ L1 , we obtain that tr(I0 I0 ) = tr([L1 , L1 ][L1 , L1 ]) = tr(L1 [L1 , [L1 , L1 ]]) = 0. By Lemma 2.1, the Lie algebra I0 is solvable. So if I0 is not commutative, Lemma 6.5 implies that I is reducible, which would be a contradiction. Therefore, I0 is commutative. Moreover, by Lemma 6.2, I0 is not scalar. Set N = [I0 , I1 ]. If N is scalar then N = 0 (since the operators in N have zero trace), and I0 is in the center of I, whence I is reducible by Lomonosov’s Theorem [L], which would be a contradiction. So N is not scalar. Since [I0 , I0 ] = 0, one has tr(I0 N ) = tr(I0 [I0 , I1 ]) = tr(I1 [I0 , I0 ]) = 0.

Vol. 63 (2009)

Invariant Subspaces

87

Furthermore, since N ⊂ I1 = L1 , we have that tr(I1 N ) = 0. Therefore we obtain that tr(IN ) = 0, and hence by Lemma 2.3, I is reducible, which is a contradiction. This proves that I is, in fact, triangularizable. Case 2. I1 consists of Engel elements of L. In this case we use the adjoint representation ad = adL of L on L. Indeed, (ad L)00 = ad I, the first component (ad L)1 of ad L equals ad I1 , and hence consists of nilpotent operators. Then by Case 1, ad I is triangularizable, and hence is solvable. This means that I is solvable, whence it is triangularizable. In either case, if I is not scalar, L is reducible by Theorem 2.6. Lemma 9.2. Let L be a Z2 -subgraded Lie algebra of finite rank operators. If L1 consists of Engel elements of L, then L00 is E-solvable. In particular, if L1 is non-scalar, then L is reducible. Proof. The lemma can be proved in the same way as Lemma 7.1, but with the use of Lemma 9.1 instead of Lemma 6.4. Lemma 9.3. Let L be a Z2 -subgraded Lie algebra of compact operators. If L1 consists of Engel elements of L, and contains a non-scalar finite rank operator, then L is reducible. Proof. The lemma follows from Lemma 9.2 in the same way that Lemma 7.2 followed from Lemma 7.1. Theorem 9.4. Let L = L0 + L1 be a Z2 -subgraded Lie algebra of compact operators. If L1 is non-scalar, and consists of Engel elements of L, then L is reducible. Proof. Suppose to the contrary, that L is irreducible. Assume first that L0 and L1 are closed. By Lemma 9.3, L1 doesn’t contain any non-scalar finite rank operators. By Theorem 7.3, L0 doesn’t consist of Engel elements of L. If there is a ∈ L0 such that ad a is not quasinilpotent on L1 , then there is a non-zero isolated point λ ∈ σ((ad a)|L1 ), and a non-zero x ∈ L1 such that x ∈ Eλ (ad a)

(9.1)

Then x is a nilpotent finite rank operator in L1 by [W, Theorem 3], which would contradict our assumption. Thus (ad L0 )|L1 consists of quasinilpotent operators. We claim that there is a ∈ L0 such that ad a is not quasinilpotent on L0 . Indeed, otherwise ad L0 consists of operators that are locally quasinilpotent on L0 ∪ L1 and hence are quasinilpotent on L0 + L1 (see the proof of Corollary 8.11), which would be a contradiction. Therefore, there exists a non-zero isolated point λ ∈ σ((ad a)|L0 ), and a non-zero x ∈ L0 such that (9.1) holds. This implies that x is

88

Kennedy, Shulman and Turovskii

IEOT

a nilpotent finite rank operator in L0 . Since L1 does not contain non-scalar finite rank operators, we have that [x, L1 ] = 0. Moreover, we must also have that [x, L00 ] = 0. Then, since x commutes with L00 , xL00 consists of nilpotent finite rank operators, and we obtain that tr(xL00 ) = 0. Let I = {b ∈ L ∩ F(X ) : tr(bL00 ) = 0}. Since x ∈ I, the set I is non-scalar. Since [L, L00 ] ⊂ L00 , it follows from tr([I, L]L00 ) = tr(I[L, L00 ]) = 0 that I is an ideal of L. Let N = I ∩ L00 . Then N is an ideal of L, and tr(N N ) = 0. By Theorem 2.2, [N, N ] consists of nilpotent operators, whence N is E-solvable by Theorem 2.5. If N is non-scalar then L is reducible by Theorem 2.6, which would be a contradiction. So N is scalar. Since I and L00 are non-scalar ideals of L with scalar intersection, L is reducible by Theorem 2.7, a contradiction. This proves that L is reducible. In the general case, consider the Z2 -subgraded Lie algebra M with M0 = L0 and M1 = L1 . It is easy to check that M1 consists of Engel elements of M and that M 0 is non-scalar. Then the previous argument applied to M shows that M is reducible, whence L is reducible. We collect our results on Z2 -subgraded Lie algebras in the following theorem. Theorem 9.5. Let L be a Z2 -subgraded Lie algebra of compact operators. Then (i) L is reducible if L1 is non-scalar, and if one of the following conditions holds. • L1 consists of Engel elements of L00 . • [L1 , L1 ] consists of Engel elements of L00 . (ii) L is E-solvable if one of the following conditions holds. • L0 consists of Engel elements of L. • L1 generates L (as a Lie algebra) and consists of Engel elements of L. • [L0 , L0 ] and [L1 , L1 ] consist of Engel elements of L. Proof. In both first cases of (ii) it suffices to show that L is reducible. These statements follow from Theorems 7.3 and 9.4. The last case of (ii) follows from Theorem 7.7. In the case (i), applying results of (ii) to L00 , we obtain that L00 is E-solvable. Since L00 is a non-scalar ideal of L, L is reducible by Theorem 2.6. Remark 9.6. Recall that each Volterra operator is an Engel element of every operator Lie algebra which contains it. Thus, it follows from Theorem 9.5 that (i) If L1 is Volterra and non-zero then L is reducible. (ii) If L1 is Volterra and generates L (as a Lie algebra) then L is triangularizable.

Vol. 63 (2009)

Invariant Subspaces

89

10. Operator Lie Triple Systems A subspace M of B(X ) is a Lie triple system if it is closed under the Lie triple product [a, [b, c]] for all a, b, c ∈ M . It is clear that every Lie algebra of operators is a Lie triple system. The following lemma shows that there is a natural embedding of Lie triple systems into Z2 -subgraded Lie algebras. Lemma 10.1. Let M be a Lie triple system in B(X ). Set L0 = [M, M ] and L1 = M . Then L = L0 + L1 is a Z2 -subgraded Lie algebra. Proof. The inclusion [L1 , L1 ] ⊂ L0 is evident; in fact, [L1 , L1 ] = L0 . The inclusion [L0 , L1 ] ⊂ L1 is also clear; it follows from the definition of a Lie triple system. Finally, we have [L0 , L0 ] = [L0 , [L1 , L1 ]] ⊂ [[L0 , L1 ], L1 ] + [[L1 , [L0 , L1 ]] ⊂ [L1 , L1 ] = L0 .

Given a subset M of B(X ), we denote by L(M ) the Lie algebra generated by M . If M is a Lie triple system then it is clear that L(M ) = M + [M, M ]. Theorem 10.2. A Lie triple system M of Volterra operators is triangularizable. Proof. Let L = L(M ). It suffices to prove that the Z2 -subgraded Lie algebra L is triangularizable. Since L1 = M and M is Volterra, this follows by Theorem 9.5. For a subset M of a Lie algebra L, put n o M [1] = M and M [k+1] = [a, b] : a ∈ M, b ∈ M [k] . It is clear that ∪M [k] is a Lie multiplicative set in L. For a subspace M of L and n > 1, we say that M is a Lie n-product system in L if M [n] ⊂ M. The following example shows that the result of Theorem 10.2 does not extend to Lie n-product systems of Volterra operators for n = 5. Example 10.3. Let 

0 a= 0 0

 1 0 0 −1  , 0 0



0 b= 1 0

0 0 1

 0 0 , 0

and let M be the linear space generated by a and b. It is not difficult to check that M is an irreducible Lie 5-product system of nilpotent finite rank operators.

90

Kennedy, Shulman and Turovskii

IEOT

As a consequence, there is an irreducible Z4 -subgraded Lie algebra L of finite rank operators such that the component L1 consists of nilpotents and generates L as a Lie algebra. Indeed, let L = L(M ). Then L = M [1] + span M [2] + span M [3] + span M [4] in virtue of M [5] ⊂ M . Take span M [4] as the zero component of L and span M [i] as the i-th component of L for i = 1, 2, 3. It is easy to check that L is Z4 -subgraded.

11. Operator Jordan Algebras Recall that a subspace J of B(X ) is a Jordan algebra if it is closed under the Jordan product a ◦ b = ab + ba for all a, b ∈ J. The equality [a, [b, c]] = (a ◦ b) ◦ c − (a ◦ c) ◦ b

(11.1)

shows that every Jordan algebra is also a Lie triple system. Corollary 11.1. A Jordan algebra of Volterra operators is triangularizable. Proof. Follows from Theorem 10.2.

For the case of a Jordan algebra of Shatten operators on a Hilbert space this result was obtained in [K]. It was a starting point of the present work. Let J be a Jordan algebra. Recall that a Jordan ideal I of J is a subspace of J such that J ◦ I ⊂ I. Here, for Lie algebras L and N , the designation N C L means that N is an ideal of L. Lemma 11.2. Let J be a Jordan algebra in B(X ) and let I be a Jordan ideal of J. Define L(J, I) = I + [J, I]. Then L(J, I) is a Lie algebra, and L(I) C L(J, I) C L(J) is a series of Lie ideals, where L(I) and L(J) are Lie algebras generated by I and J, respectively. Proof. Recall that L(J) = J + [J, J] and L(I) = I + [I, I]. As I is an ideal of J, the inclusions [[J, I], J] ⊂ I and [[J, J], I] ⊂ I easily follow from (11.1). Then, applying the Jacobi identity, we obtain that ⊂J

⊂I

z }| { z }| { [[J, J], [J, I]] ⊂ [[[J, J], J], I] + [J, [[J, J], I]] ⊂ [J, I],

Vol. 63 (2009)

Invariant Subspaces

91

and [[J, I], [I, I]] ⊂ [[[J, I], I], I] + [I, [[J, I], I]] ⊂ [I, I]. | {z } | {z } ⊂I

⊂I

It is now easy to verify that [L(J, I), L(I)] ⊂ L(I) and [L(J), L(J, I)] ⊂ L(J, I).

Theorem 11.3. A Jordan algebra J of compact operators with a non-zero Volterra ideal I is reducible. Proof. As in Lemma 11.2, we have the series of ideals of Lie algebras L(I) C L(J, I) C L(J). By Corollary 11.1, I is triangularizable. Then L(I) is triangularizable and therefore is E-solvable by Theorem 2.5. Let R be the largest E-solvable ideal of L(J, I). Since L(I) is an E-solvable ideal of L(J, I), we obtain that I ⊂ L(I) ⊂ R. Hence R contains non-scalar elements. By Theorem 2.8, R is inner-characteristic. This means that [a, R] ⊂ R for every a ∈ B(X ) with [a, L(J, I)] ⊂ L(J, I). Since [L(J), L(J, I)] ⊂ L(J, I), this implies that [L(J), R] ⊂ R. In other words, R is an ideal of Lie algebra L(J). Since, by Theorem 2.6, every Lie algebra of compact operators with a non-scalar E-solvable ideal is reducible, we obtain that L(J) is reducible, and hence that J is reducible.

References [A] [B] [Hig] [Hum] [J2] [J8] [KaR]

J. C. Alexander, Compact Banach algebras, Proc. London Math. Soc. (3) 18 (1968) 1-18. ´ ements de math´ematique, Groupes et alg´ebres de Lie, Ch. 1-3, N. Bourbaki, El´ Hermann, Paris, 1971. G. Higman, Groups and Lie rings having automorphisms without non-trivial fixed points, J. London Math. Soc. 32 (1957) 321-334. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, Springer-Verlag, New York, 1978. N. Jacobson, Lie Algebras, Wiley Interscience, New York, London, 1962. N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc., Rhode Island, 1968. A. Katavolos, H. Radjavi, Simultaneous triangularization of operators on a Banach space, J. London Math. Soc. 41 (1990) 547-554.

92 [K]

Kennedy, Shulman and Turovskii

IEOT

M. Kennedy, Triangularization of a Jordan algebra of Shatten operators, Proc. Amer. Math. Soc. (to appear). [Kl] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957) 536-537. [Kr] V. A. Kreknin, Solvability of Lie algebras with a regular automorphism of finite period, Soviet Math. Dokl. 4 (1963) 683-685. [KrK] V. A. Kreknin, A. I. Kostrikin, Lie algebras with regular automorphisms, 4 (1963) 355-358. [KhM] E. I. Khukhro, N. Yu. Makarenko, Lie rings with almost regular automorphisms, J. Algebra 264 (2003) 641-664. [L] V. Lomonosov, Invariant subspaces for operators commuting with compact operators, Funct. Anal. Appl. 7 (1973) 213-214. [MKh] N. Yu. Makarenko, E. I. Khukhro, Almost solubility of Lie algebras with almost regular automorphisms, J. Algebra 277 (2004) 370-407. [M1] G. J. Murphy, Continuity of the spectrum and spectral radius, Proc. Amer. Math. Soc. 82 (1981) 619-621. [M2] G. J. Murphy, Triangularizable algebras of compact operators, Proc. Amer. Math. Soc. 84 (1982) 354-356. [M] V. M¨ uller, Spectral theory of linear operators and spectral systems in Banach algebras, Ser: Oper. Theory. Advances and Appl. Vol 139, Birkh¨ auser Verlag, Basel, 2003. [P] A. Pietsch, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [R] H. Radjavi, On the reduction and triangularization of semigroups of operators, J.Operator Theory 13 (1985) 63-71 [RR0] H. Radjavi, P. Rosenthal, Simultaneous Triangularization, Universitext, Springer, Berlin, 2000. [Sh] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspechi Mat. Nauk, 11 (1956), 167-168 (in Russian). [S] V. S. Shulman, On invariant subspaces of Volterra operators, Funktsional. Anal. i Prilozen., 18 (1984) no. 2, 84-85 (in Russian). [ST9] V. S. Shulman, Yu. V. Turovskii, Solvable Lie algebras of compact operators have invariant subspaces, In: Spectral and Evolutionary problems, (Proc. 9-th Crimean Autumn Math. School Symp., Sep. 18-29, 1998, Sevastopol, Laspi) Vol. 9, Simferopol, (1999) 38-44. [ST0] V. S. Shulman, Yu. V. Turovskii, Joint spectral radius, operator semigroups and a problem of W. Wojty´ nski, J. Funct. Anal. 177 (2000) 383-441. [ST1] V. S. Shulman, Yu. V. Turovskii, Radicals in Banach algebras, and some problems of theory of radical Banach algebras, Funktsional. Anal. i Prilozhen. 35 (2001) no. 4, 88-91 (in Rissian). [ST2] V. S. Shulman, Yu. V. Turovskii, On the Kleinecke-Shirokov theorem for families of derivations, Studia Math, 149 (2002) 185-190. [ST5] V. S. Shulman, Yu. V. Turovskii, Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action, J. Funct. Anal. 223 (2005) 425-508.

Vol. 63 (2009)

Invariant Subspaces

93

[STW] P. Shumyatsky, A. Tamarozzi, L. Wilson, Zn -graded Lie rings, J. Algebra 283 (2005) 149-160. [Th] J. G. Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. USA 45 (1959) 578-581. [T84] Yu. V. Turovskii, Spectral properties of elements of normed algebras and invariant subspaces, Funktsional. Anal. i Prilozen., 18 (1984) no. 2, 84-85 (in Russian). [T85] Yu. V. Turovskii, Spectral properties of some Lie subalgebras and spectral radius of subsets of a Banach algebra, in: F. G. Maksudov (Ed), Spectral Theory of Operators and its Applications, Vol. 6, ”Elm”, Baku (1985) 144-181 (in Russian). [T98] Yu. V. Turovskii, Volterra semigroups have invariant subspaces, J. Funct. Anal. 162 (2) (1999) 313-322. [V] K. Vala, On compact sets of compact operators, Ann. Acad. Fenn. Ser A I 351 (1964) 1-8. [W] W. Wojty´ nski, Banach-Lie algebras of compact operators, Studia Math. 59 (1977) 263-273. Matthew Kennedy Department of Pure Mathematics University of Waterloo 200 University Avenue West Waterloo, Ontario Canada N2L 3G1 e-mail: [email protected] Victor S. Shulman Department of Mathematics Vologda State Technical University 15 Lenina Street Vologda 16000 Russian Federation e-mail: shulman [email protected] Yuri V. Turovskii Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan 9 F. Agayev Street Baku AZ1141 Azerbaijan e-mail: [email protected] Submitted: July 21, 2008. Revised: October 5, 2008.

Integr. equ. oper. theory 63 (2009), 95–102 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010095-8, published online December 22, 2008 DOI 10.1007/s00020-008-1643-0

Integral Equations and Operator Theory

On Similarity of Multiplication Operator on Weighted Bergman Space Yucheng Li Abstract. Let D be the unit disk and A2α (D) (α > −1) be the weighted Bergman space. In this paper, we prove that the multiplication operator Mzn n L is similar to Mz on A2α (D). 1

Mathematics Subject Classification (2000). 32A36, 46J40, 47A15. Keywords. Weighted Bergman space, multiplication operator, invariant subspace, similarity.

1. Introduction Let D and T be the unit disk D = {z ∈ C : |z| < 1} and the boundary of 2 D respectively. For the Hardy space HN (D), the m-th N power of the unilateral m shift Tz is unitarily equivalent to Tz Im on H 2 (D) C m , since Tzm in an isometry of multiplicity m. Let A2α (D) (α > −1) denote the weighted Bergman space of analytic functions which belong to L2α (D). It is well known that A2α (D) is a closed subspace of L2α (D), and A2α (D) is a Hilbert space. If f ∈ A2α (D), then 1 R kf kα,2 = D |f (z)|2 dAα (z) 2 , where dAα (z) = (1 + α)(1 − |z|2 )α dA(z), and we denote by dA the ordinary Lebesgue area measure on D. When α = 0, A2 (D) is the ordinary Bergman space. In studying an operator on a Hilbert space, it is of interest to characterize the operators which commute with a given operator, for such a characterization should help in understanding the structure of the operator. The commutant of an analytic Toeplitz operator on the Hardy space and Bergman space has been studied extensively in the literature, we mention here the papers (see [1–6]), and the books [7–8] include an excellent account of the knowledge of operator theory. In [9] K. H. Zhu obtained a complete description of the reducing subspaces of The author was supported in part by NSF Grant (10571041, L2007B05).

96

Li

IEOT

multiplication operators on Bergman space induced by Blaschke product with two zeros using the method of the geodesic midpoint of the two zeros of B(z). In [10], J. Y. Hu, S. H. Sun, X. M. Xu and D. H. Yu proved that the analytic Toeplitz operator with finite Blaschke product symbol on the Bergman space has at least a reducing subspace on which the restriction of the associated Toeplitz operator is unitary equivalent to Bergman shift. In 2007, Jiang and Li [11] proved that each analytic Toeplitz operator MB(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. In this paper, we prove that n L multiplication operator Mzn is similar to Mz on the weighted Bergman space 1

A2α (D)(α > −1). The following is the Main Theorem in the paper. Main Theorem. Let A2α (D) (α > −1) be the weighted Bergman space, then multin L Mz on A2α (D). plication operator Mzn is similar to 1

2. The Proof of the Main Theorem of the weighted Bergman 0, 1, . . . , n − 1), then

q

Γ(k+2+α) k k!Γ(2+α) z (k = 0, 1, . . .) be the space A2α (D) (α > −1). Set Lj =

Proof. Step 1. Let ek (z) =

orthonormal basis span{enk+j } (j =

(a) {enk+j }∞ basis of Lj . k=0 L form Lan orthonormal L (b) A2α = L0 L1 · · · Ln−1 . (c) Lj is a reducing subspace for Mzn . In fact, (a) henk+j , enm+j i s Z s Γ(nk + j + 2 + α) nk+j Γ(nm + j + 2 + α) nm+j = z z¯ (nk + j)!Γ(2 + α) (nm + j)!Γ(2 + α) D ×(1 + α)(1 − |z|2 )α dA(z); if k = m, henk+j , enm+j i Z 1 Z 2π 1 Γ(nk + j + 2 + α) (1 + α) r2(nk+j) (1 − r2 )α rdrdθ π (nk + j)!Γ(2 + α) 0 0 Γ(nk + j + 2 + α) = (1 + α)B(nk + j + 1, α + 1) = 1, (nk + j)!Γ(2 + α) =

where Γ and B stand for the usual Gamma function and Beta function, respectively. If k 6= m, then henk+j , enm+j i = 0.

Vol. 63 (2009)

Multiplication Operator on Weighted Bergman Space

97

(b) First, Lj ⊥Lt , 0 ≤ j 6= t ≤ n−1. Without loss of the generality, we assume that j > t, then henk+j , enk+t i s Z s Γ(nk + j + 2 + α) nk+j Γ(nk + t + 2 + α) nk+t z z¯ = (nk + j)!Γ(2 + α) (nk + t)!Γ(2 + α) D ×(1 + α)(1 − |z|2 )α dA(z) s 1 Γ(nk + j + 2 + α)Γ(nk + t + 2 + α) = (1 + α) πΓ(2 + α) (nk + j)!(nk + t)! Z 1 Z 2π r2(nk+t) rj−t ei(j−t)θ (1 − r2 )α rdrdθ = 0. × 0

0

Next, suppose that ∞ X

a0k enk + · · · +

k=0

∞ X

an−1 k enk+n−1 = 0.

k=0

From ∞ n−1 X

X

ajk enk+j , el = 0 (l = 0, 1, . . .),

k=0 j=0

we have ajk = 0(j = 0, . . . , n − 1, k = 0, 1, . . .), which implies that n

z }| { 0 = 0 ⊕ 0 ⊕ ··· ⊕ 0. So A2α = L0

M

L1

M

···

M

Ln−1 .

(c) It is easy to see that both Lj and L⊥ j are the invariant subspaces for Mz n . Step 2. Note that s Mz ek = z

Γ(k + 2 + α) k z = k!Γ(2 + α)

r

k+1 ek+1 . k+2+α

Set Tj = Mzn |Lj (j = 0, 1, . . . , n − 1). Then s Γ(nk + j + 2 + α) nk+j n Tj enk+j = z z (nk + j)!Γ(2 + α) s Γ(nk + j + 2 + α)(nk + n + j)! = enk+n+j . (nk + j)!Γ(nk + n + j + 2 + α)

98

Li

IEOT

Define Xj : A2α (D) −→ Lj , so that Xj ek = ckj enk+j , where ckj are given by  s  (nk + j)!(k + 1 + α)(k + α) · · · (2 + α)  ckj = (k ≥ 1), (2.1) (nk + j + 1 + α)(nk + j + α) · · · (j + 2 + α)k!j!   c0j = 1. We claim that Xj Mz ek = Tj Xj ek . Indeed, r k+1 Xj ek+1 = Tj ckj enk+j , k+2+α and s r k+1 Γ(nk + j + 2 + α)(nk + n + j)! ck+1j enk+n+j = ckj enk+n+j . k+2+α (nk + j)!Γ(nk + n + j + 2 + α) From ck+1j = ckj

s

Γ(nk + j + 2 + α)(nk + n + j)!(k + 2 + α) , Γ(nk + n + j + 2 + α)(nk + j)!(k + 1)

we can get (2.1). Step 3. In this step, we will prove that lim ckj 6= 0, ∞. k−→∞

Case 1. k ≥ 1, α ≥ 0. c2kj

(nk + j)!(k + 1 + α)(k + α) · · · (2 + α) (nk + j + 1 + α)(nk + j + α) · · · (j + 2 + α)k!j! (nk + j)(nk + j − 1) · · · (k + 1)(j + 1 + α) · · · (3 + α)(2 + α) = (nk + j + 1 + α)(nk + j + α) · · · (k + 2 + α)j! (j + 1 + α) · · · (3 + α)(2 + α)(k + 1) 1 . = α (nk + j + 1 + α)j! 1+ ··· 1 + α =

nk+j

Set

akj (α) = 1 +

k+2

α α α 1+ ··· 1 + . nk + j nk + j − 1 k+2

Then ak+1j (α) = 1 +

ak+1j (α) = akj (α)

α α α 1+ ··· 1 + , n(k + 1) + j n(k + 1) + j − 1 k+3 α α α 1 + nk+n+j 1 + nk+n+j−1 · · · 1 + nk+j+1 1+

≥

1+ 1

α nk+n+j α + k+2

n ≥

nα nk+n+j nα + nk+2n

1+ 1

α k+2

≥ 1,

which implies that {akj (α)} is an monotone increasing sequence, and α (n−1)k+j−1 α (n−1)k+j−1 ≤ akj (α) ≤ 1 + . 1+ nk + j k+2

(2.2)

Vol. 63 (2009)

Multiplication Operator on Weighted Bergman Space

Let f (k) = 1 +

α nk+j

99

(n−1)k+j−1

. Then ln f (k) = (n − 1)k + j − 1 ln(nk + j + α) − ln(nk + j) .

f 0 (k) n α n + (n − 1)k + j − 1 = (n − 1) ln 1 + − f (k) nk + j nk + j + α nk + j (n − 1)k + j − 1 nα α − = (n − 1) ln 1 + . nk + j (nk + j + α)(nk + j) Applying the Lagrange mean value theorem, we have α

f 0 (k) nα nk+j ≥ (n − 1) − (n − 1)k + j − 1 α f (k) 1 + nk+j (nk + j + α)(nk + j) n (n − 1)k + j − 1 α (n − 1)α = − nk + j + α (nk + j + α)(nk + j) n−j α ≥ 0. = nk + j + α nk + j Thus f 0 (k) ≥ 0, and f (k) is a monotone increasing function. α n+j−2 . f (k) ≥ f (1) = 1 + n+j α (n−1)k+j−1 Let g(k) = 1 + k+2 . Then ln g(k) = (n − 1)k + j − 1 ln(k + 2 + α) − ln(k + 2) . α 1 g 0 (k) 1 = (n − 1) ln 1 + + (n − 1)k + j − 1 − g(k) k+2 k+2+α k+2 α α = (n − 1) ln 1 + . − (n − 1)k + j − 1 k+2 (k + 2 + α)(k + 2) Applying the Lagrange mean value theorem, we have α α g 0 (k) ≥ (n − 1) k+2α − (n − 1)k + j − 1 g(k) 1 + k+2 (k + 2 + α)(k + 2) α 2n − j − 1 = ≥ 0. k+2+α k+2

Thus g 0 (k) ≥ 0, and g(k) is a monotone increasing function. α (n−1)k+j−1 = e(n−1)α , g(k) ≤ lim 1 + k−→∞ k+2 and we obtain

1+

α n+j−2 ≤ lim akj (α) ≤ e(n−1)α . k−→∞ n+j

(2.3)

100

Li

IEOT

Combining (2.2) and (2.3), if j ≥ 1, we obtain (j + 1 + α) · · · (2 + α) (j + 1 + α) · · · (2 + α) n + j n+j−2 2 . ≤ lim c ≤ kj k−→∞ nj! n+j+α nj!e(n−1)α If j = 0, we have 1 1 n n−2 2 . ≤ lim c ≤ k0 k−→∞ n n+α ne(n−1)α Case 2. k ≥ 1, −1 < α < 0. α α α 1 + · · · 1 + 1 + nk+n+j nk+n+j−1 nk+j+1 ak+1j (α) = α akj (α) 1 + k+2 n n α α 1 + nk+n+j 1 + nk+2n+j < < . α α 1 + k+2 1 + k+2 We consider the following inequality n α α n ln 1 + nk+2n+j ln 1 + nk+2n+j = α α ln 1 + k+2 ln 1 + k+2 < It implies that

1+

nα nk+2n+j α k+2 α 1+ k+2

=

nk + 2n + nα < 1. nk + 2n + j

n α α <1+ . nk + 2n + j k+2

So ak+1j (α) < akj (α), i.e., {ak (α)} is a monotone decreasing sequence. Note that α (n−1)k+j−1 α (n−1)k+j−1 1+ < akj (α) < 1 + . k+2 nk + j (n−1)k+j−1 α . Then Let g(k) = 1 + k+2 ln g(k) = (n − 1)k + j − 1 ln(k + 2 + α) − ln(k + 2) . g 0 (k) α 1 1 = (n − 1) ln 1 + + (n − 1)k + j − 1 − g(k) k+2 k+2+α k+2 α α = (n − 1) ln 1 + − (n − 1)k + j − 1 k+2 (k + 2 + α)(k + 2) α (n − 1)k + j − 1 α − < (n − 1) k+2 (k + 2 + α)(k + 2) α n − j + (n − 1)(1 + α) = < 0. k+2 k+2+α

Vol. 63 (2009)

Multiplication Operator on Weighted Bergman Space

101

Thus g 0 (k) < 0, and g(k) is a monotone decreasing function. α (n−1)k+j−1 = e(n−1)α . g(k) > lim 1 + k−→∞ k+2 Note that 1 α (n−1)k+j−1 lim 1 + = e(1− n )α . k−→∞ nk + j We obtain 1 e(n−1)α ≤ lim akj (α) ≤ e(1− n )α . k−→∞

Therefore (j + 1 + α) · · · (2 + α) 1 nj!e(1− n )α

≤ lim c2kj ≤ k−→∞

(j + 1 + α) · · · (2 + α) (j ≥ 1). nj!e(n−1)α

If j = 0, we have 1 1 ≤ lim c2k0 ≤ (n−1)α . 1 k−→∞ ne ne(1− n )α Step 4. From the above discussion, we know that the operator Xj (j = 0, 1, . . . , n − 1) is bounded and invertible. We obtain Tj ∼ Mz (j = 0, 1, . . . , n − 1). Moreover, Mzn |A2α (D) = T0

M

T1

M

···

M

Tn−1 ∼

n M

Mz .

1

Acknowledgment The author would like to thank the referee for his valuable comments.

References [1] M. Stessin and K. H. Zhu, Generalized factorization in Hardy spaces and the commutant of Toeplitz operators, Canad. J. Math. 55 (2003), 379–400. [2] C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1–31. [3] J. A. Deddens and T. K. Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 84 (1973), 261–273. [4] E. Nordgren, Reducing subspace of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175–181. [5] J. Thomson, The commutant of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54 (1976), 165–169. [6] K. Stroethoff and D. C. Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 169 (1999), 289–313. [7] R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. [8] K. H. Zhu, Spaces of holomorphic functions in the unit ball, GTM, Vol.226, SpringerVerlag, New York, 2005.

102

Li

IEOT

[9] K. H. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. 62 (2000), 553–568. [10] J. Y. Hu, S. H. Sun, X. M. Xu and D. H. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integr. equ. oper. theory 49 (2004), 387–395. [11] C. L. Jiang and Y. C. Li, The commutant and similarity invariant of analytic Toeplitz operators on Bergman space, Science in China Series A 50 (2007), 651–664. Yucheng Li Department of Mathematics Hebei Normal University Shijiazhuang, 050016 China e-mail: [email protected] Submitted: September 28, 2008.

Integr. equ. oper. theory 63 (2009), 103–125 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010103-23, published online December 8, 2008 DOI 10.1007/s00020-008-1639-9

Integral Equations and Operator Theory

Nevanlinna-Pick Interpolation for C + BH ∞ Mrinal Raghupathi Abstract. We study the Nevanlinna-Pick problem for a class of subalgebras of H ∞ . This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H ∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C ∗ envelopes. Mathematics Subject Classification (2000). Primary 47A57; Secondary 46E22, 30E05. Keywords. Nevanlinna-Pick interpolation, distance formulae, reproducing kernel Hilbert space.

1. Introduction Let us assume that we are given n points z1 , . . . , zn ∈ D, n complex numbers w1 , . . . , wn , and a unital subalgebra A of H ∞ . We will say that a function f ∈ A, interpolates the values z1 , . . . , zn to w1 , . . . , wn if and only if f (zj ) = wj . Such an f will also be called an interpolating function. We say that a function f ∈ A is a solution to the interpolation problem if f interpolates and kf k∞ ≤ 1. For the algebra H ∞ , the Nevanlinna-Pick theorem gives an elegant criterion for the existence of a solution to the interpolation problem. The theorem states that a holomorphic function f : D → C is a solution to the interpolation problem if and only if the Pick matrix n 1 − wi wj 1 − zi zj i,j=1 is positive (semidefinite). This research was partially supported by the NSF grant DMS 0300128. This research was completed as part of my Ph.D. dissertation at the University of Houston.

104

Raghupathi

IEOT

The operator theoretic approach to Nevanlinna-Pick interpolation has its roots in Sarason’s work [23]. Sarason used duality methods and Riesz factorization to prove a distance formula, which in turn implies the Nevanlinna-Pick theorem. Abrahamse [1] and Ball [6] extended this approach to prove a Nevanlinna-Pick type interpolation theorem for the algebra H ∞ (R) of bounded analytic functions on a multiply connected domain R. We will see in Section 2 that the algebra H ∞ (R) can be viewed as a subalgebra of H ∞ . These duality methods were used in [9] to prove an interpolation theorem for the codimension-1, weak∗ -closed, unital subalgebra of H ∞ generated by z 2 and z 3 . The duality approach has also inspired the study of the interpolation problem for noncommutative operator algebras. The work of McCullough [14] establishes a Nevanlinna-Pick type result for a dual algebra, that has broad application, and Davidson and Pitts [10] establish the result for noncommutative analytic T¨oplitz algebras. There is a clear connection in all these cases between the invariant subspaces for the algebra and the interpolation theorem. It is too much to hope for a Nevanlinna-Pick theorem for all unital, weak∗ closed subalgebras of H ∞ . In this paper we will study a certain class of subalgebras of H ∞ that arise naturally. We will call these algebras the algebras with predual factorization. We define these algebras in Section 4. ∞ := C + BH ∞ , where In Section 2 we will look at algebras of the form HB B is an inner function and C denotes the span of the constant functions in H ∞ . This algebra is easily seen to be the unitization of BH ∞ , which is a weak∗ -closed ideal of H ∞ . This is the basic construction we will manipulate in order to obtain a large class of algebras with predual factorization. In Section 3 we establish, for ∞ , an analogue of the Helson-Lowdenslager theorem on invariant the algebra HB subspaces. We also establish the analogue of the Halmos-Lax theorem. We would like to point out that our invariant subspace theorem in Theorem 3.1 generalizes the result in [19]. Our proof is also much more elementary and is in fact a simple consequence of the Helson-Lowdenslager theorem. In Section 4 we compute the distance of an element f ∈ L∞ from the weak∗ -closed ideal EH ∞ ∩ A, where E is the finite Blaschke product with zero set z1 , . . . , zn and A is an algebra with predual factorization. As a consequence we obtain a Nevanlinna-Pick type theorem for A. An important aspect of Nevanlinna-Pick interpolation is distinguishing between the scalar-valued and matrix-valued interpolation theory. We will show in Section 6 that the scalar-valued interpolation result obtained in Theorem 5.1 is not valid in the matrix-valued setting. Our work is inspired by, and considerably extends, the results in [9].

2. Notation and Examples Throughout this paper Lp , 1 ≤ p ≤ ∞, will denote the usual Lebesgue space of the circle. The subspace of elements whose negative Fourier coefficients are 0 will be denoted H p and we will freely use the identification of these spaces with spaces

Vol. 63 (2009)

∞ Nevanlinna-Pick Interpolation for HB

105

of analytic functions on the disk. We refer the reader to [11] for the relevant background. We will regard Lp as a normed space with its usual norm, but in dealing with L∞ , and its subspaces, we will work with the weak∗ -topology that L∞ inherits as the dual of L1 . Given a non-empty set S ⊆ Lp , we denote by [S]p the smallest closed subspace of Lp spanned by S, when p = 2 we will denote this as [S] and when p = ∞ we use [S]∞ to denote the weak∗ -closed subspace spanned by S. Let g ∈ H ∞ and let Hgp denote the space [C · 1 + gH p ]p . If u is an outer function, then [uH p ]p = H p . If we factor g into its inner factor B and outer factor p u, then Hgp = [C · 1 + gH p ]p = [C · 1 + BH p ]p = HB . Therefore it is enough to consider inner functions. p Our primary interest will be in the spaces HB for a Blaschke product B and for p = 1, 2, ∞. We denote by φa the elementary M¨obius transformation of the a−z , for a 6= 0, and φ(z) = z, when a = 0. We will write disk given by φa (z) = 1−a ¯z the Blaschke product B as Y |αj | φmj , B= αj αj j∈J

where J is either a finite or countably infinite set, {αj } are distinct, and mj ≥ 1. |αj | The normalizing factor is introduced to ensure convergence and is defined αj to B has at least 2 zeros, i.e., P be 1, if αj = 0. We will assume throughout that ∞ if and only if it satisfies the m ≥ 2. We point out that a function f ∈ H j B j∈J following two constraints: 1. f (αi ) = f (αj ), i, j ∈ J. 2. If mj ≥ 2, then f (i) (αj ) = 0 for i = 1, . . . , mj − 1. We will use K to denote the Szeg¨o kernel and kz to denote the Szeg¨o kernel at the point z, i.e., the element of H 2 such that f (z) = hf, kz i for all f ∈ H 2 . Note that the Szeg¨ o kernel is actually a bounded analytic function on D and so is in H ∞ . We will often abuse notation by using the letter z to represent a complex variable, the identity map on D and the identity map on T. We now provide some examples from the literature that motivate these definitions. First, consider the subalgebra H1∞ of functions for which f 0 (0) = 0, this algebra has been studied in [9]. Clearly, H1∞ = C + z 2 H ∞ and corresponds to the case where B is the Blaschke product z 2 . The algebra H1∞ is generated by z 2 and z 3 . Second, consider the algebra of functions that are equal at the points a, b ∈ D where a is different from b. Note that if f ∈ H ∞ and f (a) = f (b), then f − f (a) vanishes at both a and b. Hence, f − f (a) = Bh, where h ∈ H ∞ and B is the Blaschke product for the points a, b. Hence, this algebra is C+BH ∞ . This example can be extended to any Blaschke sequence. Interpolation questions for algebras of this type where the zeros of the Blaschke product are finite in number, and of

106

Raghupathi

IEOT

multiplicity one, were studied in [24]. In [24] the focus was the matrix-valued theory and an additional condition was imposed that reduced the result to the classical situation. When we study the matrix-valued interpolation problem in Section 6 we will need to impose a similar restriction. The two examples just mentioned are special cases of the cusp algebras studied by Agler and McCarthy [2, 3]. Cusp algebras are of finite codimension in H ∞ . ∞ We do not require HB to be of finite codimension. In dealing with cusp algebras ∞ and embedded disks we must consider not only the algebra HB but also consider finite intersections of such algebras. The interpolation theorem that we prove in Section 4 applies to both the infinite codimension case as well as an infinite inter∞ section of algebras of the form HB . Our third example provides some additional motivation as to why one may ∞ wish to study HB . If G is a group of conformal automorphisms of the disk D, then we have a natural action of the group G on H ∞ given by f 7→ f ◦ α−1 , where ∞ α ∈ G. The fixed point algebra is denoted HG . In the case where the group is a Fuchsian group there is a natural Riemann surface structure on R = D/G and ∞ is isometrically isomorphic to the algebra of bounded analytic functions on R. HG ∞ If we P assume for a moment that HG is non-trivial, then it must be the case that α∈G (1 − |α(ζ)|) < ∞ for any ζ ∈ D. Let Bζ be the Blaschke product whose zero set is the points of the orbit of ζ under G. The Blaschke product Bζ converges for each ζ ∈ D. From the constraints on page 105 we see that C + Bζ H ∞ set of T is the ∞ ∞ = ζ∈D HB functions in H ∞ that are constant on the orbit of ζ, and so HG . ζ While the interpolation results presented in Section 4 do apply to the alge∞ , the result is not as refined as Abrahamse’s theorem [1]. A result that bra HG ∞ can be found in [21, 22]. generalizes Abrahamse’s theorem to the algebra HG

3. Invariant Subspaces ∞ . In this section we examine the invariant subspaces for the algebra HB Let H be a Hilbert space, and let A be a subalgebra of B(H). We say that a subspace M ⊆ H is invariant for A if and only if A(M) ⊆ M for all A ∈ A. We first make some general comments about models for invariant subspaces for subalgebras of H ∞ . The algebra H ∞ has an isometric representation on B(H 2 ). Suppose that A is a subalgebra of H ∞ . For the purposes of invariant subspaces we may assume that A is unital and weak∗ -closed, since the lattice of invariant subspaces is unaffected by unitization and weak∗ -closure. If M ⊆ H 2 is invariant for A, then Beurling’s theorem tells us that M = [AM] ⊆ [H ∞ M] = φH 2 for some inner function φ. Hence, M = φN for some subspace N . It is the subspace N and not the inner factor φ which is relevant. In some sense the class of subspaces N that can arise are the models for invariant subspaces. A natural class of invariant subspaces for an algebra A are the cyclic subspaces. If f ∈ H 2 , then we know that it has an inner-outer factorization f = φu which is unique up to multiplication by a unimodular scalar. The cyclic subspace

Vol. 63 (2009)

∞ Nevanlinna-Pick Interpolation for HB

107

M = [Af ] = φ[Au]. We see that the cyclic subspaces generated by A and an outer function u are the models for all cyclic subspaces for A. This will be relevant to our discussion about interpolation in Section 5. Given a subalgebra A ⊆ H ∞ we say that a subspace M ⊆ Lp is invariant for A if and only if f g ∈ M, whenever f ∈ A and g ∈ M. We will assume, unless stated otherwise, that the term subspace means closed, non-trivial subspace. We begin our study with a look at the subspaces of H 2 that are invariant ∞ for HB . Beurling’s theorem proves that the shift invariant subspaces of H 2 , i.e., the subspaces of H 2 invariant for the algebra H ∞ , are of the form φH 2 , where φ is inner. The Helson-Lowdenslager theorem is an extension of Beurling’s theorem. A subspace M ⊆ L2 is called simply invariant if H ∞ M ⊆ M and the inclusion is strict. Helson and Lowdenslager proved that the simply shift invariant subspaces of L2 are of the form φH 2 where φ is unimodular, i.e., |φ| = 1 a.e. on T. Our first result is the analogue of the Helson-Lowdenslager theorem. Theorem 3.1. Let B be an inner function and let M be a subspace of Lp that ∞ . Either there exists a measurable set E such that M = is invariant under HB p χE L or there exists a unimodular function φ such that φBH p ⊆ M ⊆ φH p . In particular, if p = 2, then there exists a subspace W ⊆ H 2 BH 2 such that M = φ(W ⊕ BH 2 ). Proof. The space [BH ∞ M]p is a shift invariant subspace of Lp and since B is an inner function [BH ∞ M]p = B[H ∞ M]p . By the invariant subspace theorem for H ∞ , either [H ∞ M]p = χE Lp for some measurable subset E of the circle or [H ∞ M]p = φH p for some unimodular function φ. In the former case M ⊇ B[H ∞ M]p = BχE Lp = χE Lp ⊇ M. In the latter case we see that φBH p = B[H ∞ M]p ⊆ M ⊆ [H ∞ M]p = φH p . When p = 2, since φBH 2 ⊆ M ⊆ φH 2 , we see that M = φ(W ⊕ BH 2 ) where W ⊆ H 2 BH 2 . As a corollary we obtain: Corollary 3.2 ([9, Theorem 2.1]). Let H1∞ denote the algebra of functions in H ∞ such that f 0 (0) = 0. A subspace M of L2 is invariant under H1∞ , but not invariant under H ∞ , if and only if there exists an inner function φ, scalars α, β ∈ C with 2 2 |α| + |β| = 1, with α 6= 0, such that M = φ([α + βz] ⊕ z 2 H 2 ). Proof. From the previous result we see that M = φ(W ⊕ z 2 H 2 ) where W ⊆ H 2 z 2 H 2 = span{1, z}. Since M is not invariant under H ∞ see that W is one-dimensional and that α 6= 0. We will identify inner functions that differ only by a constant factor of modulus 1. If S = 6 {0} is a subset of H 2 , then Beurling’s theorem tells us that ∞ 2 [H S] = φH for some inner function φS . The inner function φS is called the inner

108

Raghupathi

IEOT

divisor of S. If S1 and S2 are two subsets of H 2 , then we define their greatest common divisor gcd(S1 , S2 ) to be the inner divisor of [H ∞ (S1 ∪S2 )] and the least common multiple lcm(S1 , S2 ) to be the inner divisor of [H ∞ S1 ]∩[H ∞ S2 ]. For a function f the inner divisor of {f } is clearly the inner factor of f . For functions f1 , f2 ∈ H 2 we define gcd(f1 , f2 ) := gcd({f1 }, {f2 }) and lcm(f1 , f2 ) := lcm({f1 }, {f2 }). For a more detailed description of these operations we refer the reader to [7]. Let A ⊆ B(H) be an operator algebra. Associated to this operator algebra is its lattice of invariant subspaces, which is defined as the set of subspaces of H that are invariant for A. We will denote the lattice of non-trivial, invariant subspaces of A by Lat(A). An important consequence of Beurling’s theorem is that it allows a complete description of the lattice of invariant subspaces for H ∞ . Two shift invariant subspaces φH 2 and ψH 2 are the equal if and only if φ = λψ for a unimodular constant λ. Since we have chosen to identify inner functions that differ only by a constant, we see that that the shift invariant subspaces of H 2 are parametrized by inner functions. There is a natural ordering of inner functions. If φ, ψ are inner functions, then we say that φ ≤ ψ if and only if there exists an inner function θ such that φθ = ψ. This makes the set of inner functions a lattice with meet and join given by φ ∧ ψ = gcd(φ, ψ), φ ∨ ψ = lcm(φ, ψ). In this ordering the inner function 1 is the least element of the lattice and the lattice has no upper bound. The map φ 7→ φH 2 is a bijection between the lattice of inner functions and the the lattice of non-trivial, invariant subspaces for H ∞ . This identification is a lattice anti-isomorphism, i.e., order reversing isomorphism, taking meets to joins and joins to meets. ∞ ) the situation is different. There are two parameters For the lattice Lat(HB ∞ ), an inner function φ and a that determine an invariant subspace M ∈ Lat(HB 2 2 subspace W ⊆ H BH . However, the subspace M does not uniquely determine φ and W . Conversely, different choices of φ and W can sometimes give rise to the same subspace. A simple example is obtained by setting B = z 2 , in which case zH 2 = z [1, z] ⊕ z 2 H 2 = [z] ⊕ z 2 H 2 . (1) In general, if M = φ(W ⊕ BH 2 ), then the subspace W = φM BH 2 . It is always possible to make a canonical choice of inner function and subspace W . The canonical choice is to set the inner function equal to φM , the inner divisor of M, and to let WM = φM M BH 2 . We now describe the extent to which the decomposition of the subspace M into the form φ(W ⊕ BH 2 ) fails to be unique. It is useful to keep in mind the rather trivial example in (1). Note that in addition to being invariant for Hz∞2 , the subspace zH 2 is also shift invariant. ∞ Proposition 3.3. Let M ∈ Lat(HB ), let φM be the inner divisor of M and let 2 WM = φM M BH . Let ψ be inner and V be a subspace of H 2 BH 2 such that M = ψ(V ⊕ BH 2 ). The following are true:

∞ Nevanlinna-Pick Interpolation for HB

Vol. 63 (2009)

109

1. The inner function gcd(WM , B) = 1. 2. The inner function ψ ≤ φM and φM WM = ψV ⊕ B(ψH 2 φM H 2 ).

(2)

3. If θ is such that ψθ = φM , then θ = gcd(B, V ). 4. We have φM = ψ if and only if WM = V . 5. If M 6∈ Lat(HC∞ ) for all C < B , then ψ = φM and V = WM . Proof. 1. Note that φM gcd(WM , B) is an inner function that divides M. Since φM is the inner divisor of M we get gcd(WM , B) = 1. 2. Since φ is the inner divisor of M, it follows that ψ|φ. Let θ be the inner function such that ψθ = φ. We have, ψθ(WM ⊕ BH 2 ) = φM (WM ⊕ BH 2 ) = ψ(V ⊕ BH 2 ). It follows that θWM ⊕ θBH 2 = V ⊕ BH 2 = V ⊕ B(H 2 θH 2 ) ⊕ BθH 2 . Hence, θWM = V ⊕ B(H 2 θH 2 ). (3) Multiplying by ψ gives (2). 3. From (3) we see that θ divides both V and B and so θ ≤ gcd(B, V ). From (3) we get that gcd(B, V )|θWM . Since gcd(WM , B) = 1 it must be the case that gcd(B, V ) ≤ θ. Hence, θ = gcd(B, V ). 4. The conditions ψ = φM and WM = V are equivalent. If ψ = φM , then (3) shows that WM = V . Conversely, if WM = V , then (3) shows that θWM ⊇ WM . If w ∈ WM ⊆ θWM , then there exists w1 ∈ WM such that w = θw1 . Repeating the argument we find that there exists wn ∈ WM such that θn wn = w. If θ 6= 1, then the equation θn wn = w for all n ≥ 0, contradicts the fact that θ cannot divide w with infinite multiplicity. Hence, θ = 1 and φM = ψθ = ψ. 5. If θ 6= 1, then M = ψ(V ⊕ BH 2 ) = ψθ(X ⊕ CH 2 ), where C < B. Hence, M ∈ Lat(HC∞ ).

∞ is Proposition 3.3 indicates that the lattice of invariant subspaces for HB more complicated than the lattice of shift invariant subspaces. Although the canonical choice of inner divisor seems natural, this choice does not behave as expected with respect to the lattice operations. We do not, as yet, have at our disposal a ∞ useful way to describe the lattice operation in Lat(HB ). For illustrative purposes we examine what happens to the inner divisor when we take meets and joins of ∞ ∞ elements in Lat(HB ). Note that Lat(H ∞ ) is a sublattice of Lat(HB ). Any good ∞ description of Lat(HB ) would have to take into account the fact that Lat(H ∞ ) is the lattice of inner functions.

110

Raghupathi

IEOT

∞ Let M = φM (WM ⊕ BH 2 ), N = φN (WN ⊕ BH 2 ) ∈ Lat(HB ), where φM and φN are the inner divisors of M and N respectively. Let X = M ∩ N . We have,

B lcm(φM , φN )H 2 ⊆ lcm(BφM , BφN )H 2 = (BφM H 2 ) ∩ (BφN H 2 ) ⊆M∩N =X ⊆ (φM H 2 ) ∩ (φN H 2 ) = lcm(φM , φN )H 2 . Hence, φX satisfies lcm(φM , φN ) ≤ φX ≤ B lcm(φM , φN ). These are the best general bounds we have. If we consider the case where φM = φN = 1, then we see that X =M∩N = (WM ∩ WN ) ⊕ BH 2 = gcd(WM ∩ WN , B)(WX ⊕ BH 2 ). If WM ∩ WN = {0}, then the inner divisor φX = B. However, if W1 ∩ W2 is nontrivial the situation can be different. Let B = z 5 , let M = [1+z 2 , z 3 ]⊕z 5 H 2 and let N = [1 − z 2 , z 3 ] ⊕ z 5 H 2 . It is straightforward to check that the inner divisor φX of the intersection X = M∩N is divisible by z 3 . Since the functions 1+z 2 and 1−z 2 are outer we see that φM = φN = 1. Note that gcd(φM , φN ) = 1 < φX < z 5 = B. If we consider the join of two subspaces Y = M ∨ N , then we have φY = gcd(φM , φN ). The inequality gcd(φM , φN ) ≤ φY follows from M ∨ N ⊆ (φM H 2 ) ∨ (φN H 2 ) = gcd(φM , φN )H 2 . Since φY |Y, we have φY |M and φY |N . Hence, φY | gcd(W, B)φM = φM and φY | gcd(V, B)φN = φN . Therefore, φY ≤ gcd(φM , φN ), which implies φY = gcd(φM , φN ). ∞ . We now turn our attention to the vector-valued invariant subspaces for HB It is not difficult to extend Theorem 3.1 to the vector-valued setting. If H is a 2 separable Hilbert space, then we denote by HH the H-valued Hardy space. The ∞ 2 2 natural action of H on HH is given by (f h)(z) = f (z)h(z) and this makes HH a ∞ ∞ module over H . This action obviously restricts to HB and we say that a subspace 2 ∞ ∞ ∞ M ⊆ M. We denote by HB(H) M of HH is invariant for HB if and only if HB ∞ the set of B(H)-valued bounded analytic functions. An element of HB(H) is called rigid if Φ(eiθ ) is a partial isometry a.e. on T. A subspace M is invariant under ∞ 2 H ∞ if and only if there exists a rigid function Φ ∈ HB(H) such that M = ΦHH . The proof of the scalar case carries through with the obvious modifications to give the following result. 2 ∞ Theorem 3.4. If M is a closed subspace of HH that is invariant for HB , then ∞ 2 2 there exists a rigid function Φ ∈ HB(H) and a subspace V ⊆ HH BHH such that 2 M = Φ(V ⊕ BHH ).

∞ Nevanlinna-Pick Interpolation for HB

Vol. 63 (2009)

111

2 ∞ Proof. Let M ⊆ HH be an invariant subspace for HB . As in the proof of The∞ 2 orem 3.1 we form the shift invariant subspace [H M] ⊆ HH . By the invariant ∞ 2 subspace theorem in [12] we can write [H M] = ΦHH for a rigid function Φ. Now, 2 2 2 M ⊇ [BH ∞ M] = B[H ∞ M] = BΦHH and so BΦHH ⊆ M ⊆ ΦHH . It follows 2 2 2 that M = W ⊕ BΦHH , where W ⊆ ΦHH BΦHH . If w ∈ W, then w = Φf for 2 2 2 2 some f ∈ HH BHH . Choosing V to be the subspace of elements f ∈ HH BHH such that Φf ∈ W completes the proof.

Keeping in mind our comments about models for invariant subspaces we see that H 2 serves as a model for subspaces of H 2 invariant for H ∞ . For the algebra ∞ HB the situation is more complicated. Even in the simplest cases, for example Hz∞2 , there can be infinitely many models. 2 In the vector-valued case, HH , which is a direct sum of H 2 spaces, forms the only model for an invariant subspace of H ∞ . The models for invariant subspaces ∞ 2 2 of HB are parametrized by subspaces V ⊆ HH BHH and these may fail to decompose as a direct sum of invariant subspaces contained in H 2 . Therefore, one expects the scalar theory and vector-valued theory to be fundamentally different. A first indication of this fact is given by [9, Theorem 5.3] and Theorem 6.6 of this paper provides an extension of that result.

4. A Distance Formula Let A be a weak∗ -closed, unital subalgebra of H ∞ , let z1 , . . . , zn ∈ D and let I denote the ideal I := {f ∈ A : f (z1 ) = · · · = f (zn ) = 0}. ∗

The ideal I is weak -closed and the codimension of I in A is at most n. In this section we give a formula for kf + Ik, where f ∈ L∞ . The formula relates the norm of kf + Ik to the norm of certain off-diagonal compressions of the operator Mf . The result is valid for subalgebras A ⊆ H ∞ that have a property which we call predual factorization. We identify L∞ as the dual of L1 and refer to L1 as the predual of L∞ . If X ⊆ L∞ , then the preannihilator of X by Z 1 X⊥ := {f ∈ L : f g = 0 for all g ∈ X }. We will say that a subspace X ⊆ H ∞ has predual factorization if the following two properties hold: 1. There exists a subspace (not necessarily closed) S ⊆ L1 with [S]1 = X⊥ . 2. Given f ∈ S, there exists an inner function φ such that φf ∈ H 1 . A simple consequence of Riesz factorization is that any function f ∈ S can 1/2 be written as f = ψu2 where ψ is unimodular, u is outer and |f | = |u|.

112

Raghupathi

IEOT

Proposition 4.1. Suppose that {Xj : j ∈ J} is a set of weak∗ -closed T subspaces of H ∞ . If for each j ∈ J, Xj has predual factorization, then X := j∈J Xj has predual factorization. Proof. We note that  X⊥ = 

 \

j∈J



Xj  =  ⊥

 [

j∈J

{(Xj )⊥ : j ∈ J} . 1

Set S = span{(Xj )⊥ : j ∈ J}. PmGiven fj ∈ Xj there exists an inner function φj such that φj fj ∈ H 1 . If f = i=1 ci fji ∈ S, then φf ∈ H 1 where φ = φj1 · · · φjm . Hence, S has predual factorization. Proposition 4.2. If X is a subspace of L∞ such that BH ∞ ⊆ X , then X has predual factorization. Proof. We have X⊥ ⊆ (BH ∞ )⊥ = BH01 . The inner function B multiplies X⊥ into H 1 . ∞ Corollary T 4.3. If {Bj : j ∈ J} is a set of inner functions, and Xj ⊇ Bj H , then X = j∈J Xj has predual factorization. T ∞ Corollary 4.4. If {Bj : j ∈ J} is a set of inner functions, then A = j∈J HB j has predual factorization.

Recall that a function u ∈ H 2 is called outer if [H ∞ u] = H 2 . Given an outer function u ∈ H 2 we define Mu = [Au], Ku to be the span of the kernel functions for Mu at the points z1 , . . . , zn and Nu := Mu Ku = {f ∈ Mu : f (z1 ) = · · · = f (zn ) = 0}. Given a subspace M ⊆ L2 we denote by PM the orthogonal projection of L2 onto M. Lemma 4.5. let z1 , . . . , zn be n points in D and suppose A has predual factorization. If I is the ideal of functions in A such that f (zj ) = 0, for j = 1, . . . , n, then I has predual factorization. Proof. Let I⊥ be the preannihilator of I in L1 . Since A has predual factorization there exists a subspace S ⊆ A⊥ such that 1. The closure of S in the L1 norm is A⊥ 2. For each f ∈ S, there exists an inner function φ such that φf ∈ H 1 . Note that I⊥ = A⊥ + span{kzj : 1 ≤ j ≤ n}, where kz is the Szeg¨o kernel at the point z. If E is the Blaschke product for the points z1 , . . . , zn , then Ekzj ∈ H ∞ for j = 1, . . . , n. The space S˜ = S + span{kzj : 1 ≤ j ≤ n} is dense in I⊥ . Given ˜ with h ∈ A⊥ and v ∈ span{kz : 1 ≤ j ≤ n}, there exists an inner h + v ∈ S, j function φ such that φh ∈ H 1 and so Eφ(h + v) ∈ H 1 with Eφ an inner function. Hence, I⊥ has predual factorization. Lemma 4.6. Let I and A be as in Lemma 4.5. If u is an outer function, then [Iu] = Nu .

Vol. 63 (2009)

∞ Nevanlinna-Pick Interpolation for HB

113

Proof. Since every function in I vanishes at z1 , . . . , zn , [Iu] ⊆ Nu . On the other hand given f ∈ Nu we know that there exists fm ∈ A such that kfm u − f k2 → 0. Since u does not vanish at any point of the disk we see that fm (zj ) → 0 for j = 1, . . . , n. By a construction similar to the one in Lemma 5.8 we P see that there n exists functions ej ∈ A such that ej (zi ) = δi,j . Setting gm = fm − i=1 fm (zi )ei 2 we see that gm u converges to f in H and gm ∈ I. Hence, Nu ⊆ [Iu] and our proof is complete. We will now prove our distance formula. Theorem 4.7. Let z1 , . . . , zn be n points in D, let A be a weak∗ -closed, unital subalgebra of H ∞ with predual factorization and let I be the ideal of functions in A such that f (zj ) = 0 for j = 1, . . . , n. If f ∈ L∞ , then kf + Ik = sup k(I − PNu )Mf PMu k , u

where the supremum is taken over all outer functions u ∈ H 2 . Proof. We have, Z kf + Ik = sup f g : g ∈ I⊥ , kgk1 ≤ 1 Z ˜ = sup f g : g ∈ S, kgk1 ≤ 1 , where S˜ is a dense subspace of I⊥ with the property that each function in S can ˜ and let φ be inner with be multiplied into H 1 by an inner function. Let g ∈ S, the property that φg ∈ H 1 and factor φg as g1 u where g1 , u ∈ H 2 , u is outer, and 1/2 kuk2 = kg1 k2 = kgk1 . It follows that g = g2 u where g2 ∈ L2 and u is outer with 1/2 kuk2 = kg2 k2 = kgk1 . Since g ∈ I⊥ , for all h ∈ I we get Z Z 0 = gh = g2 uh = hhu, g2 i . This shows g2 ⊥ [Iu] = Nu . Hence Z f g = |hf u, g2 i| = |hf PMu u, (I − PNu )g2 i| ≤ k(I − PNu )Mf PMu k . For the other inequality we let h ∈ I. We have Mh Mu ⊆ Nu and so (I − PNu )Mh PMu = 0. Therefore, k(I − PNu )Mf PMu k = k(I − PNu )Mf +h PMu k ≤ kMf +h k ≤ kf + hk∞ . Taking the infimum over h ∈ I yields the inequality.

114

Raghupathi

IEOT

We point out that the proof of Theorem 4.7 holdsTin the case n = 0 to give ∞ the distance of an element in L∞ from the algebra A = j∈J HB . This result can j be interpreted as a Nehari-type theorem for the algebra A. Theorem 4.8. If f ∈ L∞ , then kf + Ak = supu k(I − PMu )Mf PMu k.

5. Interpolation Let A be a subalgebra of H ∞ , unital and weak∗ -closed as before. Given n points z1 , . . . , zn in the disk D and n complex numbers w1 , . . . , wn , the interpolation problem for A is to determine conditions for the existence f ∈ A with kf k∞ ≤ 1 such that f (zj ) = wj for j = 1, . . . , n. Such an f will be called a solution. For the algebra H ∞ , the Nevanlinna-Pick theorem gives us a necessary and sufficient condition for the existence of a solution. Suppose that f1 , f2 ∈ A such that fi (zj ) = wj for j = 1, . . . , n, i = 1, 2. If I denotes the ideal of functions in A such that f (zj ) = 0 for j = 1, . . . , n, then f1 − f2 ∈ I. If we assume the existence of at least one function f ∈ A such that f (zj ) = wj , then all other solutions are of the form f + g for some g ∈ I. As A is weak∗ -closed it follows that a solution exists if and only if kf + Ik ≤ 1. In this section we will prove the interpolation theorem for algebras with predual factorization. While the interpolation theorem is a little abstract we will see that it contains as special cases the original Nevanlinna-Pick theorem and the interpolation result from [9]. It also provides us with an interpolation theorem for algebras of analytic functions on embedded disks. Theorem 5.1. Let A be a weak∗ -closed, unital subalgebra of H ∞ that has predual factorization. Let z1 , . . . , zn ∈ D and w1 , . . . , wn ∈ C. Let K u denote the kernel function for the space Mu := [Au]. There exists a function f ∈ A such that f (zj ) = wj , j = 1, . . . , n with kf k∞ ≤ 1 if and only if for all outer functions u ∈ H 2 , [(1 − wi wj )K u (zi , zj )]ni,j=1 ≥ 0. The proof of this result will follow from the distance formula obtained in Theorem 4.7. We first need to establish the fact that the multiplier algebra of Mu contains the algebra A. Proposition 5.2. Let A be a weak∗ -closed, unital subalgebra of H ∞ and let u ∈ H 2 be an outer function. If M := [Au], then A ⊆ mult(M). Proof. It is straightforward that A(M) ⊆ M. Since u does not vanish on the disk we see that none of the kernel functions in M are the zero function. If Mf denotes the multiplication operator on M induced by f , then kf kmult = kMf kB(M) ≥ kf k∞ . On the other hand if h ∈ M ⊆ L2 , then Z 2 2 2 2 kMf hk = |f h| ≤ kf k∞ khk , which proves kf kmult ≤ kf k∞ .

∞ Nevanlinna-Pick Interpolation for HB

Vol. 63 (2009)

In the special case where H = on the previous proposition.

T

j∈J

2 HB and A = j

T

j∈J

115 ∞ HB we can improve j

Proposition 5.3. be a set of inner functions. The multiplier T Let 2{Bj :T j ∈ J} ∞ algebra mult( j∈J HB ) = H . Bj j∈J j T 2 Proof. Let us denote M := j∈J HB . Let f ∈ mult(M). Since 1 ∈ M none of j the kernel functions in M can be zero. This shows that any f ∈ mult(M) must be bounded. If f ∈ mult(M), then f ∈ M, since 1 ∈ M. Hence, f = λj T + Bj kj for ∞ j ∈ J, kj ∈ H 2 . Since f is bounded so is kj and we have shown that f ∈ j∈J HB . j T ∞ On the other hand any function f ∈ j∈J HBj multiplies M into itself. It remains to be seen that kf kmult ≤ kf k∞ . This follows from Z 2 2 2 2 kMf hk = |f h| ≤ kf k∞ khk , where h ∈ M ⊆ L2 .

We now have all, but one, of the pieces required for our interpolation theorem. In order to use the distance formula to deduce the interpolation theorem we must know that there exists an interpolating function f ∈ A. The proof of this fact is contained in Lemma 5.8. Let us assume the lemma for a moment and see how the interpolation theorem follows. If f ∈ A, then Mf leaves Mu invariant and kf + Ik = sup k(I − PNu )Mf PMu k

(4)

u

= sup k(I − PMu + PKu )Mf PMu k

(5)

= sup kPKu Mf PMu k u

= sup Mf∗ PKu .

(6)

u

(7)

u

If kzu denotes the kernel function for Mu at z, then a spanning set for Ku is given by {kzu1 , . . . , kzun }. Standard results about multiplier algebras of reproducing kernel Hilbert spaces tell us that the norm of Mf∗ PKu is at most 1 if and only if [(1 − wi wj )K u (zi , zj )]ni,j=1 ≥ 0. Combining this fact with equation (4)–(7) proves the interpolation theorem. Note, when A = H ∞ , the cyclic subspace [H ∞ u] = H 2 and so we recover ∞ the classical Nevanlinna-Pick theorem. When A = HB the structure of the cyclic subspaces is given by the next Proposition. Note that we could deduce this result directly from Theorem 3.1, but a direct approach is more transparent. In general the problem of determining cyclic subspaces is easier than the problem of describing the entire lattice of invariant subspaces. Also, for the purposes of interpolation, we see that the description of cyclic subspaces is a central problem. ∞ Proposition 5.4. If B is an inner function, then [HB u] = [v] ⊕ BH 2 , where v = PH 2 BH 2 u.

116

Raghupathi

IEOT

Proof. Let u = v ⊕ Bw. If v = 0, then u ∈ BH 2 which contradicts the fact that u ∞ is outer. If λ + Bf ∈ HB , then (λ + Bf )u = λv + B(v + f v + Bf w) ∈ v ⊕ BH 2 . ∞ ∞ Conversely, let f = λv ⊕ Bh ∈ ([v] ⊕ BH 2 ) [HB u]. Since f ⊥ [HB u] we see that ∞ ∞ 0 = hλv + Bh, Bgui = hh, gui for all g ∈ H . Hence, h ⊥ [H u] = H 2 and so h = 0. Now, 2 0 = hλv, ui = λ hv, v + Bwi = λ hv, vi = λ kvk and so λ = 0.

∞ Hence, the collection of cyclic subspaces for the algebra HB is a contained in 2 the collection of subspaces of the form [v] ⊕ BH . We can assume of course, that v is a unit vector. The vector v has an additional property. If φ is an inner function that divides v and B, then φ divides u. Since u is outer, we see gcd(v, B) = 1. Let v ∈ H 2 BH 2 be a unit vector and set Hv2 = [v] ⊕ BH 2 . This is a reproducing kernel Hilbert space with kernel function,

K v (z, w) = v(z)v(w) +

B(z)B(w) . 1 − zw

Let Kv = span{kzv1 , . . . , kzvn } and let Nv = span{f ∈ Hv2 : f (zj ) = 0, j = ∞ 1, . . . , n}. Applying our distance formula to HB we get the following analogue of Nehari’s theorem. Theorem 5.5. If f ∈ L∞ , then

kf + Ik = sup (1 − PNv )Mf PHv2 , where the supremum is taken over all unit vectors v ∈ H 2 BH 2 . The interpolation result now reads as follows: Theorem 5.6. Let z1 , . . . , zn ∈ D and w1 , . . . , wn ∈ C. There exists a function ∞ f ∈ HB such that f (zj ) = wj if and only if the matrices, [(1 − wi wj )K v (zi , zj )] , are positive for all unit vectors v ∈ H 2 BH 2 . Note in both the above theorems that the collection of subspaces is really parametrized by the one-dimensional subspaces of H 2 BH 2 . The purpose of the next two lemmas is to show that if the matrix n

[(1 − wi wj )K u (zi , zj )]i,j=1 is positive for just one outer function u ∈ H 2 , then there exists an interpolating function for the algebra A. Lemma 5.7. Let H be a finite-dimensional Hilbert space, let v1 , . . . , vn beP a basis for n H and let W1 , . . . , Wn , Wn+1 ∈ Mp . Suppose that vn+1 ∈ H and vn+1 = i=1 αi vi . n+1 If the matrix Q = (I − Wi Wj∗ ) hvj , vi i i,j=1 is positive, then for all 1 ≤ i ≤ n either αi = 0 or Wi = Wn+1 .

∞ Nevanlinna-Pick Interpolation for HB

Vol. 63 (2009)

h ip (s) Proof. Let Ws = wi,j

i,j=1

117

and consider the matrix Qk that we get by compress-

ing toPthe (k, k) entry of each block in Q. The (i, j)-th entry of this matrix is (i) (j) p 1 − l=1 wk,l wk,l hvj , vi i. Let λ1 , . . . , λn+1 ∈ C and note that n+1 X

1−

i,j=1

p X

! (i) (j) wk,l wk,l

hvj , vi i λi λj ≥ 0.

l=1

By setting λj = αj for j = 1, . . . , n and λn+1 = −1 we get, ! p n X X (i) (j) 0≤ 1− wk,l wk,l hvj , vi i αi αj i,j=1

−

−

n X i=1 n X

l=1

1−

1−

j=1

p X l=1 p X

! (i) (n+1) wk,l wk,l

! (n+1) (j) wk,l wk,l

hvj , vn+1 i αj

l=1

p X (n+1) 2 1− wk,l

+

hvn+1 , vi i αi

! 2

kvn+1 k .

l=1

This simplifies to

n

2

2

p X n

X X

(n+1) (i) αi vi −

(wk,l − wk,l )αi vi ≥ 0

vn+1 −

i=1

which gives

(n+1) i=1 (wk,l

Pn

l=1

i=1

(i)

− wk,l )αi vi = 0, for 1 ≤ k, l ≤ p. If αi 6= 0, then by the (n+1)

linear independence of v1 , . . . , vn we get wk,l

(i)

= wk,l and so Wn+1 = Wi .

Lemma 5.8. Let A be a unital, weak∗ -closed subalgebra of H ∞ . Let u be an outer function and let M = [Au]. Let K be the kernel function of M, z1 , . . . , zn be n points in the disk and W1 , . . . , Wn ∈ Mk . If n (1 − Wi Wj∗ )K(zi , zj ) i,j=1 ≥ 0, (8) then there exists F ∈ Mk (A) such that F (zj ) = Wj . Proof. We may assume after reordering the points that {kz1 , . . . .kzm } is basis of span{kzj : 1 ≤ j ≤ n}, with m ≤ n. Since u is outer, u is non-zero at every point f (zj ) of D. There exists f ∈ M such that are distinct for for 1 ≤ j ≤ m. If this is u(zi ) −1

−1

not the case, then u(zj ) kzj − u(zi ) kzi = 0 is a non-trivial linear combination of kzi and kzj . Since M is the closure of [Au], we conclude that there exists g ∈ A

118

Raghupathi

IEOT

such that g(zj ) are distinct for 1 ≤ j ≤ m. By setting m Y

g − g(zr ) ∈ A, (9) g(zj ) − g(zr ) r=1,r6=j Pm we see that ei (zj ) = δi,j , for 1 ≤ i, j ≤ m. Let h = i=1 wi ei and note that for 1 ≤ j ≤ m, h(zj ) = wj . To complete the proof we need to show that h(zj ) = wj for j > m. Pm Let j > m and suppose that kzj = l=1 αl kzl . We have seen that the matrix positivity condition (8) implies that either wj = wl or αl = 0, for 1 ≤ l ≤ m. Hence, m m X X h(zj ) = wi ei (zj ) = wi u(zj )−1 (uei )(zj ) ej =

i=1

= =

m X i=1 m X

wi u(zj )−1

i=1 m X

! αl (uei )(zl )

l=1

wi u(zj )−1 αi u(zi ) = u(zj )−1

i=1

= wj u(zj )−1

m X

wi αi u(zi )

i=1 m X

αi u(zi ) = wj u(zj )−1 u(zj ) = wj .

i=1

The matrix case follows easily.

6. The C ∗ -envelope of HB∞ /I. The C ∗ -envelope of an operator algebra was defined by Arveson [4]. Loosely speaking the C ∗ -envelope of an operator algebra A, which is denoted Ce∗ (A), is the smallest C ∗ -algebra on which A has a completely isometric representation. Arveson’s work established the existence of the C ∗ -envelope in the presence of what are called boundary representations. The existence of the C ∗ -envelope of an operator algebra was established in full generality by Hamana, whose approach did not have any relation to boundary representations. Theorem 6.1 (Arveson-Hamana). Let A be an operator algebra. There exists a C ∗ -algebra, which is denoted Ce∗ (A), such that 1. There is a completely isometric representation γ : A → Ce∗ (A). 2. Given a completely isometric representation σ : A → B, where B is a C ∗ algebra and C ∗ (σ(A)) = B, there exists an onto ∗-homomorphism π : B → Ce∗ (A) such that π ◦ σ = γ. The C ∗ -envelope is unique up to ∗-isomorphism. For a detailed description of the C ∗ -envelope we refer the reader to [17]. Let A be a unital, weak∗ -closed, subalgebra of H ∞ , and let I be the ideal of functions in A that vanish at the n points z1 , . . . , zn ∈ D. In [13] McCullough and

Vol. 63 (2009)

∞ Nevanlinna-Pick Interpolation for HB

119

Paulsen showed that the structure of the C ∗ -envelope of the n-dimensional quotient algebra A/I is an indication of the complexity of the matrix-valued interpolation problem for A. For the algebra H ∞ the C ∗ -envelope of H ∞ /I is Mn . In [24] the algebras Ha∞1 ,...,am := {f ∈ H ∞ : f (a1 ) = . . . = f (am )}, were examined and the following result was obtained. Theorem 6.2 (Sollazo [24]). Let a1 = 0, a2 = 21 and let z1 = 0, z2 , z3 ∈ D with ∞ z1 , z2 , z3 distinct. The C ∗ -envelope of the algebra HB /I is M4 . ∞ Note that the quotient HB /I is a 3-idempotent algebra [18]. When we compare this to the classical case we see there has been a jump in the dimension of the C ∗ -envelope from 3 to 4. This dimension jump phenomenon has also been observed in [9, Theorem 5.3] for the algebra C + z 2 H ∞ . In this section we will show, given certain constraints on the number of zeros in the Blaschke product B, that a similar result is true for ∞ ∞ /I is to gain /I. The first step in understanding the quotient HB the algebra HB some knowledge about the structure of the ideal I. We will consider only the case where B is a finite Blaschke product. To fix notation we let α1 , . . . , αp be the zeros of B, we assume that these are distinct and have multiplicity mj ≥ 1 and we set m = m1 + . . . + mp . We arrange the points z1 , . . . , zn so that B(zj ) = 0 for j = 1, . . . , r and B(zj ) 6= 0 for j = r + 1, . . . , n. Denote by E the Blaschke product for the points z1 , . . . , zn . It is ∞ clear that I = HB ∩ EH ∞ . Since B is a finite Blaschke product we see that 2 2 W ⊆ (H BH ) ∩ H ∞ and I = E(W + BH ∞ ). This can also be seen directly ∞ . from the fact that I is invariant for HB

Theorem 6.3. Let B be a finite Blaschke product and let I be the ideal of functions ∞ in HB that vanish at the n points z1 , . . . , zn . If r = 0, then I = E([w] + BH ∞ ) for some w ∈ H ∞ ∩ (H 2 BH 2 ). If r ≥ 1, then I = lcm(B, E)H ∞ = E(W + BH ∞ ), where W is r-dimensional. Proof. Let f ∈ I and write f = λ + Bg ∈ EH ∞ , where g ∈ H ∞ . By evaluating at z1 , . . . , zn we obtain λ + B(zj )g(zj ) = 0. Pn First, consider the case where r = 0. We can write f = λ + B( j=1 cj kzj ) + P n BEh for some choice of c1 , . . . , cn ∈ C and h ∈ H ∞ . Hence, λ + B( j=1 cj kzj ) is Pn 0 at the points z1 , . . . , zn and so λ + B(zi ) j=1 cj K(zi , zj ) = 0 for i = 1, . . . , n. Rewriting this as a linear system we get       B(z1 ) K(z1 , z1 ) · · · K(z1 , zn ) c1 1        . . . . . .. .. ..     ..  = −λ  ..  . B(zn )

K(zn , z1 ) · · ·

K(zn , zn )

cn

1

120

Raghupathi

IEOT

Since r = 0, this system has a unique solution and the constants c1 , . . . , cn can be taken to depend linearly on λ. In this case W is one-dimensional. If r ≥ 1, then λ = 0 and g(zj ) = 0 for j = r + 1, . . . , n. Hence, f = Bφzr+1 · · · φzn h, f ∈ lcm(B, E)H ∞ and I ⊆ lcm(B, E)H ∞ . The reverse inclusion is straightforward. Let C = Bgcd(B, E) ∈ H 2 . From lcm(B, E)H 2 = lcm(B, E) [kz1 , . . . , kzr ] ⊕ gcd(B, E)H 2 = EBgcd(B, E) [kz1 , . . . , kzr ] ⊕ gcd(B, E)H 2 = E(C[kz1 , . . . , kzr ] ⊕ BH 2 ), we see that W is r-dimensional.

For an outer function u, [Iu] = [lcm(B, E)H ∞ u] = lcm(B, E)[H ∞ u] = lcm(B, E)H 2 . ∞ We have seen in Proposition 5.4 that [HB u] = [v] ⊕ BH 2 for some vector v and so

Ku = ([v] ⊕ BH 2 ) lcm(B, E)H 2 = [v] ⊕ B(H 2 φzr+1 · · · φzn H 2 ) = [v] ⊕ B[kzr+1 , . . . , kzn ]. The space Ku has dimension (n − r) + 1. Note that this is also the dimension of the ∞ /I. Our distance formula says that interpolation is possible quotient algebra HB if and only if the compression of Mf∗ to [v] ⊕ B[kzr+1 , . . . , kzn ] is a contraction for all v ∈ H 2 BH 2 . In the case where one or more of the points z1 , . . . , zn is a zero of B, i.e., when r ≥ 1, the distance of f ∈ L∞ from I is the distance of f from lcm(B, E)H ∞ . This is the case we will examine more closely. The objective will be to show that the scalar-valued result in Theorem 5.1 is not the correct matrix-valued interpolation result. This result also generalizes the result from [24]. A basis for K = H 2 lcm(B, E)H 2 is given by the vectors E := {z i kαi+1 : 1 ≤ j ≤ p, 0 ≤ i ≤ mj − 1} ∪ {kzr+1 , . . . , kzn }. j

(10)

We begin by computing the matrix of Mf∗ |K with respect to the basis E. It is an elementary calculation to show, for f ∈ H 2 and m ≥ 0, that f (m) (w) m m+1 = f, z kw . m! Lemma 6.4. If f ∈ H ∞ , then m+1 Mf∗ (z m kw )=

m X 1 (j) m−j+1 f (w)z m−j kw . j! j=0

∞ Nevanlinna-Pick Interpolation for HB

Vol. 63 (2009)

121

Proof. Let g ∈ H 2 and consider

(f g)(m) (w) m+1 m+1 g, Mf∗ z m kw = f g, z m kw = m! m 1 X m (j) = f (w)g (m−j) (w) m! j=0 j m

1 X m (j) m−j+1 = f (w)(m − j)! g, z m−j kw m! j=0 j * m + X 1 m−j m−j+1 = g, f (j) (w)z kw . j! j=0 From this, we see that m+1 Mf∗ (z m kw )=

m X 1 (j) m−j+1 f (w)z m−j kw . j! j=0

∞ When f ∈ HB , Lemma 6.4 and the constraints on page 105 show us that ∗ the matrix of Mf is diagonal with respect to the basis E. The matrix of Mf∗ |K is given by   f (α1 )Im1   ..   .     f (αp )Imp   ∗ Df =  ,   f (zr+1 )     .. .   f (zn ) If we partition the basis E as in (10), then the grammian matrix with respect to this basis has the form Q1 Q2 Q= , Q∗2 P where P is the Pick matrix for the points zr+1 , . . . , zn . Since Q is the grammian matrix of a linearly independent set it is invertible and positive. The matrix Q1 is m × m, positive and invertible, and the matrix Q2 is an m × (n − r) matrix of rank min{m, n − r}. For a function f ∈ H ∞ , Sarason’s generalized interpolation [23] shows that the distance of f from the ideal φH ∞ , i.e., kf + φH ∞ k, is given by the norm of the compression of Mf to H 2 φH 2 . This distance formula is also valid in the matrix-valued case. Let T be an operator on a finite-dimensional Hilbert space H, of dimension n say, let E be a Hamel basis for H and let A be the matrix of T with respect to E. The operator T is a contraction if and only if In − A∗ QA ≥ 0, where Q is the grammian matrix with respect to E.

122

Raghupathi

IEOT

∞ Using this last fact, if f ∈ HB , then

∗

Mf |K ≤ 1 ⇐⇒ Q − Df QDf∗ ≥ 0

⇐⇒ Q1/2 (I − Q−1/2 Df QDf∗ Q−1/2 )Q1/2 ≥ 0 ⇐⇒ I − Q−1/2 Df QDf∗ Q−1/2 ≥ 0 ⇐⇒ I − (Q−1/2 Df Q1/2 )(Q−1/2 Df Q1/2 )∗ ≥ 0. ∞ This induces a completely isometric embedding ρ of HB /I in Mm+n−r given by

ρ(f ) = Q−1/2 Df Q1/2 . ∞ The universal property of the C ∗ -envelope tells us that Ce∗ (HB /I) is a quo∗ ∞ tient of B := C (ρ(HB /I)). Since we are dealing with a representation on a finite-dimensional space we know that B is a direct sum of matrix algebras. In the ∞ event that B = Mm+n−r we see that B = Ce∗ (HB /I). This follows from the fact that Mm+n−r is simple. Recall that B is a finite Blaschke product, that {α1 , . . . , αp } are the distinct zeros of B, that m is the number of zeros of B counting multiplicity, and that the number of points common to {α1 , . . . , αp } and {z1 , . . . , zn } is r.

Theorem 6.5. Let r ≥ 1 and let B be the C ∗ -subalgebra of Mm+n−r generated by the image of ρ. The algebra B = Mm+n−r if and only if m ≤ n − r. Proof. We examine the commutant of B and show that B 0 contains only scalar multiples of the identity. Let R = Q1/2 and let RXR−1 ∈ B 0 . ∞ such that f (αi ) = 1 for all 1 ≤ i ≤ p and It is possible to choose f ∈ HB f (zj ) = 0 for r + 1 ≤ j ≤ n. Given j, with r + 1 ≤ j ≤ n, it is possible to choose f such that f (zj ) = 1, f (αi ) = f (zl ) = 0 for 1 ≤ i ≤ p and l 6= j. Therefore B is generated by R−1 Ej R where E0 := E1,1 + . . . + Em,m and Ej := Em+j,m+j for 1 ≤ j ≤ n − r. The matrix RXR−1 ∈ B 0 if and only if RXR−1 R−1 Ej R = R−1 Ej RRXR−1 and RXR−1 (R−1 Ej R)∗ = (R−1 Ej R)∗ RXR−1 . This happens if and only if QXQ−1 Ej = Ej QXQ−1 and XEj = Ej X. These conditions tell us that X and QXQ−1 are both block diagonal with 1 block of size m × m followed by n − r blocks of size 1. Let us write A 0 B 0 −1 X= , QXQ = , 0 D 0 E where D and E are scalar diagonal of size (n − r). We have, Q1 Q2 A 0 B 0 Q1 Q2 = . Q∗2 P 0 D 0 E Q∗2 P

∞ Nevanlinna-Pick Interpolation for HB

Vol. 63 (2009)

123

(n−r)

This tells us that P D = EP , where P = [pi,j ]i,j=1 is the Pick matrix. Since pi,j di = pi,j ej and pi,j are non-zero for 1 ≤ i, j ≤ (n − r), we get di = ej for 1 ≤ i, j ≤ (n − r). Hence, we may assume that D = E = In−r . Now comparing the off-diagonal entry we see that Q∗2 A = Q∗2 , BQ2 = Q2 and so Q∗2 A = Q∗2 B ∗ = Q∗2 . Rewriting this we get Q∗2 (A − B ∗ ) = Q∗2 (Im − A) = Q∗2 (Im − B ∗ ).

(11)

If m ≤ n − r, then Q∗2 has rank m which implies Im = A = B, X = Im+n−r and B = B 00 = {Im+n−r }0 = Mm+n−r . On the other hand if m > n − r, then there exist m − n + r linearly independent solutions to the equation Q∗2 v = 0. These can be used to construct matrices A, B 6= Im that solve equation (11). Hence, B = 6 Mm+n−r . ∞ /I is Mm+n−r if and only if Theorem 6.6. Let r ≥ 1. The C ∗ -envelope of HB m ≤ n − r.

Proof. This follows from Hamana’s theorem and the fact that Mm+n−r is simple. As a corollary we obtain the following generalization of a theorem from [9]. Corollary 6.7 ([9, Theorem 5.3]). Let z1 = 0 and n ≥ 3. The C ∗ -envelope of H1∞ /I is Mn+1 . Proof. Since r = 1 and n ≥ 3 we see that n − r = m = 2. Hence, by the previous result Ce∗ (H1∞ /I) = Mm+n−r = Mn+1 . As a corollary we also obtain Solazzo’s result [24], which we stated as Theo∞ rem 6.2, for the algebra H0, 1. 2 To close our discussion we want to make a few statements about the relevance of the result on C ∗ -envelopes in distinguishing between scalar-valued and matrixvalued problems. Note that the collection of one-dimensional subspaces of H 2 BH 2 can be identified with the complex-projective m-sphere X = P S m . For a point v ∈ X let us denote by Hv2 the subspace [v] ⊕ BH 2 and let the kernel for Hv2 be denoted K v . For a fixed pair of points z, w ∈ D, the map (z, w) 7→ K v (z, w) is continuous. Denote by Kv the span of the kernel functions at the points z1 , . . . , zn for Hv2 . The ∞ interpolation theorem tells us that there is an isometric representation of HB /I on C(X, M(n−r)+1 ) given by σ(f + I)(v) = PKv Mf PKv . If σ is a completely isometric ∞ ∞ representation, then C = C ∗ (σ(HB /I)) is a candidate for Ce∗ (HB /I). However, ∗ the C -algebra C is a subalgebra of M(n−r)+1 (C(X)) and as such its irreducible representations can be at most (n−r+1)-dimensional. The fact that m ≥ 2 tells us ∞ that m+(n−r) > (n−r)+1 and this implies that Ce∗ (HB /I) cannot be contained

124

Raghupathi

IEOT

completely isometrically in M(n−r)+1 (C(X)). This contradiction proves that the matrix-valued analogue of the interpolation result in Theorem 5.1 is generally false. Acknowledgment I would like to thank my advisor Vern I. Paulsen for his guidance and support. I would also like to thank Scott McCullough for reading a preliminary draft of this paper and his suggestions. Finally, I would like to acknowledge the reviewer for his helpful comments and careful reading of the manuscript.

References [1] M.B. Abrahamse, The Pick interpolation theorem for finitely connected domains, Michigan Math J. 26 (1979), 195–203. [2] Jim Agler and John E. McCarthy, Cusp algebras, Publicacions Matematiques (to appear). [3] , Hyperbolic algebraic and analytic curves, Indiana Univ. Math. J. 56 (2007), no. 6, 2899–2933. [4] William Arveson, Subalgebras of C ∗ -algebras, II., Acta Math. 128 (1972), 271–308. , Interpolation problems in nest algebras., J. Funct. Anal. 20 (1975), no. 3, [5] 208–233. [6] Joseph A. Ball, A lifting theorem for operator models of finite rank on multiplyconnected domains, J. Operator Theory 1 (1979), no. 1, 3–25. [7] Hari Bercovici, Operator theory and arithmetic in H ∞ , Mathematical Surveys and Monographs, vol. 26, American Mathematical Society, Providence, RI, 1988. [8] Brian Cole, Keith Lewis, and John Wermer, Pick conditions on a uniform algebra and von Neumann inequalities, J. Funct. Anal. 107 (1992), no. 2, 235–254. [9] Kenneth R. Davidson, Vern I. Paulsen, Mrinal Raghupathi, and Dinesh Singh, A constrained Nevanlinna-Pick theorem, Indiana Univ. Math. J. (to appear). [10] Kenneth R. Davidson and David R. Pitts, Nevanlinna-Pick interpolation for noncommutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), no. 3, 321–337. [11] Henry Helson, Harmonic Analysis, Addison-Wesley Publishing Company, Reading, Massachusetts, 1983. , Lectures on Invariant Subspaces., Academic Press, New York-London, 1964. [12] [13] Scott McCullough and Vern I. Paulsen, C ∗ -envelopes and interpolation theory, Indiana Univ. Math. J. 51 (2001), no. 2, 479–505. [14] Scott McCullough, Nevanlinna-Pick type interpolation in a dual algebra, J. Funct. Anal. 135 (1996), no. 1, 93–131. [15] Raymond Mortini, Amol Sasane, and Brett D. Wick, The corona theorem and stable rank for C + BH ∞ , Houston J. Math. (to appear). ¨ [16] R. Nevanlinna, Uber beschr¨ ankte Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen., Ann. Acad. Sci. Fenn. Sel A 13 (1919), no. 1, 1–72. [17] Vern I. Paulsen, Completely Bounded Maps and Operator Algebras., Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2002. [18] , Operator algebras of idempotents., J. Funct. Anal. 181 (2001), no. 2, 209– 226.

Vol. 63 (2009)

∞ Nevanlinna-Pick Interpolation for HB

125

[19] Vern I. Paulsen and Dinesh Singh, Modules over subalgebras of the disk algebra, Indiana Univ. Math. J. 55 (2006), no. 5, 1751–1766. ¨ [20] G. Pick, Uber die Beschr¨ ankungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden., Math. Ann. 77 (1916), 7–23. [21] Mrinal Raghupathi, Constrained Nevanlinna-Pick interpolation, University of Houston, 2008. [22] , Abrahamse’s interpolation theorem and Fuchsian groups, preprint, arXiv:0808.1206. [23] Donald Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. (1967), no. 127, 179–203. [24] Jim Solazzo, Interpolation and Computability, University of Houston, 2000. Mrinal Raghupathi Department of Mathematics Vanderbilt University Nashville, Tennessee, 37240-0001 U.S.A. e-mail: [email protected] URL: http://www.math.vanderbilt.edu/people/raghupathi Submitted: July 16, 2008. Revised: October 14, 2008.

Integr. equ. oper. theory 63 (2009), 127–150 c 2008 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010127-24, published online December 22, 2008 DOI 10.1007/s00020-008-1645-y

Integral Equations and Operator Theory

Anisotropic Operator Symbols Arising From Multivariate Jump Processes Nils Reich Abstract. It is shown that infinitesimal generators A of certain multivariate pure jump L´evy copula processes give rise to a class of anisotropic symbols that extends the well-known classes of pseudo differential operators of H¨ ormander-type. In addition, we provide minimal regularity convergence analysis for a sparse tensor product finite element approximation to solutions of the corresponding stationary Kolmogorov equations Au = f . The computational complexity of the presented approximation scheme is essentially independent of the underlying state space dimension. Mathematics Subject Classification (2000). Primary 45K05, 60J75, 47G30; Secondary 65N30, 47B38. Keywords. Integral operators, symbol classes, anisotropic Sobolev spaces, L´evy copulas, jump processes, sparse tensor products, wavelet finite elements.

1. Introduction On Rn , n ≥ 2, consider the integrodifferential equation Au = f,

(1.1)

where A denotes an integrodifferential operator of anisotropic order α ∈ Rn , i.e. A : H α (Rn ) → L2 (Rn ) is continuous. Here H s , s ∈ Rn , denotes the anisotropic Sobolev space

X

n

s n 0 n 2 si /2 b

H (R ) = f ∈ S (R ) : (1 + ξi ) f <∞ . i=1

L2 (Rn )

We assume that the operator A is a pseudo differential operator with symbol p : Rn × Rn → R, i.e. Z Au(x) = Ap u(x) := − eihx,ξi p(x, ξ)b u(ξ)dξ, u ∈ S(Rn ). (1.2) Rn

128

Reich

IEOT

In [21, 39], it was shown that such integral operators occur as infinitesimal generators of certain L´evy copula processes X. In this case (1.1) can be regarded as the stationary part of the Kolmogorov equation of X. Such equations occur, for instance, in the field of asset pricing in multidimensional L´evy models as introduced in [21, 34, 39, 47]. In terms of Bessel potential spaces corresponding to a continuous negative definite reference function ψ(·), symbols arising from rather general stochastic processes have been studied in [19, 20, 26, 27, 30, 45]. For an overview, we refer to the monographs [31, 32, 33]. However, classical numerical analysis of (1.1) is based on a Sobolev space characterization of the operator A. To this end, we shall see below that the infinitesimal generators of L´evy copula and certain Feller processes give rise to a new class of pseudo differential operators with symbols that extend m the classes S1,0 , m ∈ R, of H¨ ormander (cf. e.g. [28, 48]). The operators in this class act continuously on anisotropic Sobolev spaces and their symbols admit a more complex singularity structure than classical pseudo differential operators. The structure of anisotropic symbols and their corresponding distributional integral kernels has been analyzed by many authors since the 1960s: Extending the fundamental results of [8, 9] that were obtained for homogeneous singular operators, in [17, 18] a symbolic calculus is constructed for certain anisotropic operators with kernels of mixed homogeneity, spectral asymptotics are considered in [3, 5, 43] and the references therein. Furthermore, for the closely related analysis of hypo- and multi-quasi-elliptic operators we refer to [1, 3, 4, 6, 22, 25, 42, 41]. Even though the focus of this work lies on classical Sobolev- and hence L2 -based results, note that a great number of Lp -boundedness results for (different classes of) anisotropic integral operators can be found in [10, 16, 29, 40, 46] and the references there. Finally, in order to obtain numerical solutions of (1.1) we shall also extend the numerical analysis of [7, 21, 24] to obtain a minimal regularity finite element discretization of (1.1) with essentially dimension independent convergence rates for the class of anisotropic operators under consideration. For related numerical analysis we also refer to [23, 49] and the references therein. In addition, the symbol estimates provide the basis for further numerical analysis such as wavelet compression techniques, see [37, 38]. The outline of this work is as follows: In Section 2 we recall the fundamentals of L´evy copula processes and their characteristic exponents. Section 3 provides the new classes of anisotropic symbols and some examples. In Section 4 it is shown that symbols of infinitesimal generators of certain L´evy copula processes are indeed contained in these new symbol classes. These symbols are in general not contained in the classes of H¨ormander-type. Finally, in Section 5 we show that the (stationary) Kolmogorov equations for operators with such anisotropic symbols can be discretized very efficiently using a wavelet finite element scheme. Based on the symbol estimates of the previous sections, a priori convergence analysis is provided.

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

129

2. Motivation: Infinitesimal generators of L´evy copula processes Based on [21, 34, 47], in this Section we briefly introduce L´evy copula processes and characterize their infinitesimal generators. Recall that a stochastic process L = (Lt )t≥0 with state space Rn and L0 = 0 a.s. is a L´evy process if it has independent increments, is temporally homogeneous and stochastically continuous. The characteristic function ΦL and the characteristic exponent ψ L of L are defined by ΦL (ξ) = exp(−tψ L (ξ)) = E(exp(ihξ, Lt i)),

ξ ∈ Rn ,

t > 0.

The characteristic exponent ψ L (ξ) is also called L´evy symbol. The infinitesimal generator A of L and the associated bilinear form E(·, ·) are given by Z Au(x) = − eihx,ξi ψ L (ξ)b u(ξ)dξ, u ∈ C0∞ (Rn ), (2.1) Rn

n

Z

E(u, v) = hAu, vi = −(2π)

ψ L (ξ)b u(ξ)b v (ξ)dξ,

u, v ∈ S(Rn ).

(2.2)

Rn

Furthermore, the characteristic exponent ψ L admits the L´evy-Khinchin representation Z ihξ, xi L ψ (ξ) = ihγ, ξi + Q(ξ) + (1 − eihξ,xi + )ν(dx), (2.3) 1 + |z|2 Rn \{0} where Q(ξ) denotes the quadratic form 12 ξ > Qξ with a symmetric, nonnegative definite matrix Q, a drift vector γ ∈ Rn and the L´evy measure ν(dx) which satisfies Z (1 ∧ |x|2 )ν(dx) < ∞. (2.4) Rn

Any L´evy process L is completely determined by its characteristic triple (Q, γ, ν) in (2.3). We speak of a pure jump L´evy process if Q = 0 and γ = 0. We shall now define a pure jump L´evy copula process. It is denoted by X: For each i = 1, . . . , n the i-th marginal L´evy measure of X is given by νi (dxi ) = kiβi (xi ) dxi with densities kiβi : R \ {0} → R. These densities are defined by kiβi (xi ) = ci

e−βi |xi | , |xi |1+αi

(2.5)

where 0 < α1 , . . . , αn < 2 and β1 , . . . , βn ∈ R≥0 are governing the L´evy densities’ tail behavior and ci > 0 are constants. The strongest singularity of all marginal L´evy measures is given by α := |α|∞ = max {αi : i = 1, . . . , n} < 2.

(2.6)

To characterize the dependence among the margins, let F : Rn → R be a L´evy copula as defined in [21, 34] that is homogeneous of order 1, i.e. F (tξ1 , . . . , tξn ) = tF (ξ1 , . . . , ξn ) for all t > 0 and ξ ∈ Rn .

130

Reich

IEOT

By Sklar’s Theorem, [34, Theorem 3.6], we know that if the partial derivatives ∂1 . . . ∂n F exist in a distributional sense, then one can compute the L´evy density of the multivariate L´evy copula process by differentiation as follows: ν(dx1 , . . . , dxn ) = [∂1 . . . ∂n F ] (U1 (x1 ), . . . , Un (xn ))ν1 (dx1 ) . . . νn (dxn ),

(2.7)

where ν1 (dx1 ), . . . , νn (dxn ) are the marginal L´evy measures defined above and Ui , i = 1, . . . , n, denote the corresponding marginal tail integral ( νi ([xi , ∞)), if xi > 0, Ui (xi ) = − νi ((−∞, xi ]), if xi < 0. Herewith, one obtains ν(dx1 , . . . , dxn ) = [∂1 . . . ∂n F ] (U1 (x1 ), . . . , Un (xn ))k1β1 (x1 ) . . . knβn (xn ) dx1 . . . dxn , (2.8) and this can be written as ν(dx1 , . . . , dxn ) = k β (x1 , . . . , xn ) dx1 . . . dxn ,

(2.9)

with β = (β1 , . . . , βn ). To define the copula process X we specify its characteristic exponent using the L´evy-Khinchin representation (2.3). Since we are interested in pure jump processes, the characteristic exponent ψ X of X is given by Z ihξ, yi ψ X (ξ) = (1 − eihξ,yi + )ν(dy) 1 + |y|2 n R Z ihξ, yi β = (1 − eihξ,yi + )k (y)dy, (2.10) 1 + |y|2 n R with k β as in (2.8) and (2.9). Herewith the L´evy copula process X is completely determined (see e.g. [44, Section 2.11]). Definition 2.1. The L´evy copula process X is said to have α-stable margins if its marginal L´evy densities in (2.5) are of the form kiβi (xi ) = ci

1 , |xi |1+αi

for all i = 1, . . . , n,

i.e. β1 = . . . = βn = 0 in (2.5). If βi > 0 for all i = 1, . . . , n then the L´evy copula process X is said to have tempered stable margins. Lemma 2.2. For any L´evy copula process X with marginal L´evy densities as in (2.5) there holds Z (1 − coshξ, yi)k β (y)dy.

ψ X (ξ) =

(2.11)

Rn

Proof. The symmetry of (2.5) implies that the density k β is symmetric with respect to each coordinate axis. A simple change of coordinates in (2.10) implies that ψ X = ψ X , i.e. ψ X is real-valued. Thus, the result follows from [31, Corollary 3.7.9].

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

131

Since, by (2.11), the characteristic exponent ψ X is real-valued it obviously satisfies the so-called sector condition (cf. e.g. [31]). From [2, Theorem 3.7] one therefore infers that E(·, ·) defined in (2.2) is in fact a (translation invariant) Dirichlet form. In the important case that X has α-stable margins, i.e. βi = 0 for all i = 1, . . . , n in (2.5), the domain D(E) of the Dirichlet form E(·, ·) is well known: Proposition 2.3. The domain D(E) of the Dirichlet form associated to the generator of a L´evy copula process with α-stable margins can be identified with the anisotropic space H α/2 (Rn ) with α = (α1 , . . . , αn ) as in (2.5). Proof. [21, Theorem 3.7].

From Proposition 2.3 one infers Corollary 2.4. The domain D(A) of the infinitesimal generator of a L´evy copula process X with α-stable margins can be identified with H α (Rn ). We conclude this section by an example of a L´evy copula that shall be of reference throughout this work: Example. The cardinal example for our purposes is the Clayton family of L´evy copulas taken from [34, Example 5.2]: Let n ≥ 2. For θ > 0, the function Fθ defined as X −1/θ n 2−n −θ Fθ (u1 , . . . , un ) = 2 |ui | η1{u1 ···un ≥0} − (1 − η)1{u1 ···un <0} , i=1

(2.12) defines a two parameter family of L´evy copulas which resembles the Clayton family of ordinary copulas. It is a L´evy copula homogeneous of order 1, for any θ > 0 and any η ∈ [0, 1]. We shall frequently write a . b to express that a is bounded by a constant multiple of b, uniformly with respect to all parameters on which a and b may depend. Then a ∼ b means a . b and b . a.

3. Anisotropic operators and their symbol classes Recall that for any symbol p : Rn × Rn → R, the corresponding operator Ap is defined by Z Ap u(x) = − eihx,ξi p(x, ξ)b u(ξ)dξ, u ∈ S(Rn ). (3.1) Rn

Furthermore, denote the axes in Rn by Λ := {x ∈ Rn : xi = 0 for some i ∈ {1, . . . , n}}. Herewith we can define a suitable class of anisotropic symbols and corresponding operators.

132

Reich

IEOT

Definition 3.1. A function p : Rn × Rn → R is called a symbol in class Γα (Rn ), α ∈ Rn , if p(·, ξ) ∈ C ∞ (Rn ) for all ξ ∈ Rn , p(x, ·) ∈ C ∞ (Rn \Λ) ∩ C(Rn ) for all x ∈ Rn , and for any τ , τ 0 ∈ Nn0 there holds 0 Y X αk τ τ |ξi |αi −τi · (1 + |ξk |2 ) 2 , for all x, ξ ∈ Rn , (3.2) ∂x ∂ξ p(x, ξ) . i∈Iτ

k∈I / τ

where we set Iτ := {i : τi > 0}. The multiindex α is called the (anisotropic) order of the symbol p and the operator Ap . Some possible realizations of operators A with symbols p ∈ Γα (Rn ) are: Example. If for any τ ∈ Nn0 the function p ∈ C ∞ (Rn \Λ) ∩ C(Rn ) satisfies n X αi −τi τ (1 + |ξi |2 ) 2 , for all ξ ∈ Rn , ∂ξ p(ξ) . i=1

then p ∈ Γα (Rn ) and Ap is admissible in this setting. α Example. Consider a symbol p : Rn × Rn → R in the H¨ormander class S1,0 with non-negative order α, i.e. there exists some α ∈ R≥0 such that for all τ ∈ Nn0 there holds α−|τ | τ (3.3) ∂ξ p(ξ) . (1 + |ξ|2 ) 2 , for all ξ ∈ Rn .

Then p ∈ Γα (Rn ) with α1 = . . . = αn = α. To see this, one may use that for τ ∈ Nn0 there holds τ2i n n n Y Y X τi |τ | 2 2 2 1 + |ξi | ≤ 1+ |ξj | = 1 + |ξ|2 2 , i=1

i=1

j=1

and thus 1 + |ξ|2

− |τ2|

≤

n Y

1 + |ξi |2

− τ2i

.

(3.4)

i=1

Furthermore, 2

1 + |ξ|

α2

.

X n

2

1 + |ξi |

i=1

α2 .

n X

1 + |ξi |2

α2

,

(3.5)

i=1

since α ≥ 0. Clearly, (3.4) and (3.5) imply that (3.2) holds for any symbol p ∈ C ∞ (Rn ) that satisfies (3.3). Note that this statement does not remain true if α < 0 in (3.3). Example. Also, symbols of the following structure belong to Γα (Rn ) with suitable α ∈ Rn : M X p(x, ξ) = bj (x)ψj (ξ), j=1

for some M ∈ N. Here it is assumed that each ψj : Rn → R satisfies (3.2). The functions bj : Rn → R≥0 are assumed to be C ∞ -functions with bounded

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

133

derivatives. Note that similar symbols have already been studied in terms of the symbol classes S%m,ψ of [27], see e.g. [26, 30]. It is straightforward to see that if a symbol p : Rn → Rn is independent of the state variable x, then the order α ∈ Rn of p ∈ Γα (Rn ) has a natural interpretation in terms of mapping properties of the corresponding bilinear form E(u, v) := hAp u, vi: Lemma 3.2. Let p ∈ Γα (Rn ) be independent of x and let Ap be the corresponding pseudo differential operator. Then the bilinear form E(·, ·) = hAp ·, ·i corresponding to Ap acts continuously on the anisotropic space H α/2 (Rn ), i.e. there exists some constant c > 0 such that for all u, v ∈ H α/2 (Rn ).

|E(u, v)| ≤ ckukH α/2 (Rn ) kvkH α/2 (Rn ) , Proof. For u, v ∈ H α/2 (Rn ) there holds Z n E(u, v) = (2π)

(3.6)

p(ξ)b u(ξ)b v (x)dξ.

Rn

Thus, by (3.2), the Cauchy-Schwarz inequality yields (2π)−n |E(u, v)| Z X n = (1 + |ξk |2 )αk /2 b u(ξ)b v (x) dξ Rn k=1

Z ≤

n X

! 12 2

(1 + |ξk |2 )αk /2 |b u(ξ)| dξ

Rn k=1

Z

n X

! 12 2

(1 + |ξk |2 )αk /2 |b v (ξ)| dξ

Rn k=1

= kukH α/2 (Rn ) kvkH α/2 (Rn ) .

From Lemma 3.2 one immediately infers Corollary 3.3. Let p ∈ Γα (Rn ) be independent of x and let Ap be the corresponding pseudo differential operator. Then Ap maps the anisotropic space H α (Rn ) continuously into L2 (Rn ), i.e. there exists some constant c0 > 0 such that kAp ukL2 (Rn ) ≤ c0 kukH α (Rn ) ,

for all u ∈ H α (Rn ).

Remark 3.4. In order to prove the continuity of general operators Ap , with xdependent symbol p ∈ Γα (Rn ), further smoothness assumptions on p are required. For instance, the Calder´ on-Vaillancourt Theorem can be employed to obtain the τ0 τ desired estimates if the partial derivatives ∂x ∂ξ p, |τ 0 |, |τ | ≤ 3, exist and are continuous on the whole Rn × Rn , see e.g. [32, Theorem 2.5.3]. However, since in this work we are mainly interested in symbols arising from L´evy processes (which are stationary) we omit such considerations here.

134

Reich

IEOT

4. Anisotropic symbol estimates In this section, we prove anisotropic symbol estimates for the characteristic exponent ψ X : Rn → R of a L´evy copula process defined by (2.10). We will see that indeed ψ X ∈ Γα (Rn ) with αi , i = 1, . . . , n, given by (2.5). 4.1. Symbol estimates for processes with stable margins At first, we consider the generator A of a L´evy copula process X 0 with α-stable 0 margins. Its symbol is denoted by ψ X . The following two lemmas provide the necessary estimates: Lemma 4.1. There holds, 0

ψ X (ξ1 , . . . , ξn ) .

n X

(1 + |ξi |2 )

αi 2

,

for all (ξ1 , . . . , ξn ) ∈ Rn .

i=1 0

Proof. By [21, Theorem 3.3], ψ X : Rn → R is an anisotropic distance function such that for any t > 0, 0

1

1

0

ψ X (t α1 ξ1 , . . . , t αn ξn ) = t · ψ X (ξ1 , . . . , ξn ),

for all ξ ∈ Rn .

(4.1)

Since all anisotropic distance functions of the same homogeneity are equivalent, 0

ψ X (ξ1 , . . . , ξn ) ∼ |ξ1 |α1 + . . . + |ξn |αn , and the result follows.

To state the following lemma, recall that for τ ∈ Nn0 we denote Iτ := {i ∈ {1, . . . , n} : τi > 0} ,

(4.2)

and let S n−1 be the unit sphere in Rn . Lemma 4.2. Let τ ∈ Nn0 . Suppose there exists some constant c > 0 such that Y X αk τ X0 |ξi |αi −τi · (1 + |ξk |2 ) 2 , for all ξ ∈ S n−1 . (4.3) ∂ξ ψ (ξ) ≤ c · i∈Iτ

k∈I / τ

Then there holds, Y X αk τ X0 |ξi |αi −τi · (1 + |ξk |2 ) 2 , ∂ξ ψ (ξ) . i∈Iτ

(4.4)

k∈I / τ

for all ξ ∈ Rn such that |ξi | ≥ 1 if i ∈ Iτ . Proof. Without loss of generality one may assume that τi ≥ 1 for at least one i ∈ {1, . . . , n}. Otherwise, the claim in (4.4) coincides with Lemma 4.1. By differentiation of (4.1) one obtains, t1 tn 1 τ X0 −1 τ X 0 α1 +...+ α n ∂ξ ψ (ξ) = t α1 ∂ξ ψ (t 1 ξ1 , . . . , t αn ξn ) , t > 0, ξ ∈ Rn .

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes 1

135

1

By [15, Lemma 2.1, (iv)], the mapping t → |(t α1 ξ1 , . . . , t αn ξn )|, ξ 6= 0, maps (0, ∞) onto itself. Thus, one can choose t = t(ξ), such that 1

1

|(t α1 ξ1 , . . . , t αn ξn )| = 1. By (4.3) one obtains τ X0 ∂ξ ψ (ξ) t1

tn

t1 α1

tn +...+ α n

≤ c · t α1 +...+ αn −1 ·

Y

1

|t αi ξi |αi −τi ·

i∈Iτ

≤c·t

t−1 ·

Y

X

= c · t|Iτ |−1 ·

t |ξi |αi −τi t

|ξi |αi −τi ·

i∈Iτ

αk 2

k∈I / τ τ

− αi

i

i∈Iτ

Y

1

(1 + |t αk ξk |2 )

X αk 1 · (1 + |t αk ξk |2 ) 2 k∈I / τ

X

(1 + |t

1 αk

ξk |2 )

αk 2

.

k∈I / τ 2

2

Since there exists some i ∈ {1, . . . , n} with |ξi | ≥ 1, t α1 ξ12 +. . .+t αn ξn2 = 1 implies 1 t αi ≤ |ξ1i | ≤ 1. Thus, t ≤ 1 and the result follows. Remark 4.3. The technical assumption (4.3) is satisfied by all common examples of anisotropic distance functions (cf. e.g. [15]). Furthermore, using the L´evy-Khinchin representation (2.11) it can be shown that (4.3) is satisfied if the underlying L´evy copula is of Clayton-type as in (2.12). Nonetheless, to prove the validity of (4.3) in general, one requires further analytical properties of the L´evy copula. 0

The combination of Lemmas 4.1 and 4.2 implies ψ X ∈ Γα (Rn ) with αi , i = 1, . . . , n, given by (2.5). In the following section, we extend this result to the case of tempered stable margins. 4.2. Symbol estimates for processes with tempered stable margins Let X be a L´evy copula process as defined in Section 2. Suppose that the marginal densities of X are given by (2.5) with β1 , . . . , βn > 0. The structure of the density 0 k β of X is illustrated in Figure 1. Throughout, we denote by ψ X : Rn → R the 0 symbol of a L´evy copula process X with α-stable margins corresponding to X. In particular, X and X 0 share the same α1 , . . . , αn in (2.5). The L´evy density of X 0 is denoted by k 0 : Rn → R≥0 . Denote by k β : Rn → R≥0 the L´evy density of X defined in Section 2. Since τ for any τ ∈ Nn0 there holds (1 − coshx, ξi)∂x (xτ11 . . . xτnn k β (x)) ∈ L1 (Rn ) for all ξ ∈ Rn , one may apply integration by parts to obtain, Z τ1 τ1 τ1 τn τ X τn τn β f (hx, ξi)x1 . . . xn k (x)dx ξ1 . . . ξn ∂ξ ψ (ξ) = ξ1 . . . ξn Rn Z (4.5) τ1 τ τn β = (1 − coshx, ξi)∂x x1 . . . xn k (x) dx , Rn

136

Reich

IEOT

Figure 1. Illustration of a two-dimensional density k β under a Clayton-type L´evy copula with marginal densities defined by (2.5) with α1 = 1, α2 = 1, β1 = 2, β2 = 2. where f is either cos or sin depending on whether |τ | is even or odd. By the Riemann-Lebesgue Lemma, the singularity structure (and strength) of β

kτ (x) := ∂xτ (xτ11 . . . xτnn k β (x)) governs the behavior of |ξ1τ1 . . . ξnτn ∂ξτ1 ψ X (ξ)| as |ξ| → ∞. To study this structure, from now on, we make the following technical assumption on the underlying copula F . Assumption 4.4. Assume for any τ ∈ Nn0 the underlying L´evy copula F satisfies n Y ∂xτ ∂1 . . . ∂n F (x) = ∂1 . . . ∂n F (x) ·

1 · bτ (x), τ |x i| i i=1

for all x ∈ Rn ,

(4.6)

where bτ : Rn → R is uniformly bounded. Herewith, one obtains the following crucial result: Proposition 4.5. Under Assumption 4.4, for any τ ∈ Nn0 and x ∈ Rn , |x| ≤ 1, there holds τ τ ∂x x 1 . . . xτnn k β (x1 , . . . , xn ) . k 0 (x1 , . . . , xn ). (4.7) 1 The proof of Proposition 4.5 is long and technical. It is detailed in Appendix A. Remark 4.6. Assumption 4.4 is often satisfied in practice. For instance, in dimension n = 2, the Clayton-type L´evy copulas Fθ given by (2.12) satisfy (4.6) for any θ > 0 with bounded function bτ (x1 , x2 ) of the form τX 1 −1 τX 2 −1 |x1 |k1 θ |x2 |k2 θ (bk2 |x2 |θ − bk1 |x1 |θ ) dk1 |x1 |k1 θ dk2 |x2 |k2 θ ak + c , k (|x1 |θ + |x2 |θ )k1 +k2 +1 (|x1 |θ + |x2 |θ )k1 +k2 k1 =0 k2 =0

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

137

where k = (k1 , k2 ) and ak , bki , ck , dki 6= 0 for ki = 0, . . . , τi − 1, i = 1, 2, are some suitable coefficients depending only on θ and ki . With Proposition 4.5 one obtains the desired symbol estimates. Theorem 4.7. If the L´evy copula F satisfies Assumption 4.4 then there holds n X X αi ψ (ξ) . (1 + |ξi |2 ) 2 , for all ξ ∈ Rn . (4.8) i=1

Furthermore, for τ ∈ Nn0 there holds, Y X αk τ X |ξi |αi −τi · (1 + |ξk |2 ) 2 , ∂ξ ψ (ξ) . i∈Iτ

(4.9)

k∈I / τ

for all ξ ∈ Rn such that |ξi | > 1 if i ∈ Iτ . Here, as above, Iτ = {i : τi > 0}. 0

Proof. Let ψ X be the characteristic exponent of the α-stable copula process X 0 corresponding to X, i.e. the margins of both processes share the same α1 , . . . , αn in (2.5). We split the integral Z β τ1 τ X τn (1 − coshξ, xi)kτ (x)dx ξ1 . . . ξn · ∂ξ ψ (ξ) ≤ B1 (0) Z β + (1 − coshξ, xi)kτ (x)dx , Rn \B1 (0) β

where B1 (0) denotes the unit ball in Rn . Since kτ ∈ L1 (Rn \B1 (0)), by the Riemann-Lebesgue Lemma, for each τ ∈ Nn0 there exists some constant D > 0 such that Z β (1 − coshξ, xi)kτ (x)dx ≤ D, for all ξ ∈ Rn . (4.10) Rn \B1 (0) Thus, using Proposition 4.5, there exists some constant C1 ≥ 0 such that Z τ1 τ (1 − coshξ, xi)k 0 (x)dx + D ξ1 . . . ξnτn · ∂ξ ψ X (ξ) ≤ C1 · B1 (0) 0

≤ C1 · ψ X (ξ) + D ≤ C1 · C2 ·

n X

(1 + |ξi |2 )

αi 2

+ D,

i=1

where the last line follows from Lemma 4.1 with some suitable constant C2 ≥ 0. Merging the constants thus implies n X αi τ1 τ X τn (1 + |ξi |2 ) 2 , for all ξ ∈ Rn . ξ1 . . . ξn · ∂ξ ψ (ξ) . i=1

Hence, setting τ = 0 ∈ Nn0 implies (4.8). For any τ ∈ Nn0 , estimate (4.9) follows from division by |ξ1 |τ1 . . . |ξn |τn .

138

Reich

IEOT

5. Sparse Tensor Product Approximation of Anisotropic Operators In this section we study the numerical solution of the original integrodifferential equation (1.1), Au = f, with A = Ap , p ∈ Γα (Rn ) for some α ∈ Rn . For the numerical solution of (1.1), we restrict the state space Rn to a bounded subdomain := [0, 1]n , say, and employ the Galerkin finite element method with respect to a hierarchy of conforming trial spaces VbJ ⊂ VbJ+1 ⊂ . . . ⊂ H α/2 (), where n o H α/2 () := u| : u ∈ H α/2 (Rn ), u|Rn \ = 0 . For an analysis of the error introduced by the localization of Rn to , we refer to [39, Section 4.5]. Now, the variational problem of interest reads: Find uJ ∈ VbJ such that, E(uJ , vJ ) := hAuJ , vJ i = hf, vJ i for all vJ ∈ VbJ . (5.1) The index J represents the meshwidth of order 2−J . In order to ensure that there exists a unique solution to (5.1), in addition to the continuity (3.6) of E(·, ·) we assume that the bilinear form satisfies a G˚ arding inequality in H α/2 , i.e. there 0 exist constants c > 0, c ≥ 0 such that E(u, u) ≥ ckuk2H α/2 − c0 kuk2L2 ,

for all u ∈ H α/2 .

(5.2)

The nested trial spaces VbJ ⊂ VbJ+1 we employ in (5.1) shall be sparse tensor product spaces based on a wavelet multiresolution analysis described in the next sections. 5.1. Wavelets on the unit interval On the unit interval [0, 1] we shall employ scaling functions and wavelets based on the construction of [12, 13, 35] and the references therein. The trial spaces Vj are spanned by single-scale bases Φj = {φj,k : k ∈ ∆j }, where ∆j denote suitable index sets. The approximation order of the trial spaces we denote by d, i.e. inf vj ∈Vj kv − vj k0 s d = sup s ∈ R : sup < ∞ , ∀ v ∈ H ([0, 1]) . (5.3) 2−js kvks j≥0 To these single-scale bases there exist biorthogonal complement or wavelet bases Ψj = {ψj,k : k ∈ ∇j }, where ∇j := ∆j+1 \∆j . Denoting by Wj the span of Ψj , there holds Vj+1 = Wj+1 ⊕ Vj , for all j ≥ 0, (5.4) and Vj = W0 ⊕ . . . ⊕ Wj , for all j ≥ 0.

(5.5)

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

139

Crucial for the following analysis is that the wavelets on [0, 1] satisfy the following norm estimates (cf. e.g. [13, 14], for the one-sided estimates we refer to [50]): For an arbitrary u ∈ H t ([0, 1]), 0 ≤ t ≤ d, with wavelet decomposition u=

∞ X X

uj,k ψj,k ,

j=0 k∈∇j

there holds the norm equivalence, X 2 22tj |uj,k |2 ∼ kukH t ([0,1]) , if 0 ≤ t < d − 1/2,

(5.6)

(j,k)

or the one-sided estimate, X 2 22tj |uj,k |2 . kukH t ([0,1]) , if d − 1/2 ≤ t < d.

(5.7)

(j,k)

In case t = d there only holds, X 2 22tj |uj,k |2 . J kukH t ([0,1]) , if t = d.

(5.8)

(j,k) j≤J

For concrete examples of wavelet bases we refer to [11, 21]. 5.2. Sparse tensor product spaces For x = (x1 , . . . , xn ) ∈ [0, 1]n , we denote, ψj,k (x) := ψj1 ,k1 ⊗ . . . ⊗ ψjn ,kn (x1 , . . . , xn ) = ψj1 ,k1 (x1 ) . . . ψjn ,kn (xn ). On [0, 1]n =: , we define the subspace VJ ⊂ H α/2 () as the (full) tensor product of the spaces defined on [0, 1] n O VJ := VJ , (5.9) i=1

which can be written using (5.5) as VJ = span {ψj,k : ki ∈ ∇ji , 0 ≤ ji ≤ J, i = 1, . . . , n} =

J X

Wj1 ⊗ . . . ⊗ Wjn .

j1 ,...,jn =0

We define the regularity γ > |α|∞ /2 of the trial spaces by γ = sup {s ∈ R : VJ ⊂ H s ()} .

(5.10)

The sparse tensor product spaces VbJ are defined by, VbJ :=span {ψj,k : ki ∈ ∇ji , i = 1, . . . , n; 0 ≤ |j|1 ≤ J} X = Wj1 ⊗ . . . ⊗ Wjn . 0≤|j|1 ≤J

(5.11)

140

Reich

IEOT

bJ := dim(VbJ ) = One readily infers that NJ := dim(VJ ) = O(2nJ ) whereas N O(2J J n−1 ) as J tends to infinity. However, both spaces have similar approximation properties in terms of the finite element meshwidth h = 2−J , provided the function to be approximated is sufficiently smooth. To characterize the necessary extra smoothness we introduce the spaces Hs ([0, 1]n ), s ∈ Nn0 , of all measurable functions u : [0, 1]n → R, such that the norm, X 1/2 α1 αn 2 kukHs () := k∂1 . . . ∂n ukL2 () , 0≤αi ≤si , i=1,...,n

is finite. That is s

n

H ([0, 1] ) =

n O

H si ([0, 1]).

(5.12)

i=1

For arbitrary s ∈ Rn≥0 , we define Hs by interpolation. By (5.9), one may decompose any u ∈ L2 () into X X X X u(x) = uj,k ψj,k (x) = uj,k ψj1 ,k1 (x1 ) . . . ψjn ,kn (xn ). ji ≥0 ki ∈∇ji i=1,...,n

ji ≥0 ki ∈∇ji i=1,...,n

In this style, the sparse grid projection PbJ : L2 () → VbJ is defined by truncation of the wavelet expansion: X X (PbJ u)(x) := uj,k ψj,k (x), (5.13) 0≤|j|1 ≤J k∈∇j

where ∇j = ∇(j1 ,...,jn ) := ∇j1 × . . . × ∇jn . 5.3. Convergence rates Denoting by u and uJ the solutions of (1.1) and the corresponding variational problem (5.1), we need to analyze the error ku − uJ kE ∼ ku − uJ kH α/2 () . For this, at first we derive an anisotropic version of the approximation property of the sparse tensor product projection PbJ , see [49, Proposition 3.2] for its isotropic properties. Theorem 5.1. For i = 1, . . . , n suppose 0 ≤ α2i < γ and let α2i < ti ≤ d with γ and d given by (5.10) and (5.3). For u ∈ H α/2 () there holds  α 6= 0 or   2( α2 −t)J kukHt () if ti 6= d for all i, ku − PbJ ukH α/2 () . (5.14)   ( α2 −t)J n−1 2 J 2 kukHt () otherwise, where we denote t = (t1 , . . . , tn ) and ( α2 − t) = max{ α21 − t1 , . . . , α2n − tn }.

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

141

Proof. At first recall that, as shown in [36], in contrast to the tensor product structure of n O Hs = H si ([0, 1]), i=1

for each s ∈ Rn the spaces H s () admit an intersection structure n \ H s () = Hisi (), i=1

in the sense of equivalent norms. Therefore, due to the norm equivalences (5.6) one infers that if 0 ≤ si < γ, i = 1, . . . , n, there holds for each v ∈ H s , ∞ X (5.15) kvk2H s () ∼ (1 + 22s1 j1 + . . . + 22sn jn )kQj1 ⊗ . . . ⊗ Qjn vk2 , j1 ,...,jn =0

where the mappings Qji : L2 ([0, 1]) → Wji , i = 1, . . . , n, denote the projections onto the increments spaces Wji defined in Section 5.1. Furthermore, because of the tensor product structure of Ht (), for each v ∈ Ht () there also holds the one-sided estimate ∞ X Pd 22 i=1 ti ji kQj1 ⊗ . . . ⊗ Qjn vk2 . kvk2Ht () , (5.16) j1 ,...,jn =0

provided that ti < d for all i = 1, . . . , n. Combining (5.15) and (5.16), setting s = α/2, and writing wj = Qj1 ⊗ . . . ⊗ Qjn u, one obtains in case ti < d for all i = 1, . . . , n, ku − PbJ uk2H α/2 X . (1 + 2α1 j1 + . . . + 2αn jn )kwj k2 |j|1 >J

.

X

(2−2

Pn

i=1 ti ji

+ 2(α1 −2t1 )j1 + . . . + 2(αn −2tn )jn )22

Pn

i=1 ti ji

kwj k2

|j|1 >J Pn . max (2−2 i=1 ti ji + 2(α1 −2t1 )j1 + . . . + 2(αn −2tn )jn ) kuk2Ht () |j|1 >J

. 2(α−2t)J kuk2Ht () , with (α − 2t) = max{α1 − 2t1 , . . . , αn − 2tn }. In case the set I := {i ∈ {1, . . . , n} : ti = d} , is non-empty, one may assume without loss of generality that for each i ∈ I there holds αi = α and α − 2t = α − 2d = αi − 2ti

for all i ∈ I, t0i

(5.17)

because otherwise one can replace ti with some suitable < ti = d and argue as above to obtain the same convergence rate and smoothness requirements on u, 0 since H ti ([0, 1]) ⊂ H ti ([0, 1]).

142

Reich

IEOT

Because for each coordinate direction i ∈ I, i.e. ti = d, there only holds the weaker one-sided norm estimate (5.8), instead of (5.16) one obtains

2 O P

O

2 i∈I ti ji 2 2 Qji ⊗ id[0,1] v (5.18)

. kvkHτ () , i∈I

i∈I /

with τi := ti if i ∈ I and τi := 0 otherwise. Here id[0,1] denotes the identity on L2 ([0, 1]). Employing the stronger norm estimates deduced from (5.6) and (5.7) in all directions i ∈ / I first, one infers exactly as above, ku − PbJ uk2H α/2 X . (1 + 2α1 j1 + . . . + 2αn jn )kQj1 ⊗ . . . ⊗ Qjn uk2 |j|1 >J

P / tk jk . max 2αi ji −2 k∈I |j|1 >J ji : i∈I /

(5.19)

2 X O

O

maxk {αk jk } × 2 Qjk ⊗ id[0,1] u

t−τ jk :k∈I

H

k∈I /

k

()

X P P / tk jk . max 2αi ji −2 k∈I 2α maxk {jk } 2−2d k jk kuk2Ht () , |j|1 >J ji : i∈I /

jk :k∈I

where in the last line (5.17) was employed in conjunction with (5.18). To estimate the remaining sum one may now proceed as in the proof of [49, Proposition 3.2]. If α > 0, herewith one obtains n o X P P 2α maxk {jk } 2−2d k jk . max 2(α−2d) k jk , (5.20) jk :k∈I

jk :k∈I

where the jk run through the set of all indices that are admissible in the last sum of (5.19). Finalizing the argument one obtains ku − PbJ uk2H α/2 n o P P / {αi ji }−2 k∈I / tk jk 2(α−2d) k∈I jk . max 2maxi∈I kuk2Ht () |j|1 >J n P o P P / αk jk −2 k∈I / tk jk 2(α−2d) k∈I jk . max 2 k∈I kuk2Ht () |j|1 >J n o Pn . max 2(α−2t) k=1 jk kuk2Ht () |j|1 >J

. 2(α−2t)J kuk2Ht () . In case α = 0, instead of (5.20) one obtains X X n−1 P P 2α max{i} {ji } 2−2d i ji . max 2(α−2d) i ji ji . ji :i∈I

ji :i∈I

i

(5.21)

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

143

Then analogous arguments as in the case α > 0 yield the required result.

Herewith one immediately obtains the desired minimal regularity sparse tensor product convergence result: Proposition 5.2. For a L´evy copula process with tempered stable margins defined by (2.5) and α as in (2.6) the solutions u and uJ of (1.1) and (5.1) satisfy α

ku − uJ kE ∼ ku − uJ kH α/2 () . 2−(d− 2 )J kukHρ () ,

(5.22)

provided u ∈ Hρ (). The smoothness parameter ρ ∈ Rn>0 is given by ρi = d − (

α αi − ), 2 2

(5.23)

for each i = 1, . . . , n. Proof. With this choice of ρ there holds (α − 2ρ) = αi − 2ρi for all i ∈ {1, . . . , n}. Hence the smoothness requirement on u in each coordinate direction is minimal and the result follows from Theorem 5.1. Remark 5.3. In case αi = α for all i = 1, . . . , n, Proposition 5.2 coincides with the sparse tensor product convergence result for isotropic operators (cf. [49]).

Appendix A. Proof of Proposition 4.5 The goal of this Section is the proof of Proposition 4.5. Suppose for any τ ∈ Nn0 the underlying L´evy copula F satisfies n Y ∂xτ ∂1 . . . ∂n F (x) = ∂1 . . . ∂n F (x) ·

1 · bτ (x), τ |x i| i i=1

for all x ∈ Rn ,

(A.1)

where bτ : Rn → R is uniformly bounded. Then for any τ ∈ Nn0 and x ∈ Rn , |x| ≤ 1, there holds τ τ ∂x x 1 . . . xτnn k β (x1 , . . . , xn ) . k 0 (x1 , . . . , xn ). (A.2) 1 By the quasi self-reproductive structure of the derivatives of F in (A.1), it suffices to show that for any i = 1, . . . , n there holds τ τ β ∂xi x i k (x1 , . . . , xn ) . k 0 (x1 , . . . , xn ), |x| ≤ 1. i i Without loss of generality we assume i = 1. The proof comprises of the following lemmas. Throughout, we assume x1 6= 0. Since we are only interested in derivatives with respect to x1 , we simplify some notation and assume that x2 , . . . , xn ∈ R are fixed unless indicated otherwise. With the tail integrals U1β1 , . . . , Unβn as in (2.7), we set G(x1 ) := G(x1 , . . . , xn ) := ∂1 . . . ∂n F (x1 , . . . , xn ), H(x1 ) := H(x1 , . . . , xn ) := G(U1β1 (x1 ), . . . , Unβn (xn )).

144

Reich

IEOT

Furthermore, we denote by G(k) , H (k) the k-th derivative of G and H with respect to x1 . In order to estimate the derivatives of k β (x1 , . . . , xn ) = H(x1 , . . . , xn )k1β1 (x1 ) . . . knβn (xn ), we begin by analyzing the marginal tail integral U1β1 : Lemma A.1. Let s ∈ N. For any νj ∈ N, pj ∈ N0 , j = 1, . . . s, the derivative Y s ∂x1 (∂ νj U1β1 )pj , j=1

of U1β1 is a linear combination of terms of the form 0

s X

πj =

j=1

s X

Q s0

0

pj ,

j=1

s X

µj πj = 1 +

j=1

j=1 (∂

s X

µj

U1β1 )πj , with

νj pj .

j=1

Proof. The claim is proved by induction on s. For s = 1 there holds µ µ−1 ∂x1 (∂ ν U1β1 ) = µ (∂ ν U1β1 ) · (∂ ν+1 U1β1 ), which proves the basis. To show that the validity of the hypothesis for some s ∈ N implies its validity for s + 1 one finds s+1 Y s Y νj β1 pj νj β1 pj ∂x1 (∂ U1 ) = ∂x1 (∂ U1 ) (∂ νs+1 U1β1 )ps+1 j=1

j=1

+

s Y

(∂ νj U1β1 )pj · ps+1 (∂ νs+1 U1β1 )ps+1 −1 (∂ νs+1 +1 U1β1 ).

j=1

(A.3) Since the hypothesis is valid for s, one obtains that the first summand in (A.3) is indeed a linear combination of terms of the required form. The sum of its powers satisfies s0 s s+1 X X X πj + ps+1 = pj + ps+1 = pj , j=1

j=1

j=1

as required. For the weighted sums there holds 0

s X

µj πj + νs+1 ps+1 = 1 +

j=1

s X

νj pj + νs+1 ps+1 = 1 +

j=1

s+1 X

νj pj .

j=1

One readily infers that the second summand of (A.3) can be represented as a suitable product of derivatives of g. The powers of these derivatives satisfy s X j=1

pj + (ps+1 − 1) + 1 =

s+1 X j=1

pj .

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

145

For the weighted sums one finally obtains s X

νj pj + (ps+1 − 1)νs+1 + (νs+1 + 1) =

s+1 X

j=1

νj pj + 1.

j=1

Lemma A.1 enables us to show Lemma A.2. For any k ∈ N there holds H

(k)

(x1 ) =

∂xk1 H(x1 , . . . , xn )

=

k X

cl,k G(l) (U1β1 (x1 ), . . . , Unβn (xn ))Jl,k (x1 ),

l=1

where s(l,k)

Jl,k =

X

cl,k,m

m

Y

(∂ νj,m U1β1 )pj,m ,

(A.4)

j=1

with suitable νj,m ∈ N, pj,m ∈ N0 and constants cl,k , cl,k,m ∈ R. Furthermore, for each m there holds s(l,k) s(l,k) X X pj,m = l, νj,m pj,m = k. (A.5) j=1

j=1

Proof. We proceed by induction on k. For k = 1, with J1,1 = ∂U1β1 the induction basis is obvious. Assuming the validity of the hypothesis for some k ∈ N one obtains its validity for k + 1 as follows: H (k+1) (x1 ) =

k X

cl,k G(l+1) (U1β1 (x1 ), . . . , Unβn (xn )) · (∂U1β1 )Jl,k (x1 )

l=1

+

k X

(A.6) c0l,k G(l) (U1β1 (x1 ), . . . , Unβn (xn ))

· ∂x1 (Jl,k (x1 )) ,

l=1

where cl,k , c0l,k denote some suitable constants. By the hypothesis, Jl,k is a linear combination of products as in (A.4). Thus, any “pure” summand (i.e. it does not contain any further sub-summands) in the first summand of (A.6) is of the form s Y c · G(l+1) (U1β1 (x1 ), . . . , Unβn (xn )) · (∂U1β1 ) (∂ νj U1β1 )pj , j=1

|

{z

=:A

}

where c denotes some constant. Using the validity of the hypothesis for k, the additional factor A defines Jl+1,k+1 and satisfies (A.5) for k + 1. For the second summand of (A.6) one needs to show that for each l = 1, . . . , k the factor ∂x1 (Jl,k (x1 )) provides a suitable additive contribution to Jl,k+1 . By the hypothesis, each “pure” summand of Jl,k is of the form F :=

k Y j=1

(∂ νj U1β1 )pj .

146

Reich

IEOT

By Lemma A.1, its derivative ∂x1 F is a linear combination of terms of the form Q µj β1 πj with j (∂ U1 ) X

πj =

j

X

µj πj =

j

k X

νj = l,

j=1 k X

νj pj + 1 = k + 1,

j=1

where in both equations the induction hypothesis was applied to obtain the last equality. Thus, ∂x1 (Jl,k (x1 )) indeed provides an additional additive term to the representation of Jl,k+1 that satisfies (A.5). The following lemma will finally enable us to give the proof of Proposition 4.5 below. Lemma A.3. If (A.1) holds then k ∂x H(x1 , . . . , xn ) . 1

1 H(x1 , . . . , xn ), |x1 |k

for all (x1 , . . . , xn ) ∈ Rn , |x1 | ≤ 1. Proof. Denoting by cl,k some suitable constants, Lemma A.2 implies k ∂x H(x1 , . . . , xn ) 1

≤

k X

cl,k · G(l) (U1β1 (x1 ), . . . , Unβn (xn )) · |Jl,k (x1 )|

l=1

≤

k X

s(l,k) (l) β1 Y νj β1 pj βn cl,k · G (U1 (x1 ), . . . , Un (xn )) · (∂ U1 ) , j=1

l=1

where the powersPpj and the orders P of differentiation νj still depend on l and k in such a way that j pj = l and j νj pj = k. Note that e−β1 x1 · Pνj (x1 ), |x1 |νj +α1 where Pνj is some suitable polynomial of degree νj − 1 in x1 that does not vanish at x1 = 0. One therefore obtains (∂ νj U1β1 )(x1 ) =

k ∂x H(x1 , . . . , xn ) 1 ≤

k X

c0l,k

s(l,k) (l) β1 Y βn · G (U1 (x1 ), . . . , Un (xn )) ·

l=1

≤

k X l=1

c0l,k · G(l) (U1β1 (x1 ), . . . , Unβn (xn )) ·

1

pj (νj +α1 ) j=1 x1

1 . |x1 |k+lα1

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

147

By (A.1) there holds G(U β1 (x ), . . . , U βn (x )) (l) β1 1 n n 1 , G (U1 (x1 ), . . . , Unβn (xn )) . β1 l (U1 (x1 )) for all (x1 , . . . , xn ) ∈ Rn . Thus, since for each x1 ∈ R with |x1 | ≤ 1 there holds (U1β1 (x1 ))−l . |x1 |lα1 , one obtains k ∂x H(x1 , . . . , xn ) . G(U β1 (x1 ), . . . , Unβn (xn )) · 1 , for |x1 | ≤ 1. 1 1 |x1 |k Using the above lemmas one can now prove Proposition 4.5: Proof. Using Leibniz’ rule, τ X xτ1 k β (x1 , . . . , xn ) = cj ∂xj 1 k β (x1 , . . . , xn ) ∂ τ −j (xτ1 )

τ ∂x

1

j=0

τ X 0 j j β = cj x1 ∂x1 k (x1 , . . . , xn ) . j=0

∂xτ1 (k1β1 (x1 ))

Since implies

τ ∂x

1

≤

τ X

τ X

. |x1 |

for all x1 ∈ R with |x1 | ≤ 1, Lemma A.3

xτ1 k β (x1 , . . . , xn )

c0j |x1 |j

j=0

≤

−(τ +1+α1 )

j X i=0

c0j |x1 |j

j=0

j X i=0

0

ci |x1

|i+1+α1

·

n Y

1 · ∂xj−i H(x1 , . . . , xn ) 1 1+α s |xs | s=2

n Y ci 1 1 · · G(U1β1 (x1 ), . . . , Unβn (xn )) · |x1 |1+α1 s=2 |xs |1+αs |x1 |j

≤ c · k (x1 , . . . , xn ).

References [1] A. A. Albanese, A. Corli, and L. Rodino. Hypoellipticity and local solvability in Gevrey classes. Math. Nachr., 242:5–16, 2002. [2] C. Berg and G. Forst. Non-symmetric translation invariant Dirichlet forms. Inventiones Math., 21:199–212, 1973. [3] P. Boggiatto and E. Buzano. Spectral asymptotics for multi-quasi-elliptic operators in Rn . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24(3):511–536, 1997. [4] P. Boggiatto, E. Buzano, and L. Rodino. Multi-quasi-elliptic operators in Rn . In Partial differential operators and mathematical physics (Holzhau, 1994), volume 78 of Oper. Theory Adv. Appl., pages 31–42. Birkh¨ auser, Basel, 1995. [5] P. Boggiatto, E. Buzano, and L. Rodino. Spectral asymptotics for hypoelliptic operators. In Differential equations, asymptotic analysis, and mathematical physics (Potsdam, 1996), volume 100 of Math. Res., pages 40–46. Akademie Verlag, Berlin, 1997.

148

Reich

IEOT

[6] P. Boggiatto and F. Nicola. Non-commutative residues for anisotropic pseudodifferential operators in Rn . J. Funct. Anal., 203(2):305–320, 2003. [7] H.-J. Bungartz and M. Griebel. A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives. J. Complexity, 15(2):167–199, 1999. [8] A. P. Calder´ on and A. Zygmund. On the existence of certain singular integrals. Acta Math., 88:85–139, 1952. [9] A.-P. Calder´ on and A. Zygmund. Singular integral operators and differential equations. Amer. J. Math., 79:901–921, 1957. [10] Y. Chen and Y. Ding. Lp bounds for the commutator of parabolic singular integral with rough kernels. Potential Anal., 27(4):313–334, 2007. [11] A. Cohen. Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam, 2003. [12] A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45(5):485–560, 1992. [13] W. Dahmen, A. Kunoth, and K. Urban. Biorthogonal spline wavelets on the interval—stability and moment conditions. Appl. Comput. Harmon. Anal., 6(2):132– 196, 1999. [14] W. Dahmen and R. Schneider. Wavelets with complementary boundary conditions— function spaces on the cube. Results Math., 34(3-4):255–293, 1998. [15] H. Dappa. Quasiradiale Fouriermultiplikatoren. PhD Thesis, TU Darmstadt, 1982. [16] E. B. Fabes, W. Littman, and N. M. Rivi`ere. Commutators of singular integrals with C 1 -kernels. Proc. Amer. Math. Soc., 48:397–402, 1975. [17] E. B. Fabes and N. M. Rivi`ere. Singular intervals with mixed homogeneity. Studia Math., 27:19–38, 1966. [18] E. B. Fabes and N. M. Rivi`ere. Symbolic calculus of kernels with mixed homogeneity. In Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pages 106–127. Amer. Math. Soc., Providence, R.I., 1967. [19] W. Farkas, N. Jacob, and R. L. Schilling. Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces. Dissertationes Math. (Rozprawy Mat.), 393:62, 2001. [20] W. Farkas and H.G. Leopold. Characterisations of function spaces of generalised smoothness. Annali di Mat. Pura ed. Appl., 185:1–62, 2006. [21] W. Farkas, N. Reich, and C. Schwab. Anisotropic stable L´evy copula processes— analytical and numerical aspects. Math. Models Methods Appl. Sci., 17(9):1405–1443, 2007. [22] J. Friberg. Multi-quasielliptic polynomials. Ann. Scuola Norm. Sup. Pisa (3), 21:239–260, 1967. [23] M. Griebel and S. Knapek. Optimized general sparse grid approximation spaces for operator equations. Mathematics of Computations, 2008. Submitted. Also available as SFB611 preprint No 402. [24] M. Griebel, P. Oswald, and T. Schiekofer. Sparse grids for boundary integral equations. Numer. Math., 83(2):279–312, 1999. [25] S. Hofmann. Parabolic singular integrals of Calder´ on-type, rough operators, and caloric layer potentials. Duke Math. J., 90(2):209–259, 1997.

Vol. 63 (2009)

Anisotropic Symbols Arising From Jump Processes

149

[26] W. Hoh. The martingale problem for a class of pseudo differential operators. Math. Ann., 300:121–147, 1994. [27] W. Hoh. Pseudo Differential Operators generating Markov Processes. Habilitationsschrift, University of Bielefeld, 1998. [28] L. H¨ ormander. Linear partial differential operators. Grundlehren der mathematischen Wissenschaften, Vol. 116, Springer Verlag, Berlin, 1963. [29] T. P. Hyt¨ onen. Anisotropic Fourier multipliers and singular integrals for vectorvalued functions. Ann. Mat. Pura Appl. (4), 186(3):455–468, 2007. [30] N. Jacob. A class of Feller semigroups generated by pseudo differential operators. Math. Z., 215:151–166, 1994. [31] N. Jacob. Pseudo Differential Operators and Markov Processes, Vol. 1: Fourier Analysis and Semigroups. Imperial College Press, London, 2001. [32] N. Jacob. Pseudo Differential Operators and Markov Processes, Vol. 2: Generators and their potential theory. Imperial College Press, London, 2002. [33] N. Jacob. Pseudo Differential Operators and Markov Processes, Vol. 3: Markov processes and applications. Imperial College Press, London, 2005. [34] J. Kallsen and P. Tankov. Characterization of dependence of multidimensional L´evy processes using L´evy copulas. Journal of Multivariate Analysis, 97:1551–1572, 2006. [35] H. Nguyen and R. Stevenson. Finite Elements on manifolds. IMA J. Numer. Math., 23:149–173, 2003. [36] S.M. Nikolskij. Approximation of functions of several variables and embedding theorems. Springer Verlag, Berlin, 1975. [37] N. Reich. Wavelet Compression of Anisotropic Integrodifferential Operators on Sparse Tensor Product Spaces. PhD Thesis 17661, ETH Z¨ urich, 2008. http://e-collection.ethbib.ethz.ch/view/eth:30174. [38] N. Reich. Wavelet Compression of Anisotropic Integrodifferential Operators on Sparse Tensor Product Spaces. In preparation. Research report, Seminar for Applied Mathematics, ETH Z¨ urich, 2008. [39] N. Reich, C. Schwab, and C. Winter. On Kolmogorov Equations for Anisotropic Multivariate L´evy Processes. submitted. Research report 2008-3, Seminar for Applied Mathematics, ETH Z¨ urich, 2008. [40] N. M. Rivi`ere. On singular integrals. Bull. Amer. Math. Soc., 75:843–847, 1969. [41] L. Rodino. Polysingular integral operators. Ann. Mat. Pura Appl. (4), 124:59–106, 1980. [42] L. Rodino and P. Boggiatto. Partial differential equations of multi-quasi-elliptic type. Ann. Univ. Ferrara Sez. VII (N.S.), 45(suppl.):275–291 (2000), 1999. Workshop on Partial Differential Equations (Ferrara, 1999). [43] L. Rodino and F. Nicola. Spectral asymptotics for quasi-elliptic partial differential equations. In Geometry, analysis and applications (Varanasi, 2000), pages 47–61. World Sci. Publ., River Edge, NJ, 2001. [44] K.-I. Sato. L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, 1999. [45] R.L. Schilling and T. Uemura. On the Feller property of Dirichlet forms generated by pseudo differential operators. Tohoku Math. J. (2), 59(3):401–422, 2007.

150

Reich

IEOT

ˇ Strkalj ˇ [46] Z. and L. Weis. On operator-valued Fourier multiplier theorems. Trans. Amer. Math. Soc., 359(8):3529–3547 (electronic), 2007. [47] P. Tankov. Dependence structure of L´evy processes with applications to risk man´ agement. Rapport Interne No. 502, CMAPX Ecole Polytechnique, Mars 2003. [48] M.E. Taylor. Pseudodifferential operators. Princeton University Press, Princeton, 1981. [49] T. von Petersdorff and C. Schwab. Numerical solution of parabolic equations in high dimensions. M2AN Math. Model. Numer. Anal., 38(1):93–127, 2004. [50] T. von Petersdorff, C. Schwab, and R. Schneider. Multiwavelets for second-kind integral equations. SIAM J. Numer. Anal., 34(6):2212–2227, 1997. Nils Reich ETH Z¨ urich Seminar for Applied Mathematics 8092 Z¨ urich Switzerland e-mail: [email protected] Submitted: June 11, 2008. Revised: November 1, 2008.

Integr. equ. oper. theory 63 (2009), 151–163 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020151-13, published online January 27, 2009 DOI 10.1007/s00020-008-1650-1

Integral Equations and Operator Theory

Compact and Finite Rank Perturbations of Closed Linear Operators and Relations in Hilbert Spaces Tomas Ya. Azizov1 , Jussi Behrndt, Peter Jonas2 and Carsten Trunk Abstract. For closed linear operators or relations A and B acting between Hilbert spaces H and K the concepts of compact and finite rank perturbations are defined with the help of the orthogonal projections PA and PB in H ⊕ K onto the graphs of A and B. Various equivalent characterizations for such perturbations are proved and it is shown that these notions are a natural generalization of the usual concepts of compact and finite rank perturbations. Mathematics Subject Classification (2000). Primary 47A55; Secondary 47A06. Keywords. Closed linear operator, closed linear relation, finite rank perturbation, compact perturbation, Stone-de Snoo formula.

1. Introduction Let H and K be Hilbert spaces and assume first that A and B are bounded linear operators defined on H with values in K. Then A is said to be a compact (finite rank) perturbation of B if the operator A − B is compact (finite dimensional, respectively). If A and B are unbounded closed operators these notions in general make no sense since the domains dom A and dom B may not coincide and hence A − B can only be defined on the (possibly trivial) subspace dom A ∩ dom B of H. However, if in the special case H = K the operators A and B have a common point in their resolvent sets, then a natural generalization of the above notions of compact and finite rank perturbations is defined via the resolvent difference of A and B. Namely, A is said to be a compact (finite rank) perturbation of B if (A − λ)−1 − (B − λ)−1 , 1 2

λ ∈ ρ(A) ∩ ρ(B),

The research of Tomas Ya. Azizov is partially supported by RFBR grant 05-01-00203-a. Sadly, our colleague and friend Peter Jonas passed away on July, 18th 2007.

(1.1)

152

Azizov, Behrndt, Jonas and Trunk

IEOT

is a compact operator (finite rank operator, respectively). Such types of compact and finite rank perturbations play an important role in pure and applied linear functional analysis and have been studied extensively for a long time, see, e.g., [8]. The main objective of this paper is to introduce the notions of compact and finite rank perturbations of closed linear operators and, more generally, closed linear relations A and B acting between H and K, and to give various equivalent characterizations. The key idea here is to use the orthogonal projections PA and PB in H ⊕ K onto the closed graphs or subspaces A and B of H ⊕ K. We shall say that A is a compact (finite rank) perturbation of B if PA − PB is a compact operator (finite dimensional operator, respectively). It is shown in Theorem 3.1 that A is a finite rank perturbation of B if and only if A and B are both finite dimensional extensions of their common part A ∩ B. Furthermore, it is verified in Theorem 4.2 that the linear relation A is a compact perturbation of the linear relation B if and only if for every ε > 0 there exists a closed linear relation F from H in K such that PB − PF is a finite rank operator and kPA − PF k < ε. This characterization of compact perturbations is very convenient and useful, see, e.g., [2] for stability investigations of sign type properties of spectral points of closed linear operators and relations in indefinite inner product spaces under compact perturbations and perturbations small in gap. The paper is organized as follows. In Section 2 we first recall some basic definitions (cf. [1, 4]) and decompositions of linear relations in Hilbert spaces. The orthogonal projection PA in H ⊕ K onto a closed linear relation A from H in K is expressed in terms of the operator part Aop of A in Proposition 2.1. This representation of PA coincides in essence with the Stone-de Snoo formula, cf. [5, 7, 10, 13]. Sections 3 and 4 are devoted to the concepts of finite rank and compact perturbations of closed linear operators and relations and contain our main results. Here we introduce the corresponding notions and prove various equivalent formulations. Moreover, we show that theses notions are natural generalizations of the usual ones for bounded and unbounded operators.

2. The orthogonal projection onto a closed linear relation Let throughout this paper H and K be Hilbert spaces. We study linear relations from H in K, that is, linear subspaces of H×K. The set of all closed linear relations e e from H in K will be denoted by C(H, K). If K = H we write C(H). For a linear relation A we write dom A, ran A, ker A and mul A for the domain, range, kernel and multivalued part of A, respectively. The elements in a linear relation A will usually be written as column vectors ( xx0 ), where x ∈ dom A and x0 ∈ ran A. For the usual definitions of the linear operations with relations, the inverse etc., we refer to [1, 4, 6]. The set of all densely defined closed linear operators from H to K will be denoted by C(H, K), we write C(H) if H = K. For the set of everywhere defined bounded linear operators from H in K we write L(H, K) and L(H) if H

Vol. 63 (2009)

Compact and Finite Rank Perturbations

153

and K coincide. Linear operators are identified as linear relations via their graphs e and hence the inclusions L(H, K) ⊂ C(H, K) ⊂ C(H, K) hold. Let A be a linear relation from H in K. Then the adjoint relation A∗ ∈ e H) is defined by C(K, y x 0 0 ∈ A . A∗ = : (x , y) = (x, y ) for all y0 x0 Note that this definition extends the usual definition of the adjoint of a densely defined operator. If A is a linear relation in H, then A is said to be symmetric e (selfadjoint) if A ⊂ A∗ (A = A∗ , respectively). Let A ∈ C(H, K). As mul A = (dom A∗ )⊥

and

mul A∗ = (dom A)⊥

it is clear that A (A∗ ) is a densely defined closed operator if and only if dom A∗ (dom A, respectively) is dense. Observe that the orthogonal complement of A in H ⊕ K is the relation (−A∗ )−1 , that is, H ⊕ K = A ⊕ (−A∗ )−1 . e Let A ∈ C(H, K). In the following the Hilbert spaces H and K will be decomposed in the form H = mul A∗ ⊕ H1 ,

where H1 := (mul A∗ )⊥ = dom A,

(2.1)

and K = K1 ⊕ mul A, where K1 := (mul A)⊥ = dom A∗ , respectively. The operator part Aop of A is defined by Aop := A ∩ H1 × K1 .

(2.2)

It is easy to see that in fact mul Aop = {0} holds, and hence it follows that Aop is a densely defined closed operator from H1 in K1 , that is, Aop ∈ C(H1 , K1 ). Furthermore, dom Aop = dom A and x A= : x ∈ dom Aop , z ∈ mul A . (2.3) Aop x + z e H) is defined as Analogously the operator part (A∗ )op of the relation A∗ ∈ C(K, (A∗ )op := A∗ ∩ K1 × H1 ∈ C(K1 , H1 ) and it is straightforward to check that the adjoint (Aop )∗ ∈ C(K1 , H1 ) of the operator part Aop of A coincides with the operator part (A∗ )op of the adjoint relation A∗ , that is, (Aop )∗ = (A∗ )op . In the sequel we simply write A∗op . The next proposition will be useful for the considerations in Section 3 and Section 4. e Proposition 2.1. Let A ∈ C(H, K) be a closed linear relation from H in K and let H1 , K1 and the operators Aop and A∗op be defined as above. Then the operator   0 0 0 0 0 (I + A∗op Aop )−1 A∗op (I + Aop A∗op )−1 0  PA =  (2.4) ∗ −1 ∗ ∗ −1 0 Aop (I + Aop Aop ) Aop Aop (I + Aop Aop ) 0 0 0 0 I

154

Azizov, Behrndt, Jonas and Trunk

IEOT

is the orthogonal projection in H ⊕ K onto the linear relation A with respect to the decomposition mul A∗ ⊕ H1 ⊕ K1 ⊕ mul A of H ⊕ K. In the operator case Proposition 2.1 reduces to the following well-known statement, see, e.g., [3, 11, 13]. Corollary 2.2. Let A ∈ C(H, K) be a closed densely defined linear operator from H in K. Then the orthogonal projection PA in H ⊕ K onto A is given by (I + A∗ A)−1 A∗ (I + AA∗ )−1 PA = . A(I + A∗ A)−1 AA∗ (I + AA∗ )−1 Proof of Proposition 2.1. Recall that A∗op Aop and Aop A∗op are nonnegative selfadjoint operators in the Hilbert spaces H1 and K1 , respectively, cf. [8, § V. Theorem 3.24]. In particular, the entries in the matrix representation of PA are everywhere defined bounded operators. For x ∈ dom Aop we have Aop x = Aop (I + A∗op Aop )(I + A∗op Aop )−1 x = (I + Aop A∗op )Aop (I + A∗op Aop )−1 x and hence (I + Aop A∗op )−1 Aop x = Aop (I + A∗op Aop )−1 x. As dom Aop = H1 and (I + A∗op Aop )−1 ∈ L(H1 ) we conclude (I + Aop A∗op )−1 Aop = Aop (I + A∗op Aop )−1 .

(2.5)

PA2

Making use of (2.5) it follows without difficulties that = PA holds. Furthermore, from (2.5) we conclude ∗ A∗op (I + Aop A∗op )−1 = (I + Aop A∗op )−1 Aop = Aop (I + A∗op Aop )−1 . (2.6) Now relation (2.6) together with the selfadjointness of A∗op Aop and Aop A∗op imply PA = PA∗ . Therefore PA is an orthogonal projection in H ⊕ K. It remains to show ran PA = A. According to [8, §V. Theorem 3.24] the (graph of the) restriction Aop dom A∗op Aop is dense in Aop and hence it follows that the range of the orthogonal projection (I + A∗op Aop )−1 A∗op (I + Aop A∗op )−1 H1 H1 PAop := : → Aop (I + A∗op Aop )−1 Aop A∗op (I + Aop A∗op )−1 K1 K1 is a dense subspace of Aop . On the other hand ran PAop is closed, therefore ran PAop = Aop and with (2.3) we have ran PA = A. Proposition 2.1 is proved. For a closed linear relation A the following matrix representation of PA is due to H.S.V. de Snoo and can be obtained from Proposition 2.1, see also [5] and [7]. e e Proposition 2.3. Let A ∈ C(H, K) be a closed linear relation. Then A∗ A ∈ C(H) ∗ e and AA ∈ C(K) are nonnegative selfadjoint relations and the orthogonal projection PA in Proposition 2.1 has the form (I + A∗ A)−1 ιH1 [A∗ (I + AA∗ )−1 ]op H H PA = : → , (2.7) ιK1 [A(I + A∗ A)−1 ]op I − (I + AA∗ )−1 K K

Vol. 63 (2009)

Compact and Finite Rank Perturbations

155

where ιH1 and ιK1 denote the canonical embeddings of H1 in H and K1 in K, respectively. e Let again A ∈ C(H, K). Following [12] we define the operators cos A and sin A by cos A := (I + A∗ A)−1/2 ∈ L(H) and sin A := ιK1 A(I + A∗ A)−1/2 op ∈ L(H, K), where ιK1 is the canonical embedding of K1 into K. Now Propositions 2.1 and 2.3 yield the following corollary, which is a slight generalization of the main result in [12], see also [9]. e Corollary 2.4. Let A ∈ C(H, K) be a closed linear relation. Then the orthogonal projection PA in H ⊕ K onto A has the form cos2 A cos A sin A∗ cos2 A sin A∗ cos A∗ PA = = . cos A∗ sin A I − cos2 A∗ sin A cos A I − cos2 A∗ Proof. It is clear from Proposition 2.3 that the diagonal entries of PA are given by cos2 A and I − cos2 A∗ . In order to see the form of the off-diagonal entries denote by EA∗op Aop (·) and EAop A∗op (·) the spectral functions of the selfadjoint operators A∗op Aop ∈ C(H1 ) and Aop A∗op ∈ C(K1 ), respectively. Then Aop EA∗op Aop (·)x = EAop A∗op (·)Aop x,

x ∈ dom Aop ,

A∗op EAop A∗op (·)y

y ∈ dom A∗op ,

=

EA∗op Aop (·)A∗op y,

imply that the identities Aop (I + A∗op Aop )−1 = (I + Aop A∗op )−1/2 Aop (I + A∗op Aop )−1/2 , A∗op (I + Aop A∗op )−1 = (I + A∗op Aop )−1/2 A∗op (I + Aop A∗op )−1/2 hold. Now the statement follows from Proposition 2.1, 0 0 mul A∗ mul A∗ cos A = : → , H1 H1 0 (I + A∗op Aop )−1/2 K1 K1 (I + Aop A∗op )−1/2 0 ∗ cos A = : → , mul A mul A 0 0 and 0 Aop (I + A∗op Aop )−1/2 0 0

0 A∗op (I + Aop A∗op )−1/2

sin A = ∗

sin A =

mul A∗ K1 : → , H1 mul A 0 K1 mul A∗ : → . mul A H1 0

156

Azizov, Behrndt, Jonas and Trunk

IEOT

3. Finite rank perturbations of closed linear operators and relations In this section we are concerned with finite dimensional perturbations of closed linear operator and, more generally, closed linear relations in Hilbert spaces. The notion of finite rank perturbations introduced below is compatible with the usual notions for unbounded and bounded operators, cf. Corollary 3.4 and Corollary 3.5. Roughly speaking, a linear relation is a finite rank perturbation of another linear relation if both differ by finitely many dimensions. This is made precise in the following theorem, where also an alternative description in terms of orthogonal projections is given. e Theorem 3.1. Let A, B ∈ C(H, K) be closed linear relations from H in K and let PA and PB be the orthogonal projections in H ⊕ K onto A and B, respectively. Then the following assertions are equivalent: (i) PA − PB is a finite rank operator; (ii) dim A/(A ∩ B) < ∞ and dim B/(A ∩ B) < ∞. If (i) or (ii) holds, then A is said to be a finite rank perturbation of B and B is said to be a finite rank perturbation of A. Proof. The identities dim ran (PA − PA∩B ) = dim A/(A ∩ B), dim ran (PB − PA∩B ) = dim B/(A ∩ B) together with PA − PB = (PA − PA∩B ) − (PB − PA∩B ) show that (ii) implies (i). Assume now that (i) holds. We can assume B = H × {0} since H ⊕ K = B ⊕ B ⊥ and A can also be regarded as a closed linear relation from B to B ⊥ . Hence in the following we consider the case B = 0 ∈ L(H, K). Then       I 0 0 0 mul A∗ mul A∗ 0 I 0 0  H1   H1       PB =  (3.1) 0 0 0 0 :  K1  →  K1  , 0 0 0 0 mul A mul A where H1 = dom A and K1 = dom A∗ , and by Proposition 2.1 we have  −I 0 0 ∗ ∗ −1  0 −A∗op Aop (I + A∗op Aop )−1 A (I + A op Aop ) op PA − PB =  ∗ −1 ∗ 0 Aop (I + Aop Aop ) Aop Aop (I + Aop A∗op )−1 0 0 0

 0 0 . 0 I

Thus, the assumption dim ran (PA − PB ) < ∞ implies dim mul A∗ < ∞ A∗op Aop

and

dim mul A < ∞.

Moreover, as Aop dom is dense in Aop it follows that Aop ∈ C(H1 , K1 ) is an operator of finite rank. Therefore dim H1 / ker Aop < ∞ and also dim H/ ker A < ∞. From A ∩ B = ker A × {0} we conclude dim A/(A ∩ B) < ∞. Replacing the

Vol. 63 (2009)

Compact and Finite Rank Perturbations

157

roles of A and B it follows that also dim B/(A ∩ B) < ∞ holds. Hence (i) implies (ii) and Theorem 3.1 is proved. e Proposition 3.2. Let A, B ∈ C(H, K) and T ∈ L(H, K). Then A is a finite rank perturbation of B if and only if A − T is a finite rank perturbation of B − T . Proof. Assume that A is a finite rank perturbation of B. Then it follows from Theorem 3.1 (ii) that there exists a finite dimensional subspace N ⊂ A such that each element ( uv ) ∈ A can be written as u u1 u2 u1 u2 = + , where ∈ A ∩ B, ∈ N. v v1 v2 v1 v2 Hence

u u1 u2 = + , v − Tu v1 − T u1 v2 − T u2 that is, A − T = ((A ∩ B) − T ) M , where u2 u2 M= : ∈N v2 − T u2 v2 and

denotes the sum of two linear manifolds. Since (A − T ) ∩ (B − T ) = (A ∩ B) − T

and dim M < ∞ it follows that dim(A − T )/ (A − T ) ∩ (B − T ) < ∞. Similarly, we get dim(B − T )/ (A − T ) ∩ (B − T ) < ∞, so that, by Theorem 3.1 (ii) A − T is a finite rank perturbation of B − T . The converse implication follows when A, B and T are replaced by A − T , B − T and −T , respectively. e For A, B ∈ C(H, K) we define ρ(A, B) := T ∈ L(H, K) : (A − T )−1 , (B − T )−1 ∈ L(K, H) .

(3.2)

Observe that if in the special case H = K the intersection of the resolvent sets ρ(A) and ρ(B) is nonempty, then {λI : λ ∈ ρ(A) ∩ ρ(B)} is a subset of ρ(A, B). e Proposition 3.3. Let A, B ∈ C(H, K) and ρ(A, B) 6= ∅. Then A is a finite rank perturbation of B if and only if (A − T )−1 − (B − T )−1 is a finite rank operator for some (and hence for all) T ∈ ρ(A, B). Proof. Suppose that A is a finite rank perturbation of B and let T ∈ ρ(A, B). By Proposition 3.2 A − T is a finite rank perturbation of B − T and Theorem 3.1 (ii) implies that the closed linear relations A−T and B −T are both finite dimensional extensions of the linear relation (A − T ) ∩ (B − T ). Hence the same holds for the inverses, that is, dim(A − T )−1 /((A − T )−1 ∩ (B − T )−1 ) < ∞

(3.3)

158

Azizov, Behrndt, Jonas and Trunk

IEOT

and dim(B − T )−1 /((A − T )−1 ∩ (B − T )−1 ) < ∞. (3.4) −1 −1 Now the statement follows from (A − T ) − (B − T ) ∈ L(K, H). Conversely, if for some T ∈ ρ(A, B) the operator (A − T )−1 − (B − T )−1 ∈ L(K, H) is of finite rank, then (3.3) and (3.4) hold. This implies that A − T and B − T are finite dimensional extensions of (A − T ) ∩ (B − T ) and therefore A is a finite rank perturbation of B by Theorem 3.1 (ii) and Proposition 3.2. We complete this section with two corollaries. The first one shows that for closed linear operators and relations in the same Hilbert space and a common point in their resolvent sets the notion of finite rank perturbations suggested above is compatible with the usual definition via resolvent differences. e Corollary 3.4. Let A, B ∈ C(H) and ρ(A) ∩ ρ(B) 6= ∅. Then A is a finite rank perturbation of B if and only if (A − λ)−1 − (B − λ)−1 is a finite rank operator for some (and hence for all) λ ∈ ρ(A) ∩ ρ(B). Corollary 3.5. Let A, B ∈ L(H, K). Then A is a finite rank perturbation of B if and only if A − B is a finite rank operator. Proof. Suppose that A is a finite rank perturbation of B, i.e., PA − PB is a finite rank operator. From Corollary 2.2 it follows that the entries in the first column of PA − PB are given by (I + A∗ A)−1 − (I + B ∗ B)−1 and A(I + A∗ A)−1 − B(I + B ∗ B)−1 , respectively, and are finite rank operators. Multiplying the first operator from the left with B and subtracting the second one yields that A − B is a finite rank operator. Conversely, if A − B is a finite rank operator, then also A∗ − B ∗ , A∗ A − B ∗ B and AA∗ − BB ∗ are finite rank operators. Making use of Corollary 2.2 it is not difficult to see that PA − PB is a finite rank operator and hence A is a finite rank perturbation of B

4. Compact perturbations of closed linear operators and relations Recall that the gap between two closed subspaces M and N of a Hilbert space is defined by n o ˆ δ(M, N ) := max sup dist (u, N ), sup dist (v, M ) . u∈M,kuk=1

v∈N,kvk=1

If PM and PN denote the orthogonal projections onto M and N , respectively, then the gap between M and N is ˆ δ(M, N ) = kPM − PN k, cf. [8]. The following lemma is known for the special case that A and B are closed operators, see [8, Theorem IV.2.17]. The proof for the relation case is very similar, however, for the convenience of the reader we present the details.

Vol. 63 (2009)

Compact and Finite Rank Perturbations

159

e Lemma 4.1. Let A, B ∈ C(H, K), T ∈ L(H, K) and γ := 2(1 + kT k2 ). Denote by PA , PB , PA−T and PB−T the orthogonal projections in H ⊕ K onto A, B, A − T and B − T , respectively. Then the following estimate holds: 1 kPA−T − PB−T k ≤ kPA − PB k ≤ γkPA−T − PB−T k. (4.1) γ Proof. It suffices to verify the first estimate in (4.1), the second estimate follows when A − T , B − T and T are replaced by A, B and −T , respectively. Let ϕ ∈ A − T , kϕk = 1, and choose ( uv ) ∈ A such that u ϕ= ∈ A − T, kϕk2 = kuk2 + kv − T uk2 = 1. (4.2) v − Tu Then r2 := kuk2 + kvk2 > 0, and r−1 ( uv ) belongs to the unit sphere of A. ˆ Therefore, for any δ 0 > kPA − PB k = δ(A, B) the element r−1 ( uv ) has a distance 0 less than δ from B. Hence there exists an element ( xy ) ∈ B with 2

kr−1 u − xk2 + kr−1 v − yk2 < δ 0 , i.e., 2

ku − rxk2 + kv − ryk2 < r2 δ 0 . We define an element ψ of B − T by ψ :=

(4.3)

rx . ry − rT x

With the help of (4.3) we find kϕ − ψk2 = ku − rxk2 + k(v − ry) − T (u − rx)k2 ≤ ku − rxk2 + 2kv − ryk2 + 2kT k2 ku − rxk2 ≤ 2(1 + kT k2 )(ku − rxk2 + kv − ryk2 ) < 2(1 + kT k2 )r2 δ 0

2

and on the other hand r2 = kuk2 + kv − T u + T uk2 ≤ kuk2 + 2kv − T uk2 + 2kT k2 kuk2 . Then (4.2) implies r2 ≤ 2(1 + kT k2 ) and hence 2

kϕ − ψk2 ≤ 4(1 + kT k2 )2 δ 0 .

(4.4) 0

As ϕ is an element of the unit sphere of A − T , ψ ∈ B − T and δ is an arbitrary number greater than kPA − PB k it follows that sup

dist (ϕ, B − T ) ≤ 2(1 + kT k2 )kPA − PB k

ϕ∈A−T,kϕk=1

holds. The estimate sup ϕ∈B−T,kϕk=1

dist (ϕ, A − T ) ≤ 2(1 + kT k2 )kPA − PB k

160

Azizov, Behrndt, Jonas and Trunk

IEOT

is obtained by interchanging A and B in the above considerations and therefore kPA−T − PB−T k ≤ γkPA − PB k, where γ = 2(1 + kT k2 ). In the next theorem, which is the main result in this section, two equivalent notions for compact perturbations of closed linear operators and relations are introduced. In analogy to Theorem 3.1 a linear relation is a compact perturbation of another linear relation if the difference of the corresponding orthogonal projections is compact. This notion was already used in [11, Proposition 18] in connection with the (semi-)Fredholm theory of linear relations. e Theorem 4.2. Let A, B ∈ C(H, K) be closed linear relations from H in K and let PA and PB be the orthogonal projections in H ⊕ K onto A and B, respectively. Then the following assertions are equivalent: (i) PA − PB is a compact operator; e (ii) For every ε > 0 there exists a linear relation F ∈ C(H, K) such that PB − PF is a finite rank operator and ˆ δ(A, F ) = kPA − PF k < ε. If (i) or (ii) holds, then A is said to be a compact perturbation of B and B is said to be a compact perturbation of A. Proof. Since PA −PB = PA −PF −(PB −PF ) it is clear that (ii) implies (i). Suppose that (i) holds. As in the proof of Theorem 3.1 we can assume that B = 0, B ∈ L(H, K). Let H1 = dom A and K1 = dom A∗ . Then the orthogonal projections PA and PB are given by (2.4) and (3.1), respectively. Since   −I 0 0 0  0 −A∗op Aop (I + A∗op Aop )−1 A∗op (I + Aop A∗op )−1 0  PA − PB =  ∗ −1 0 Aop (I + Aop Aop ) Aop A∗op (I + Aop A∗op )−1 0 0 0 0 I is compact it is clear that mul A∗ and mul A are finite dimensional and the nonnegative selfadjoint operator A∗op Aop (I + A∗op Aop )−1 ∈ L(H1 ) is also compact. Therefore σ A∗op Aop (I + A∗op Aop )−1 \{0} consists only of isolated eigenvalues with finite multiplicity and zero is the only possible accumulation point. It follows from the spectral mapping theorem that σ(A∗op Aop ) has the same properties, hence A∗op Aop is a compact operator. Using the polar decomposition of Aop it follows that Aop ∈ L(H1 , K1 ) is compact. Therefore, for each ε > 0 there exists a decomposition Aop = Fop + Gop such that Fop ∈ L(H1 , K1 ) is a finite rank operator, Gop ∈ L(H1 , K1 ) is sufficiently small, and kPAop − PFop k < ε, where PAop and PFop are the orthogonal projections in H1 ⊕ K1 onto Aop and Fop , respectively. The norm estimate kPAop − PFop k < ε can easily be verified with the help of Corollary 2.2.

Vol. 63 (2009)

Compact and Finite Rank Perturbations

161

e Define the linear relation F ∈ C(H, K) by x 0 F := : x ∈ H , x ∈ mul A . 1 Fop x + x0 Then mul F = mul A and mul F ∗ = (dom F )⊥ = (dom A)⊥ = mul A∗ imply that the orthogonal projection PF in H ⊕ K onto F is given by   0 0 0 0 ∗ ∗ ∗ −1 0 (I + Fop Fop )−1 Fop (I + Fop Fop ) 0  PF =  (4.5) ∗ −1 ∗ ∗ −1 0 Fop (I + Fop Fop ) Fop Fop (I + Fop Fop ) 0 0 0 0 I with respect to the decomposition mul A∗ ⊕ H1 ⊕ K1 ⊕ mul A, cf. Proposition 2.1. Hence, by (2.4) and Corollary 2.2 we have kPA − PF k = kPAop − PFop k < ε. As mul A∗ and mul A are finite dimensional and Fop is a finite rank operator it follows from (3.1) and (4.5) that   −I 0 0 0 ∗ ∗ ∗ ∗ −1  0 Fop Fop (I + Fop Fop )−1 Fop (I + Fop Fop ) 0  PB − PF =  ∗ −1 ∗ ∗ −1 0 Fop (I + Fop Fop ) Fop Fop (I + Fop Fop ) 0 0 0 0 I is a finite rank operator. This completes the proof of Theorem 4.2.

The next proposition is the analogue of Proposition 3.2 for compact perturbations. For the case of closed linear operators in H = K the statement reduces to [10, Proposition 3.1]. e Proposition 4.3. Let A, B ∈ C(H, K) and T ∈ L(H, K). Then A is a compact perturbation of B if and only if A − T is a compact perturbation of B − T . Proof. Assume that A is a compact perturbation of B and let T ∈ L(H, K). e According to Theorem 4.2 (ii) for given ε > 0 there exists F ∈ C(H, K) such that PB − PF is a finite rank operator and kPA − PF k < ε. According to Theorem 3.1 and Proposition 3.2 also PB−T − PF −T is a finite rank operator and by Lemma 4.1 kPA−T − PF −T k ≤ 2(1 + kT k2 )kPA − PF k < 2(1 + kT k2 )ε holds. This implies that PA−T − PB−T is compact and hence A − T is a compact perturbation of B − T by Theorem 4.2 (i). By replacing A, B and T with A − T , B − T and −T , respectively, it follows that A is a compact perturbation B when A − T is a compact perturbation of B − T . Next we formulate an analogue of Proposition 3.3 for the case of compact perturbations. The set ρ(A, B) was introduced in (3.2). e Proposition 4.4. Let A, B ∈ C(H, K) and ρ(A, B) 6= ∅. Then A is a compact perturbation of B if and only if (A − T )−1 − (B − T )−1 is a compact operator for some (and hence for all) T ∈ ρ(A, B).

162

Azizov, Behrndt, Jonas and Trunk

IEOT

Proof. Assume that A is a compact perturbation of B and let T ∈ ρ(A, B). By Proposition 4.3, A − T is a compact perturbation of B − T and hence the operator PA−T − PB−T is compact. Observe that PA−T is connected with the orthogonal projection P(A−T )−1 ∈ L(K ⊕ H) in K ⊕ H onto (A − T )−1 in the following manner: Let h ∈ H and k ∈ K. Then h z PA−T = ∈A−T k z0 if and only if 0 k z = ∈ (A − T )−1 . h z The projections PB−T and P(B−T )−1 are connected in the same way. Therefore, since the compact operator PA−T −PB−T maps bounded sequences onto sequences with a convergent subsequence, the same is true for P(A−T )−1 − P(B−T )−1 , and hence this operator is compact. From Corollary 2.2 it follows that the entries in the first column of P(A−T )−1 − P(B−T )−1 are given by −1 −1 I + (A∗ − T ∗ )−1 (A − T )−1 − I + (B ∗ − T ∗ )−1 (B − T )−1 (4.6) P(A−T )−1

and (A − T )−1 I + (A∗ − T ∗ )−1 (A − T )−1

−1

− (B − T )−1 I + (B ∗ − T ∗ )−1 (B − T )−1

−1

(4.7)

and both are compact operators. Multiplying (4.6) from the left with (B − T )−1 and subtracting (4.7) then implies that (A − T )−1 − (B − T )−1 is a compact operator. Conversely, suppose that (A − T )−1 − (B − T )−1 is compact for some T ∈ ρ(A, B). Then also the operators in (4.6) and (4.7) are compact and with the help of Corollary 2.2 it follows that P(A−T )−1 − P(B−T )−1 is compact. Therefore the above considerations imply that also PA−T − PB−T is compact. Hence A − T is a compact perturbation of B − T and Proposition 4.3 yields that A is a compact perturbation of B. The following two corollaries show that the notion of compact perturbations introduced in Theorem 4.2 reduces to the usual notions if, e.g., both operators or relations act in the same Hilbert space and have a common point in their resolvent sets. e Corollary 4.5. Let A, B ∈ C(H) and ρ(A) ∩ ρ(B) 6= ∅. Then A is a compact perturbation of B if and only if (A − λ)−1 − (B − λ)−1 is a compact operator for some (and hence for all) λ ∈ ρ(A) ∩ ρ(B). Corollary 4.6. Let A, B ∈ L(H, K). Then A is a compact perturbation of B if and only if A − B is a compact operator. Corollary 4.6 can be proved in the same way as Corollary 3.5; simply replace the expression “finite rank” in the proof of Corollary 3.5 by “compact”.

Vol. 63 (2009)

Compact and Finite Rank Perturbations

163

References [1] R. Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 9–23. [2] T.Ya. Azizov, J. Behrndt, P. Jonas, and C. Trunk, Spectral points of definite type and type π for linear operators and relations in Krein spaces, submitted. [3] H.O. Cordes and J.P. Labrousse, The invariance of the index in the metric space of closed operators, J. Math. Mech. 12 (1963), 693–719. [4] R. Cross, Multivalued Linear Operators, Monographs and Textbooks in Pure and Applied Mathematics, 213. Marcel Dekker, Inc., New York, 1998. [5] M. Fernandez Miranda and J.P. Labrousse, The Cayley transform of linear relations, Proc. Amer. Math. Soc. 133 (2005), no. 2, 493–499. [6] M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkh¨ auser, Basel, 2006. [7] S. Hassi, Z. Sebestyen, H.S.V. de Snoo, and F.H. Szafraniec, A canonical decomposition for linear operators and linear relations, Acta Math. Hungar. 115 (2007), no. 4, 281–307. [8] T. Kato, Perturbation Theory for Linear Operators, Second Edition, Springer– Verlag, Berlin–Heidelberg–New York, 1976. [9] J.P. Labrousse, Inverses g´en´eralis´es d’op´erateurs non born´es (French), Proc. Amer. Math. Soc. 115 (1992), no. 1, 125–129. ´ [10] J.P. Labrousse and B. Mercier, Equivalences compactes entre deux op´erateurs ferm´es sur un espace de Hilbert, Math. Nachr. 133 (1987), 91-105. [11] Y. Mezroui, Le compl´et´e des op´erateurs ferm´es ´ a domaine dense pour la m´etrique du gap (French), J. Oper. Theory 41 (1999), 69-92. [12] Y. Mezroui, Projection orthogonale sur le graphe d’une relation lin´eaire ferm´ee (French), Trans. Amer. Math. Soc. 352 (2000), 2789-2800. [13] M.H. Stone, On unbounded operators on a Hilbert space, J. Indian Math. Soc. 15 (1951), 155–192. Tomas Ya. Azizov Department of Mathematics, Voronezh State University, Universitetskaya pl. 1 394006 Voronezh, Russia e-mail: [email protected] Jussi Behrndt and Peter Jonas Institut f¨ ur Mathematik, MA 6-4, Technische Universit¨ at Berlin, Straße des 17. Juni 136 10623 Berlin, Germany e-mail: [email protected] Carsten Trunk Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau, Postfach 10 05 65 98684 Ilmenau, Germany e-mail: [email protected] Submitted: February 26, 2008. Revised: December 23, 2008.

Integr. equ. oper. theory 63 (2009), 165–180 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020165-16, published online January 27, 2009 DOI 10.1007/s00020-008-1646-x

Integral Equations and Operator Theory

The First Order Asymptotics of the Extreme Eigenvectors of Certain Hermitian Toeplitz Matrices A. B¨ottcher, S. Grudsky, E. A. Maksimenko and J. Unterberger Abstract. The paper is concerned with Hermitian Toeplitz matrices generated by a class of unbounded symbols that emerge in several applications. The main result gives the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the extreme eigenvectors of the matrices as their dimension increases to infinity. Mathematics Subject Classification (2000). Primary 47B35; Secondary 15A18, 41A80, 46N30. Keywords. Toeplitz matrix, Fisher-Hartwig symbol, eigenvalue, eigenvector.

1. Introduction and main results The n × n Toeplitz matrix Tn (a) generated by a function a ∈ L1 (−π, π) is the matrix (aj−k )nj,k=1 constituted by the Fourier coefficients Z π 1 a` = a(θ)e−i`θ dθ (` ∈ Z) 2π −π of the function a. We assume that a is real-valued, which implies that all the (n) (n) matrices Tn (a) are Hermitian. Let λ1 ≤ . . . ≤ λn denote the eigenvalues of (n) (n) Tn (a) and let {v1 , . . . , vn } be an orthonormal basis of eigenvectors, such that (n) (n) (n) Tn (a)vj = λj vj . Suppose a is bounded from below, denote by m the essential infimum of a on (−π, π), and assume a is not identically equal to m. It is well known that then, for each fixed k, (n)

λk

> m for all n ≥ 1

and

(n)

lim λk

n→∞

= m.

This work was partially supported by CONACYT projects 60160 and 80504, Mexico.

(1)

166

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

(n)

This is the reason for calling λk extreme eigenvalues if k is fixed. In several applications in probability theory and physics one encounters the function a given by x −2α a(x) = |1 + eix |−2α = 2−2α cos , (2) 2 where α ∈ (0, 1/2) is a parameter. In that case (1) holds with m = 2−2α , but this is not sufficient for even modest purposes and what one is actually interested (n) in is more precise results on the asymptotics of the eigenvalues λk and also of (n) the eigenvectors vk as k remains fixed and n goes to infinity. MATLAB gives (n) normalized eigenvectors vk for the Toeplitz matrices generated by (2) which are very close to r n 2 jkπ (n) ψk := sin . (3) n+1 n + 1 j=1 Note that

n X j=1

sin2

jkπ n+1 = . n+1 2

(4)

(n)

We will prove that the eigenvectors vk are indeed asymptotically equal to these (n) ψk . Let k · k be the `2 norm on Cn . If we multiply an eigenvector of unit length by a scalar factor of modulus 1 we get again an eigenvector of unit length. Therefore it is convenient to use the distance

|hu, vi|

p

= 2 − 2|hu, vi| %(u, v) := min kτ u − vk = u − v

hu, vi |τ |=1 between unit vectors u, v ∈ Cn . Here is our main result. Theorem 1.1. Let a be a real-valued function in L1 (−π, π) whose essential infimum m is finite and suppose there are positive numbers A, B, σ such that m + Ax2 ≤ a(x) for all x ∈ (−π, π)

(5)

a(x) ≤ m + Ax2 + Bx4 for all x ∈ (−σ, σ).

(6)

and Then there exist positive constants γ, β1 , β2 , . . . , N1 , N2 , . . . depending only on a such that m + 4A sin2

kπ kπ γk 3 (n) ≤ λk ≤ m + 4A sin2 + 2(n + 1) 2(n + 1) (n + 1)3

for all k ≥ 1 and all n ≥ Nk and βk (n) (n) % vk , ψk ≤√ n+1 for all k ≥ 1 and all n ≥ k.

(7)

(8)

Vol. 63 (2009)

Extreme Eigenvectors of Hermitian Toeplitz Matrices

167

(n)

Note that (8) means that there exist numbers τk on the complex unit circle √ (n) (n) (n) T such that kτk vk − ψk k ≤ βk / n + 1. Theorem 1.1 gives the third order asymptotics kπ 1 Aπ 2 k 2 1 (n) λk = m + 4A sin2 +O = m + + O (9) 2(n + 1) n3 n2 n3 and the first order asymptotics (n) vk

=

1

(n) ψ (n) k τk

+O

1 √ n

,

(10)

√ (1) (2) (n) where O(1/ n) stands for a sequence δk , δk , . . . of vectors δk ∈ Cn such that √ (n) kδk k = O(1/ n) as n → ∞. We remark that (9) implies that, for fixed k, we (n) (n) have the strict inequality λk < λk+1 for all sufficiently large n. Thus, the extreme eigenvalues are all simple if only the matrix dimension is sufficiently large. Here is a consequence of Theorem 1.1. Theorem 1.2. If a is given by (2), then (7) and (8) hold with m = 2−2α and A = α · 2−2α−2 . The asymptotic behavior of the extreme eigenvalues of Hermitian Toeplitz matrices has been studied since the 1950s [4], [6], [8], [9], [13]. For example, Widom [13] showed that if a is the restriction to (−π, π) of a real-valued, even, and continuous 2π-periodic function on R, if a(0) = m and a(x) > m for x ∈ [−π, π] \ {0}, if a is four times continuously differentiable in a neighborhood of 0 and a00 (0) > 0, then a00 (0)π 2 k 2 ω 1 (n) λk = m + 1+ +o (11) 2 2(n + 1) n+1 n3 with some completely identified constant ω. This result is not applicable to the function (2) because that function is unbounded. Serra Capizzano [10], [11] proved that ν1,k ν2,k (n) ≤ λk − m ≤ 2s 2s n n with constants 0 < ν1,k ≤ ν2,k under assumptions much weaker than Widom’s; here 2s is the maximal order of the zeros of a − m. See also [7] and [12]. This result is applicable to the function (2) but not as precise as Theorem 1.2. We remark that in recent time there is a revival of the interest in spectral properties of Toeplitz matrices generated by unbounded functions in connection with several applications; see, for instance, [2], [3], [5]. We have not found results of the type of (8) or (10) in the literature. In situations where eigenvectors of Toeplitz matrices are needed, one often replaces Toeplitz matrices by circulant matrices, since for the latter the eigenvalues and eigenvectors are explicitly available. As such a replacement is critical, rigorous

168

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

results like (8) or (10) are of value. We also emphasize that it does not suffice to show that (n) (n) (n) kTn (a)ψk − λk ψk k = o(1) (12) (n)

in order to conclude that ψk is close to an eigenvector of Tn (a). This was pointed out in [2], where we proved that if a is given by (2), then (n)

(n)

(n)

kTn (a)ϕk − λk ϕk k = o(1) whenever (n)

(n) ϕk

=

fk

(n) fk

n jkπ = f n+1 j=1

, (n) kfk k with an arbitrary trigonometric polynomial f (x) = c0 +

N X

(c` cos `x + d` sin `x).

`=1 (n)

In [2], we called sequences {ψk }∞ n=1 of unit vectors satisfying (12) asymptotic (n) pseudomodes and referred to sequences {ψk }∞ n=1 of unit vectors for which (n) (n) % vk , ψk = o(1) as asymptotic eigenvectors. Connections and differences between these two concepts are discussed in [2]. The paper is organized as follows. In Section 2 we cite some general results on the problem of deciding whether a given vector is close to an eigenvector. The results of Section 3 are technical and serve the preparation of the proofs of Theorems 1.1 and 1.2, which are given in Section 4. Section 5 reports on some numerical experiments.

2. Vectors close to eigenvectors Let A ∈ Cn×n be a Hermitian matrix. We denote by λ1 ≤ . . . ≤ λn the eigenvalues of A and we let {v1 , . . . , vn } be an orthonormal basis of eigenvectors of A, assuming that vj is an eigenvector for λj . The following two propositions provide us with conditions in terms of the Rayleigh quotient hAu, ui/hu, ui which ensure that u is close to an eigenvector of A. Proposition 2.1. Let u ∈ Cn , kuk = 1, and suppose (i) hAu, ui ≤ λ1 + ε and (ii) λ2 ≥ λ1 + δ. Then %2 (v1 , u) ≤ 2ε/δ. Pn Pn Proof. Let u = j=1 αj vj with j=1 |αj |2 = 1. Write α1 = τ |α1 | with |τ | = 1 Pn and put w = j=2 αj vj . Then u = |α1 |τ v1 + w. From (ii) we obtain that hAu, ui − λ1 =

n X j=2

(λj − λ1 )|αj |2 ≥ δkwk2 ,

Vol. 63 (2009)

Extreme Eigenvectors of Hermitian Toeplitz Matrices

169

which in conjunction with (i) gives kwk2 ≤ ε/δ. Since %(v1 , u)2 = 2 − 2|hv1 , ui| = 2 − 2|α1 | ≤ 2 − 2|α1 |2 = 2kwk2 , we therefore get %(v1 , u)2 ≤ 2ε/δ.

Proposition 2.2. Fix a number k ∈ {2, . . . , n − 1}. If u ∈ Cn , kuk = 1, and (i) λk −ε ≤ hAu, ui ≤ λk +ε, (ii) λk+1 ≥ λk +δ, (iii) λk ≤ λ1 +ξ, (iv) |hu, vj i| ≤ ηj for all j ∈ {1, . . . , k − 1}, then ! k−1 ε ξ X 2 2 % (vk , u) ≤ 2 + 1+ |ηj | . δ δ j=1 Pn Pn Proof. We have u = j=1 αj vj with j=1 |αj |2 = 1. Let αk = τ |αk | with |τ | = 1 Pn and denote j=k+1 αj vj by w. Thus, u=

k−1 X

αj vj + |αk |τ vk + w.

j=1

From (iv) we infer that k−1 X

|αj |2 =

j=1

k−1 X

|hu, vj i|2 ≤

j=1

k−1 X

ηj2 .

(13)

j=1

This estimate together with (ii) and (iii) yields that n X |hAu, ui − λk | = (λj − λk )|αj |2 j=1

≥

n X

(λj − λk )|αj |2 −

(λk − λj )|αj |2

j=1

j=k+1

≥ δkwk2 − ξ

k−1 X

k−1 X

ηj2 ,

j=1

and taking into account (i) we arrive at the inequality k−1 ε ξX 2 kwk ≤ + η . δ δ j=1 j 2

(14)

From (13) and (14) we get 1 − |αk |2 =

k−1 X j=1

|αj |2 + kwk2 ≤

k−1 ε ξ X + 1+ |ηj |2 =: µ δ δ j=1

and consequently, %(vk , u)2 = 2 − 2|hvk , ui| = 2 − 2|αk | ≤ 2 − 2|αk |2 ≤ 2µ.

170

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

3. Estimates for Rayleigh quotients (n)

Recall that ψk ∈ Cn is given by (3). This section is devoted to estimates for the Rayleigh quotients (n)

(n)

hTn (a)ψk , ψk i (n) (n) hψk , ψk i

(n)

(n)

= hTn (a)ψk , ψk i

and the results of this section will be used in the proofs of Section 4. (n) (n) The key observation is that {ψ1 , . . . , ψn } is just an orthonormal basis of eigenvectors of the tridiagonal Hermitian Toeplitz matrix Tn (b) generated by x b(x) = |1 − eix |2 = 2 − eix − e−ix = 2 − 2 cos x = 4 sin2 . (15) 2 (n)

(n)

In fact we have Tn (b)ψk (n)

µk

(n)

= µk ψk where kπ kπ =b = 4 sin2 n+1 2(n + 1)

(1 ≤ k ≤ n);

(16)

(n)

see, for example, [1] or [4]. Throughout what follows, b and µk are always given by (15) and (16). For u = (u1 , . . . , un ) ∈ Cn , we denote by F u the trigonometric polynomial (F u)(x) = u1 + u2 eix + . . . + un ei(n−1)x . It is well known and easily seen that hTn (a)u, ui =

1 2π

Z

π

a(x)|u(x)|2 dx

(17)

−π

for every a ∈ L1 (−π, π). It follows in particular that if a ∈ L1 (−π, π) is real-valued and a(x) ≥ Ab(x) for all x ∈ (−π, π), then (n)

(n)

(n)

(n)

(n)

hTn (a)ψk , ψk i ≥ AhTn (b)ψk , ψk i = Aµk . (n)

(18) (n)

The aim of this section is to derive upper bounds for hTn (a)ψk , ψk i under the assumption that a ∈ L1 (−π, π) is real-valued and a(x) ≤ Ab(x) + Db2 (x) for |x| ≤ x0 , where A and D are positive constants. Lemma 3.1. If n ≥ 1 and k ∈ {1, . . . , n}, then (n)

(n)

4 kπ kπ sin2 + 16 sin4 n+1 n+1 2(n + 1) π4 k2 k2 ≤ 4π 2 + . n + 1 (n + 1)3

hTn (b2 )ψk , ψk i =

Proof. This was in principle already shown in [4, Section 5.4(d)]. For the convenience of the reader, we cite the proof here.

Vol. 63 (2009)

Extreme Eigenvectors of Hermitian Toeplitz Matrices (n)

From (17) with a(x) = b2 (x) = |1 − eix |4 and u = ψk (n)

(n)

hTn (b2 )ψk , ψk i =

1 2π

π

Z

−π

171

=: (uj )nj=1 we get

2 (1 − eix )2 (u1 + u2 eix + · · · + un ei(n−1)x ) dx.

Expanding the products, sorting the terms by powers of eix , and using the orthog(n) (n) 2 onality of the system {eijx }∞ j=−∞ , we see that hTn (b )ψk , ψk i equals |u1 |2 + |u2 − 2u1 |2 +

n X

|uj − 2uj−1 + uj−2 |2 + |un−1 − 2un |2 + |un |2 .

(19)

j=3

Put q =

kπ n+1 .

Then 2 | sin jq − 2 sin(j − 1)q + sin(j − 2)q|2 n+1 2 q = 16 sin4 sin2 (j − 1)q, n+1 2

|uj − 2uj−1 + uj−2 |2 =

which together with (4) gives n X

n−1 2 qX 2 16 sin4 sin jq n+1 2 j=2 2 q n+1 = 16 sin4 − sin2 q − sin2 nq . n+1 2 2

|uj − 2uj−1 + uj−2 |2 =

j=3

Since sin2 q = sin2 nq and |u1 |2 = |un |2 =

2 sin2 q, n+1

|u2 − 2u1 |2 = |un−1 − 2un |2 =

2 q 16 sin4 sin2 q, n+1 2

we see that the sum (19) is 4 q sin2 q + 16 sin4 . n+1 2

(n)

(n)

Lemma 3.2. If x ∈ (−π, π) and |x| 6= kπ/(n + 1), then Ψk (x) := (F ψk )(x) is given by (n−1)x+(k+1)π

2 sin kπ ei sin (n+1)x+kπ (n) 2 Ψk (x) = p n+1 . kπ kπ x− n+1 x+ n+1 2(n + 1) sin 2 sin 2

172

B¨ ottcher, Grudsky, Maksimenko and Unterberger

Proof. Put q =

IEOT

kπ n+1 .

We have r n 2 X (n) Ψk (x) = sin(jq)ei(j−1)x n + 1 j=1 r n 2 X 1 ijq = e − e−ijq ei(j−1)x n + 1 j=1 2i n h i X e−ix = p eij(x+q) − eij(x−q) i 2(n + 1) j=1 in(x−q) 1 1 − ein(x+q) −iq 1 − e = p eiq − e . 1 − ei(x+q) 1 − ei(x−q) i 2(n + 1)

The term in brackets is (eiq − eix )(1 − ein(x+q) ) − (e−iq − eix )(1 − ein(x−q) ) r =: . i(x+q) i(x−q) s (1 − e )(1 − e ) Since ei(n+1)q = (−1)k and einq = (−1)k e−iq , we get r = eiq − e−iq + ei(n+1)x einq − e−inq

= eiq − e−iq + ei(n+1)x (−1)k e−iq − (−1)k eiq = eiq − e−iq 1 − ei(n+1)x+ikπ = 2i sin q ei

(n+1)x+kπ 2

(−2i) sin

(n + 1)x + kπ . 2

Using that s = ei

x+q 2

(−2i) sin

x + q i x−q x−q e 2 (−2i) sin 2 2

we finally obtain that (n+1)x+kπ sin q sin (n+1)x+kπ 1 r 2 2 = i ei e−ix x−q . i s sin x+q sin 2 2

Lemma 3.3. Let x0 ∈ (0, π). If x ∈ (−π, −x0 ) ∪ (x0 , π) and n ≥ 2kπ/x0 , then (n)

|Ψk (x)| ≤

C1 k x20 (n + 1)3/2

√ where C1 = 2 2 π 3 . Proof. Let q =

kπ n+1 .

Lemma 3.2 gives

(n) |Ψk (x)| ≤ √

kπ 1 . x−q 3/2 sin 2 sin x+q 2 (n + 1) 2

Vol. 63 (2009)

Extreme Eigenvectors of Hermitian Toeplitz Matrices

173

Assume without loss of generality that x ∈ (x0 , π). The requirement n ≥ 2kπ/x0 ensures that q < x0 /2. Hence x − q > x0 /2 and thus x−q 2 x−q x0 sin ≥ > . 2 π 2 2π If x + q ≤ π, we obtain analogously that 2 x+q x0 x0 x+q sin ≥ > > . 2 π 2 π 2π So let x + q > π. Then x = π − q + y with 0 < y ≤ q. It follows that x0 x0 x0 x0 y+ ≤q+ < + = x0 < π, 2 2 2 2 that is, π − y > x0 /2 and consequently, x+q π+y π−y 2 π−y x0 sin = sin = sin ≥ > . 2 2 2 π 2 2π In summary, 2 kπ 2π (n) |Ψk (x)| ≤ √ . 3/2 x0 2 (n + 1) Theorem 3.4. Let a ∈ L1 (−π, π) be real-valued and suppose there exist positive numbers x0 , A, D such that a(x) ≤ Ab(x) + Db2 (x)

x ∈ (−x0 , x0 ).

for

(20)

2 2

Then if n ≥ 2kπ/x0 and n ≥ π k , (n)

(n)

(n)

hTn (a)ψk , ψk i ≤ Aµk +

C2 k 2 (n + 1)3

where C2 = 5π 2 D + 8π 6 kak1 /x40 and kak1 is the L1 norm of a. Proof. By virtue of (17) and (20), (n) (n) hTn (a)ψk , ψk i

1 = 2π

Z

π

−π

(n)

a(x)|Ψk (x)|2 dx ≤ I1 + I2 + I3

where Z A x0 (n) b(x)|Ψk (x)|2 dx, 2π −x0 Z 1 (n) I3 = a(x)|Ψk (x)|2 dx. 2π |x|>x0 I1 =

I2 =

D 2π

Z

x0

−x0

(n)

b2 (x)|Ψk (x)|2 dx,

Clearly, I1 ≤

A 2π

Z

π

−π

(n)

(n)

(n)

(n)

b(x)|Ψk (x)|2 dx = A hTn (b)ψk , ψk i = Aµk . 4 2

k If n ≥ π 2 k 2 , then 4π 2 + πn+1 ≤ 5π 2 and hence Lemma 3.1 implies that Z D π 2 k2 (n) (n) (n) I2 ≤ b (x)|Ψk (x)|2 dx = D hTn (b2 )ψk , ψk i ≤ 5π 2 D . 2π −π (n + 1)3

174

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

From Lemma 3.3 we finally infer that if n ≥ 2πk/x0 , then (n)

I3 ≤ kak1 sup |Ψk (x)|2 ≤ |x|>x0

with

C12

C12 kak1 k2 4 x0 (n + 1)3

6

= 8π .

4. Asymptotics of eigenvalues and eigenvectors We first establish a result for eigenvalues. Theorem 4.1. Let a ∈ L1 (−π, π) be real-valued and suppose there are positive numbers x0 , A, D such that Ab(x) ≤ a(x) 2

a(x) ≤ Ab(x) + Db (x)

for

x ∈ (−π, π),

for

x ∈ (−x0 , x0 ).

Then for n ≥ 2kπ/x0 and n ≥ π 2 k 2 , (n)

Aµk

(n)

≤ λk

(n)

≤ Aµk +

C2 k 3 (n + 1)3

(21)

where C2 = 5π 2 D + 8π 6 kak1 /x40 . Proof. We proceed as in [4, Section 5.4(c)]. By the Fischer-Courant-Weyl theorem, (n)

λk

= max min hTn (a)u, ui S∈Mk u⊥S, kuk=1

(22)

where Mk is the set of all k − 1 dimensional subspaces of Cn . From (17), (22), (n) (n) and the inequality a(x) ≥ Ab(x) we deduce that λk ≥ Aµk . To prove the upper bound in (21), let S be any subspace in Mk . We construct a vector u ∈ Cn such that u ⊥ S, kuk = 1, and hTn (a)u, ui does not exceed the right-hand side of (21). This will give the desired upper bound. (n) (n) We take u in the form u = ξ1 ψ1 + · · · + ξk ψk with |ξ1 |2 + · · · + |ξk |2 = 1. Since dim S = k − 1 < k, we can find ξ1 , . . . , ξk such that u ⊥ S. Put U = F u. By (17), it remains to prove that Z π 1 a(x)|U (x)|2 dx (23) 2π −π is not larger than the right-hand side of (21). As in the proof of Theorem 3.4, the integral (23) is at most I1 + I2 + I3 with Z Z A x0 D x0 2 I1 = b(x)|U (x)|2 dx, I2 = b (x)|U (x)|2 dx, 2π −x0 2π −x0 Z 1 I3 = a(x)|U (x)|2 dx. 2π |x|>x0

Vol. 63 (2009)

Extreme Eigenvectors of Hermitian Toeplitz Matrices

175

We have Z A π b(x)|U (x)|2 dx = A hTn (b)u, ui 2π −π * k + k X (n) X (n) (n) =A µj ξj ψj , ξ j ψj

I1 ≤

j=1

=A

k X

j=1

(n)

(n)

µj |ξj |2 ≤ Aµk

j=1

k X

(n)

|ξj |2 = Aµk .

j=1

The Cauchy-Schwarz inequality gives k 2 k X X (n) (n) |U (x)|2 = ξj Ψj (x) ≤ |Ψj (x)|2 . j=1

(24)

j=1

Combining (24) and Lemma 3.1 with the term in parentheses replaced by 5π 2 , we get I2 ≤

D 2π

Z

π

b2 (x)|U (x)|2 dx ≤ D

−π

k X

(n)

(n)

hTn (b2 )ψj , ψj i ≤ 5π 2 D

j=1

k3 , (n + 1)3

while (24) in conjunction with Lemma 3.3 shows that I3 ≤

k X

(n)

kak1 sup |Ψj (x)|2 ≤ |x|>x0

j=1

kak1 C12 k3 . 4 x0 (n + 1)3

We can finally turn to eigenvectors. (n)

(n)

Theorem 4.2. Let a be as in Theorem 4.1 and let {v1 , . . . , vn } be an orthonormal (n) (n) (n) basis of eigenvectors of Tn (a) such that Tn (a)vj = λj vj for all j. Then there exist positive numbers β1 , β2 , . . . depending only on a such that βk (n) (n) % vk , ψk ≤√ (25) n+1 for all k ≥ 1 and all n ≥ k. Proof. We prove the theorem by induction on k. (n) Let first k = 1. We employ Proposition 2.1 with A = Tn (a) and u = ψk . Condition (i) follows from Theorems 3.4 and 4.1, because these show that if n ≥ n1 with some sufficiently large n1 , then (n)

(n)

(n)

hTn (a)ψ1 , ψ1 i ≤ Aµ1 +

C2 (n) ≤ λ1 + ε (n + 1)3

176

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

with ε = C2 /(n + 1)3 . From Theorem 4.1 we see that C2 (n) (n) (n) (n) λ2 − λ1 ≥ A µ2 − µ1 − (n + 1)3 π 2π C2 = 2A cos − cos − n+1 n+1 (n + 1)3 3π π C2 = 4A sin sin − 2(n + 1) 2(n + 1) (n + 1)3 2 2 3π 2 C2 ≥ 4A − , π 4(n + 1)2 (n + 1)3 and this is at least δ = 6A/(n + 1)2 if n ≥ n2 ≥ n1 . Consequently, for n ≥ n2 (n) condition (ii) is also satisfied. Proposition 2.1 therefore shows that there is a τ1 on T such that (n) (n)

kτ1 v1

(n)

− ψ1 k2 ≤

2C2 (n + 1)2 C2 1 = 3 (n + 1) 6A 3A n + 1

whenever n ≥ n2 . Choosing β1 large enough, we can guarantee (25) also for k = 1 and 1 ≤ n < n2 . (n) Now let k ≥ 2 and assume for j ∈ {1, . . . , k−1} there are βj > 0 and τj ∈ T such that βj2 (n) (n) (n) kτj vj − ψj k2 ≤ (26) n+1 for all n ≥ j. We prove (26) for j = k and n ≥ k using Proposition 2.2 with (n) A = Tn (a) and u = ψk . From (18) and Theorem 3.4 we know that (n)

Aµk

(n)

(n)

(n)

≤ hTn (a)ψk , ψk i ≤ Aµk +

C2 k 2 (n + 1)3

for all n ≥ n1 , and Theorem 4.1 tells us that (n)

Aµk

(n)

≤ λk

(n)

≤ Aµk +

C2 k 3 (n + 1)3

for all n ≥ n1 . This gives condition (i) of Proposition 2.2 with ε = 2C2 k 3 /(n + 1)3 . From Theorem 4.1 we infer that C2 k 3 (n) (n) (n) (n) λk+1 − λk ≥ A µk+1 − µk − (n + 1)3 (2k + 1)π π C2 k 3 = 4A sin sin − 2(n + 1) 2(n + 1) (n + 1)3 2 2 (2k + 1)π π C2 k 3 4A ≥ 4A − ≥ π 2(n + 1) 2(n + 1) (n + 1)3 (n + 1)2

Vol. 63 (2009)

Extreme Eigenvectors of Hermitian Toeplitz Matrices

177

if only n ≥ n3 ≥ n1 . Thus, we have condition (ii) of Proposition 2.2 with δ = 4A/(n + 1)2 for n ≥ n3 . Again by Theorem 4.1, C2 k 3 (n) (n) (n) (n) λ1 − λk ≥ A µ1 − µk − (n + 1)3 (k + 1)π (k − 1)π C2 k 3 4A = −4A sin sin − ≥− 2(n + 1) 2(n + 1) (n + 1)3 (n + 1)2 if n ≥ n4 ≥ n3 . This yields condition (iii) of Proposition 2.2 with ξ = 4A/(n + 1)2 for n ≥ n4 . Finally, (26) implies that if j ≤ k − 1 and j ≤ n, then (n)

(n)

(n)

(n) (n)

(n)

(n) (n)

(n)

(n)

(n)

(n) (n)

|hψk , vj i| = |hψk , τj vj i| = |hψk , ψj i + hψk , τj vj = |hψk , τj vj

(n)

(n) (n)

− ψj i| ≤ kτj vj

(n)

− ψj i|

(n) − ψj k ≤ √

βj , n+1

√ which delivers condition (iv) of Proposition 2.2 with ηj = βj / n + 1. In summary, (n) we can use Proposition 2.2 to see that there is a τk ∈ T such that the number (n) (n) (n) kτk vk − ψk k2 does not exceed   k−1 2 2 3 X βj2 2C k (n + 1) 4A (n + 1) 2  ≤ Mk 2 + 1+ (n + 1)3 4A (n + 1)2 4A n + 1 n+1 j=1 for n ≥ n4 . This is (26) for j = k and n ≥ n4 . Taking βk large enough, we can ensure (26) also for j = k and k ≤ n < n4 . Proof of Theorem 1.1. We may without loss of generality assume that m = 0. Since x2 ≥ 4 sin2 (x/2) = b(x), we see that if a(x) ≥ Ax2 for x ∈ (−π, π), then a(x) ≥ Ab(x) for x ∈ (−π, π). Now consider the function g(x) = A(2 − 2 cos x) + D(2 − 2 cos x)2 − Ax2 − Bx4 . We have g 0 (x) = 2A sin x + 2D(2 − 2 cos x)2 sin x − 2Ax − 4Bx3 = (2A + 8D) sin x − 4D sin 2x − 2Ax − 4Bx3 , g 00 (x) = (2A + 8D) cos x − 8D cos 2x − 2A − 12Bx2 , g 000 (x) = −(2A + 8D) sin x + 16D sin 2x − 24Bx, g (4) (x) = −(2A + 8D) cos x + 32D cos 2x − 24B. Obviously, g(0) = g 0 (0) = g 00 (0) = g 000 (0) = 0. Furthermore, g (4) (0) = 24D −24B − 2A > 0 if only D > B + A/12. Thus, for D > B + A/12 we get g(x) =

g (4) (0) 4 x + O(x5 ) 24

178

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

and this is non-negative for all x ∈ (−x1 , x1 ). We therefore see that if a(x) ≤ Ax2 + Bx4 for |x| < σ, then a(x) ≤ A(2 − 2 cos x) + D(2 − 2 cos x)2 = Ab(x) + Db2 (x) for |x| < x0 := min(σ, x1 ) and D > b + A/12. Consequently, if a satisfies the hypothesis of Theorem 1.1, then it also satisfies the hypothesis of Theorems 4.1 and 4.2. Inequalities (7) with Nk = max(2πk/x0 , 2π 2 k 2 ) and γ = C2 and inequality (8) now follow from Theorems 4.1 and 4.2. Proof of Theorem 1.2. Put x −2α α g(x) = cos and f (x) = 1 + x2 . 2 4 We have to show that f (x) ≤ g(x) for x ∈ [0, π) and g(x) ≤ f (x) + Bx4 for x ∈ [0, σ). Obviously, x −2α−1 x g 0 (x) = α cos sin , 2 2 −2α−2 α(2α + 1) x x α x −2α g 00 (x) = cos sin2 + cos 2 2 2 2 2 00 Since g is monotonically increasing on [0, π), it follows that g 000 (x) ≥ 0 = f 000 (x) for x ∈ [0, π), and as α g(0) = f (0) = 1, g 0 (0) = f 0 (0) = 0, g 00 (0) = f 00 (0) = , 2 000 it results that g(x) ≥ f (x) for x ∈ [0, π). Because g (0) = 0 (note that g is even), Taylor expansion yields g(x) = f (x) + O(x4 ), which gives the desired inequality g(x) ≤ f (x) + Bx4 for x ∈ [0, σ).

5. Numerical experiments We considered Tn (a) for a as in (2) with α = 1/4 and computed (n)

Xk

(n)

:= λk − 2−2α − 4 α · 2−2α−2 sin2

kπ , 2(n + 1)

(n)

Yk

(n) (n) := % vk , ψk

using the computer algebra system PARI/GP [14]. Tables 1 and 2 show some values.

k=1 k=2 k=3

n = 50 0.488 0.532 0.596

n = 100 n = 150 n = 200 n = 250 n = 300 0.478 0.475 0.473 0.472 0.472 0.500 0.490 0.484 0.481 0.479 0.532 0.511 0.500 0.494 0.489 (n)

Table 1. Values of Xk

· (n + 1)3 /k 2

Vol. 63 (2009)

k=1 k=2 k=3

Extreme Eigenvectors of Hermitian Toeplitz Matrices

n = 50 0.547 0.499 0.484

179

n = 100 n = 150 n = 200 n = 250 n = 300 0.551 0.553 0.554 0.554 0.554 0.502 0.503 0.504 0.504 0.504 0.490 0.491 0.492 0.493 0.493 (n)

Table 2. Values of Yk

· (n + 1)/k

The tables lead to the conjectures that (n)

Xk

≤

CX k 2 , (n + 1)3

(n)

Yk

≤

CY k n+1

with constants CX and CY depending only on α. The first √ conjecture does perfectly fit with (11), while the second gives a hint that the βk / n + 1 in (8) could by more powerful techniques perhaps be shown to be actually βk/(n+1) with some constant β depending only on a.

References [1] A. B¨ ottcher and S. Grudsky, Spectral Properties of Banded Toeplitz Matrices, SIAM, Philadelphia 2005. [2] A. B¨ ottcher, S. Grudsky, and J. Unterberger, Asymptotic pseudomodes of Toeplitz matrices, Operators and Matrices, in press. [3] C. Estatico and S. Serra-Capizzano, Superoptimal approximation for unbounded symbols, Linear Algebra Appl. 428 (2008), 564–585. [4] U. Grenander and G. Szeg¨ o, Toeplitz Forms and Their Applications, University of California Press, Berkeley 1958. [5] C. M. Hurvich and Yi Lu, On the complexity of the preconditioned conjugate gradient algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity, SIAM J. Matrix Anal. Appl. 27 (2005), 638–653. [6] M. Kac, W. L. Murdock, and G. Szeg¨ o, On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal. 2 (1953), 767–800. [7] A. Yu. Novosel’tsev and I. B. Simonenko, Dependence of the asymptotics of extreme eigenvalues of truncated Toeplitz matrices on the rate of attaining an extremum by a symbol, St. Petersburg Math. J. 16 (2005), 713–718. [8] S. V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations, Trans. Amer. Math. Soc. 99 (1961), 153–192. [9] S. V. Parter, On the extreme eigenvalues of Toeplitz matrices, Trans. Amer. Math. Soc. 100 (1961), 263–276. [10] S. Serra, On the extreme spectral properties of Toeplitz matrices generated by L1 functions with several minima/maxima, BIT 36 (1996), 135–142. [11] S. Serra, On the extreme eigenvalues of Hermitian (block ) Toeplitz matrices, Linear Algebra Appl. 270 (1998), 109–129.

180

B¨ ottcher, Grudsky, Maksimenko and Unterberger

IEOT

[12] S. Serra Capizzano and P. Tilli, Extreme singular values and eigenvalues of nonHermitian block Toeplitz matrices, J. Comput. Appl. Math. 108 (1999), 113–130. [13] H. Widom, On the eigenvalues of certain Hermitian operators, Trans. Amer. Math. Soc. 88 (1958), 491–522. [14] PARI/GP, version 2.3.3, Bordeaux 2006, http://pari.math.u-bordeaux.fr/ A. B¨ ottcher Fakult¨ at f¨ ur Mathematik TU Chemnitz D-09107 Chemnitz Germany e-mail: [email protected] S. Grudsky and E. A. Maksimenko Departamento de Matem´ aticas CINVESTAV del I.P.N. Apartado Postal 14-740 07000 M´exico, D.F. M´exico e-mail: [email protected] [email protected] J. Unterberger ´ Cartan Institut Elie Universit´e Henri Poincar´e Nancy I B.P. 239 54506 Vandœuvre-l`es-Nancy Cedex France e-mail: [email protected] Submitted: September 18, 2008.

Integr. equ. oper. theory 63 (2009), 181–215 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020181-35, published online February 2, 2009 DOI 10.1007/s00020-009-1657-2

Integral Equations and Operator Theory

On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions I.Yu. Domanov and M.M. Malamud Abstract. Let Jkα be a real power of the integration operator Jk defined on the Sobolev space Wpk [0, 1]. We investigate the spectral properties of the operator L Ln α k Ak = n j=1 λj Jk defined on j=1 Wp [0, 1]. Namely, we describe the commu0 tant {Ak } , the double commutant {Ak }00 and the algebra Alg Ak . Moreover, we describe the lattices Lat Ak and HypLat Ak of invariant and hyperinvariant subspaces of Ak , respectively. We also calculate the spectral multiplicity µAk of Ak and describe the set Cyc Ak of its cyclic subspaces. In passing, we present a simple counterexample for the implication HypLat(A ⊕ B) = HypLat A ⊕ HypLat B ⇒ Lat(A ⊕ B) = Lat A ⊕ Lat B to be valid. Mathematics Subject Classification (2000). Primary 47A15, 47A16, 47L80; Secondary 47L10. Keywords. Riemann-Liouville operator, invariant subspace, hyperinvariant subspace, commutant, double commutant.

1. Introduction It R xis well known [9, 20, 33, 36] that theαVolterra integration operator J : f (x) → f (t) dt as well as its real powers J play an exceptional role in the spectral 0 theory of nonselfadjoint operators in L2 [0, 1]. The paper is devoted to the spectral analysis of direct sums of multiples of powers J α of the integration operator J in Sobolev spaces. To describe its content we first briefly recall basic facts on the integration operator. It is well known [9, 20, 33, 36] that J is unicellular on Lp [0, 1] for p ∈ [1, ∞) and the lattice Lat J of its invariant subspaces is anti-isomorphic to the segment

182

Domanov and Malamud

IEOT

[0, 1]. The same is also true (see [20, 36]) for the simplest Volterra operators Z x (x − t)α−1 f (t) dt, α > 0, J α : f (x) → Γ(α) 0 being the positive powers of the integration operator J. More precisely, it is known (see [9, 20, 33, 36]) that Lat J α = HypLat J α = {Ea : a ∈ [0, 1]}, Ea := {f ∈ Lp [0, 1] : f (x) = 0 for a.a. x ∈ [0, a]}.

(1.1)

Description (1.1) yields (and, in fact, is equivalent to) [9, 20, 36] the following description of cyclic vectors of J α Z ε f is a cyclic vector for J α ⇔ |f (x)|p dx > 0 for all ε > 0. (1.2) 0

This condition is called the ε-condition. Description (1.1) of HypLat J α is closely connected with the description of the commutant {J α }0 . The commutant {J}0 of the operator J defined on L2 [0, 1] as well as the (weakly closed) algebra Alg J generated by J and I were originally described by D. Sarason [44] (see also a simple proof in [18]). Another description of Alg J for J acting in Lp [0, 1] has also been obtained in [29, 30]. Namely, it was shown in [29, 30] that if J is defined on Lp [0, 1] (1 < p < ∞), then {J α }0 = Alg J α and K ∈ {J α }0 if and only if it is bounded and admits a representation Z x d (Kf )(x) = k(x − t)f (t) dt, k ∈ Lp0 [0, 1], (1.3) dx 0 −1

where p0 + p−1 = 1. Using a criterion of boundedness of K defined on L2 [0, 1] (see [30, Proposition 3.1’]) it can easily be shown that for p = 2 description (1.3) is equivalent to that obtained in [44]. Ln α Now, let A = J α ⊗ B(= j=1 λj J ) be a tensor product of the operα ator J defined on Lp [0, 1] and the n × n nonsingular diagonal matrix B = diag(λ1 , . . . , λn ) ∈ Cn×n . The investigation of such operators with B = B ∗ was initiated by G. Kalisch [24]. He has extended the known Livsic theorem (see [9, 20]) to the case of (abstract) Volterra operators with finite-dimensional real part and characterized those of them that are unitarily equivalent to A with B = B ∗ and α = 1 (see also [9, 20]). Rx Later, sufficient conditions for a Volterra operator K : f → 0 K(x, t)f (t) dt defined on Lp [0, 1]⊗Cn to be similar to the operator A have been indicated in [32]. So, A may be treated as a similarity model for a wide class of Volterra operators. This result has been applied in [32] to the problem of unique recovery of a Dirac type system by its monodromy matrix (see also references therein). Further, one of the authors [29, 31] described the lattices Lat A and LnHypLat A and the set Cyc A of cyclic subspaces of the operator A = J α ⊗ B(= j=1 λj J α ) defined on Lp [0, 1] ⊗ Cn , p ∈ (1, ∞). In particular, in [29, 31] necessary and sufficient conditions for a sequence {λi }ni=1 guaranteeing the splitting of each of

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

183

the lattices Lat A and HypLat A, as well as of the commutant {A}0 and double commutant {A}00 of A were found. More precisely, it was proved in [29, 31] that each of the following relations Lat

HypLat

n M j=1 n M

λj J α = λj J α =

j=1

M n j=1

λj J

α

0 =

n M

α 0

{λj J } =

j=1

n M j=1 n M

Lat λj J α ,

(1.4)

HypLat λj J α ,

(1.5)

j=1 M n

λj J

j=1

α

00 =

n M

{λj J α }00

(1.6)

j=1

is equivalent to the condition arg λi 6= arg λj

(mod 2π)

1 6 i < j 6 n.

(1.7)

Some partial cases of the equivalence (1.4)⇔(1.7) have been obtained earlier in [23, 39, 40] (see Remark 2.20). It is easily seen that (1.6) is equivalent to the following fact : for any λ 6∈ (0, +∞) an operator equation J α X = λXJ α

(1.8)

has only zero bounded solution X. Moreover, in [29, 31] a description of all nonzero solutions X of (1.8) with λ ∈ (0, +∞) was obtained. Recently, equation (1.8), and even more general ones with a bounded A in place of J α , has attracted attention of several mathematicians (see, for instance, [5, 6, 26], and [8, 10, 45]). In particular, some results from [29] on equation (1.8) were rediscovered in [5] and [26] (the case α = 1) and in [6] (the case α ∈ Z+ \ {0}). These authors treat any solution X of AX = λXA as an extended eigenvector of A (see Remark 2.22 (2)). Note also that if (1.7) is not fulfilled then A is not cyclic. The set Cyc A of cyclic subspaces of A was described in [29, 31] by using a notion of ∗-determinant (see Definition 2.15). For example, vectors f1 := (f11 , f12 ), f2 := (f21 , f22 ) generate a cyclic subspace of the operator A = J ⊕J defined on Lp [0, 1]⊕Lp [0, 1] if and only f11 f12 if the function ∗-det := f11 ∗f22 −f12 ∗f21 satisfies the ε-condition (1.2) f21 f22 (here R x f ∗ g stands for the convolution of functions f, g ∈ L1 [0, 1] : (f ∗ g)(x) := f (x − t)g(t) dt). 0 Passing to the case of the Sobolev space we should mention the pioneering work of E. Tsekanovskii [46]. More precisely, it is shown in [46] (see also [41]) that Rx the integration operator Jk : f (x) → 0 f (t) dt defined on Wpk [0, 1] is unicellular too and Lat Jk consists of a continuous part Latc Jk and a discrete part Latd Jk , Lat Jk = Latc Jk ∪ Latd Jk . Here k Latc Jk = Ea,0 : a ∈ (0, 1] ∪ E0,0 , (1.9) k k Ea,0 := f ∈ Wpk [0, 1] : f (x) = 0 for x ∈ [0, a] , E0,0 := Wp,0 [0, 1],

184

Domanov and Malamud

IEOT

is a continuous chain and Latd Jk = {Elk }kl=0 with Ekk := Wpk [0, 1] and Elk = f ∈ Wpk [0, 1] : f (0) = · · · = f (k−l−1) (0) = 0 , l ∈ {1, . . . , k − 1}, (1.10) is a discrete chain. It is clear that, for 0 6 a1 6 a2 6 1, k k {0} = E1,0 ⊂ Eak2 ,0 ⊂ Eak1 ,0 ⊂ E0,0 k = Wp,0 [0, 1] = E0k ⊂ E1k ⊂ · · · ⊂ Ekk = Wpk [0, 1].

In [16] we investigated the spectral properties of the complex powers Jkα of the integration operator Jk defined on the Sobolev space Wpk [0, 1]. Namely, in [16] were described the lattices Lat Jkα and HypLat Jkα , the set of cyclic subspaces Cyc Jkα , the operator algebra Alg Jkα , the commutant {Jkα }0 and the double commutant {Jkα }00 . In particular, it turns out that {Jkα }0 = {Jkα }00 and {Jkα }0 and Alg Jkα can be described as follows: Z x α 0 R ∈ {Jk } ⇔ (Rf )(x) = cf (x) + r(x − t)f (t) dt, r ∈ Wpk−1 [0, 1], (1.11) 0

R ∈ Alg Jkα ( (1.12) R ∈ {Jkα }0 , r(l) (0) = 0, l 6= mα − 1, m 6 [ k−1 α ], 1 6 α 6 k − 1, ⇔ k−1 1 α 0 R ∈ {Jk } , r ∈ Wp,0 [0, 1], 2 6 k 6 α + p. It was also shown in [16] that the operator Jkα is unicellular on Wpk [0, 1] if and only if either k = 1 or α = 1. Moreover, the unicellularity of Jkα is equivalent to the validity of the “Neumann-Sarason” identity Alg Jkα = {Jkα }00 . In this paper we extend the main results from [16] and [31] to the case of the operator Ak := Jkα ⊗ B defined on the Soblev space Wpk [0, 1] ⊗ Cn of vectorfunctions. Moreover, we investigate the spectral properties of the operator Ak := Ln α λ J j=1 j kj . The paper is organized as follows. In Section 2, we collect some auxiliary results about invariant subspaces for C0 contractions and accretive operators. Here present and Ln we also Lncomplete some results from [31] for the operator A = α j=1 λj J defined on j=1 Lp [0, 1]. Ln In Section 3, it is shown that the operator A = J α defined on j=1 λjL Ln Ln n α k j=1 Lp [0, 1] and the operator Ak,0 = j=1 λj Jk,0 defined on j=1 Wp,0 [0, 1] are isometrically equivalent. Hence all results on the operator A presented in Section 2 are immediately extended to the case of the operator Ak,0 . Ln α In Section 4, we provide a spectral analysis of the operator Ak = j=1 λj Jk,0 Ln k defined on j=1 Wp [0, 1]. A descriptions of the (weakly closed) algebra Alg Ak , commutant {Ak }0 and double commutant {Ak }00 is presented in Subsection 4.1, Subsection 4.2 and Subsection 4.3, respectively. In Subsection 4.4, we obtain a description of the lattice Lat Ak assuming Ln α that Ak := j=1 λj Jkj satisfies condition (1.7). This description is essentially based on a description of Lat T (Theorem 2.1) for a finite-dimensional operator

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

185

Ln kj T in j=1 C . In Subsection 4.5, a description of the lattice HypLat Ak is contained. We emphasize that HypLat Ak,0 = HypLatc Ak and the “continuous part” of HypLat Ak does not depend on α. It turns out that under condition (1.7) HypLat Ak as well as the commutant {Ak }0 of the operator Ak splits, that is, relations (1.5)-(1.6) remain valid with HypLat A and {A}0 replaced by HypLat Ak and {Ak }0 , respectively. On the other hand, under condition (1.7) Lat Ak does not split for k > 1 in contrast to (1.4). In this connection we recall (see [11]) that for a direct sum T1 ⊕ T2 of two operators on a Banach space the relations (1.5)-(1.6) are equivalent to each other and both are implied by (1.4). Thus, the operator Ak presents a simple counterexample to the validity of the implication HypLat(T1 ⊕ T2 ) = HypLat T1 ⊕ HypLat T2 =⇒ Lat(T1 ⊕ T2 ) = Lat T1 ⊕ Lat T2 . Other counterexamples can be found in [11]. In Subsection 4.6, we compute the spectral multiplicity and present a description of the cyclic subspaces Cyc Ak for the operator Ak . It should be emphasized that descriptions of the sets Cyc Ak and Cyc Ak,0 essentially differ. Namely, the first description does not depend on a choice of a n sequence {λj }n1 , though Ln the second one depends on {arg λj }1 and is similar to that obtained in [31] for j=1 Lp [0, 1]. Lm A description of the set of cyclic subspaces of the operator A = j=1 λj Jkαj ⊕ Ln Lm Ln kj kj α j=m+1 λj Jkj ,0 acting in the mixed space j=1 Wp [0, 1] ⊕ j=m+1 Wp,0 [0, 1] is presented too. Main results of the paper have been announced (without proofs) in [15]. 1.1. Notations and agreements 1. X, X1 , X2 stand for Banach spaces; 2. [X1 , X2 ] is the space of bounded linear operators from X1 to X2 ; [X] := [X, X]; 3. I and Ik denote the identity operators on X and on Ck , respectively; O := 0·I, Ok := 0 · Ik ; 4. J(0; k) denotes the Jordan nilpotent cell of order k; 5. ker T = {x ∈ X : T x = 0} is the kernel of T ∈ [X]; 6. ran T = {T x : x ∈ X} is the range of T ∈ [X]; 7. Cyc T denotes the set of cyclic subspaces of an operator T ∈ [X] (see Definition 2.12); 8. {T }0 and {T }00 denote the commutant and the double commutant ( or bicommutant) of an operator T ∈ [X], respectively; 9. Alg{T1 , . . . , Tn } stands for a weakly closed subalgebra of [X] generated by T1 , . . . , Tn ∈ [X] and the identity I; 10. Lat A denotes the lattice of invariant subspaces of the algebra A; 11. Lat T (:= Lat(Alg T )) and HypLat T (:= Lat({T }0 )) denote the lattices of invariant and hyperinvariant subspaces of T ∈ [X], respectively; 12. span E is the closed linear span of the set E ⊂ X;

186

Domanov and Malamud

IEOT

13. rR ∗ f stands for the convolution of functions r, f ∈ L1 [0, 1] : (r ∗ f )(x) := x r(x − t)f (t) dt; 0 14. Z+ := {n ∈ Z : n > 0}; R+ := {x ∈ R : x > 0}. As usual, Wpk [0, 1] (p ∈ (1, ∞), k ∈ Z+ \ {0}) stands for the Sobolev space consisting of functions f having k−1 absolutely continuous derivatives and f (k) ∈ Lp [0, 1]. Wpk [0, 1] is a Banach space equipped with the norm  1/p Z 1 k−1 X kf kWpk [0,1] =  |f (j) (0)|p + |f (k) (t)|p dt . j=0

0

k Wp,0 [0, 1] := {f ∈ Wpk [0, 1] : f (0) = · · · = f (k−1) (0) = 0}. 0 We set Wp0 [0, 1] := Lp [0, 1] and Wp,0 [0, 1] = Lp [0, 1]. α α α k Let Jk,0 and Jk := Jk,k stand for the operator J α defined on Wp,0 [0, 1] and k α k Wp [0, 1], respectively. The operator Jk,0 is well defined on Wp,0 [0, 1] for any α > 0. The operator Jkα is well defined on Wpk [0, 1] if either α ∈ Z+ \ {0} or α > k − p1 . Therefore throughout the paper we assume that Ln Ln 1. the operator A := j=1 λj J α is defined on j=1 Lp [0, 1] for α > 0; Ln Ln k 2. the operator Ak,0 := j=1 λj Jkαj ,0 is defined on j=1 Wp,0j [0, 1] with kj > 0 and α > 0; Ln Ln kj α 3. the operator Ak := j=1 λj Jkj is defined on j=1 Wp [0, 1] with kj > 1 and for α ∈ Z+ \ {0} or α > max16j6n kj − p1 .

We will also assume that λj 6= 0 for j ∈ {1, . . . , n}.

2. Preliminaries 2.1. Invariant subspaces of some operators Here we present some known results on invariant subspaces of finite-dimensional nilpotent operators and C0 contractions.We also recall a condition about splitting of Alg(A ⊕ B), where A, B ∈ [X]. Theorem 2.1. [7, 21] If Q is nilpotent on a finite-dimensional vector space V, then [ Lat(Q) = [M, Q−1 M ] : M ∈ Lat(Q QV ) , (2.1) M

where [M, Q−1 M ] is an interval in the lattice of all subspaces of V . Each interval satisfies the equation dim Q−1 M − dim M = dim ker Q.

(2.2)

The following result was first discovered by P. Halmos [22] for operators defined on finite-dimensional spaces. The generalization to C0 contractions on Hilbert spaces belongs to H. Bercovici [2, Proposition 5.33], [3, Corollary 2.11] and P. Wu [48, Theorem 1.2], and [49, Theorem 5])(see also references therein).

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

187

Theorem 2.2. Let T be a C0 -contraction defined on a separable Hilbert space. Then every invariant subspace of T is the closure of the range and the kernel of some bounded linear transformation that commutes with T , that is, Lat T = ker C : C ∈ {T }0 = ran C : C ∈ {T }0 . Definition 2.3 (see [33],[36]). Let A and B be bounded operators defined on a Banach space X1 and X2 respectively. A is said to be quasisimilar to B if there exist deformations K : X1 → X2 and L : X2 → X1 (i.e. ran K = X2 , ker K = {0}, ran L = X1 , ker L = {0}) such that AL = LB and KA = BK. Remark 2.4. (i) Standard manipulations with Cayley transform implies that Theorem 2.2 holds also for quasinilpotent accretive operators with finitedimensional real part. (ii) Let operator A be defined on a Banach space. Let also A be quasisimilar to a C0 contraction T . Then, obviously the statement of Theorem 2.2 is true for A, that is, Lat A = ker C : C ∈ {A}0 = ran C : C ∈ {A}0 . Let X be a Banach space and let n be a positive integer. Then X (n) denotes the direct sum of n copies of X. If A is an operator on X, then A(n) denotes the direct sum of n copies of A (regarded as an operator on X (n) ). The following theorem is implicitly contained in [43] (see also [42, Theorem 7.1, Theorem 7.2]) Theorem 2.5. Let T1 , . . . , Tr ∈ [X] and (n)

Lat(T1

(n)

⊕ · · · ⊕ Tr(n) ) = Lat T1

⊕ · · · ⊕ Lat Tr(n) ,

n = 1, 2, . . .

Then Alg(T1 ⊕ · · · ⊕ Tr ) = Alg T1 ⊕ · · · ⊕ Alg Tr . Ln Ln 2.2. Spectral analysis of the operator A = i=1 λi J α defined on i=1 Lp [0, 1] Throughout this subsection X stands for Lp [0, 1], with p ∈ (1, ∞). present LnHere we α some results from [31] on spectral analysis of the operator A = λ J defined i i=1 Ln on 1 X. Moreover, we obtain a description of Alg A and investigate its properties. We begin with the following simple statement. Ln Lemma 2.6. Let Ai , Mi , Ni ∈ [X] for i ∈ {1, . . . , n} and A = i=1 Ai . Assume also that the following identities are satisfied m Am i = Mi A1 Ni ,

m ∈ Z+ ,

i ∈ {1, . . . , n}.

(2.3)

Then Alg A =

M n

Ri : R1 ∈ Alg A1 , Ri = Mi R1 Ni , i ∈ {2, . . . , n} .

(2.4)

i=1

Ln Ln Proof. Let M := i=1 Mi and N := i=1 Ni . Then for any polynomial p(·) the identities (2.3) yield p(Ai ) = Mi p(A1 )Ni . Hence, M n n n M M p(A) = p(Ai ) = Mi p(A1 )Ni = M p(A1 ) N. i=1

i=1

i=1

188

Domanov and Malamud

IEOT

On the other hand, by definition of Alg A polynomials p(A) are dense inL Alg A in the n weak operator topology. The last identities thus imply Alg A = M Alg( i=1 A1 )N . Ln Ln To complete the proof it remains to note that Alg( i=1 A1 ) = i=1 Alg(A1 ). Ln Next we apply Lemma 2.6 to describe Alg A for the operator A = i=1 λi J α with factors λi having equal arguments, λi = λ1 /sα i ∈ {1, . . . , n}. (2.5) i , 1 = s1 6 s2 6 . . . 6 sn , Ln Ln α Theorem 2.7. Let the operator A = be defined on i=1 λi J i=1 X with λi satisfying condition (2.5). Then Alg A is Z x n d ri (x − t)f (t) dt, Alg A = R = diag(R1 , . . . , Rn ) : (Ri f )(x) = dx 0 (2.6) o r1 ∈ Lp0 [0, 1], ri (x) = r1 (s−1 x), R ∈ [L [0, 1]] . 1 p i Proof. To apply Lemma 2.6 we introduce the operators Mi and Ni by setting ( f (si x), x ∈ [0, s−1 −1 i ], (Mi f )(x) := f (si x), (Ni f )(x) := (2.7) −1 0, x ∈ [si , 1]. Clearly, ker Ni = {0}, ran Ni = χ[0,s−1 ] Lp [0, 1], ker Mi = χ[s−1 ,1] Lp [0, 1] and i i ran Mi = Lp [0, 1]. It can easily be checked that Mi Ni = ILp [0,1] and, moreover, (λi J α )m = Mi (λ1 J α )m Ni ,

m ∈ Z+ ,

i ∈ {1, . . . , n}.

Setting Ai := λi J α and applying Lemma 2.6 we obtain n n M Alg A = R = Ri : R1 ∈ Alg(λ1 J α ), Ri = Mi R1 Ni ,

o i ∈ {2, . . . , n} .

i=1

(2.8) On the other hand, according to (1.3), any (bounded) R1 ∈ Alg(λ1 J α ) admits a representation Z x d R1 : f (x) → r1 (x − t)f (t) dt, r1 ∈ Lp0 [0, 1]. (2.9) dx 0 Straightforward calculations show that that Z x d (Mi R1 Ni f )(x) = r1 (s−1 i (x − t))f (t) dt, dx 0

i ∈ {2, . . . , n}.

Combining the last equality with (2.8) we complete the proof.

To state the results on {A}0 we need some additional notations. For any a ∈ R+ \{0} we define an operator La : X → X by  0 < a 6 1,  f((ax), −1 La : f (x) → g(x) = (2.10) 0, x ∈ [0, 1 − a ],  a > 1.  −1 f (ax − a + 1), x ∈ [1 − a , 1],

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

189

We set also La {J α }0 := {La K : K ∈ {J α }0 },

{J α }0 La := {KLa : K ∈ {J α }0 }.

It is easily checked that La {J α }0 = {J α }0 La . Ln Ln Theorem 2.8. [31, Proposition 4.6] Suppose A = i=1 λi J α is defined on i=1 X and λi satisfy condition (2.5). Set also aij := s−1 i sj for i, j ∈ {1, . . . , n}. Then the commutant {A}0 is of the form {A}0 = {K : K = (Kij )ni,j=1 , Kij ∈ Laij {J α }0 }. Next we complete Theorem 2.7 by establishing the Neumann type identity, {A}00 = Alg A. Note, that for the case p = 2 and α = 1 it follows from a general result of B.S.-Nagy and C. Foias [34] on a dissipative operator with finite dimensional imaginary part. Ln Ln Theorem 2.9. Suppose A = i=1 λi J α is defined on i=1 X and λi satisfy condition (2.5). Then {A}00 = Alg A. Proof. It is known (and easily seen) that if T1 and T2 are boundedL operators on a n 00 00 00 00 α 00 Banach space Y , then {T ⊕T } ⊂ {T } ⊕{T } . Hence {A} = { 1 2 1 2 i=1 λi J } ⊂ Ln α 00 00 {λi J } . It follows that any R ∈ {A} admits a direct sum decomposition i=1L n R = i=1 Ri with Ri ∈ {λi J α }00 = {λiRJ α }0 , i ∈ {1, ..., n}. According to (1.3) Ri x d admits a representation (Ri f )(x) = dx r (x − t)f (t) dt, where ri ∈ Lp0 [0, 1] and 0 i it is such that Ri ∈ [X]. Further, let K = (Kij )ni,j=1 be an operator matrix with entries Kij = Laij for i > j and Kij = O for i ≤ j. Let also aij := s−1 i sj for i, j ∈ {1, . . . , n}. Then, by Theorem 2.8, K ∈ {A}0 . Clearly, relation RK = KR yields Ri Lai1 = Lai1 R1 ,

i ∈ {2, . . . , n}.

(2.11)

It is easily seen that d (Ri Lai1 f )(x) = dx

Z

x

ri (x − t)f (s−1 i t) dt,

i ∈ {2, . . . , n}.

(2.12)

0

On the other hand, Z x1 d r1 (x1 − t)f (t) dt dx1 0 x1 =s−1 i x Z s−1 Z x x i d d −1 −1 = si r1 (si x − t)f (t) dt = r1 (s−1 i (x − t))f (si t) dt. dx 0 dx 0

(Lai1 R1 f )(x) =

Comparing this relation with (2.12) and taking into account (2.11) and the obvious relation ran(Lai1 ) = X, we obtain ri (x) = r1 (s−1 i x), i ∈ {2, . . . , n}. By Theorem 2.7, this means that R ∈ Alg A, that is {A}00 ⊂ Alg A. Since the inclusion {A}00 ⊃ Alg A is obvious, we get {A}00 = Alg A.

190

Domanov and Malamud

IEOT

In the following theorem we obtain a description of Lat A similar to that of Lat T for C0 -contractions T described in Theorem 2.2. It is interesting to note that though a description is completely the same, the operator A in not accretive in L2 [0, 1] for α > 1 (cf. Remark 2.4 (i)). Ln Ln Theorem 2.10. Let A = i=1 λi J α be defined on i=1 X and λi satisfy conditions (2.5). Then every invariant subspace of A is the closure of the range (the kernel) of a bounded linear transformation that commutes with A. Ln Proof. Alongside the operator A we consider the operator A1 := i=1 λ1 s−1 i J. By Ln L n −1 α λ s Theorem 2.7, Alg A = Alg( i=1 λ1 s−α J ) = Alg( J) = Alg A 1 . Hence i i=1 1 i Lat A = Lat A1 and {A}0 = {A1 }0 . So we can assume that λ1 = 1 and α = 1. We put M n n n M M K := J∈ Lp [0, 1], L2 [0, 1] , L :=

i=1

i=1

i=1

n M

M n

n M

J∈

i=1

B :=

n M

si J ∈

i=1

L2 [0, 1],

i=1 M n

Lp [0, 1] ,

i=1

L2 [0, 1] .

i=1

Ln It i=1 L2 [0, 1], ran L = Lnis clear that ker K = {0}, ker L = {0}, ran K = L [0, 1], KA = BK and A L = LB. Hence A is quasisimilar to B. So, we 1 1L 1 i=1 p n can assume that A1 is defined on i=1 L2 [0, 1]. Note that A1 is accretive, since si > 0 for i ∈ {1, ..., n}. Now the assertions of the theorem follow from Theorem 2.2 (see also Remark 2.4 (i)). Next, we recall a description of HypLat A. Ln Ln Theorem 2.11. [31, Proposition 4.8] Suppose A = i=1 λi J α is defined on i=1 X and λi satisfy condition (2.5). Then the lattice HypLat A is of the form M n HypLat A = Eai : (a1 , . . . , an ) ∈ P (s1 , . . . , sn ) , i=1

where P (s1 , . . . , sn ) := (a1 , . . . , an ) ∈ [0, 1]n : si ai+1 6 si+1 ai 6 si+1 − si + si ai+1 , 1 6 i 6 n − 1 . Definition 2.12 (cf. [36]). (1) A subspace E of a Banach space X1 is called a cyclic subspace for an operator T ∈ [X1 ] if span{T n E : n > 0} = X1 ; (2) a vector f (∈ X1 ) is called cyclic for T if span{T n f : n > 0} = X1 ; (3) the set of all cyclic subspaces of an operator T is denoted by Cyc T . Definition 2.13. (1) The number µT := inf {dim E : E is a cyclic subspace of the operator T on X1 } E

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

191

is called the spectral multiplicity of an operator T on X1 ; (2) The operator T is called cyclic if µT = 1. It is well known that the concept of spectral multiplicity plays an important role in control theory (see for instance [47]). Investigating some other problems of control theory, N.K. Nikol’skii and V.I. Vasjunin [38] introduced one more “cyclic” characteristic of an operator. Definition 2.14. [38] Let T ∈ [X]. Then disc T :=

sup min{dim E 0 : E 0 ⊂ E, E 0 ∈ Cyc T }. E∈Cyc T

disc T is called a disc-characteristic of an operator T . (“disc” is the abbreviation of “Dimension of the Input Subspace of Control”.) Clearly, disc T > µT . To present a description of Cyc A we recall the following definition. Definition 2.15. [29, 31, 35] The determinant of a functional matrix F (x) = (fij (x))ni,j=1 (fij ∈ X) calculated with respect to the convolution product Z x Z x (f ∗ g)(x) = f (x − t)g(t) dt = g(x − t)f (t) dt = (g ∗ f )(x) 0

0

is called ∗-determinant and is denoted by ∗-det F (x). Similarly, ∗-minors of F (x) are the minors calculated with respect to the convolution product. ∗-rank F (x) will be the highest order of ∗-minors of F (x) satisfying ε-condition (1.2). Next we complete [31, Theorem 2.3] by computing disc A. Ln Ln α Theorem 2.16. Suppose A = is defined on i=1 λi J i=1 X and λi satisfy condition (2.5). Then the system {fl }N of vectors l=1 fl = fl1 ⊕ · · · ⊕ fln ∈

n M

X,

l ∈ {1, . . . , N }, i ∈ {1, . . . , n}

i=1

generates a cyclic subspace for the operator A if and only if (i) N > n; (ii) the matrix   f11 (s1 x) f12 (s2 x) . . . f1n (sn x)   .. .. .. Fn (x) =   . . . fN 1 (s1 x) fN 2 (s2 x) . . .

fN n (sn x)

is of maximal ∗-rank , namely, ∗-rank Fn (x) = n; (iii) disc A = µA = n. Proof. (i), (ii) and the equality µA = n were proved in [31, Theorem 2.3](see also [14, Proposition 3.2] for another proof). (iii) Let us prove that disc A = n. Let E = span{f1 , . . . , fN } be an N dimensional subspace cyclic for the operator A. It is necessary to show that this

192

Domanov and Malamud

IEOT

space contains an n-dimensional subspace which is also cyclic for the operator A. Since ∗-rank Fn (x) = n, it follows that there exists an n × n submatrix Gn (x) of Fn (x) such that ∗-rank Gn (x) = n. Hence we can choose n vectors fi1 , . . . , fin (i1 , . . . , in ∈ {1, . . . , N }) such that span{fi1 , . . . , fin } is a cyclic subspace for A. Ln Ln Corollary 2.17. Let K ∈ {J α }0 and Kn = i=1 K be defined on i=1 L2 [0, 1]. Then µKn ≥ n. Ln Proof. It follows from Theorem 2.7 that Kn ∈ Alg A, where A = i=1 J is defined Ln i=1 L2 [0, 1]. Hence, by Theorem 2.16 µKn ≥ µA = n. Remark 2.18. In the recent paper [4, Proposition 7.6] Corollary 2.17 was proved for the case n = 2. Next we recall the following notation. Let Tj ∈ [Xj ] (j = 1, 2) and R ∈ Cyc(T1 ⊕ T2 ). It is clear that Pj R ∈ Cyc Tj , where Pj is the projection from X1 ⊕ X2 onto Xj , j ∈ {1, 2}. Following [38], we write Cyc(T1 ⊕ T2 ) = Cyc T1 ∨ Cyc T2 if Pj R ∈ Cyc Tj (j = 1, 2) yields R ∈ Cyc(T1 ⊕ T2 ) for every R ⊂ X1 ⊕ X2 . In particular, if Lat(T1 ⊕ T2 ) = Lat T1 ⊕ Lat T2 then Cyc(T1 ⊕ T2 ) = Cyc T1 ∨ Cyc T2 . Next we complete [31, Proposition 4.1, Proposition 4.2]. Lr Lr Theorem 2.19. Suppose A = j=1 λj J α is defined on j=1 X and arg λi 6= arg λj

(mod 2π),

1 6 i < j 6 r.

(2.13)

r r M M {J α }0 = {J α }00 ,

(2.14)

Then Alg A = {A}0 = {A}00 =

r M

Alg J α =

j=1

Lat A = HypLat A =

r M j=1

Cyc A =

r _

Lat J α =

j=1 r M

j=1

HypLat J α ,

(2.15)

j=1

Cyc J α ,

(2.16)

j=1

disc A = µA = 1.

(2.17)

Proof. (2.15)-(2.17) and the splitting of {A}0 and {A}00 were proved in [29], [31]. We present two different proofs of the splitting of Alg A due to the first and to the second author, respectively. First proof. We will derive the splitting of Alg A from the splitting of Cyc A. α−1 α−1 By (2.16) g := xΓ(α) ⊕ · · · ⊕ xΓ(α) ∈ Cyc A. Hence, there exists a sequence {Pn (x)}∞ n=1 such that s-limn→∞ Pn (A)g = 0 ⊕ · · · ⊕ 0 ⊕

xα−1 Γ(α) .

s-lim APn (A) = O ⊕ · · · ⊕ O ⊕ λr J α . n→∞

We claim that (2.18)

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

Indeed, for any f = f1 ⊕ · · · ⊕ fr ∈

Lr

j=1

193

X one has

s-lim APn (A)f = s-lim (λ1 J α Pn (λ1 J α )f1 ⊕ · · · ⊕ λr J α Pn (λr J α )fr ) n→∞ n→∞ xα−1 xα−1 α α = s-lim λ1 ∗ (Pn (λ1 J )f1 )(x) ⊕ · · · ⊕ λr ∗ (Pn (λr J )fr )(x) n→∞ Γ(α) Γ(α) α−1 α−1 α x α x = s-lim λ1 f1 ∗ (Pn (λ1 J ) ) ⊕ · · · ⊕, λr fr ∗ (Pn (λr J ) n→∞ Γ(α) Γ(α) α−1 x = λ1 f1 ∗ 0 ⊕ λ2 f2 ∗ 0 ⊕ · · · ⊕ λr fr ∗ = diag(O, . . . , O, λr J α )f. Γ(α) So (2.18) is proved. A similar argument shows that for any j ∈ {1, . . . , r} there exists a sequence of polynomials {Pj,n }∞ n=1 such that s-lim APj,n (A) = O ⊕ · · · ⊕ O ⊕ λj J α ⊕ O ⊕ · · · ⊕ O. n→∞

Hence the splitting of Alg A is proved. Second proof. Keeping in mind notations of Theorem 2.24 (see below), for any j ∈L {1, . . . , r} we let nj = n and λj1 := · · · := λjn := λj . Then setting nj A(j) := i=1 λji J α we rewrite A(j) and A as A(j) =

n M i=1

λj J α = (λj J α )(n)

and A =

r M j=1

A(j) =

r M

(λj J α )(n) ,

j=1

where the factors λj have different arguments, λj 6= λk for j 6= k. Therefore Lr Lr by Theorem 2.24 the lattice Lat( j=1 (λj J α )(n) ) splits, Lat( j=1 (λj J α )(n) ) = Lr α (n) . One completes the proof by applying Theorem 2.5 with Tj = j=1 Lat(λj J ) α λj J , j ∈ {1, . . . , n}. Remark 2.20. Some particular statements of Theorem 2.19 were obtained in [1, 23, 39, 40] for the case p = 2. Namely, A. Atzmon [1] proved that for every integer k ≥ 2, the operator πi iJ 1−1/k ⊕ e 2k J 1−1/k is cyclic. In [39, 40] L B.P. Osilenker and V.S. Shulman proved that (2.13) implies the r splitting of Lat( j=1 λj J). Their proof cannot be extended to the case α 6= 1. L.T. Hill [23] showed that if α ∈ (0, 1) and λ is a nonzero complex number, then Lat(J α ⊕ λJ α ) splits if and only if λ is not positive. His proof cannot be extended neither to the case of α > 1 nor to the number of summands n > 2. The following result is easily implied by combining Theorems 2.8 and 2.19. Corollary 2.21. [28, 31] Let c ∈ C and let R ∈ [X] be a solution of the equation RJ α = cJ α R. Then the following statements hold: (i) if c 6∈ R+ , then R = O; (ii) if c = aα > 0, a > 0, then R ∈ La {J α }0 , where La is defined by (2.10).

194

Domanov and Malamud

IEOT

Remark 2.22. (i) It was shown in [19] that the operators J and cJ are similar if and only if c = 1. Corollary 2.21 implies that operators J α and cJ α are not even quasisimilar for any c 6= 1. (ii) In particular cases Corollary 2.21 (i) was recently reproved by another method in [5], [26] (the case α = 1, p = 2) and in [6] (the case α ∈ Z+ \ {0}, p = 2). Some solutions R of the equation RJ α = cJ α R in the case c > 0, α ∈ Z+ were also indicated in [5], [6], [26]. We need the following lemma in the sequel. Lemma 2.23. Suppose that A ∈ [X1 ] is quasisimilar to B ∈ [X2 ] with intertwining deformations L and K. That is, AL = LB and KA = BK. Let also LK = A2 and KL = B 2 . Then (i) E ∈ Cyc A ⇔ KE ∈ Cyc B; (ii) F ∈ Cyc B ⇔ LF ∈ Cyc A; (iii) disc A = disc B. Proof. The proof is left for the reader.

Now we can consider the case of any diagonal nonsingular matrix B. Next we complete [31, Proposition 3.2, Theorem 3.4, Corollary 3.5, Theorem 4.10, Theorem 4.11]. Lnj Lnj α Theorem 2.24. Lr Suppose A(j) := i=1 Lrλji JLnisj defined on i=1 X, j ∈ {1, . . . , r} and A := j=1 A(j) is defined on j=1 ( i=1 X). Let also arg λj1 = arg λji

(mod 2π),

1 6 j 6 r,

1 6 i 6 nj ,

arg λi1 6= arg λj1

(mod 2π),

1 6 i < j 6 r.

Then Alg A = {A}0 = {A}00 =

Lat A =

HypLat A =

r M j=1 r M j=1 r M j=1 r M j=1 r M j=1

Alg A(j),

(2.19)

{A(j)}0 ,

(2.20)

{A(j)}00 ,

(2.21)

Lat A(j),

(2.22)

HypLat A(j),

(2.23)

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

Cyc A =

r _

Cyc A(j),

195

(2.24)

j=1

disc A = µA = max µA(j) . 16j6r

Proof. Relations (2.20)-(2.24) and the equality µA = max16j6r µA(j) were proved in [31]. Let us prove (2.19). By Theorem 2.19, for any j ∈ {1, . . . , r} O ⊕ · · · ⊕ O ⊕ λj1 J α ⊕ O ⊕ · · · ⊕ O ∈ Alg(λ11 J α ⊕ · · · ⊕ λr1 J α ). Thus, by Theorem 2.7 we have that O ⊕ · · · ⊕ O ⊕ A(j) ⊕ O ⊕ · · · ⊕ O ∈ Alg A, and hence (2.19) is proved. Lr Let Lnusj prove that disc A = µA . Assume that A1 := A is defined on the space ( i=1 L2 [0, 1]). Then [37, Statement 1.13] and [38, Corollary 13] imply the j=1 equality disc A1 = µA1 . We define M nj nj nj r M r M r M M M α K := λji J ∈ L2 [0, 1] , Lp [0, 1] , j=1 i=1

L :=

nj r M M

j=1

λji J α ∈

j=1 i=1

i=1

M nj r M j=1

j=1

i=1

nj r M M Lp [0, 1] , L2 [0, 1] ,

i=1

j=1

i=1

and A2 := A. It is clear that K and L are deformations and A1 L = LA2 , KA1 = A2 K. Now an application of Lemma 2.23 completes the proof.

3. The operator Ak,0 α Let Jk,l stand for the operator Jkα acting on the subspace Elk of Wpk [0, 1] defined by (1.10) (l 6 k − 1) and Ekk := Wpk [0, 1]. α Next we establish isometric equivalence of Jk,0 and J α . α Lemma 3.1. The operator Jk,l defined on Elk is isometrically equivalent to the opα l α k erator Jl defined on Wp [0, 1]. In particular, the operator Jk,0 defined on Wp,0 [0, 1] α α 0 is isometrically equivalent to the operator J0 =: J defined on Wp [0, 1] = Lp [0, 1].

Proof. It is clear that the operator U = Elk on Wpl [0, 1]. Moreover,

dk−l dxk−l

: Elk → Wpl [0, 1] isometrically maps

U −1 = U ∗ = J k−l : Wpl [0, 1] → Elk . α The assertion follows now from the identity Jk,l = U −1 Jlα U .

Ln Ln α Corollary 3.2. The operator Ak,0 := W ki [0, 1] is i Jki ,0 defined on i=1 λL i=1 Ln p,0 n α isometrically equivalent to the operator A := i=1 λi J defined on i=1 Lp [0, 1].

196

Domanov and Malamud

IEOT

Corollary the operator A Ln 3.2 makes it possible to translate all results on L n ki defined on i=1 Lp [0, 1] to the results on operator Ak,0 defined on i=1 Wp,0 [0, 1]. For instance, Theorem 2.10 takes the following form Ln Ln ki Theorem 3.3. Let Ak,0 := i=1 λi Jkαi ,0 be defined on i=1 Wp,0 [0, 1] and λi satisfy condition (2.5). Then every invariant subspace of Ak,0 is the closure of the range (the kernel) of a bounded linear transformation that commutes with Ak,0 .

4. The operator Ak This section contains the main results of L the paper. Namely, we described Lnthe n α (n) spectral properties of the operator Ak := λ J defined on X = j k j=1 1 X where X = Wpk [0, 1]. 4.1. The algebra Alg Ak Ln Ln Theorem 4.1. Suppose Ak = i=1 λi Jkα is defined on i=1 Wpk [0, 1] and λi = λ1 /sα i ,

1 = s1 6 s2 6 . . . 6 sn ,

Let also M n n M k R := Ri ∈ Wp [0, 1] , i=1

i ∈ {1, . . . , n}.

(4.1)

(Ri f )(·) = ci f (·) + (ri ∗ f )(·), i ∈ {1, . . . , n}.

i=1

(4.2) Then the following is true: (1) if 1 6 α 6 k − 1, then −1 Alg Ak = R : c1 = · · · = cn ∈ C; r1 ∈ Wpk−1 [0, 1]; ri (x) = s−1 i r1 (si x), (4.3) (l) 1 6 i 6 n; r1 (0) = 0, l 6= mα − 1, 1 6 m 6 [(k − 1)/α] ; (2) if 2 6 k 6 α + p1 , then k−1 Alg Ak = R : c1 = · · · = cn ∈ C; r1 ∈ Wp,0 [0, 1]; −1 ri (x) := s−1 i r1 (si x),

16i6n .

(4.4)

Proof. Let (Mi f )(x) := f (s−1 i x),

 x ∈ [0, s−1 f (si x), i ], k−1 (Ni f )(x) := P (xsi −1)m (m)  f (1), x ∈ [s−1 i , 1]. m! m=0

It can easily be checked that (λi Jkα )m = Mi (λ1 Jkα )m Ni ,

m ∈ Z+ ,

i ∈ {1, . . . , n}.

Setting Ai := λi Jkα and applying Lemma 2.4 we obtain n n o M Alg A = R = Ri : R1 ∈ Alg(λ1 Jkα ), Ri = Mi R1 Ni , i ∈ {2, . . . , n} . (4.5) i=1

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

197

Next we confine ourselves to the case 1 6 α 6 k − 1. The case 2 6 k 6 α + p1 is considered similarly. By (1.12), R1 ∈ Alg(λ1 Jkα ) if and only if Z x R1 : f (x) → c1 f (x) + r1 (x − t)f (t) dt, c1 ∈ C, r1 ∈ Wpk−1 [0, 1], 0 (4.6) (l) r1 (0) = 0, l 6= mα − 1, 1 6 m 6 [(k − 1)/α]. Straightforward calculations show that Z x −1 (Mi R1 Ni f )(x) = c1 + s−1 i r1 (si (x − t))f (t) dt,

i ∈ {2, . . . , n}.

0

Combining the last relations with (4.5) we arrive at the required description.

In the proof of the following theorem we need a concept of the weak operator topology in the algebra [X]. Recall the following definition. N ∗ Definition 4.2. Let {fi }N i=1 and {gi }i=1 be the sets of unit vectors in X and X , respectively, and let ε be a positive number. For any R ∈ B[X] define V := V(ε; {fi , gi }N i=1 ) to be the set of all operators T satisfying

|(T − R)fi , gi )| < ε,

i ∈ {1, . . . , N }.

Then V is a weak neighborhood of R and the family of all such sets V is a base of weak neighborhoods of R. Lr Lr Theorem 4.3. Suppose Ak = j=1 λj Jkα is defined on j=1 Wpk [0, 1] and arg λi 6= arg λj

(mod 2π),

1 6 i < j 6 r.

Let also R :=

r M

Rj ∈

j=1

M r

,

Wpk [0, 1]

(Rj f )(·) = cj f (·) + (rj ∗ f )(·),

j ∈ {1, . . . , r}.

j=1

Then the following are true: (1) if 1 6 α 6 k − 1, then n Alg Ak = R : c1 = · · · = cr ∈ C; rj ∈ Wpk−1 [0, 1], (αm−1)

(αm−1)

m (0) = (λj λ−1 1 ) r1

hk − 1i

, 1 6 j 6 r; (4.7) o (l) rj (0) = 0, l 6= αm − 1, m 6 [(k − 1)/α], 1 6 j 6 r ; rj

(0), m 6

α

(2) if 2 6 k 6 α + p1 , then k−1 Alg Ak = R : c1 = · · · = cr ∈ C; rj ∈ Wp,0 [0, 1],

16j6r .

198

Domanov and Malamud

IEOT

α Proof. (i) Theorem 2.19 and Corollary 3.2 imply that O ⊕ · · · ⊕ O ⊕ λj Jk,0 ⊕ Lr α O ⊕ · · · ⊕ O ∈ Alg( j=1 λj Jk,0 ) for any j ∈ {1, . . . , r}. It easily implies that Mj := O ⊕ · · · ⊕ O ⊕ (λj Jkα )k+1 ⊕ O ⊕ · · · ⊕ O ∈ Ak Alg Ak . Thus Mj ∈ Alg Ak and (1.12) implies that if either α ∈ Z+ \ {0} or α > k − p1 , then k−1 Alg Ak ⊃ R : c1 = · · · = cr ∈ C; rj ∈ Wp,0 [0, 1], 1 6 j 6 r . (4.8)

(ii) Let 2 6 k 6 α + p1 . Then combining the obvious inclusion Alg Ak ⊂ α j=1 Alg λj Jk with (1.12) we arrive at opposite inclusion in (4.8). Thus, (2) is proved. (iii) L Let us prove the inclusion “⊂” in (4.7). Description (1.12) and inclusion r Alg Ak ⊂ j=1 Alg λj Jkα imply that Alg Ak ⊂ R : cj ∈ C; rj ∈ Wpk−1 [0, 1], (l) rj (0) = 0, l 6= αm − 1, m 6 [(k − 1)/α], 1 6 j 6 r . Lr

For j ∈ {1, . . . , r} and m ∈ {1, . . . , [ k−1 α ]} by definition, put xjm := 0 ⊕ · · · ⊕ 0 ⊕ 1 ⊕ 0 ⊕ · · · ⊕ 0, | {z } j

xαm yjm := 0 ⊕ · · · ⊕ 0 ⊕ ⊕ 0 ⊕ · · · ⊕ 0. Γ(αm) | {z } j

Let R :=

Lr

j=1

Rj ∈ Alg Ak . Choose ε1 > 0 and put

ε := min |2−1 λm j ε1 | : 1 ≤ j ≤ r, 0 ≤ m ≤ k1

and

k1 :=

hk − 1i α

.

r,k1 1 k Next, choose vectors {xjm }r,k j,m=1 and {yjm }j,m=1 belonging to the spaces Wp [0, 1] and (Wpk [0, 1])∗ = Wpk0 [0, 1], respectively and define a weak neighborhood V := r,k1 1 V(ε; {xjm }r,k j,m=1 , {yjm }j,m=1 ) of R according to Definition 4.2. Then by definition PN of Alg Ak there exists a polynomial p(x) := l=0 al xl such that p(Ak ) belongs to the weak neighborhood V of R, p(Ak ) ∈ V, that is

|((R − p(Ak ))xjm , yjm )| < ε,

j ∈ {1, . . . , r},

m ∈ {0, . . . , k1 }.

It is clear that (4.9) is equivalent to the following system xαm Rj − p(λj Jkα ) 1, < ε, j ∈ {1, . . . , r}, Γ(αm)

(4.9)

m ∈ {0, . . . , k1 }.

After simple computations this systems reduces to the following one (αm−1) rj (0) ε |cj − a0 | < ε, − a j ∈ {1, . . . , r}, m ∈ {0, . . . , k1 }. m < m , m λj λj Finally, the triangle inequality implies that (αm−1) (αm−1) r (0) (0) rj < 2ε ≤ ε1 , |c1 − cj | < 2ε ≤ ε1 , 1 m − λm λ1 λm i j

m ∈ {0, . . . , k1 }.

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

199

Hence, cj = c1 , (αm−1)

rj

(αm−1)

m (0) = (λj λ−1 1 ) r1

(0),

m ∈ {1, . . . , k1 },

j ∈ {1, . . . , r}.

Thus, the inclusion “⊂” in (4.7) is proved. (iv) Let R belong to the algebra defined by the right side of (4.7). Since rj ∈ Wpk−1 [0, 1], it follows that ! k−2 k−2 X (i) xi X (i) xi rj (x) = rj,0 + rj,k−2 := rj (x) − rj (0) + rj (0) , j ∈ {1, . . . , r}. i! i! i=0 i=0 According to this decomposition we can write R = R0 + Rk−2 , where R0 = R1,0 ⊕ · · · ⊕ Rr,0 ,

Rk−2 = R1,k−2 ⊕ · · · ⊕ Rr,k−2 ,

and (Rj,0 f )(·) := (rj,0 ∗ f )(·),

(Rj,k−2 f )(·) := (rj,k−2 ∗ f )(·),

j ∈ {1, . . . , r}.

Furthermore, R0 ∈ Alg Ak by (4.8) and Rk−2 ∈ Alg Ak by (iii). Thus (1) is proved. Combining Theorems 4.3 and 4.1 we arrive at Lnj Ln j α k Theorem 4.4. Suppose Ak (j) := is defined on i=1 λji Jk L i=1 Wp [0, 1], j ∈ Lr L nj r k {1, . . . , r} and Ak := j=1 A(j) is defined on j=1 ( i=1 Wp [0, 1]). Let also λji = λj1 /sα ji , 1 = sj1 6 sj2 6 . . . 6 sjnj ,

1 6 j 6 r, 1 6 i 6 nj ,

arg λi1 6= arg λj1

1 6 i < j 6 r.

(mod 2π),

Let also R :=

nj r M M j=1 i=1

Rji ∈

M nj r M

Wpk [0, 1] , (Rji f )(·) = (rji ∗ f )(·), 1 6 j 6 r.

j=1 i=1

Then the following are true: (1) if 1 6 α 6 k − 1, then n Alg Ak = cI + R : c ∈ C; rj1 ∈ Wpk−1 [0, 1], 1 6 j 6 r; −1 rji (x) = s−1 ji rj1 (sji x), 1 6 j 6 r, 1 6 i 6 nj ; hk − 1i (αm−1) m (αm−1) rj1 (0) = (λj1 λ−1 (0), m 6 , 1 6 j 6 r; 11 ) r11 α hk − 1i o (l) rj1 (0) = 0, l 6= αm − 1, m 6 , 16j6r ; α (2) if 2 6 k 6 α + p1 , then k−1 Alg Ak = cI + R : c ∈ C; rj1 ∈ Wp,0 [0, 1], 1 6 j 6 r; −1 −1 rji (x) = sji rj1 (sji x), 1 6 j 6 r, 1 6 i 6 nj .

200

Domanov and Malamud

IEOT

Remark 4.5. In this paper we do not consider questions about the reflexivity of the operator Ak . Such results are contained in [17]. 4.2. The commutant {Ak }0 As in Section 2 we define operator La ∈ [Wpk [0, 1]] for a ∈ (0, 1] and La ∈ k [Wp,0 [0, 1], Wpk [0, 1]] for a ∈ (1, ∞) by  0 < a 6 1,  f((ax) −1 La : f (x) → g(x) = (4.10) 0, x ∈ [0, 1 − a ],  a > 1.  −1 f (ax − a + 1), x ∈ [1 − a , 1], Next we investigate solvability of the equation RJkα = cJkα R

(4.11)

Wpk [0, 1]

in the space X = and describe the set of its solutions. The following proposition plays a crucial role in the sequel. Its proof is based on Corollary 2.21 and uses some ideas from [16]. Proposition 4.6. Let c ∈ C and let R ∈ [X] be a solution of equation (4.11) where X = Wpk [0, 1]. Then (1) if c 6∈ R+ , then R = 0; (2) if 0 < c = aα 6 1, a > 0, then R ∈ La {Jkα }0 = {Jkα }0 La , that is, Z x d (Rf )(x) = r(x − t)f (at) dt, r ∈ Wpk [0, 1]; dx 0 (3) if 1 < c = aα , a > 0, then R ∈ La {Jkα }0 , that is, d (Rf )(x) = La (r ∗ f ) (x) dx   0,  ax−a+1 R = −1 d k  r(ax − a + 1 − t)f (t) dt, r ∈ Wp,0 [0, 1], a dx

x ∈ [0, 1 − a−1 ], x ∈ [1 − a−1 , 1].

0

Proof. Let c ∈ C and RJkα = cJkα R. Consider the block matrix representations of the operators Jkα and R with respect to the direct sum decomposition Wpk [0, 1] = k k Wp,0 [0, 1] u Xk , where Xk := span{1, x, . . . , xk−1 }. Since Wp,0 [0, 1] ∈ Lat Jkα , one has α α J11 J12 R11 R12 Jkα = , R = . α O J22 R21 R22 Now the equality RJkα = cJkα R splits into

α R21 J12 α R11 J12

+ +

α α α R11 J11 = cJ11 R11 + cJ12 R21 ,

(4.12)

α R21 J11 α R22 J22 α R12 J22

(4.13)

= = =

α cJ22 R21 , α cJ22 R22 , α cJ11 R12 +

α cJ12 R22 .

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

201

α αk It is clear that J22 is a nilpotent operator on Xk and consequently J22 = 0. αk αk Therefore one derives from (4.13) that R21 J11 = cJ22 R21 = O. It follows that αk k R21 = O since ran J11 is dense in Wp,0 [0, 1]. Now equation (4.12) takes the form α α α α R11 J11 = cJ11 R11 , that is, R11 intertwines the operators J11 and cJ11 . (1) Let c 6∈ R+ . Then Corollary 2.21 (i) yields R11 = O. Furthermore, since k J αk xm ∈ Wp,0 [0, 1], m ∈ {0, . . . , k − 1}, one has

0 = R11 Jkαk xm = RJkαk xm = cJkαk Rxm . It follows that Rxm = 0 for m ∈ {0, . . . , k − 1}, hence R = O. (2) Let 0 0. Then Corollary 2.21 (ii) yields x d (R11 f )(x) = dx r(x − t)f (at) dt, where r ∈ Lp0 [0, 1]. Let us prove that r ∈ 0 k Wp [0, 1]. We have aαk (Jkαk R1)(x) = (RJkαk 1)(x) = (R11 Jkαk 1)(x) Z x d (at)αk = r(x − t) dt = aαk (J αk r)(x). dx 0 Γ(αk + 1) Hence r = R1 ∈ Wpk [0, 1]. k So, the operator R11 defined on Wp,0 [0, 1] admits a continuation T as an k operator defined on Wp [0, 1] by

T : Wpk [0, 1] → Wpk [0, 1],

T : f (x) →

d dx

Z

x

r(x − t)f (at) dt. 0

k k k Since T Wp,0 [0, 1] = R Wp,0 [0, 1] = R11 and Jkαk xm ∈ Wp,0 [0, 1] for m ∈ {0, . . . , k − 1}, we obtain

Jkαk T xm = a−αk T Jkαk xm = a−αk RJkαk xm = Jkαk Rxm . It follows that T xm = Rxm for m ∈ {0, . . . , k − 1}. Thus R = T . (3) Since c = aα > 1, Corollary 2.21 (ii) yields d (R11 f )(x) = La (r ∗ f ) (x) dx   x ∈ [0, 1 − a−1 ], 0, ax−a+1 R = −1 d  r(ax − a + 1 − t)f (t) dt, x ∈ [1 − a−1 , 1], a dx 0

202

Domanov and Malamud

IEOT

k where r ∈ Lp0 [0, 1]. Let us prove that r ∈ Wp,0 [0, 1].

aαk (Jkαk R1)(x) = (RJkαk 1)(x) = (R11 Jkαk 1)(x)   x ∈ [0, 1 − a−1 ], 0, ax−a+1 R = tαk −1 d  r(ax − a + 1 − t) Γ(αk+1) dt, x ∈ [1 − a−1 , 1], a dx 0

( 0, = d a−1 dx (J αk+1 r)(ax − a + 1), ( 0, = (J αk r)(ax − a + 1),

x ∈ [0, 1 − a−1 ], x ∈ [1 − a−1 , 1], x ∈ [0, 1 − a−1 ], x ∈ [1 − a−1 , 1].

Hence ( 0, x ∈ [0, 1 − a−1 ], (R1)(x) = r(ax − a + 1), x ∈ [1 − a−1 , 1]. k Since R1 ∈ Wpk [0, 1], it follows that r ∈ Wp,0 [0, 1]. k So, the operator R11 defined on Wp,0 [0, 1] admits a continuation T on Wpk [0, 1] defined by   x ∈ [0, 1 − a−1 ], 0, ax−a+1 R (T f )(x) = −1 d  r(ax − a + 1 − t)f (t) dt, x ∈ [1 − a−1 , 1]. a dx 0

k k k Since T Wp,0 [0, 1] = R Wp,0 [0, 1] = R11 and J αk xm ∈ Wp,0 [0, 1] for m ∈ {0, . . . , k − 1}, one deduces

Jkαk T xm = a−αk T Jkαk xm = a−αk RJkαk xm = Jkαk Rxm . It follows that T xm = Rxm for m ∈ {0, . . . , k − 1}. Thus R = T .

Corollary 4.7. [16, Theorem 3.4] R ∈ {Jkα }0 if and only if Z x Z x d (Rf )(x) = r(x − t)f (t) dt = r(0)f (x) + r0 (x − t)f (t) dt, r ∈ Wpk [0, 1]. dx 0 0 Ln Ln α Theorem 4.8. Suppose Ak = i=1 λi Jk is defined on X (n) = i=1 Wpk [0, 1] and λi = λ1 /sα i ,

1 = s1 6 s2 6 . . . 6 sn ,

aij = s−1 i sj ,

1 6 i, j 6 n.

0

Then the commutant {Ak } is of the form {Ak }0 = {R : R = (Rij )ni,j=1 , Rij = Laij Kij }, where d (Kij f )(x) = dx

Zx kij (x − t)f (t) dt, 0

( Wpk [0, 1], kij ∈ k Wp,0 [0, 1],

aij 6 1, aij > 1.

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

203

Proof. Let R = (Rij )ni,j=1 be the block matrix partition of the operator R with Ln respect to the direct sum decomposition X (n) = i=1 Wpk [0, 1]. Then the equality RAk = Ak R is equivalent to the following system −1 α α α α α Rij Jkα = λi λ−1 j Jk Rij = (si sj ) Jk Rij = aij Jk Rij ,

1 6 i, j 6 n.

To complete the proof it remains to apply Proposition 4.6. Lr L r Theorem 4.9. Suppose Ak = j=1 λj Jkα is defined on j=1 Wpk [0, 1] and arg λi 6= arg λj (mod 2π) for 1 6 i < j 6 r. Then the commutant {Ak }0 splits, that is, {Ak }0 =

r M

{λj Jkα }0 .

j=1

Proof. Following the proof of Theorem 4.8, one arrives at the relations α Rij Jkα = λi λ−1 j Jk Rij ,

1 6 i, j 6 r.

(4.14)

{Jkα }0

The latter results with i = j yield Rii ∈ for i ∈ {1, . . . , r}, hence by Proposition 4.6 (2) Z x d Rii : f → pii (x − t)f (t) dt, rii ∈ Wpk [0, 1], i ∈ {1, . . . , r}. dx 0 Since arg λi 6= arg λj (mod 2π) (1 6 i < j 6 r), it follows that λi λ−1 j 6∈ R+ , hence by Proposition 4.6 (1) Rij = 0 (1 6 i 6= j 6 r). This completes the proof. Combining Theorems 4.8 and 4.9, we arrive at L nj Ln j α Theorem 4.10. Suppose Ak (j) := on Wpk [0, 1] j ∈ i=1 λji Jk is defined Lr Lr Lni=1 j k {1, . . . , r} and Ak := i=1 Wp [0, 1]). Let j=1 A(j) is defined on W = j=1 ( also arg λj1 = arg λji

(mod 2π),

1 6 j 6 r,

arg λi1 6= arg λj1

(mod 2π),

1 6 i < j 6 r.

Then {Ak }0 =

r M

1 6 i 6 nj ,

{Ak (j)}0 ,

j=1

where the algebras {Ak (j)}0 are described in Theorem 4.8. 4.3. The double commutant {Ak }00 Ln Ln Theorem 4.11. Suppose Ak = i=1 λi Jkα is defined on W = i=1 Wpk [0, 1] and λi = λ1 /sα i ,

1 = s1 6 s2 6 . . . 6 sn ,

aij = s−1 i sj ,

1 6 i, j 6 n.

Then (1) {Ak }00 = cI + R : c ∈ C, R = diag(R1 , . . . , Rn ), (Ri f )(·) = (ri ∗ f )(·), −1 k−1 ri (x) = s−1 [0, 1], 1 6 i 6 n . i r1 (si x), ri ∈ Wp

204

Domanov and Malamud

IEOT

(2) The dimension dk,α of the quotient space {Ak }00 / Alg Ak is dk,α = k − 1 − [(k − 1)/α]. In particular, Alg Ak = {Ak }00 if and only if either α = 1 or k = 1. Proof. Let us set Eij := eTi ej ,

ei := (0, . . . , 0, 1, 0, . . . , 0), | {z }

1 6 i, j 6 n.

i

Then Theorem 4.8 implies {Ak }0 = Alg Jk ⊗ Eii , 1 6 i 6 n; Laij ⊗ Eij , 1 6 j 6 i 6 n; Laij Jkk ⊗ Eij , 1 6 i < j 6 n . Ln Ln Since { i=1 λi Jkα }00 ⊂ i=1 {λi Jkα }00 , it follows from (1.11) {Ak }00 ⊂ T := (c1 I + R1 ) ⊕ · · · ⊕(cn I + Rn ) : ci ∈ C, (Ri f )(·) = (ri ∗ f )(·), ri ∈ Wpk−1 [0, 1] . It is clear that T (Jk ⊗Eii ) = (Jk ⊗Eii )T for i ∈ {1, . . . , n}. It can easily be checked that T (Laij ⊗ Eij ) = (Laij ⊗ Eij )T, T (Laij Jkk

⊗ Eij ) =

(Laij Jkk

⊗ Eij )T,

1 6 j 6 i 6 n,

(4.15)

16i<j6n

(4.16)

−1 if and only if c1 = · · · = cn and ri (x) = s−1 i r1 (si x) for 1 6 i 6 n. Indeed, (4.15) and (4.16) are equivalent to the first and the second of the following relations

(cj I + Rj )Laij f = Laij (ci I + Ri )f, (cj I +

Rj )Laij Jkk f

=

Laij Jkk (ci I

+ Ri )f,

f ∈ Wpk [0, 1],

1 6 j 6 i 6 n,

Wpk [0, 1],

1 6 i < j 6 n,

f∈

respectively. According to the definition of Laij (see (4.10)), we obtain a Zij x

Zx rj (x − t)f (aij t) dt = ci f (aij x) +

cj f (aij x) + 0

for f ∈

Wpk [0, 1],

cj (Jkk f )(aij x

ri (aij x − t)f (t) dt 0

x ∈ [0, 1] and 1 6 j 6 i 6 n, and Zx − aij + 1) +

rj (x − t)(Jkk f )(aij t − aij + 1) dt

1−a−1 ij

= ci (Jkk f )(aij x − aij + 1) +

aij x−a Z ij +1

ri (aij x − aij + 1)(Jkk f )(t) dt

0

for f ∈

Wpk [0, 1],

(4.17)

x ∈ [1 −

a−1 ij , 1]

and 1 6 j < i 6 n.

(4.18)

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

205

After simple computations with (4.17)-(4.18), we get Zx

rj (x − t) − aij ri (aij (x − t)) f (aij t) dt = (ci − cj )f (aij x),

0

Zx

k −1 k ri (x − t) − a−1 ij rj (aij (x − t)) (Jk f )(t) dt = (cj − ci )(Jk f )(x).

0

Now it is easy to see that any of the latter equations is equivalent to c1 = · · · = cn −1 and ri (x) = s−1 i r1 (si x) for i ∈ {1, . . . , n}. Thus, (1) is proved. l k−1 (2) It is clear that Wpk−1 [0, 1] ≈ Wp,0 [0, 1] u span{ xl! : l = 1, . . . , k − 2}. Hence (1) implies that n xl o k−1 {Ak }00 ≈ C1 u Wpk−1 [0, 1] ≈ C1 u Wp,0 [0, 1] u span : l = 0, . . . , k − 2 . (4.19) l! Further, Theorem 4.1 yields Ak is isomorphic αm−1 x k−1 k−1 1 Alg Ak ≈ C u Wp,0 [0, 1] u span :16m6 . (4.20) (αm − 1)! α Combining (4.19) with (4.20) we easily arrive at (2). Lr L r Theorem 4.12. Suppose Ak = j=1 λj Jkα is defined on j=1 Wpk [0, 1] and arg λi 6= arg λj (mod 2π) for 1 6 i < j 6 r. Then Lr (1) {Ak }00 = j=1 {Jkα }00 . (2) The dimension dk,α of the quotient space {Ak }00 / Alg Ak is dk,α = rk − 1 − [(k − 1)/α]. In particular, Alg Ak = {Ak }00 if and only if either (a) r = 1 and α = 1, or (b) r = 1 and k = 1. Proof. (1) is implied by Theorem 4.9. Furthermore, (1) and Theorem (4.3) imply that r M {Ak }00 ≈ (C1 ⊕ Wpk−1 [0, 1]) j=1 r M

r M

n xl o span : l = 0, . . . , k − 2 l! j=1 j=1 αm−1 r M x k−1 k−1 Alg Ak ≈ C1 u Wp,0 [0, 1] u span :16m6 . (αm − 1)! α j=1 ≈ Cr u

k−1 Wp,0 [0, 1] u

Now it is easy to see that dk,α = r + r(k − 1) − 1 − (2) is proved.

k−1

Combining Theorems 4.11 and 4.12, we obtain

α

= rk − 1 −

k−1 α . Thus

206

Domanov and Malamud

IEOT

Theorem 4.13. Under the conditions of Theorem 4.10, we have r M {Ak }00 = {Ak (j)}00 , j=1 00

where the algebras {Ak (j)} are described in Theorem 4.11. Remark 4.14. Recall that according to celebrated von Neumann theorem {T }00 = Alg T whenever T is a normal operator. B. Sz.-Nagy and C. Foias [33]-[34] generalized this result to the wide class of accretive (disssipative) operators. In particular, this result holds for the accretive operator A = J ⊗ B defined on L2 [0, 1] ⊗ Cn , where B is a diagonal positive matrix, B = B ∗ > 0. By Theorem 4.1 this result also valid for non-accretive operator T := Ak = Jkα ⊗ B defined on Ln remains k j=1 W2 [0, 1], with the same B. 4.4. Invariant subspaces In [16] we proved that every subspace invariant under Jkα belongs either to the “continuous chain” Latc Jkα or to the “discrete chain” Latd Jkα . It turns out that Latc Jkα does not depend on α: Latc Jkα = Latc Jk (see (1.9)). We proved also that the description of Latd Jkα easily follows from that of Lat J(0, k)α . This description is extracted from Theorem 2.1. In this section we prove that every Ak -invariant subspace can be decomposed into a direct sum of two invariant subspaces: the first one belongs to the “continuous part” of Lat Ak and the second one belongs to the “discrete part” of Lat Ak . We show also that the “continuous part” does not depend on α. Moreover, a description of the “discrete part” is deduced from Theorem 2.1. Let χs stand for the characteristic function of an arbitrary nonempty subset cS the canonical projections from S ⊂ Zn := {1, . . . , n}. We denote by PS and P Ln Ln L Ln kj k n kj onto j=1 χs (j)Wp j [0, 1] and j=1 χs (j)C kj , j=1 Wp [0, 1] and from j=1 C respectively. Next we let n n M M cS [ Ak,S := χs (j)λj Jkαj ran PS , A χs (j)λj J(0; kj )α ran P k,S := j=1

j=1

cS . and denote by πS the quotient mapping from ran PS onto ran P Ln L k n Theorem 4.15. Suppose that Ak = j=1 λj Jkαj is defined on j=1 Wp j [0, 1] and arg λi 6= arg λj (mod 2π) for 1 6 i < j 6 n. Then E ∈ Lat Ak if and only if there exists S ⊂ Zn and a1 , . . . , an ∈ [0, 1] such that E = Lat Ak,S

n M

k

χsc (j)Eajj ,0 ,

j=1

where Lat Ak,S =

[ M

n o −1 d πS−1 [M, (Ad M ] : M ∈ Lat Ad k,S ) k,S Ak,S M

(4.21)

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

207

−1 and S c is the complement for S in Zn (S ∪ S c = Zn ). Here [M, (Ad M ] is a k,S ) c closed interval in the lattice of all subspaces of ran PS . Each interval satisfies the equation X −1 dim(Ad M − dim M = min −[−α], kj . (4.22) k,S ) j∈S k

Proof. For every E ∈ Lat Ak , we put j in S := SE if Pj E 6⊂ Wp,0j [0, 1] and put j in S c otherwise. Next we introduce the subspaces ES := span{Am k,S PS E : m > 0} Ln kj m and ES c := span{Ak,S c PS c E : m > 0} ⊂ j=1 χsc (j)Wp,0 [0, 1]. It is clear that E ⊂ ES ⊕ ES c . Let M = max16j6n kj . Then the subspace F := AM k E is invariant for the Ln Ln k k operator Ak,0 := Ak j=1 Wp,0j [0, 1] and, by Theorem 2.19, F = j=1 Eajj ,0 for some aj ∈ [0, 1]. By the construction of S, it is clear that aj = 0 for j ∈ S and hence ! ! n n M M kj kj F = χs (j)Wp,0 [0, 1] ∪ χsc (j)Eaj ,0 . (4.23) j=1

j=1

It is clear that E ⊃ F ⊃ ES c . Hence E ⊃ PS c E and, therefore, E ⊃ PS E. The latter inclusion yields E ⊃ ES and consequently E splits : E = ES ⊕ ES c . Ln k In turn, by Theorem 2.19, ES c splits: ES c = j=1 χsc (j)Eajj ,0 . On the other hand, combining (4.23) with the relations E = ES ⊕ ES c ⊃ F , one gets ES ⊃ Ln kj c j=1 χs (j)Wp,0 [0, 1]. Therefore, πS (ES ) ∈ Lat AS . The quotient map πS estabL k lishes a bijective correspondence between ES ∈ Lat AS with ES ⊃ j∈S Wp,0j [0, 1] and πS (ES ). Hence one derives ES = πS−1 (πS ES ). One completes the proof by applying Theorem 2.1. Furthermore, relations (4.21) and (4.22) are implied by the relations (2.1) and (2.2), respectively. Corollary 4.16. [16] Let π be the quotient map k π : Wpk [0, 1] → Xk := Wpk [0, 1]/Wp,0 [0, 1] c α d α α α and Jc k be the quotient operator on Xk . Then Lat Jk = Lat Jk ∪ Lat Jk , where (a) k k k Latc Jkα = Ea,0 : 0 6 a 6 1 , Ea,0 := f ∈ Wp,0 [0, 1] : f (x) = 0, x ∈ [0, a]

is the “continuous part” of Lat Jkα ; (b) α Latd Jkα = π −1 (Lat Jc k)=

[

n o α −1 M ] : M ∈ Lat(J cα Jcα M ) π −1 [M, (Jc k) k k

M α −1 M ] is a closed interval in is the “discrete part” of Lat Jkα . Here [M, (Jc k) the lattice of all subspaces of Xk . Each interval satisfies the equation α −1 M − dim M = d, dim(Jc k)

where d = min{−[−α], k}.

208

Domanov and Malamud

IEOT

Corollary 4.17. [16] The operator Jkα is unicellular if and only if either α = 1 or k = 1. Example. Suppose that the operator A = λ1 Jkα1 ⊕λ2 Jkα2 (arg λ1 6= arg λ2 ) (mod 2π) is defined on Wpk1 [0, 1] ⊕ Wpk2 [0, 1]. By Theorem 4.15, one has the following description of its lattice of invariant subspaces: [ [ −1 k [ Lat A = (Eak11,0 ⊕ Eak22,0 ) ∪ π{1} (Lat A {1} ) ⊕ Ea,0 [a1 ,a2 ]∈[0,1]×[0,1]

[

∪

k Ea,0

⊕

a∈[0,1] −1 [ π{2} (Lat A {2} )

∪

[

−1 \ π{1,2} (Lat A {1,2} ),

a∈[0,1] −1 −1 where the lattices π{1} (Lat A{1} ) = Latd Jkα1 and π{2} (Lat A{2} ) = Latd Jkα2 are described in Corollary 4.16. For example, if k1 = 1, k2 = 2, λ1 = i, λ2 = 1 and d 1 −1 −1 1 1 [ [ α = 1, one has π{1} (Lat A {1} ) = Lat J1 = Wp,0 [0, 1] ∪ Wp [0, 1], π{2} (Lat A{2} ) = 2 \ Latd J21 = Wp,0 [0, 1] ∪ E12 ∪ Wp2 [0, 1]. It is easily seen that A {1,2} = 0 ⊕ J(0; 2), 3 \ \ hence, A {1,2} ran(A{1,2} ) : e3 → 0 (here {e1 , e2 , e3 } is the standard basis in C ). Thus, by Theorem 2.1, [ −1 \ \ Lat A [M, (A M ] = [0, {e1 , e3 }] ∪ [{e3 }, {e1 , e2 , e3 }] {1,2} = {1,2} ) M ⊂{e3 }

= {0} ∪

[ α,β∈C

≈ {0} ∪

[

[

{αe1 + βe3 } ∪

{αe1 + βe2 , e3 } ∪ {e1 , e2 , e3 }

α,β∈C

{(α, βx)} ∪

α,β∈C

[

{(α, β), (0, x)} ∪ {(1, 0), (0, 1), (0, x)}.

α,β∈C

Hence −1 1 2 \ π{1,2} ( Lat A {1,2} ) = Wp,0 [0, 1] ⊕ Wp,0 [0, 1] [ ∪ {f1 , f2 } : f1 ∈ Wp1 [0, 1], f2 ∈ E12 , αf1 (0) + βf20 (0) = 0 α,β∈C

∪

[

{f1 , f2 } : f1 ∈ Wp1 [0, 1], f2 ∈ Wp2 [0, 1], αf1 (0) + βf2 (0) = 0

α,β∈C

∪ Wp1 [0, 1] ⊕ Wp2 [0, 1] . Remark 4.18. (i) An alternative description of Latd Jkα might be obtained from the Halmos description of Lat T for T ∈ [Cn ] (see Theorem 2.2). (ii) A quite different proof of the description of Lat Jk has been originally obtained by E.Tsekanovskii [46]. 4.5. Hyperinvariant subspaces To present a description of HypLat Ak we keep the notation from Subsection 4.4.

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

209

Theorem 4.19. Let the conditions of Theorem 4.8 hold. Then [ HypLat Ak = {ES c ⊕ ES }. S⊂Zn

Here (a) the “continuous part” ES c is of the form M n ES c = χS c (j)Eakj ,0 : a = {aj }j∈S c ∈ P ({sj }j∈S c ) , j=1

where n P ({si }i∈S c ) := P (sn1 , . . . , sn|Sc | ) = (an1 , . . . , an|Sc | ) ∈ |S c | : o snj anj+1 6 snj+1 anj 6 snj+1 − snj + snj anj+1 , 1 6 j 6 |S c | − 1 . Ln (b) the “discrete part” ES is of the form ES = j=1 χS (j)Elkj , where 1 6 lj 6 k − 1 and lj 6 li if sj 6 si for 1 6 i, j 6 n; In particular, if λ1 = · · · = λn , then ( n ) n [ M M k k HypLat Ak = χs (j)El χsc (j)Ea,0 . S⊂Zn , a∈[0,1], 16l6k−1

j=1

i=j

Ln Ln Proof. It is clear that HypLat Ak = HypLat( j=1 λj Jkα ) ⊂ j=1 HypLat λj Jkα = Ln Ln j=1 Lat λj Jk . Hence if E ∈ HypLat Ak then E = j=1 Ej , where Ej ∈ Lat Jk . k For each Ej ∈ Lat Jk (1 6 j 6 n) we put j in S if Ej ∈ Latd Jk \Wp,0 [0, 1] c c c , where and put j in S otherwise (i.e., if E ∈ Lat J ). Thus E = E ⊕ E j k S S Ln Ln ES c = j=1 χS (j)Eakj ,0 and ES = j=1 χS c (j)Elkj . Now ES c is described in TheLn orem 2.11 and Corollary 3.2. Let us prove that ES = j=1 χS c (j)Elkj ∈ HypLat Ak if and only if lj 6 li whenever sj 6 si for 1 6 i, j 6 n. Let sj 6 si and P ∈ {Ak }0 be such that the block L matrix partition of the n operator P with respect to the direct sum decomposition j=1 Wpk [0, 1] contains the only non-zero element Pij := Laij . Then the inclusion P ES ⊂ ES yields Elj = Pij Elj ⊂ Eli . So sj 6 si yields Elj ⊂ Eli or lj 6 li . The opposite statement may be obtained using routine matrix calculations, which we omit. Theorem 4.20. Under the conditions of Theorem 4.9, the lattice HypLat Ak splits: n n M M HypLat Ak = HypLat λj Jkα = Lat Jk . j=1

j=1

Remark 4.21. It is well known (see [11]) that for two bounded operators T1 and T2 the splitting of Lat(T1 ⊕T2 ) implies the splitting of HypLat(T1 ⊕T2 ). In other words, the relation Lat(T1 ⊕ T2 ) = Lat T1 ⊕ Lat T2 yields the relation HypLat(T1 ⊕ T2 ) = HypLat T1 ⊕ HypLat T2 . Theorem 4.20 demonstrates that the converse implication

210

Domanov and Malamud

IEOT

is not true. Nevertheless the converse implication is true for C0 contractions T1 and T2 defined on Hilbert space ([11]). Summing up Theorems 4.19 and 4.20, we obtain Theorem 4.22. Under the conditions of Theorem 4.10, we have r M HypLat Ak = HypLat Ak (j), j=1

where the lattices HypLat Ak (j) are described in Corollary 4.19. 4.6. Cyclic subspaces Some results of this subsection were announced in [13]. First, we present the following simple Lemma 4.23. Let A ∈ [Ck ], σ(A) = {0} and Pker A∗ be the orthoprojection from Ck onto ker A∗ . Then (1) µA = disc A = dim(ker A∗ ) = dim(ker A); (2) E ∈ Cyc A if and only if P E = ker A∗ . Proof. Necessity. Note that span{E, ran A} ⊃ span{Aj E : j > 0} and (Ik − Pker A∗ )E = Pran A E ⊂ ran A. Therefore, since E ∈ Cyc A, we have Ck = span{Aj E : j = 0, 1, . . . , k − 1} ⊂ span{Pker A∗ E, (Ik − Pker A∗ )E, ran A} = span{Pker A∗ E, ran A} ⊂ span{ker A∗ , ran A} = ker A∗ ⊕ ran A = Ck . Hence Pker A∗ E = ker A∗ . Sufficiency. Let P E = ker A∗ . Then Ck = span{Pker A∗ E, ran A} ⊂ span{E, (Ik − Pker A∗ )E, ran A} = span{E, ran A}. Applying the operator Aj , we obtain ran Aj = span{Aj E, ran Aj+1 } (1 6 j 6 k − 1). Hence Ck = span{E, ran A} = span{E, AE, ran A2 } = · · · = span{E, . . . , Ak−1 E}. It means E ∈ Cyc A.

− N − → → For every system φ = { φl }1 , φl ∈ Cn , we denote by W (φ) the n × N matrix → − − → −→ consisting of the columns φl : W (φ) = (φ1 , . . . , φN ). Ln Ln kj Corollary 4.24. Suppose that A = j=1 λj J(0; kj )α is defined on and j=1 C mj := min(−[−α], kj ) for 1 6 j 6 n. Then Pn (1) µA = disc A = j=1 mj ; (2) the following system → − φl = col(φl11 , . . . , φl1k1 , φl21 , . . . , φl2k2 , . . . . . . φln1 , . . . , φlnkn ), 16l6N generates aPcyclic subspace for the operator A if and only if n (i) N > j=1 mj ;

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

211

(ii) P the matrix W0 = Pker A∗ W (φ) is of maximal rank, that is, rank W0 = n j=1 mj . Ln Ln k Theorem 4.25. Suppose Ak = j=1 λj Jkαj is defined on j=1 Wp j [0, 1] and mj := min(−[−α], kj ) for 1 6 j 6 n. Then Pn (1) µAk = disc Ak = j=1 mj ; (2) the system {fl (x)}N l=1 of vectors fl (x) = {fl1 (x), . . . , fln (x)} generates a cyclic subspace for A k if and only if the following conditions hold: Pn (i) N > j=1 mj ; (ii) the matrix   f11 (0) f21 (0) ... fN 1 (0) 0 0 0   f11 (0) f21 (0) ... fN 1 (0)     . . . .. .. ..     (m1 −1) (m1 −1)  (m1 −1)  (0) f21 (0) . . . fN 1 (0)   f11   .. .. ..  W (0) =  . . .     f2n (0) ... fN n (0)   f1n (0)   0 0 0 (0) f2n (0) ... fN  f1n n (0)    .. .. ..   . . .   (mn −1) (mn −1) (mn −1) f1n (0) f2n (0) . . . fN n (0) Pn is of maximal rank, i.e., rank W (0) = j=1 mj . ck . To prove the converse Proof. It is clear that E ∈ Cyc Ak implies πE ∈ Cyc A Ln kj ck and assertion we choose a subspace E ⊂ j=1 Wp [0, 1] such that πE ∈ Cyc A L n denote by F := span{Aj E : j > 0}. Since πF = j=1 Ckj , one gets that F ⊃ Ln kj j=1 Wp,0 [0, 1]. Therefore, just in the same way as in Theorem 4.15, we obtain Ln Ln k that F = π −1 (πF ) = π −1 ( j=1 Ckj ) = j=1 Wp j [0, 1], that is, E ∈ Cyc Ak . To complete the proof it suffices to apply Corollary 4.24. Remark Ln4.26. For α = 1 and k1 = · · · = kn =: k > 1, that is, for the operator Ak = j=1 λj Jk , Theorem 4.25 has been established in [12] by another method. We emphasize that the description of the set Cyc Ak,0 essentially differs from that of Cyc Ak . Namely, in contrast to the operator Ak,0 , the description of the set Cyc Ak does not depend on the choice of λj . A= Ln

Summing up, we obtain a description of the cyclic subspaces for the operator Lm Ln Lm kj α α j=1 λj Jkj ⊕ j=m+1 λj Jkj ,0 acting on the mixed space j=1 Wp [0, 1] ⊕

j=m+1

k

Wp,0j [0, 1].

Theorem 4.27. Suppose that the operators m m M M Ak (1) := λj Jkαj , Ak,0 (1) := λj Jkαj ,0 j=1

j=1

212

Domanov and Malamud

and

n M

Ak,0 (2) :=

IEOT

λj Jkαj ,0 and A := Ak (1) ⊕ Ak,0 (2)

j=m+1

are defined on X(1) :=

m M

Wpkj [0, 1], X0 (1) :=

j=1

and X0 (2) :=

n M

m M

k

Wp,0j [0, 1]

j=1

k

Wp,0j [0, 1] and X := X(1) ⊕ X0 (2),

j=m+1

respectively. Furthermore, let P (1) be the canonical projection from X = X(1) ⊕ X0 (2) onto X(1). Then (1) µA = max{µAk (1) , µAk,0 (1)⊕Ak,0 (2) }; (2) E ∈ Cyc A if and only if (a) P (1)E ∈ Cyc Ak (1), (b) AM E ∈ Cyc(Ak,0 (1) ⊕ Ak,0 (2)), where M := max16j6m kj . Furthermore, the set Cyc(Ak,0 (1) ⊕ Ak,0 (2)) is described in Theorems 2.24 and 2.16 and the set Cyc Ak (1) is described in Theorem 4.25. We express our gratitude to Professor Pei Yuan Wu for giving precise references concerning Theorem 2.2. We are also grateful to the referee for a number of helpful suggestions for improvement in the article.

References [1] A. Atzmon, Unicellular and non-unicellular dissipative operators, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 45–54. [2] H. Bercovici, On the Jordan model of C0 operators II, Acta Sci. Math. (Szeged) 42 (1980), no. 1-2, 43–56. [3] H. Bercovici, Operator theory and arithmetic in H ∞ , Mathematical Surveys and Monographs, 26, Amer. Math. Soc., Providence, RI, 1988. [4] S. Bermudo, A. Montes Rodriguez, S.A. Shkarin, Orbits of Operators Commuting With the Volterra Operator, Journal de Math´ematiques Pures Et Appliquees. 89 (2008), no. 2, 145–173. [5] A. Biswas, A. Lambert, A. Petrovic, Extended eigenvalues and the Volterra operator, Glasgow Math. J. 44 (2002), 521–534. [6] A. Biswas, S. Petrovic, On Extended Eigenvalues of Operators, Integr. Equ. Oper. Theory 55 (2006), no. 2, 233–248. [7] L. Brickman, P.A. Fillmore, The invariant subspace lattice of a linear transformation, Canad. J. Math. 19 (1967), 810–822. [8] P.S. Bourdon, J.H. Shapiro, Intertwining relations and extended eigenvalues for analytic Toeplitz operators, eprint arXiv:0801.1972.

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

213

[9] M.S. Brodskii, Triangular and Jordan Representations of Linear Operators, Translational Mathematical Monographs, 32, AMS, Providence, RI, 1971. [10] J.B. Conway, G. Prajitura, On λ-commuting operators, Studia Math. 166 (2005), 1–9. [11] J.B. Conway, P.Y. Wu, The splitting of α(T1 ⊕ T2 ) and related questions, Indiana Univ. Math. J. 26 (1977), no. 1, 41–56. [12] I.Yu. Domanov, On Cyclic and Invariant Subspaces of an Operator J ⊗ B in the Sobolev Spaces of Vector-Functions, Methods Funct. Anal. Topology 5 (1999), no. 1, 1–12. [13] I.Yu. Domanov, On the spectral multiplicity of some Volterra operators in Sobolev spaces, Math. Notes 72 (2002), no. 2, 275–280. [14] I.Yu.RDomanov, On cyclic subspaces and the unicellularity of the operator (V f )(x) = x q(x) 0 w(t)f (t) dt, Ukr. Math. Bull. 1 (2004), no. 2, 177–219. [15] I.Yu. Domanov, M.M. Malamud, Invariant and Hyperinvariant Subspace Lattices of Operator J α ⊗ B in Sobolev Spaces, Math. Notes 70 (2001), no. 3, 508–514. [16] I.Yu. Domanov, M.M. Malamud, Invariant and hyperinvariant subspaces of an operator J α and related operator algebras in sobolev spaces, Linear Algebra Appl. 348 (1-3) (2002), 209–230. α [17] I.Yu. Domanov, V.V. Surovtseva, On the reflexivity of the operator Jkα ⊕ Jk+s , Dopovidi NANU, (2004), no. 9, 26–30 (in Russian).

[18] J.A. Erdos, The commutant of the Volterra operator, Integr. Equ. Oper. Theory 5 (1982), 127–130. [19] C. Foias, J.P. Williams, Some Remarks on the Volterra Operator, Proc. Amer. Math. Soc. 31 (1972), no. 1, 177–184. [20] I.C. Gohberg, M.G. Krein, Theory and Applications of Volterra operators in Hilbert space, Translational Mathematical Monographs 24, AMS, Providence, RI, 1970. [21] I. Gohberg, P. Lancaster, L. Rodman, Invariant Subspaces of Matrices with Applications, Wiley-Interscience, New-York, 1986. [22] P.R. Halmos, Eigenvectors and Adjoints, Linear Algebra Appl. 4 (1971), 11–15. [23] L.T. Hill, Invariant subspaces of direct sum of finite convolution operators, Integr. Equ. Oper. Theory 6 (1983) 525–535. [24] G.K. Kalisch, Characterizations of direct sums and commuting sets of Volterra operators, Pacific J. Math. 18 (1966), 545–552. [25] V.V. Kapustin, A.V. Lipin, Operator algebras and lattices of invariant subspaces I, (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), Issled. Linein. Oper. Teorii Funktsii. 18, 23–56, 184; translation in J. Soviet Math. 61 (1992), no. 2, 1963–1981. [26] M.T. Karaev, Invariant subspaces, cyclic vectors, commutant and extended eigenvectors of some convolution operators, Methods of Functional Anal. Topol. 11 (2005), no. 1, 48–59. [27] M.M. Malamud, Remarks on the spectrum of one-dimensional perturbations of Volterra operators, (Russian), Matematicheskaya Fizika 32 (1982), 99–105.

214

Domanov and Malamud

IEOT

[28] M.M. Malamud, Similarity of Volterra operators and related questions of the theory of differential equations of fractional order, Trans. Moscow Math. Soc. 55 (1994), 57–122. [29] M.M. Malamud, On Reproducing Subspaces of Volterra Operators, Doklady Mathematics 54 (1996), no. 3, 901–906. [30] M.M. Malamud, Spectral Analysis of Volterra Operators and Inverse Problems for Systems of Ordinary Differential Equations, SFB 288, Differentialgeometrie und Quantenphysik, Preprint no. 269, 1997. [31] M.M. Malamud, Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators, Operator Theory: Adv. Appl., Integral Differential Oper. 102 (1998), 143–167. [32] M.M. Malamud, Uniqueness Questions in Inverse Problems for Systems of Differential Equations on a Finite Interval, Trans. Moscow Math. Soc. 60 (1999), 173–224. [33] B. Sz.-Nagy, C. Foias, Harmonic Analysis of operators on Hilbert space, Academiai Kiado, Budapest, 1970. [34] B. Sz.-Nagy, C. Foias, Mod` ele de Jordan pour une classe d’op´ erateurs de l’espace de Hilbert, Acta Sci. Math. (Szeged) 31 (1970), no. 1-2, 91–115. [35] M.S. Nikol’ski˘ı, On Systems of Linear Integral Equations of Volterra Type in Convolutions, Proc. of the Steklov Inst. of Math. 220 (1988), 210–216. [36] N.K. Nikol’ski˘ı, Treatise on the Shift Operator, Springer, Berlin, 1986. [37] N.K. Nikol’ski˘ı, V.I. Vasjunin, Control subspaces of minimal dimension, Elementary introduction, Discotheca. (Russian), Investigations on linear operators and the theory of functions, XI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 113 (1981), 41–75. [38] N.K. Nikol’ski˘ı, V.I. Vasjunin, Control subspaces of minimal dimension. Unitary and model operators, J. Operator Theory 10 (1983), no. 2, 307–330. [39] B.P. Osilenker, V.S. Shul’man, On lattices of invariant subspaces of some operators, Funct.analiz i ego prilozheniya 17 (1983), no. 1, 81–82. (in Russian) [40] B.P. Osilenker, V.S. Shul’man, Lattices of invariant subspaces of certain operators, (Russian), Studies in the theory of functions of several real variables, Yaroslav. Gos. Univ., Yaroslavl’ 156 (1984), 105–113. [41] P.V. Ostapenko, V.G. Tarasov, Unicellularity of the integration operator in certain function spaces, (Russian), Teor. Funkci˘ıFunkcional. Anal. i Priloˇzen. 27 (1977), 121– 128. [42] H. Radjavi, P. Rosenthal, Invariant subspaces, Springer, Berlin, 1973. [43] D. Sarason, Invariant subspaces and unstarred operator algebras, Pacif. J. Math. 17 (1966), no. 3, 511–517. [44] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967), 179–203. [45] S. Shkarin, Compact operators without extended eigenvalues, J. Math. Anal. Appl. 332 (2007), 445–462. [46] E.R. Tsekanovskii, On description of invariant subspaces and unicellularity of the (p) integration operator in the space W2 , Uspehi Mat. Nauk. 6 (126) (1965), 169–172. (in Russian)

Vol. 63 (2009)

On the Direct Sums of Riemann-Liouville Operators

215

[47] M.W. Wonham, Linear Multivariable Control, Springer, Berlin, 1974. [48] P.Y. Wu, On a conjecture of Sz.-Nagy and Foia¸s, Acta Sci. Math. (Szeged) 42 (1980), no. 3-4, 331–338. [49] P.Y. Wu, Which C· 0 -contraction is quasisimilar to its Jordan model? Acta Sci. Math. (Szeged) 47 (1984), no. 3-4, 449–455. I.Yu. Domanov Mathematical Institute, AS CR Zitna 25 CZ - 115 67, Praha 1 Czech Republic and Institute of Applied Mathematics and Mechanics Roza-Luxemburg 74 Donetsk 83114 Ukraine e-mail: [email protected] M.M. Malamud Institute of Applied Mathematics and Mechanics Roza-Luxemburg 74 Donetsk 83114 Ukraine e-mail: [email protected] Submitted: November 23, 2007. Revised: September 17, 2008.

Integr. equ. oper. theory 63 (2009), 217–225 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020217-9, published online February 2, 2009 DOI 10.1007/s00020-009-1658-1

Integral Equations and Operator Theory

A Note on Toeplitz Operators in Schatten-Herz Classes Associated with Rearrangement-invariant Spaces Liliana Gabriela Gheorghe Abstract. In recent papers, B. Choe, H. Koo, K. Na (see [3]) and Loaiza, M. Lopez-Garcia e S. Perez-Esteva (see [5]) studied conditions in order to a Toeplitz operator, acting on the harmonic Bergman space over the unit ball in Rn and on analytic Bergman space on the unit disk in the complex plane, respectively, belong to the so-called Schaten-Herz class. The purpose of this note is to prove necessary and sufficient conditions in order to a Toeplitz operator Tµ with positive symbol, acting on the harmonic F Bergman space on the unit ball in Rn belong to a Schatten-Herz class SE , associated with a pair of rearrangement invariant sequence spaces E and F. The conditions involve the Berezin transform µ e of its symbol and the average function µ ˆδ on some euclidian discs. The main point is the characterization of Toeplitz operators, that belong to Schatten ideals SE associated with an arbitrary rearrangement invariant sequence space E. Mathematics Subject Classification (2000). Primary 47B35, 47B10; Secondary 46E30, 46B10. Keywords. Toeplitz operators, Schatten ideals, Bergman kernel, Berezin transform, rearrangement-invariant space, interpolation.

1. Introduction Let b2 (B) be the space of all the harmonic functions on the open unit ball B in Rn that belongs to L2 (B), called the harmonic Bergman space. Let K(x, y) be the Bergman reproducing kernel for b2 (B) and dλ(x) = K(x, x)dV (x),

218

Gheorghe

IEOT

where dV is the Lebesgue measure on B be the analogue of the M¨obius-invariant measure on B. Let µ be a positive Borel measure on H; the Toeplitz operator of symbol µ defines as Z Tµ f (x) = K(x, y)f (y)dµ(x), f ∈ b2 (B), x ∈ B. B

In what concerns Toeplitz operators, basic proprieties in harmonic case are similar to those in the analytic case. For instance, the characterizations of boundedness, compactness and membership in Schatten ideals acting on harmonic Bergman space are completely analogous to those in the analytic setting. See e.g [2], for the harmonic case and [4], [7], for the analytic case. Recently, M. Loaiza, M. Lopez-Garcia e S. Perez-Esteva (see [5]) studied the so-called Schatten-Herz Toeplitz operators Hpq over the analytic Bergman space. In [3], B. Choe, H. Koo and K. Na solved a similar problem, in the harmonic setting. In this paper, we obtain necessary and sufficient conditions in order to a Toeplitz operator acting on the Bergman space b2 (B) be in the Schatten-Herz F class SE associated with a a pair of rearrangement-invariant spaces E and F. The b F and the Berezin-Herz conditions are given in terms of averaging-Herz classes H E,r e F associated with a pair of rearrangement invariant spaces E and F. In classes H E the classic case of E = lp and F = lq , we re-obtain the results in [3]. Our main result is the following (we postpone the notations and the definitions to the next section): Theorem 1.1 (see Theorem 4.4). Let E and F be two rearrangement invariant sequence spaces and let E(dλ) the corresponding rearrangement invariant function space over (B, dλ). Then there exists δ0 > 0 such that for all 0 < δ < δ0 and for all positive Borel measure µ on B, the following statements are equivalent: F 1) Tµ ∈ SE ; F e ; 2) µ ∈ H E bF . 3) µ ∈ H E,δ

This result give a new proof and extend Theorem 4.4 in [3]. The main ingredient of the description, in terms of Berezin transform and averaging function, of those Toeplitz operators that belong to Schatten ideal SE , associated to an arbitrary rearrangement invariant sequence space (see Theorem 3.2).

2. Some estimations on Berezin transform and averaging function Let µ be a positive Borel measure on B. The Berezin transform of µ is Z µ ˜(x) = |kx (y)|2 dµ(y), x ∈ B, B

Vol. 63 (2009)

Schatten-Herz Classes

219

where kx (y) is the L2 −normalised reproducing Bergman kernel for the harmonic Bergman space b2 (B). It is easy to see that µ ˜(x) =< Tµ kx , kx >,

x ∈ B,

where Tµ is the Toplitz operator of symbol µ. In particular, when µ = |f |dV, the Berezin transform of the measurable function f defines as Z ^ fe(x) := |f |dV (x) = |kx (y)|2 |f (y)|dV (y). B

Let 0 < δ < 1; the averaging function of µ is µ ˆδ (x) =

µ[E(x, δ)] , x ∈ B, δ > 0, |E(x, δ)|

where E(x, δ) is the euclidian disc centered at x and of radius δ(1 − |x|) : E(x, δ) = {y ∈ B : |x − y| < δ(1 − |x|)}, y ∈ B. Denote by r(x) = 1 − |x|, the distance from a point x ∈ B to the boundary of the unit ball. We shall list some known results and prove some of its consequences, that we shall need later on. The main ingredient of all this estimations is the following Covering Lemma, due to V.L. Oleinik (see [6]): Lemma 2.1 (see Lemma of Covering in [6]; see also Lemma 3.4 [2]). For all 0 < δ < 1, there exists a sequence {ai }i∈N satisfying the following conditions: S∞ 1) 1 E(ai , 9δ ) = B. 2) There exists a positive integer N (that neither depends on δ nor on {ai }i∈N ) such that for any k ∈ N, each disc E(ak , δ) intersects at most N of the disks {E(ai , δ)}i∈N . The next result is classic by now. Lemma 2.2 ( see e.g. Proposition 2.1, [3]). Let µ be a positive finite Borel measure on B. Then there exists δ0 > 0 such that for all 0 < δ < δ0 , there exists a constant C = Cδ > 0 such that µ cδ (x) ≤ C µ e(x),

∀x ∈ B.

Despite of probably being known, we did not find a precise reference for the following: Lemma 2.3. There exists 0 < δ0 < 1 such that for all 0 < δ < δ0 and for all f ∈ b2 (B), Z Z 2 |f (x)| dµ(x) ≤ |f (x)|2 µ cδ (x)dV (x). B

B

220

Gheorghe

IEOT

Proof. Let {ai }i as in the Lemma of Covering (Lemma 2.1); then Z XZ 2 |f (x)| dµ(x) ≤ |f (x)|2 dµ(x). B

i

E(ai , δ9 )

Fix i ≥ 1; we shall prove that there exists C > 0, that not depends on i, such that Z Z |f (x)|2 dµ(x) ≤ C |f (x)|2 µ bδ (x)dV (x), ∀f ∈ b2 (B). E(ai , δ9 )

E(ai , δ3 )

In fact, if f ∈ b2 (B), then |f |2 is subharmonic, hence Z

Z

2

|f (x)| dµ(x) ≤ E(ai , δ9 )

E(ai , δ9 )

1 δ |E(x, 27 )|

Z

|f (y)|2 dV (y)dµ(x).

δ E(x, 27 )

According to Lemma 3.1 in [2], for all 0 < δ < δ0 , r(x) ≈ r(ai ), ∀x ∈ E(ai , δ), with a constant that does not depend on i. In particular, |E(x,

δ δ δ )| ≈ |E(ai , )|, ∀x ∈ E(ai , ). 27 27 27

On the other hand, E(x,

δ δ δ ) ⊂ E(ai , ), ∀x ∈ E(ai , ). 27 3 9

All these estimates allow us to apply Fubini’s theorem and to switch the order of integration: Z

Z 1 |f (y)|2 dV (y)dµ(x) δ δ E(ai , δ9 ) |E(x, 27 )| E(x, 27 ) Z Z 1 ≤C |f (y)|2 dV (y)dµ(x) δ δ δ |E(a , )| E(ai , 9 ) E(ai , 3 ) i 9 Z Z 1 2 =C |f (y)| dµ(x)dV (y) |E(ai , 9δ )| E(ai , δ9 ) E(ai , δ3 ) Z =C |f (y)|2 µ b δ (ai )dV (y). E(ai , δ3 )

9

(2.1) (2.2) (2.3) (2.4)

A very similar reasoning leads to δ µ ˆ δ (ai ) ≤ C µ ˆδ (y), ∀ y ∈ E(ai , ), ∀i ∈ N. 9 3

(2.5)

Vol. 63 (2009)

Schatten-Herz Classes

Thus, for all f ∈ b2 (B), Z XZ 2 |f (x)| dµ(x) ≤ B

≤

i

XZ i

E(ai , δ3 )

221

|f (y)|2 dµ(y)

(2.6)

E(ai , δ9 )

|f (y)|2 µ bδ (y)dV (y) ≤ CN

Z B

|f (y)|2 µ bδ (y)dV (y).

(2.7)

The proof is finished.

3. Schatten ideals associated with rearrangement invariant spaces The Schatten ideal associated with E consists in all compact operators on b2 (B) whose sequence of singular numbers are in E : SE = {T : H → H, T compact : kT kSE = k{sn (T )}n∈N kE < ∞} where sn (T ) = inf{kT − Rk, rankR ≤ n} is the nth singular number of T. Obviously, when E = lp , then Slp = Sp . A well-known result states that a compact operator A ∈ SE if and only if {< Aen , en >}n∈N ∈ E, for all orthonormal base {en }n∈N of H. Among remarkable examples of rearrangement-invariant spaces, we cite the Lorenz spaces Lp,q , the Lorenz-Zygmund spaces LlogL and Lexp, the Orlicz spaces Lψ , the Lorenz spaces Mϕ and Λϕ , etc. See [1] for details on this topic. By Calderon’s theorem, the interpolation spaces for the couple (L1 , L∞ ) over resonant measure spaces (i.e. spaces with a sigma-finite measure, that is either completely atomic, with atoms having the same measure, or without atoms) are precisely the rearrangement invariant one (see e.g.[1], III Theorem 2.12). Since (L1 , L∞ ) is a Calderon couple (see [1], chap. V for details), a space E is an interpolation space for the couple (L1 , L∞ ) if and only if there exists a monotone Riesz-Fischer norm ρ such that E = (L1 , L∞ )ρ . In other words, if E is a given rearrangement invariant space, there exists a monotone Riesz-Fischer norm such that E = (L1 , L∞ )ρ . In view of these facts, if E is a given rearrangement invariant space and if E = (l1 , l∞ )ρ , then we shall denote by E(dλ) = (L1 (dλ), L∞ (dλ))ρ . For notations, unexplained definitions and interpolation proprieties of rearrangement invariant spaces and of monotone Riesz-Fischer norms, we refer to [1]. For notations and basic proprieties of Schatten ideals Sp we refer to [7]. The following result is implicit in [4]; for convenience of the reader, we shall prove it. Proposition 3.1 (see also [4]). Let E(dλ) be a monotone Riesz-Fischer space over (B, dλ). Then if f ∈ E(dλ), the Toeplitz operator Tf is in the Schatten ideal SE . Proof. Fix an orthonormal base {en }n∈N of b2 (B) and let T : L1 (dλ) + L∞ (dλ) → l1 + l∞ ,

T (f ) = {< Tf en , en >}n∈N ,

1

If f ∈ L (dλ), then Z | < Tf en , en > | ≤ B

|en (z)|2 |f (x)|dV (x)

∀ n ∈ N.

222

Gheorghe

IEOT

hence, using the well-known fact that K(x, y) =

∞ X

en (x)en (y) ∀ x, y ∈ B

0

and for any orthonormal base {en }n∈N of b2 (B), we get Z ∞ X | < Tf en , en > | ≤ K(x, x)|f (x)|dV (x) = kf kL1 (dλ) . B

0 ∞

On the other hand, if f ∈ L (dλ), then obviously | < Tf en , en > | ≤ kf k∞ , so T is bounded from L (dλ) to l1 and from L∞ (dλ) to l∞ . By interpolation, T : E(dλ) → E is bounded, too. 1

We are ready to prove the characterization of those Toeplitz operators acting on Bergman space of the unit ball, whose singular numbers belongs to a given rearrangement invariant spaces. Theorem 3.2. Let E a given rearrangement invariant sequence space, let E = (l1 , l∞ )ρ and denote by E(dλ) the corresponding rearrangement invariant function space over B (E(dλ) = (L1 (dλ), L∞ (dλ))ρ .) Let 0 < δ < δ0 and µ a positive Borel measure. The following statements are equivalent. 1) Tµ ∈ SE ; 2) µ e ∈ E(dλ); 3) µ cδ ∈ E(dλ). Moreover, kTµ kSE ≈ ke µkE(dλ) ≈ kc µδ kE(dλ). Proof. 1) ⇒ 2). Let U : S1 + S∞ → L1 (dλ) + L∞ (dλ) be the linear operator U (T )(x) =< T kx , kx >, x ∈ B. Since U (Tµ )(x) =< Tµ kx , kx >= µ e(x), x ∈ B is the Berezin transform of the positive measure µ, by interpolation, it will suffice to show that U maps boundedly L1 (dλ) to S1 P and L∞ (dλ) to S∞ . Fix {en }n∈N an orthonormal base of b2 (B). Let ∞ T ∈ S1 , T = n=0 λn < , en > en , where {λn }n∈N ∈ l1 is the sequence of singular numbers of T. Then Z Z ∞ X | < T kx , kx > |dλ(x) ≤ |λn | | < kx , en > |2 dλ(x) ≤ kT kS1 , B

0

B

where the last step is due to the reproducing property of the Bergman kernel: Z Z | < kx , en > |2 dλ(x) = |en (x)|2 dV (x) = 1. B

B

On the other hand, if T ∈ S∞ , then |U (T )(x)| = | < T kx , kx > | ≤ kT k∞ , ∀x ∈ B.

Vol. 63 (2009)

Schatten-Herz Classes

223

2) ⇒ 3) is obvious, since, by Lemma 2.2 0≤µ bδ (x) ≤ µ e(x), ∀ x ∈ B, ∀0 < δ < δ0 and E(dλ) is order ideal. 3) ⇒ 1) Let 0 < δ < 1 and consider the covering associated to 9δ , as in the Covering Lemma (Lemma 2.1). Then, by Lemma 2.3, there exists a constant C > 0 such that Z 0 ≤ Tµ f, f >= |f (x)|2 dµ(x) (3.1) B Z ≤C |f (x)|2 µ cδ (x)dV (x) = C| < Tµcδ f, f > |, ∀f ∈ b2 (B). (3.2) B

Invoking again the fact that Schatten ideals are ordered, the prof is finished.

Let Lp,q (dλ) be the Lorentz functions space over the measure space (B, dλ) and let lp,q be the corresponding Lorentz sequence space. An immediate consequence of Theorem 3.2, is the following Corollary 3.3 (see Proposition 2.4 [3]). With notations as above, if 1 ≤ p ≤ q ≤ ∞, and 0 < δ < δ0 , then the following are equivalent: 1) Tµ ∈ Sp,q ; 2) µ e ∈ Lp,q (dλ); 3) µ cδ ∈ Lp,q (dλ). In particular, when p = q, we re-obtain Proposition 2.4 [3].

4. Toeplitz operators in Schatten-Herz class Let Am be the diadic annulus 1 1 ≤ |x| < 1 − m+1 }, m ≥ 0 2m 2 and χm the characteristic function of the set Am . Denote by µχm , the restriction of the Borel measure µ to Am . Let E and F be arbitrary rearrangement invariant sequence spaces and let E(dλ) be the corresponding rearrangement invariant space over the measure space (B, dλ). Am = {x ∈ B : 1 −

b F consists of all positive finite Borel Definition 4.1. The averaging-Herz class H E,δ measures µ that satisfies the following two conditions: \ 1) for all m ≥ 0, the averaging function, (µχ m )δ belong to E(dλ) and \ 2) the sequence {k(µχm ) kE(dλ) }m∈N ∈ F. δ

Then, by definition, kµkHbF

E,δ

\ = k{k(µχ m )δ kE(dλ) }m∈N kF .

b F consists of all positive finite Borel Definition 4.2. The Berezin-Herz class H E,δ measures µ that satisfy the following conditions:

224

Gheorghe

IEOT

^ 1) for all m ≥ 0, the Berezin transform (µχ m )δ belong to E(dλ) and ^ 2) the sequence {k(µχm )kE(dλ) }m∈N ∈ F. By definition, ^ kµkHeF = k{k(µχ m )kE(dλ) }m∈N kF . E

We remark here that the definitions above are slightly different of those in [3] or [5], even for the classic setting of E(dλ) = Lp (dλ) and F = lq ; nevertheless, the equivalence of the two definitions, in the classic case, is, as we shall see, a direct consequence of the description of Schatten-Herz Toeplitz operators. Definition 4.3. Let µ be a positive finite Borel measure on B. The Toeplitz operator F of symbol µ is said to be in the Schatten-Herz class SE if for all m ≥ 0, the operator Tµχm ∈ SE and the sequence {kTµχm kSE }m∈N ∈ F. By definition kTµ kSEF := k{kTµχm kSE }m∈N kF . q

In particular, if E = lp and F = lq , then Sllp = Spq , as defined in [3]. Now we are ready to proove our main result. Theorem 4.4. With notations as above, for all rearrangement invariant sequence spaces E and F and for all 0 < δ < δ0 , the following conditions are equivalent: F 1) Tµ ∈ SE ; F e 2) µ ∈ HE ; bF . 3) µ ∈ H E,δ

Proof. By Theorem 3.2, \ 0 ≤ k(µχ gm kE(dλ) , m )δ kE(dλ) ≈ kTµχm kSE ≈ kµχ

∀m ∈ N,

with a constant that does not depend on m. Since rearrangement invariant spaces are norm ideals, by taking the F -norm, the result follows. In [2], as well as in [5], the characterization of Schatten-Herz classes Toeplitz operators is in terms of the so-called Herz spaces Hpq . Definition 4.5. Let 1 ≤ p, q ≤ ∞. The Herz space Hpq consists in all measurable functions on B such that: 1) f χm ∈ Lp (dλ), ∀m ∈ N; 2) {kf χm kLp (dλ) }m∈N ∈ lq . By definition, kf kHqp = k{kf χm kLp (dλ) }m∈N klq . By Corollary 3.3, Theorem 4.4 above and Theorem 4.4 in [3], we deduce that our definitions on Herz classes (Definitions 4.1 and 4.2) agree to those in [3], at least in the classic setting.

Vol. 63 (2009)

Schatten-Herz Classes

225

Corollary 4.6 (see [3], Theorem 4.4). Let 1 ≤ p, q ≤ ∞. With notations as above, the following conditions are equivalent: 1) Tµ ∈ Spq ; 2) µ e ∈ Hpq ; 3) µ bδ ∈ Hpq ; epq ; 4) µ ∈ H bq . 5) µ ∈ H p,δ

References [1] C. Benett, R. Sharpley, Interpolation of operators Academic Press Inc., 1988. [2] B.R.Choe, K. Na, Toeplitz operators on Harmonic Bergman Spaces. Nagoya Math. J. 174 (2004), 165–186. [3] B.R. Choe, H. Koo, K. Na, Positive Toeplitz Operators of Schatten-Herz Type, Nagoya Math. J. 185 (2007), 31–62. [4] L.G. Gheorghe, Schatten Ideals Toeplitz operators, Math. Reports 1 (51) (1999), 351–357. [5] M. Loaiza, M. Lopez Garcia, S. Perez-Esteva, Herz Classes and Toeplitz Operators in the disk, Integr. Equ. Oper. Theory 53 (2005), 287–296. [6] V.L. Oleinik, B.S. Pavlov, Embeding theorems for weighted classes of harmonic and analytic functions, J. Soviet. Math. 2 (1974), 135–142. [7] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc. 1990. Liliana Gabriela Gheorghe Departamento de Matem´ atica Universidade Federal de Pernambuco 50 740-540 Recife PE Brazil e-mail: [email protected] Submitted: April 30, 2007.

Integr. equ. oper. theory 63 (2009), 227–247 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020227-21, published online January 27, 2009 DOI 10.1007/s00020-008-1651-0

Integral Equations and Operator Theory

The Integral Equation Method and the Neumann Problem for the Poisson Equation on NTA Domains Dagmar Medkov´a Abstract. The Neumann problem for the Poisson equation is studied on a general open subset G of the Euclidean space. The right-hand side is a distribution F supported on the closure of G. It is shown that a solution is the Newton potential corresponding to a distribution B ∈ E(cl G), where E(cl G) is the set of all distributions with finite energy supported on the closure of G. The solution is looked for in this form and the original problem reduces to the integral equation T B = F . If the equation T B = F is solvable, then the solution is constructed by the Neumann series. The necessary and sufficient conditions for the solvability of the equation T B = F is given for NTA domains with compact boundary. Mathematics Subject Classification (2000). Primary 31B10; Secondary 35J05; 35J25; 65N99. Keywords. Poisson equation, Neumann problem, integral equation method, NTA domain, successive approximation method.

1. Introduction We study the Neumann problem for the Poisson equation on general open subsets G of Rm , m > 2, using the integral equation method. The application of the integral eqution method for the Neumann problem for the Laplace equation is classical. The solution is looked for in the form of the single layer potential with an unknown density on the boundary of the domain. Originally it was study the classical Neumann problem on bounded domains with smooth boundary and smooth boundary conditions. Later it was studied the weak Neumann problem for the Laplace equation and boundary conditions given by real measures on the The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.

228

Medkov´ a

IEOT

boundary on open sets with compact boundary and bounded cyclic variation (see [13] and [16]). The solution is looked for in the form of the single layer potential corresponding to an unknown real measure ν supported on the boundary, i.e. in the form of the Newton potential corresponding to ν. The Neumann problem for the Laplace equation has been studied also for Lipschitz domains G with connected boundary and boundary conditions from Lp (∂G) (see [12]). The integral equation method has been used for the study of this problem on Lipschitz domains for other classes of boundary conditions lately (see [3], [10], [25]). The Neumann problem for the Poisson equation has been studied with the help of the integral equation method on Lipschitz domains lately, too ([5] ). If we study the Neumann problem for the Laplace equation using the boundary integral equation method, then we look for a solution in the form of a single layer potential (i.e. in the form of the Newton potential with a density supported on the boundary). The original problem reduces to the integral equation T B = F . Here F is the boundary condition, B is an unknown density and T is a bounded integral operator on an appropriate space of densities on the boundary. Now, we need to calculate the solution B of this equation. The classical result is that we can expresss B by the Neumann series for a convex domain G (see [21], [14], [13], [6]). D. Medkov´ a proved this result on wider class of open sets including sets with piecewise-smooth boundary in R3 (see [18]). O. Steinbach and W. L. Wendland studied in 2001 the weak Neumann problem for the Laplace equation in W 1,2 (G) with boundary conditions from W −1/2,2 (∂G) on bounded Lipschitz domains with connected boundary in R3 (see [25]). They proved that the corresponding integral equation T B = F is solvable by the Neumann series B=

∞ X

(I − T )j F,

(1.1)

j=0

where I is the identity operator. D. Medkov´a studied in 2007 the weak Neumann problem for the Laplace equation on a general open set G with nonempty boundary in Rm , m > 2 (see [19]). It was shown that each solution of this problem is the Newton potential with a density B from E(∂G). (If K is a closed subset of Rm , then E(K) denotes the space of distributions with finite energy supported on K - see §3.) If we look for a solution in this form, then the original problem reduces to the integral equation T B = F , where the boundary condition F is a distribution supported on ∂G. It was shown that T is a bounded operator on E(∂G) for arbitrary open set G. This gives that F ∈ E(∂G) is a necessary condition for the solvability of the Neumann problem for the Laplace equation with the boundary condition F . Under the assumption that G is a bounded W 1,2 extension open set (see §7) the necessary and sufficient conditions for the solvability of the Neumann problem were given and it was proved that the solution of the equation T B = F is given by the Neumann series (1.1). The relation between the results from [25] and the results from [19] is established in this paper: If G is a bounded Lipschitz domain, then G is a W 1,2 exstension set, E(∂G) = W −1/2,2 (∂G) (see Remark 7.11),

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

229

the function u is a weak solution of the Neumann problem if and only if it is a weak solution in W 1,2 (G) (see Remark 7.10). This paper gerenalizes the results from [19]. We study the weak Neumann problem for the Poisson equation on a general nonempty open subset G of Rm , m > 2. (If G is a bounded W 1,2 exstension set, then the weak solution of the Neumann problem for the Poisson equation is a weak solution of the Neumann problem for the Poisson equation in W 1,2 (G) (see Remark 7.10).) It is shown that each solution of this problem is the Newton potential with a density B from E(cl G), where cl G is the closure of G. If we look for a solution in this form, then the original problem reduces to the integral equation T B = F , where the boundary condition F is a distribution supported on cl G. It is shown that T is a bounded operator on E(cl G) for arbitrary open set G. This gives that F ∈ E(cl G) is a necessary condition for the solvability of the Neumann problem for the Poisson equation with the boundary condition F . Under the assumption that G is a W 1,2 extension open set with compact boundary the necessary and sufficient conditions for the solvability of the Neumann problem for the Poisson equation are given and it is proved that the solution of the equation T B = F is given by the Neumann series (1.1). It is also shown how quickly this series converges. (Remark that the class of W 1,2 extendible open sets contains not only Lipschitz domains but also the wider class of NTA domains, which may have fractal boundary.) Let now G be general. It is proved that 0 ≤ T ≤ I and for such types of operators the series (1.1) converges whenever the equation T B = F is solvable. Moreover, it is proved that the equation T B = F can be solved using the successive approximation method inspite the spectral radius of I − T might be equal to 1.

2. Formulation of the problem We study the Neumann problem for the Poisson equation ∂u = g on ∂G (2.1) ∂n for a general open subset G of Rm , m > 2. (Here n is the outward unit normal of G.) Suppose first that G is a bounded domain with smooth boundary and u ∈ C 2 (cl G) is a classical solution of the problem. (Here cl G denotes the closure of G.) Denote by Hk the k-dimensional Hausdorff measure normalized so that Hk is the Lebesgue measure in Rk . Then Green’s formula yields Z Z Z ∇u · ∇ϕ dHm = f ϕ dHm + gϕ dHm−1 (2.2) −∆u = f

G

in G,

G

∂G

for ϕ ∈ D (= the space of all compactly supported infinitely differentiable real functions in Rm ). Let G ⊂ Rm be a nonempty open set. We denote by L2loc (G) the class of all complex measurable functions in G that are in L2 (K) for every compact subset

230

Medkov´ a

IEOT

K of G. Denote by L1,2 (G) the space of all functions in L2loc (G) for which all generalized derivatives of order 1 are in L2 (G). Denote W 1,2 (G) = L1,2 (G)∩L2 (G). The space W 1,2 (G) is equipped with the norm sZ (|u|2 + |∇u|2 ) dHm .

kukW 1,2 (G) = G

Let now G be a bounded domain with Lipschitz boundary, f ∈ L2 (G), g ∈ L2 (∂G). We say that u is a weak solution of the problem (2.1) if u ∈ W 1,2 (G) and the iquality (2.2) holds for each ϕ ∈ W 1,2 (G) (see [20], Exemple 2.8). Since D is dense in W 1,2 (G) (see [20], Chapitre 2 Th´eorem 3.1) we can suppose only that (2.2) is true for ϕ ∈ D. If we denote by F the distribution Z Z F(ϕ) = f ϕ dHm + gϕ dHm−1 , G

∂G

then Z ∇u · ∇ϕ dHm = F(ϕ)

(2.3)

G

for each ϕ ∈ D. We now want to define a weak solution of the problem (2.1) for a general open set G so, that it will coincide with the weak solution in W 1,2 for bounded domains with Lipschitz boundary. Let now G ⊂ Rm be a nonempty open set and F be a distribution supported on cl G. We can try to define a solution of the Neumann problem for the Poisson equation with the right-hand side F as a function from W 1,2 (G) such that (2.3) holds for each ϕ ∈ D. If we use this definition, then there exist a bounded domain G and a nonconstant solution of the Neumann problem for the Poisson equation in G with zero right-hand side. In [19] the author constructed a bounded domain V ⊂ R2 and a nonconstant function v ∈ W 1,2 (V ) such that Z ∇v · ∇ψ dH2 = 0 V

for each ψ ∈ D (see [19], Example 2.1). Put G = {[x, y, z]; [x, y] ∈ V, z ∈ (0, 1)}, u(x, y, z) = v(x, y). Then G is a bounded domain, u ∈ W 1,2 (G) is nonconstant and Fubini’s theorem yields that u is a solution of the Neumann problem for the Poisson equation in G with zero right-hand side. So, we must define a solution of the Neumann problem for general open sets by another way. We use the fact that for a bounded domain G with Lipschitz boundary we have W 1,2 (G) ⊂ L1,2 (Rm ) (see for example [20], Th´eor`eme 3.9). Let G ⊂ Rm be a nonempty open set and F be a distribution supported on cl G. We say that a function u is a weak solution of the Neumann problem for the Poisson equation with the right-hand side F if u can be extended as a function from L1,2 (Rm ) and (2.3) holds for each ϕ ∈ D. If F is supported on ∂G, then we solve the Neumann problem for the Laplace equation (2.1) with f ≡ 0 and g = F. If F is supported in G we solve the Poisson

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

231

equation with the homogeneous Neumann boundary condition (f = F, g ≡ 0 in (2.1)). If u, v are two weak solutions of the Neumann problem for the Poisson equation in G with the right-hand side F, then u − v is constant on each component of G by [19], Theorem 3.1.

3. Representability of solutions by potentials For x, y ∈ Rm denote 1 (m−2)A |x

hx (y) =

− y|2−m

∞

for x 6= y, for x = y,

where A is the area of the unit sphere in R . For a closed set F denote by C 0 (F ) the space of all finite complex Borel measures with support in F . For µ ∈ C 0 (Rm ), µ ≥ 0, denote Z m

Uµ(x) =

x ∈ Rm

hx (y) dµ(y),

(3.1)

Rm

the Newton potential corresponding to µ. According to [15], Theorem 1.11 Z Uµ(x) = lim (Hm (Ωr (x)))−1 Uµ dHm , (3.2) r&0

Ωr (x)

where Ωr (x) is the ball with the centre x and the radius r. Now we define the Newton potential for some class of distributions. Denote by S the linear space of all f ∈ C ∞ (Rm ) such that lim |x|n Dα f (x) = 0

|x|→∞

for each integer n and each multiindex α. The sequence fk converges to f in S if (1 + |x|n )Dα fk (x) converges uniformly to (1 + |x|n )Dα f (x) for each integer n and each multiindex α. Denote by S ∗ the dual space of S. If F ∈ S ∗ is a real measure absolutely continuous with respect to the Lebesgue measure, then we identify its density with F. For f ∈ S define the Fourier transformation fˆ of f by Z fˆ(x) = f (y)e−2πx·y dHm (y), Rm

where y · x denotes the scalar product of x and y. Then the mapping f 7→ fˆ is an isomorpism on S. For F ∈ S ∗ denote by Fb the Fourier transformation of F. Then b Fb ∈ S ∗ and F(ϕ) = F(ϕ) ˆ for each ϕ ∈ S. Denote by E the space of all complex distribution F = F1 + iF2 , where F1 , F2 ∈ S ∗ , such that the Fourier transform Fb = Fb1 + iFb2 of F is absolutely continuous with respect to the Lebesgue measure and s Z b |F(x)|2 kFkE = dHm (x) < ∞. |x|2

232

Medkov´ a

IEOT

kFkE is so called energy of F. Then E equipped with the energy kFkE as a norm is a complex Hilbert space with the scalar product (F, G)E =

Z b b F(x)G(x) dHm (x). |x|2

0 m b b (Here G(x) R denotes complex conjugate of G(x).) If µ ∈ C (R ), then µ ∈ E if and only if U|µ|(x) d|µ|(y) < ∞. (Here |µ| denotes the variarion of µ.) The space E ∩ C 0 (Rm ) is dense in E. (Remark that E = {∆u; u ∈ L1,2 (Rm )}.) For each F ∈ E there is only complex distribution UF = G1 + iG2 , where −2 b G1 , G2 ∈ S ∗ , such that UbF (x) = F(x)|x| . The complex distribution UF is a complex measure which is absolutely continuous with respect to the Lebesgue measure and UF ∈ L1,2 (Rm ) by [15], Theorem 6.4. Denote by UF the representation of UF /(4π 2 ) such that Z UF(x) = lim (Hm (Ωr (x)))−1 UF dHm (3.3) r&0

Ωr (x)

at each x ∈ Rm for which the limit on the right-hand side exists (so called the Newton potential of F). Then UF is determined Hm almost everywhere on Rm and Z (F, ν)E = 4π 2

UF dν

(3.4)

for each ν ∈ E ∩ C 0 (Rm ), where ν denotes the complex conjugate of ν (see [15], Theorem 6.2). According to [15], Theorem 6.4 sZ kFkE =

|∇UF|2 dHm .

If spt F, the support of F, is compact, then UF = h0 ∗ F, where h0 ∗ F denotes the convolution of the distributions h0 and F, (see [15], p. 434) and UF(x) = F(hx )

for x ∈ Rm \ spt F.

If F ∈ C 0 (Rm ), F ≥ 0, then UF is given by (3.1) (see (3.2), [4], p. 155 and [4], chap. II, §2). For the closed set K denote by E(K) the space of all distribution from E supported on K with the energy k kE as a norm. Then E(K) is a complex Hilbert space (see [4], p. 121). Theorem 3.1. Let u be a solution of the weak Neumann problem for the Poisson equation in G. Then there are B ∈ E(cl G) and a complex number a such that u = UB + a in G.

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

233

Proof. We may suppose that u ∈ L1,2 (Rm ). According to [4], p. 155 there are G ∈ E and a complex number a such that u = UG + a allmost everywhere. Denote by B the orthogonal projection of G to E(cl G). Then UB = UG on G by [4], Chapitre I, Theorem 4 and u = a + UB in G.

4. Integral operator According to Theorem 3.1 we can look for a weak solution of the Neumann problem for the Poisson equation in G in the form UB with B ∈ E(cl G). If M ⊂ Rm is a Borel set, F ∈ E, then there is unique JM F ∈ E such that Z ∇UG · ∇UF dHm = (G, JM F)E M

for each G ∈ E. The operator JM : F 7→ JM F is a bounded linear nonnegative operator on E with kJM k ≤ 1. Moreover, JM (E) ⊂ E(cl M ). (See [19], Lemma 5.2.) We use the following lemma proved in [19] (Lemma 5.1): Lemma 4.1. Let F ∈ E, ϕ ∈ D. Then ∆ϕ ∈ E and F(ϕ) = −(F, ∆ϕ)E . Theorem 4.2. Let G ⊂ Rm be a nonempty open set, B ∈ E, F be a distribution supported on cl G. Then Z JG B(ϕ) = ∇ϕ · ∇UB dHm . G

Therefore UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F if and only if F ∈ E(cl G) and JG B = F. Proof. Fix ϕ ∈ D. Then ϕ = −U(∆ϕ) by [15], p.100. According to Lemma 4.1 Z Z JG B(ϕ) = −(JG B, ∆ϕ)E = ∇[−U(∆ϕ)] · ∇UB dHm = ∇ϕ · ∇UB dHm . G

G

Thus UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F if and only if JG B = F. Since JG (E) ⊂ E(cl G) we obtain F ∈ E(cl G). Corollary 4.3. Let G ⊂ Rm be a nonempty open set. Then JG (E) = JG (E(cl G)) ⊂ E(cl G). Proof. Let F ∈ E. Then JG F ∈ E(cl G) by [19], Lemma 5.2 and UF is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side JG F by Theorem 4.2. Denote by B the orthogonal projection of F to E(cl G). Then UB = UF on G by [4], Chapitre I, Theorem 4 and UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side JG F. Theorem 4.2 gives JG B = JG F. Remark 4.4. Let G ⊂ Rm be a nonempty open set. It was shown in [19] that JG (E(∂G)) ⊂ E(∂G) (see Lemma 5.3).

234

Medkov´ a

IEOT

Remark 4.5. If G = Rm \ K with Hm (K) = 0, then JG = I, where I is the identity operator. If F is a distribution, then the Neumann problem for the Poisson equation with the right-hand side F is solvable if and only if F ∈ E and UF is a solution of the problem. Remark 4.6. Suppose that G is a Lipschitz domain with compact boundary. Denote by H the restriction of Hm−1 onto ∂G. Then there is the exterior unit normal nG (x) of G at H-a.a. x ∈ ∂G. Let F = f H ∈ E(∂G), where f ∈ Lp (H), 1 < p < ∞. Then for H-a.a. x ∈ ∂G exists the limit Z g(x) = lim nG (x) · ∇hy (x)f (y) dH(y) &0

and JG UF =

( 12 f

∂G\Ω (x)

+ g)H (see [26]).

Remark 4.7. Suppose that G is an open set with compact boundary and finite perimeter. (If Hm−1 (∂G) < ∞, then G has finite perimeter.) If z ∈ Rm and θ is a unit vector such that the symmetric difference of G and the half-space {x ∈ Rm ; (x − z) · θ < 0} has m-dimensional density zero at z, then nG (z) = θ is termed the exterior normal of G at z in Federer’s sense. If there is no exterior normal of G at z in this sense, we denote by nG (z) the zero vector in Rm . (The exterior normal of G at z in ordinary sense is the exterior normal of G at z in Federer’s sense.) For x ∈ Rm denote Z v G (x) = |nG (y) · ∇hx (y)| dHm−1 (y) ∂G

the cyclic variation of G at x. Suppose that the cyclic variation of G is bounded. (This is true for G convex or for G with ∂G ⊂ L1 ∪ · · · ∪ Lk , where Li are (m -1)-dimensional Ljapunov surfaces, i.e., of class C 1+α .) If F ∈ E(∂G) ∩ C 0 (∂G), then JG UF ∈ E(∂G) ∩ C 0 (∂G) and Z Z Z JG UF(M ) = dG (x) dF(x) + nG (y) · ∇hx (y) dHm−1 (y) dF(x) M

∂G

∂G∩M

for each Borel set M (see [13]). Here dG (x) = lim

r&0

Hm (G ∩ Ωr (x)) Hm (Ωr (x))

is the density of G at x. Remark 4.8. Suppose that G is an open set with compact boundary such that Hm−1 (∂G) < ∞. Denote by H the restriction of Hm−1 onto ∂G. Let F ∈ E(cl G) has compact support in G. For x ∈ ∂G denote g(x) =

m X j=1

Then JG F = F + gH.

nG j (x)F(∂j hx ).

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

235

Proof. Denote C = Rm \ cl G. Since Hm (∂G) = 0 we have JG + JC = I and thus JG F = F − JC F. Fix ϕ ∈ D. According to [4], p. 158 we have ∆UF = −F. This means that UF is a harmonic function in Rm \ spt F. Since the support of F is compact we have UF = h0 ∗ F. Thus ∂j UF = (∂j h0 ) ∗ F. The Divergence theorem (see [13], p. 49) and Theorem 4.2 give Z Z Z ϕg dH = − ϕnC ·∇UF dH = − [ϕ∆UF +∇ϕ·∇UF] dHm = −JC F(ϕ). ∂G

∂C

C

5. Solution of the problem We have shown that the Neumann problem of the Poisson equation in G with the right-hand side F is solvable if and only if there is B ∈ E(cl G) such that JG B = F. If it is true, then UB is a weak solution of the problem. We prove that B is given by the Neumann series corresponding to JG and F. Notation 5.1. Let X be a complex Hilbert space. If M ⊂ H denote by M ⊥ the orthogonal complement of M in H. If T is a bounded linear operator in X denote by Ker T the kernel of T , by σ(T ) the spectrum of T and by T ∗ the adjoint operator of T . Lemma 5.2. Let X be a complex Hilbert space and T be a bounded linear operator on X such that 0 ≤ T ≤ I, where I is the identity operator. Denote by P the orthogonal projection onto Ker(I − T ). If x ∈ X, then T n x → P x as n → ∞. Proof. If x ∈ Ker(I − T ), then T x = x and lim T n x = x = P x. It suffices to prove ˜ ≡ [Ker(I − T )]⊥ . that T n x = 0 for x ∈ X Since T is nonnegative it is selfadjoint by [22], p. 309. Since I −T is selfadjoint, ˜ ⊂X ˜ (see [8], Satz 70.3). Denote by T˜ the restriction of T onto X. ˜ Then (I −T )(X) n 0 ≤ T˜ ≤ I. Since T˜ ≥ 0 by [22], Chapter XIII, Corollary 2.5, we obtain I −T˜n ≤ I. Since (I − T˜n+1 ) − (I − T˜n ) = T˜n (I − T˜) ≥ 0 by [22], Chapter XIII, Corollary 2.5, we have I − T˜n ≤ I − T˜n+1 . According to [22], Chapter XIII, Lemma 2.4 there is a bounded linear operator S on X such that (I − T˜n )x → (I − S)x as n → ∞ for ˜ then Sx = lim T˜n+1 x = each x ∈ X. Thus T˜n x → Sx for each x ∈ X. If x ∈ X, n ˜ ˜ ˜ T lim T x = T Sx. Thus Sx ∈ Ker(I − T ). Since Sx ∈ [Ker T ]⊥ we deduce that lim T n x = Sx = 0. Theorem 5.3. Let X be a complex Hilbert space and T be a bounded linear operator on X such that 0 ≤ T ≤ I. If y ∈ T (X), then the series x=

∞ X j=0

converges and T x = y.

(I − T )j y

236

Medkov´ a

IEOT

Proof. Since X is the direct sum of Ker T and (Ker T )⊥ there is z ∈ (Ker T )⊥ such T z = y. Then n X j=0

j

(I−T ) y =

n X

j

(I−T ) T z =

j=0

n X

j

(I−T ) z−

j=0

n X

(I−T )j (I−T )z = z−(I−T )n+1 z.

j=0

Since 0 ≤ I − T ≤ I and z ∈ (Ker T )⊥ , Lemma 5.2 yields that (I − T )n+1 z → 0. Thus ∞ X (I − T )j y = z. j=0

Proposition 5.4. Let X be a complex Hilbert space and T be a bounded linear operator on X such that 0 ≤ T ≤ I. Fix y ∈ T (X), x0 ∈ X. Put xn = (I − T )xn−1 + y

(5.1)

for positive integer n. Then there is x ∈ X such that xn → x as n → ∞ and T x = y. Proof. Choose z ∈ Ker T and v ∈ (Ker T )⊥ such that x0 = z + v. Then xn = (I − T )n v + z +

n−1 X

(I − T )j y

j=0

for each positive integer n. According to Theorem 5.3 the series ∞ X

(I − T )j y

j=0 n

converges. Since (I − T ) v → 0 as n → ∞ by Lemma 5.2 there is x ∈ X such that xn → x as n → ∞. Using a limit in the equation (5.1) we get x = (I − T )x + y and thus T x = y. Theorem 5.5. Let G ⊂ Rm be a nonempty open set, F be a distribution supported on cl G. If the Neumann problem for the Poisson equation in G with the right-hand side F is solvable, then F ∈ E(cl G) and the series B=

∞ X

(I − JG )j F

(5.2)

j=0

is convergent (in E(cl G)) and UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F. Proof. Since the Neumann problem for the Poisson equation in G with the righthand side F is solvable, Theorem 3.1 and Theorem 4.2 give that F ∈ JG (E) ⊂ E(cl G). Since 0 ≤ JG ≤ I, Theorem 5.3 shows that the series (5.2) converges and JG B = F. Thus UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F by the Theorem 4.2.

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

237

Proposition 5.6. Let G ⊂ Rm be a nonempty open set, F ∈ E(cl G). Suppose that the Neumann problem for the Poisson equation in G with the right-hand side F is solvable. Fix B0 ∈ E. Put Bn = (I − JG )Bn−1 + F

(5.3)

for positive integer n. Then there is B ∈ E such that Bn → B as n → ∞ and UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F. If B0 ∈ E(cl G), then Bn , B ∈ E(cl G). If F, B0 ∈ E(∂G), then Bn , B ∈ E(∂G). Proof. Since the Neumann problem for the Poisson equation in G with the righthand side F is solvable, Theorem 3.1 and Theorem 4.2 give that F ∈ JG (E). Since 0 ≤ JG ≤ I, Proposition 5.4 gives that there is B ∈ E such that Bn → B as n → ∞ and JG (B) = F. According to the Theorem 4.2 the potential UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F. If B0 ∈ E(cl G), then Bn ∈ E(cl G), because JG (E) ⊂ E(cl G) by Corollary 4.3. Since E(cl G) is a bounded subspace of E we obtain B ∈ E(cl G). If F, B0 ∈ E(∂G), then Bn ∈ E(∂G), because JG (E(∂G)) ⊂ E(∂G) by Remark 4.4. Since E(∂G) is a bounded subspace of E we obtain B ∈ E(∂G).

6. Drazin inverse Lemma 6.1. Let G ⊂ Rm be a nonempty open set. Then Ker JG ⊂ E(Rm \ G), cl JG (E) = (Ker JG )⊥ = E(cl G) ∩ [E(∂G) ∩ Ker JG ]⊥ . If JG (E) is closed, then JG (E) = (Ker JG )⊥ . Let F ∈ E. Then F ∈ Ker JG if and only if UF is constant on each component of G. If G is an unbounded set with compact boundary, then UF = 0 on the unbounded component of G. Proof. F ∈ Ker JG if and only if UF is constant on each component of G by Theorem 4.2 and [19], Theorem 3.1. Suppose now that F ∈ Ker N G U. Since ∆UF = −F (see [4], p. 158), we deduce F ∈ E(Rm \ G). If G is unbounded, ∂G is compact and H is the unbounded component of G, then there is a constant c such that UF = c on H. According to [15], Theorem 6.4 Z 1 UF(x) dHm−1 (x) = lim cHm−1 (∂Ω1 (0))rm−3 . 0 = 2 lim r→∞ r r→∞ ∂Ωr (0) Therefore c = 0. Since JG is positive, it is selfadjoint (see [22], p. 309). Since JG is selfadjoint, we have Ker JG = JG (E)⊥ (see [8], Satz 70.3). Since cl JG (E) ⊂ E(cl G), we have cl JG (E) = (Ker JG )⊥ = E(cl G) ∩ [E(cl G) ∩ Ker JG ]⊥ = [E(∂G) ∩ Ker JG ]⊥ ∩ E(cl G). Definition 6.2. Let X be a Banach space, T , S be bounded linear operators on X. The operator S is a Drazin inverse of T , written S = T d , if T S = ST,

S = ST S,

T k = T k ST,

238

Medkov´ a

IEOT

for some nonnegative integer k. The least nonnegative integer k for which these equations hold is the Drazin index of T . Theorem 6.3. Let G ⊂ Rm be a nonempty open set such that JG (E) is a closed subset of E. Denote by P the orthogonal projection of E onto JG (E). Then (I−JG )P is a contractive operator, i.e. q ≡ k(I − JG )P k < 1. The series ∞ X S= (I − JG )j P (6.1) j=1

converges and it is a Drazin inverse of JG of index 0 or 1. Proof. Denote by J˜G the restriction of JG onto JG (E). Then 0 ≤ J˜G ≤ I. Therefore σ(J˜G ) ⊂ h0; 1i by [27], Chapter XI, §8, Theorem 2. Since JG (E) = (Ker JG )⊥ (see Lemma 6.1) the operator J˜G is injective and JG (JG (E)) = JG (E). Since J˜G is injective and surjective we deduce 0 6∈ σ(J˜G ) by [27], Chapter II, §6. The spectral mapping theorem gives that σ(I − J˜G ) ⊂ h0; 1)(see [22], Theorem 9.5). Since σ(I − J˜G ) ⊂ [0, 1) the spectral radius of I − J˜G is smaller than 1 (see [27], Chapter VIII, §2, Theorem 4). Since the operator I − J˜G is selfadjoint it is hyponormal. Since the norm of a hyponormal operator is equal to its spectral radius (see [24], Theorem 1) we have kI − J˜G k < 1. Since kP k ≤ 1 by [27], Chapter III, Theorem 3, the series (6.1) converges. Easy calculation yields that JG S = SJG , S = SJG S, JG = JG SJG . Corollary 6.4. Let G ⊂ Rm be a nonempty open set such that JG (E) is a closed subset of E. Denote by P the orthogonal projection of E onto JG (E) and q = k(I − JG )P k. Then q < 1. Let F ∈ JG (E), B0 ∈ E. Put Bn = (I − JG )Bn−1 + F

(6.2)

for positive integer n. Then there is B ∈ E such that Bn → B as n → ∞ and kB − Bn kE ≤ q n [kB0 kE + (1 − q)−1 kFkE ]. for each positive integer n. Proof. According to Lemma 6.1 there is Z ∈ Ker JG and V ∈ JG (E) such that B0 = Z + V. Since Bn = [(I − JG )P ]n V + Z +

n−1 X

[(I − JG )P ]j F

j=0

for each positive integer n, we obtain ∞ X B=Z+ [(I − JG )P ]j F j=0

and thus kB − Bn kE ≤ q n kVkE +

∞ X j=n

q j kFkE ≤ q n [kB0 kE + (1 − q)−1 kFkE ].

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

239

Proposition 6.5. Let G ⊂ Rm be an open set with nonempty ∂G. If F ∈ E(cl G), then JG F − F ∈ E(∂G). The space JG (E) is closed if and only if JG (E(∂G)) is closed. Proof. Put C = Rm \ G. If F ∈ E(cl G), then JG F − F = −JC (F) ∈ E(C). Since JG (F), F ∈ E(cl G) we deduce JG F − F ∈ E(cl G) ∩ E(C) = E(∂G). Suppose first that JG (E) is closed. Let F ∈ cl JG (E(∂G)). Since F ∈ JG (E) = JG (E(cl G)) (see Corollary 4.3) there is B ∈ E(cl G) such that JG B = F. Since B − JG B ∈ E(∂G) and JG B = F ∈ cl JG (E(∂G)) ⊂ E(∂G) (see [19], Lemma 5.3), we have B = (B − JG B) + JG B ∈ E(∂G) and F = JG B ∈ JG (E(∂G)). Let now JG (E(∂G)) be closed. Let F ∈ cl JG (E). Since F ∈ E(cl G) we have F − JG F ∈ E(∂G). Since F, JG F ∈ E(cl G) ∩ [E(∂G) ∩ Ker JG ]⊥ by Lemma 6.1, we obtain F − JG F ∈ E(∂G) ∩ [E(∂G) ∩ Ker JG ]⊥ . Since JG (E(∂G)) is closed, [19], Theorem 7.6 gives that there is B ∈ E(∂G) such that JG B = F − JG F. Thus F = JG (B + F) ∈ JG (E). Example. Denote x0 = (x1 , . . . , xm−1 ). Suppose that G = {(x0 , xm ); xm > 0}. Fix F ∈ E(cl G). Then F −JG F ∈ E(∂G) by Lemma 6.5. Since JG B = 12 B for each B ∈ (E(∂G)) by [19], Example 7.1, we obtain JG (3F − 2JG (F)) = JG (2(F − JG (F)) + F) = [F − JG (F)] + JG (F) = F. According to Theorem 5.5 and Theorem 4.2 the Neumann problem for the Poisson equation for G with the right-hand side F is solvable if and only if F ∈ E(cl G) and U(3F − 2JG (F)) is a solution of this problem. Example. Denote x0 = (x1 , . . . , xm−1 ). Let V = {(x0 , xm ); xm > 0}, G = {(x0 , xm ); ˜ = {(x0 , xm ); xm > 0, xm−1 < 0}. JV (E(cl V )) = E(cl V ) by xm > 0, xm−1 > 0}, G ˜ = 0, we deduce J ˜ (E(cl V )) = the previous example. Since Hm (V \ (G ∪ G)) G∪G ˜ ⊂ E(cl V ). Then there is B ∈ E(cl V ) such that E(cl V ) . Fix F ∈ E(∂G ∩ ∂ G) F = JG∪G˜ B. According to Corollary 4.3 we have JG (B) ∈ E(cl G), JG˜ (B) ∈ ˜ Thus J ˜ (B) = F − JG (B) ∈ E(cl G). Therefore J ˜ (B) ∈ E(∂G ∩ ∂ G). ˜ E(cl G). G G ˜ Since JG˜ (B) ∈ JG˜ (E) ∩ E(∂G ∩ ∂ G) we get from the symmetry that JG˜ B ∈ JG (E). ˜ = J ˜ (B). Hence F = JG (B + B) ˜ ∈ JG (E). So, there is B˜ ∈ E such that JG (B) G Let now F ∈ E(∂G). Then there is B ∈ E(cl V ) such that F = JG∪G˜ B. ˜ Thus According to Corollary 4.3 we have JG (B) ∈ E(cl G), JG˜ (B) ∈ E(cl G). ˜ there is B˜ ∈ E such that JG˜ (B) = F − JG (B) ∈ E(cl G). Since JG˜ B ∈ E(∂G ∩ ∂ G), ˜ Hence F = JG (B + B) ˜ ∈ JG (E). JG˜ B = JG (B). Let now F ∈ E(cl G). Then F − JG F ∈ E(∂G) by Proposition 6.5. Therefore there is B ∈ E such that JG B = F − JG F and JG (B + F) = F ∈ JG (E). According to Theorem 5.5 and Theorem 4.2 the Neumann problem for the Poisson equation for G with the right-hand side F is solvable if and only if F ∈ E(cl G) and UB, where B is given by (5.2), is a solution of this problem.

240

Medkov´ a

IEOT

7. Extension open sets Definition 7.1. An open set G ⊂ Rm is said to be W 1,2 extension open set if there is a bounded linear operator T : W 1,2 (G) → W 1,2 (Rm ) such that T u = u on G for each u ∈ W 1,2 (G). The following result has been proved recently (see [7], Theorem 2). Lemma 7.2. If G is a W 1,2 extension domain, then there is a positive constant C such that Hm (G ∩ Ωr (x)) ≥ CHm (Ωr (x)) (7.1) for all balls Ωr (x) with x ∈ cl G and 0 < r ≤ 1. Definition 7.3. A domain G is an (, δ) domain, > 0, 0 < δ ≤ ∞, if whenever x, y ∈ G and |x − y| < δ, then there is a rectifiable arc γ ⊂ G with length l(γ) joining x to y and satisfying |x − y| l(γ) < , |x − z| · |y − z| dist(z, ∂G) ≥ for all z ∈ γ. |x − y| Remark 7.4. If G is an (, δ) domain, then it is W 1,2 extendsion (see [11], Theorem 1). S. Jerison and C. E. Kenig studied in [9] so called nontangentially accessible domains. As was noticed by P. W. Jones in [11], p. 73, these domains are precisely (, ∞) domains. Remark that Lipschitz domains and polyhedral domains are nontangentially accessible domains. If G is an (, δ) domain, then Hm (∂G) = 0 (see [11], Lemma 2.3). The boundary of an (, δ) domain can be highly nonrectifiable and no regularity condition on the boundary can be inferred from the (, δ) property. If m − 1 ≤ α < m, one can construct a nontangentially accessible domain G ⊂ Rm such that Hα (V ∩ ∂G) > 0 for all open sets V satisfying V ∩ ∂G 6= ∅. In general (, δ) domains are not sets of finite perimeter. Lemma 7.5. Let G be an unbounded open set such that ∂G ⊂ ΩR (0). Put H = G ∩ ΩR (0). Then Ker JG ∩ E(∂G) is a subset of Ker JH ∩ E(∂H) of codimension 1. Proof. If F ∈ Ker JG ∩ E(∂G), then F ∈ E(∂H) and UF is constant on each component of H by Lemma 6.1. Thus F ∈ Ker JH ∩ E(∂H). Denote by W the unbounded component of G. Denote by F1 the restriction of Hm−1 onto ∂ΩR (0). Then UF1 = (m − 2)−1 R on ΩR (0) ⊃ H and UF1 (x) = (m − 2)−1 Rm−1 |x|2−m on Rm \ΩR (0) and thus F1 ∈ [Ker JH ∩E(∂H)]\[Ker JG ∩E(∂G)]. Let G ∈ Ker JH ∩ E(∂H). Then there is c such that UG = c on H ∩ W . Since UG is harmonic in Rm \ cl ΩR (0) and |∇UG| ∈ L2 (Rm \ cl ΩR (0)), [23], Chapter I, Theorem 3.5 yields that there are constants a, b and a harmonic function u in Ω1/R (0) with u(0) = 0 such that UG(x) = a + b|x|2−m + |x|2−m u(x/|x|2 ) on Rm \ cl ΩR (0). According to [15], Theorem 6.4 Z 1 0 = lim 2 UG(x) = a lim Hm−1 (Ω1 (0))rm−3 . r→∞ r r→∞ ∂Ωr (0)

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

241

Thus a = 0. Since u(x/|x|2 ) ∈ W 1,2 (Ω2R (0) \ ΩR (0)) and x 7→ x/|x|2 is biLipschitzian mapping from Ω2R (0) \ ΩR (0) onto Ω1/R (0) \ Ω1/2R (0) the function u is in W 1,2 (Ω1/R (0) \ Ω1/2R (0)) by [28], Theorem 2.2.2. Since u is harmonic in Ω1/R (0) it is in W 1,2 (Ω1/R (0)). Since the trace of UG on ∂ΩR (0) is c the trace of |x|2−m u(x) on ∂ΩR (0) must be c − bR2−m . Thus u is a harmonic function from W 1,2 (Ω1/R (0)) with the trace c(R)m−2 − b on ∂(Ω1/R (0)). From the uniqueness of the weak solution of the Dirichlet problem for the Laplace equation in W 1,2 ((Ω1/R (0)) we get that u = cRm−2 − b. From u(0) = 0 we get that u ≡ 0 and b = cRm−2 . Therefore UG = c on H ∩ W , UG(x) = cRm−2 |x|2−m on W \ H. Put F = G −c(m−2)R−1 F1 . Then F ∈ E(∂H) and UF is constant on each component of G. According to Lemma 6.1 we have F ∈ Ker JG . Since ∆UF = −F and UF is harmonic on W we deduce that spt F ⊂ ∂H \ W = ∂G. Thus Ker JG ∩ E(∂G) is a subset of Ker JH ∩ E(∂H) of codimension 1. Theorem 7.6. Let G, V ⊂ Rm be open sets such that V ⊂ G and ∂G ⊂ ∂V . Suppose that Ker JG ∩ E(∂G) is a subset of Ker JV ∩ E(∂V ) of finite codimension. If JV (E) is closed, then JG (E) is closed. Proof. Denote by J˜V the restriction of JV onto JV (E). Then J˜V is invertible (compare Lemma 6.1). Since 0 ≤ J˜V ≤ I, the spectrum of J˜V is a subset of the interval h0, 1i (see [27], Chapter XI, §8, Theorem 2). Since J˜V is invertible, there is α > 0 so that the spectrum of J˜V is a subset of the interval hα, 1i. Thus (JV F, F)E ≥ α(F, F)E for each F ∈ JV (E) = E(cl V ) ∩ [E(∂V ) ∩ Ker JV ]⊥ (see [27], Chapter XI, §8, Theorem 2 and Lemma 6.1). Since JG\H ≥ 0, we have kJG FkE · kFkE ≥ (JG F, F)E = (JV F, F)E + (JG\V F, F)E ≥ (JG F, F)E ≥ α(F, F)E for each F ∈ E(∂G) ∩ [E(∂V ) ∩ Ker JV ]⊥ ⊂ E(cl V ) ∩ [E(∂V ) ∩ Ker JV ]⊥ . If Fn ∈ E(∂G) ∩ [E(∂V ) ∩ Ker JV ]⊥ , JG Fn → B, then Fn is Cauchy’s sequence because kFn − Fk kE ≤ α−1 kJG Fn − JG Fk kE . Since E(∂G) ∩ [E(∂V ) ∩ Ker JV ]⊥ is closed, there is F ∈ E(∂G) ∩ [E(∂V ) ∩ Ker JV ]⊥ such that Fn → F. From the continuity of JG we get that B = JG F ∈ JG (E(∂G) ∩ [E(∂V ) ∩ Ker JV ]⊥ . Thus JG (E(∂G)∩[E(∂V )∩Ker JV ]⊥ ) is closed. Since JG (E(∂G)∩[E(∂V )∩Ker JV ]⊥ ) is a subset of JG (E(∂G)∩[E(∂G)∩Ker JG ]⊥ ) of finite codimension, the set JG (E(∂G)∩ [E(∂G) ∩ Ker JG ]⊥ ) is closed by [22], Lemma 5.2. Thus JG (E(∂G)) = JG (E(∂G) ∩ [E(∂G) ∩ Ker JG ]⊥ ) is closed. Proposition 6.5 gives that JG (E) is closed. Theorem 7.7. If G is W 1,2 extension open set with compact boundary, then JG (E) is closed. Proof. If G is bounded, then JG (E(∂G)) is closed by [19], Theorem 8.8. Proposition 6.5 gives that JG (E) is closed. Suppose now that G is unbounded. Fix R > 0 such that ∂G ⊂ ΩR (0). Put H = G ∩ Ω2R (0). Since G, Ω2R (0) are W 1,2 extension open set there are bounded linear operators L1 : W 1,2 (G) → W 1,2 (Rm ), L2 : W 1,2 (Ω2R (0)) → W 1,2 (Rm ) such that L1 f = f on G and L2 f = f on Ω2R (0). Fix ϕ ∈ D such that ϕ = 1 on ΩR (0) and spt ϕ ⊂ Ω2R (0). If f ∈ W 1,2 (H) put f1 = f ϕ, f2 = (1 − ϕ) on H, f1 = 0 on

242

Medkov´ a

IEOT

G \ H, f2 = 0 on Ω2R (0) \ H. Then Lf = L1 f1 + L2 f2 represents a bounded linear operator from W 1,2 (H) to W 1,2 (Rm ) such that Lf = f on H. Thus H is a bounded W 1,2 extension open set. We have proved that JH (E) is closed. Lemma 7.5 says that Ker JG ∩ E(∂G) is a subset of Ker JH ∩ E(∂H) of codimension 1. So, JG (E) is closed by Theorem 7.6. Theorem 7.8. Let ∂G be bounded. Suppose that there is a W 1,2 extension open set H such that Hm ((G \ H) ∪ (H \ G)) = 0. Then JG (E) is closed. Put W = G ∪ H. Then W has finitely many components W1 , . . . , Wk . Let W1 , . . . , Wn be all bounded components of W . Then there are ϕ1 , . . . , ϕn ∈ W 1,2 (Rm ) with compact support such that ϕj = 1 in Wj , ϕj = 0 in W \ Wj , j = 1, . . . , n. Let F ∈ E(cl G). Then there is a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F if and only if Z ∇UF · ∇ϕj dHm = 0 (7.2) Rm

for j = 1, . . . , n. If cl Wj ∩ cl Wk = ∅ for k 6= j we can take ϕ1 , . . . , ϕn ∈ D and there is a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F if and only if F(ϕj ) = 0 for j = 1, . . . , n. Suppose now that the Neumann problem for the Poisson equation on G with the right-hand side F is solvable. Let B be given by (5.2). Then UB +

k X

cj χWj ,

c1 , . . . , ck ∈ C,

(7.3)

j=1

is the general form of a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F. (Here χA is the characteristic function of the set A.) Proof. The set W is W 1,2 extension set by [19], Lemma 9.2. Since G, H are open sets and Hm ((G\H)∪(H \G)) = 0, we deduce ∂W ⊂ ∂H and thus ∂W is bounded. Theorem 7.7 shows that JW (E) is closed. Since Hm ((G \ W ) ∪ (W \ G)) = 0 we have JG = JW and thus JG (E) = JW (E) is closed. If W is bounded put W0 = ∅. If W is unbounded denote by W0 the unbounded component of W . Since W \ W0 is a bounded W 1,2 extension open set, it has finitely many components W1 , . . . , Wn by [19], Lemma 8.6. Put ϕj = 1 in Wj , ϕj = 0 in W \ Wj , j = 1, . . . , n. Since ϕj ∈ W21 (W ) there is an extension of ϕj from W 1,2 (Rm ). According to [4], p. 155 there are Bj ∈ E and complex numbers aj such that ϕj = UBj + aj allmost everywhere. Since ϕj is a weak solution of the Neumann problem for the Poisson equation in W with zero right-hand side, we obtain JW Bj = 0. Since JG = JW , we deduce Bj ∈ Ker JG . Let F ∈ E(cl G) = E(cl W ). Suppose first that there is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F. Then F ∈ (Ker JG )⊥ by Theorem 3.1, Theorem 4.2 and Lemma 6.1. Since Bj ∈ Ker JG , we obtain (7.2).

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

243

Suppose now that (7.2) is true for j = 1, . . . , n. Denote by Gj the orthogonal projection of Bj onto E(cl W ). Then UGj = UBj = ϕj on W by [4], Chapitre I, Theorem 4. Since ∆UGj = −Gj by [4], p. 158 and ∆UGj = ∆ϕj = 0 on W , we obtain Gj ∈ E(∂W ). Since UGj = 1 on Wj and UGj = 0 on W \ Wj , the distributions G1 , . . . , Gn form a linearly independent subset of Ker JW ∩ E(cl W ). Let G ∈ Ker JW ∩ E(cl W ). Then there are cj ∈ C such that UG = cj on Wj for j = 1, . . . , n and UG = 0 on W0 (see [19], Lemma 6.1). Put G˜ = G −

n X

cj Gj .

j=1

Then U G˜ = 0 in W . If x ∈ ∂W , then W has not zero density at x by Lemma 7.2. According to [19], Remark 6.6 and [19], Proposition 6.7 we obtain that G˜ = 0. This means that G1 , . . . , Gn form a basis of Ker JW ∩ E(cl W ). Since F ∈ E(cl W ) and Gj is the orthognal projection of Bj onto E(cl W ) we have Z (F, Gj )E = (F, Bj )E = ∇UF · ∇ϕj dHm = 0. Rm

Since G1 , . . . , Gn form a basis of Ker JW ∩ E(cl W ), we deduce F ∈ (Ker JW ∩ E(cl W ))⊥ . Lemma 6.1 gives that F ∈ JW (E). Since F ∈ JW (E) = JG (E), Theorem 4.2 gives that there is a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F. Suppose now that there is a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F. Let B be given by (5.2). Then UB is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F by Theorem 5.5. If W is unbounded, put ϕn+1 = 1 −

n X

ϕj .

j=1

Since ϕ1 , . . . , ϕk ∈ L1,2 (Rm ) and ϕ1 , . . . , ϕk are constant on each component of G, the function k k X X UB + cj χWj = UB + cj ϕj j=1

j=1

is a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F. Let now u ∈ L1,2 (Rm ) be a weak solution of the Neumann problem for the Poisson equation on G with the right-hand side F. Then v = u−UB is a weak solution of the Neumann problem for the Poisson equation on G with zero right-hand side. Since v is constant on each component of G by Lemma 3.1 we obtain ∇v = 0 in G. Since Hm (W \ G) = 0 the vector function ∇v = 0 Hm -a.e. in W . According to [17], Lemma on the page 11 there are constants c1 , . . . , cn such that v = cj Hm -a.e.in Wj for j = 1, . . . , n. Since v is continuous in G we obtain v = cj in G ∩ Wj . Thus u has a form (7.3).

244

Medkov´ a

IEOT

Lemma 7.9. Let G ⊂ Rm be a bounded open set. Denote by (W 1,2 (G))0 the dual space of W 1,2 (G). Then (W 1,2 (G))0 ⊂ E(cl G) and the identity is a bounded linear operator from (W 1,2 (G))0 to E(cl G). Proof. Since G is bounded there is a positive constant K such that kUBkW 1,2 (G) ≤ KkBkE for each B ∈ E (see [19], Lemma 8.4). Fix F ∈ (W bution supported on cl G. Put

1,2

(7.4) 0

(G)) . Clearly, F is a distri-

F (B) = F(UB). for B ∈ E. Using (7.4) we get |F (B)| ≤ kFk(W 1,2 (G))0 kUBkW 1,2 (G) ≤ kFk(W 1,2 (G))0 KkBkE . Thus F is a bounded linear functional on E and kF k ≤ KkFk(W 1,2 (G))0 .

(7.5)

The Riesz representation theorem (see [27], Chapter III, §6) gives that there is G ∈ E such that F (B) = (B, G)E for each B ∈ E and kGkE = kF k. If ϕ ∈ D we have ϕ = −U∆ϕ and ∆ϕ ∈ E by [15], p. 100 and [15], Theorem 6.2. Using [19], Lemma 5.1, we get F(ϕ) = −F (∆ϕ) = −(∆ϕ, G)E = −(G, ∆ϕ)E = G(ϕ). Therefore F = G ∈ E and kFkE = kGkE = kF k ≤ KkFk(W 1,2 (G))0 .

Remark 7.10. Let G be a bounded W 1,2 extension domain. Then (W 1,2 (G))0 = E(cl G) and both norms are equivalent. Let F be a distribution supported on cl G. Then there is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F if and only if F ∈ (W 1,2 (G))0 and F(1) = 0. If u is a solution of this problem, then u ∈ W 1,2 (G) and (2.3) holds for every ϕ ∈ W 1,2 (G). On the other hand, if F ∈ (W 1,2 (G))0 , u ∈ W 1,2 (G) and (2.3) holds for every ϕ ∈ W 1,2 (G), then u is a weak solution of the Neumann problem for the Poisson equation in G with the right-hand side F. Proof. Denote E0 (cl G) = {F ∈ E(cl G); F(1) = 0}. Then JG (E) = E0 (cl G) by Theorem 7.8 and Theorem 4.2. Let S be the Drazin inverse of JG defined by (6.1). Denote V = {f ∈ W 1,2 (G); f = ϕ in G for some ϕ ∈ D}. Fix F ∈ E0 (cl G). Then U(SF) is a weak solution of the Neumann problem for the Poisson equation with the right-hand side F (see Theorem 5.5). If ϕ ∈ D, then Z F(ϕ) = ∇ϕ · ∇U(SF) dHm (7.6) G

and F reprerents a linear operator on V . H¨older’s inequality yields |F(ϕ)| ≤ k|∇ϕ|kL2 (G) k|∇U(SF)|kL2 (G) ≤ kϕkW 1,2 (G) kSFkE ≤ kϕkW 1,2 (G) kSkkFkE .

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

245

Thus F can be extended onto a bounded linear functional on W 1,2 (G) such that kFk(W 1,2 (G))0 ≤ kSkkFkE (see [22], Theorem 2.6). Since V is dense in W 1,2 (G) (see [28], Remark 2.5.2 and [28], Lemma 2.1.3) this extension is unique. If F is the zero functional on W 1,2 (G), then F ≡ 0 by (7.6). Thus the identity operator is a bounded injective operator from E0 (cl G) to (W 1,2 (G))0 . According to Lemma 7.9 we have E0 (cl G) = {F ∈ (W 1,2 (G))0 ; F(1) = 0}. Denote by µ the restriction of Hm onto G. Then µ ∈ E(cl G) ∩ (W 1,2 (G))0 . Since E(cl G) is spanned by E0 (cl G) ∪ {µ} and (W 1,2 (G))0 spanned by {F ∈ (W 1,2 (G))0 ; F(1) = 0} ∪ {µ}, we deduce that E(cl G) = (W 1,2 (G))0 . Since the identity is a bounded linear operator from (W 1,2 (G))0 onto E(cl G), the norms on E(cl G) and (W 1,2 (G))0 are equivalent (see Lemma 7.9 and [22], Theorem 3.12). Remark 7.11. Let G ⊂ Rm be a bounded Lipschitz domain. For u ∈ W 1,2 (G) denote by tr u its trace (see [20], Chapitre I, §1.3). Denote W 1/2,2 (∂G) = {tr u; u ∈ W 1,2 (G)} equipped with the norm kvkW 1/2,2 (∂G) = inf{kukW 1,2 (G) ; u ∈ W 1,2 (G); v = tr u}. Denote by W −1/2,2 (∂G) the dual space of W 1/2,2 (∂G). Then we have E(∂G) = W −1/2,2 (∂G) and both norms are equivalent. Proof. Let F ∈ W −1/2,2 (∂G). Then F is a distribution supported on ∂G. Denote F (v) = F(tr v) for v ∈ W 1,2 (G). Then F ∈ (W 1,2 (G))0 . Remark 7.10 gives that F ∈ E(cl G). Since F(ϕ) = F (ϕ) for each ϕ ∈ D we see that F = F ∈ E(cl G). Since F is a distribution supported on ∂G, we obtain F ∈ E(∂G). Since kF k(W 1,2 (G) = kFkW −1/2,2 (∂G) , Remark 7.10 gives that c1 kFkW −1/2,2 (∂G) ≤ kFkE ≤ c2 kFkW −1/2,2 (∂G)

(7.7)

with positive constants c1 , c2 independent on F. Since W −1/2,2 (∂G) is a dual space, it is a Banach space (see [22], Theorem 2.10). The relation (7.7) gives that W −1/2,2 (∂G) is a closed subspace of E(∂G). Let now µ ∈ E(∂G) ∩ C 0 (∂G). Then µ ∈ (W 1,2 (cl G))0 by Remark 7.10 and Z µ(v) = tr v dµ ∂G

for each v ∈ W (G) (compare (3.4) and [4], p. 155). Thus µ ∈ W −1/2,2 (∂G). This gives that E(∂G) ∩ C 0 (∂G) ⊂ W −1/2,2 (∂G). But E(∂G) ∩ C 0 (∂G) is a dense subset of E(∂G) (see [4], p. 143). It forces E(∂G) = W −1/2,2 (∂G). 1,2

246

Medkov´ a

IEOT

References [1] D. R. Adams, L. I. Hedberg, Function spaces and Potential Theory. Springer-Verlag, 1996. [2] D. H. Armitage, S. J. Gardiner, Classical Potential Theory. Springer-Verlag, 2001. [3] R. M. Brown, The Neumann problem on Lipschitz domains in Hardy spaces of order less than one. Pacific Journal of Mathematics 171 (1995), 389–407. [4] J. Deny, Les potentiels d’´energie finie. Acta Math. 82(1950), 107–183. [5] E. Fabes, O. Mendez, M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. Journal of Functional Analysis 159 (1998), 323–368. [6] E. Fabes, M. Sand, J. K. Seo, The spectral radius of the classical layer potentials on convex domains. IMA Vol. Math. Appl. 42 (1992), 129-137. [7] P. Hajlasz, P. Koskela, H. Tuominen, Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254 (2008), 1217–1234. [8] H. Heuser, Funktionalanalysis. Teubner, 1975. [9] S. Jerison, C. E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains. Advances in Mathematics. 47 (1982), 80–147. [10] H. Jia, H. Wang, Harmonic Sobolev-Besov spaces, layer potentials and regularity for the Neumann problems in Lipschitz domains. J. Differential Equations 204 (2004), 123–138. [11] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Mathematica. 147 (1981), 71–88. [12] C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. American Mathematical Society, 1994. [13] J. Kr´ al, Integral Operators in Potential Theory. Springer-Verlag, 1980. [14] J. Kr´ al, I. Netuka, Contractivity of C. Neumann’s operator in potential theory. J. Math. Anal. Appl. 61 (1977), 607–619. [15] N. L. Landkof, Fundamentals of Modern Potential Theory. Izdat. Nauka, 1966. (Russian). [16] V. G. Maz’ya, Boundary Integral Equations. Analysis IV. Encyclopaedia of Mathematical Sciences, vol 27. Springer-Verlag, 1991, 127–222. [17] V. G. Maz’ya, S. V. Poborchi, Differentiable Functions on Bad Domains. World Scientific Publishing, 1997. [18] D. Medkov´ a, Solution of the Neumann problem for the Laplace equation. Czechoslov. Math. J. 48 (1998), 768–784. [19] D. Medkov´ a, The Neumann problem for the Laplace equation on general domains. Czech. Math. J. 57 (2007), 1107–1139. [20] J. Neˇcas, Les m´ethodes directes en th´eorie des ´equations ´elliptiques. Academia, 1967. [21] J. Plemelj, Potentialtheoretische Untersuchungen. B. G. Teubner, 1911. [22] M. Schechter, Principles of Functional Analysis. Academic press, 1973. [23] Ch. G. Simader, H. Sohr, The Dirichlet problem for the Laplacian in bounded and unbounded domains. Pitman Research Notes in Mathematics Series 360. Addison Wesley Longman Inc., 1996.

Vol. 63 (2009)

Neumann Problem for the Poisson Equation

247

[24] J. Stampfli, Hyponormal operators. Pacific J. Math. 12 (1962), 1453–1458. [25] O. Steinbach, W. L. Wendland, On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries. Journal of Mathematical Analysis and Applications 262 (2001), 733–748. [26] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. Journal of Functional Analysis. 59 (1984), 572–611. [27] K. Yosida, Functional Analysis. Springer-Verlag, 1965. [28] W. P. Ziemer, Weakly Differentiable Functions. Springer-Verlag 1989. Dagmar Medkov´ a Mathematical Institute of the Academy of Sciences of the Czech Republic ˇ a 25 Zitn´ 115 67 Praha 1 Czech Republic e-mail: [email protected] Submitted: March 25, 2008. Revised: December 22, 2008.

Integr. equ. oper. theory 63 (2009), 249–261 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020249-13, published online February 2, 2009 DOI 10.1007/s00020-009-1656-3

Integral Equations and Operator Theory

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C0-Semigroups Ciprian Preda and Petre Preda Abstract. In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup {T (t)}t≥0 , in terms of the admissibility of the pair (Lp (R+ , X), Lq (R+ , X)). It is already known the equivalence between the (Lp (R+ , X), Lq (R+ , X))admissibility condition (1 ≤ p ≤ q ≤ ∞ and (p, q) 6= (1, ∞)) and the hyperbolicity of a C0 -semigroup {T (t)}t≥0 , when we assume a priori that the kernel of the dichotomic projector (denoted here by X2 ) is T (t)-invariant and T (t)|X2 is an invertible operator. We succeed to prove in this paper that the admissibility of the pair (Lp (R+ , X), Lq (R+ , X)) still implies the existence of an exponential dichotomy for a C0 -semigroup T = {T (t)}t≥0 even in the general case where the kernel of the dichotomic projector, X2 , is not assumed to be T (t)-invariant. Mathematics Subject Classification (2000). Primary 34D05; Secondary 34D20, 47D06. Keywords. C0 -semigroup, exponential dichotomy, admissibility.

1. Introduction and Preliminaries Let X be a Banach space and B(X) be the Banach algebra of all bounded linear operators acting on X. The norm on X and on B(X) will be denoted by k · k. Consider the abstract Cauchy problem du(t, x) = Au(t, x), u(0, x) = x ∈ X, t ≥ 0 () dt with a closed densely defined operator A on X. By dichotomy we understand the existence of a bounded projection, P , such that the solutions which start in Im P decay to zero and the solutions which start in Im(I −P ) are unbounded. Dichotomy

250

C. Preda and P. Preda

IEOT

and, in particular exponential dichotomy is one of the principal instruments in the analysis of linear differential equations in Banach spaces, linearized instability for nonlinear equations, existence of the invariant and center manifolds and we can continue the list with many other applications. The very first consideration of the exponential dichotomy of linear differential equations was done by O. Perron in [14]. Roughly speaking, Perron established an equivalence between the condition that the non-homogeneous equation has some bounded solution for every bounded “second member” on the one hand and a certain form of conditional stability of the solutions of the homogeneous equation on the other. After seminal results of O. Perron, important results concerning the extension of Perron’s problem in the more general framework of infinite-dimensional Banach spaces were obtained by M. G. Krein [3], J. L. Daleckij [3], J. L. Massera [8] and J. J. Sch¨affer[8], and recently by van Neerven [12], van Minh [10, 11], A. Pogan, C. Preda, P. Preda [15, 16, 17], F. R¨ abiger [11], R. Schnaubelt [11]. Assume for a moment that () is well-posed; that is A generates a C0 semigroup T = {T (t)}t≥0 on X. We recall that a B(X)-valued function T = {T (t)}t≥0 is named a semigroup of linear operators if the identity on X can be obtained as T (0) and the semigroup property T (t + s) = T (t)T (s) is also satisfied for all t, s ≥ 0. If in addition T is strongly continuous (i.e. there exists limt→0+ T (t)x = x, for all x ∈ X) then we will call T a C0 -semigroup. It is widely known that each C0 -semigroup is exponentially bounded i.e. kT (t)k ≤ M eωt

for all t ≥ 0

for some M, ω > 0. See for instance [9, 10]. Therefore it makes sense to define ω(T) = inf{α ∈ R : there exists β ≥ 1 such that kT (t)k ≤ βeαt , for all t ≥ 0}. There is an important connection between ω(T) and the spectral radius of each operator T (t), in the sense that r(T (t)) = etω(T) , (see for instance Proposition 1.2.2. in [12]). We denote by X1 the space of all x ∈ X with the property that T (·)x is bounded. In what follows X1 will be assumed complemented (i.e. X1 is closed and there exists X2 a closed subspace such that X = X1 ⊕ X2 ). Also we denote by P a projection onto X1 along X2 (that is P ∈ B(X), P 2 = P and ker(P ) = X2 ) and by Q = I − P . It is easy to see that X1 is T (t)-invariant for all t ≥ 0 (that is equivalent to P T (t)P = T (t)P for each t ≥ 0) and so the application T1 : R+ → B(X1 ), T1 (t) = T (t)|X1 is also a C0 -semigroup, acting on X1 . It can be seen that if Qx 6= 0 then T (t)Qx 6= 0, for all t ≥ 0. Definition 1.1. The C0 -semigroup T = {T (t)}t≥0 is exponentially dichotomic if there exist the constants N1 , N2 , ν > 0 such that (d1 ) kT (t)xk ≤ N1 e−νt kxk, for all t ≥ 0 and all x ∈ X1 ; (d2 ) kT (t)xk ≥ N2 eνt kxk, for all t ≥ 0 and all x ∈ X2 . Remark 1.2. The condition (d1 ) is equivalent with ω(T1 ) < 0.

Vol. 63 (2009) Admissibility and Exponential Dichotomy for Semigroups

251

We want to mention that there exists another way to see the exponential dichotomy as introduced by Bart et al. [1] and elaborated by Cornelis van der Mee in [9]. In [1, 9] so-called bisemigroups tS e (I − P ), t > 0, E(t; S) = −etS P, t ≤ 0, are introduced, where {etS |ker P }t≥0 and {e−tS |Im P }t≥0 are exponentially decaying C0 -semigroups and S is exponentially dichotomous. In this case there exists > 0 such that Z ∞ (λ − S)−1 x = e−λt E(t; S)x dt, | Re λ| < , x ∈ X. −∞

If we compare our definition of exponential dichotomy with the one introduced by Bart et al. [1], we have to admit that our definition looks more “asymmetric”. As it can be seen above for us a C0 -semigroup is exponentially dichotomic if its restriction to the subspace X1 = {x ∈ X : kT (t)xk = O(1), t → ∞} is uniformly exponentially stable and there exists a closed complement X2 of X1 such that inf x∈X2 ,kxk=1 kT (t)xk is exponentially increasing as t → ∞, without assuming that X2 is invariant under T (t). The bisemigroups in [1, 9] can be subsumed under the exponential dichotomic semigroups defined in this article by setting T (t) = E(t; S)(I − P ) − E(−t; S)P. In a more restrictive situation the concept of exponential dichotomy has a spectral interpretation through the notion of hyperbolicity. We recall that a C0 -semigroup T = {T (t)}t≥0 is called hyperbolic if σ(T (t)) does not intersect the unit circle for t 6= 0, where we write σ(·) for the spectrum. Suppose we know that A generates a hyperbolic semigroup. Then () has dichotomy (and even exponential dichotomysee definition above) where P can be cast as being the Riesz projection for T (t), t > 0, that corresponds to the part of T (t) in the unit disk. Moreover an equivalence was established between hyperbolicity and a particular case of exponential dichotomy. Theorem 1.3 (M. A. Kaashoek and S. M. Verduyn Lunel [7, Theorem 1.1]). A C0 -semigroup {T (t)}t≥0 is hyperbolic if and only if there exists a projector P on X (so-called dichotomic projection) such that the following statements hold: (i) P T (t) = T (t)P for all t ≥ 0. (ii) There are positive constants N, ν such that kT (t)xk ≤ N e−νt kxk for all x ∈ P (X) and t ≥ 0. (iii) The restriction T (t)|ker(P ) is an invertible operator (so extends to a C0 -group) and kT −1 (t)xk ≤ N e−νt kxk for all x ∈ ker(P ) and t ≥ 0. Thus, in this spirit, the concept of hyperbolicity is in fact double exponential stability, first for the restriction T1 (t) and second for T2−1 (t), where T2 (t) = T (t)|ker(P ) . It is trivial to see now that hyperbolicity implies exponential dichotomy for strongly continuous semigroups. The converse implication does not hold, and

252

C. Preda and P. Preda

IEOT

with a small effort an example of a C0 -semigroup can be built, which is exponentially dichotomic but is not hyperbolic. For instance, consider A to be a p × p matrix with real entries, whose spectrum is enclosed in the open left-half plane, and take the right shift semigroup on L1 (R+ , R), given by 0, 0≤s
L∞ (R+ , X) = {f ∈ M(R+ , X) : ess sup kf (t)k < ∞}. t∈R+ p

∞

We note that L (R+ , X), L (R+ , X) are Banach spaces endowed with the respective norms: Z ∞ p1 kf kp = kf (t)kp dt 0

kf k∞ = ess sup kf (t)k t∈R+

We will denote also by C(R+ , X) the space of all bounded continuous functions from R+ to X, which is a Banach space endowed with the norm |||f ||| = sup kf (t)k. t≥0

Vol. 63 (2009) Admissibility and Exponential Dichotomy for Semigroups

253

Definition 1.4. The pair (Lp (R+ , X), Lq (R+ , X)), p, q ∈ [1, ∞] is said to be admissible to the C0 -semigroup T = {T (t)}t≥0 if for each f ∈ Lp (R+ , X) there exists x ∈ X such that u(·; x, f ) ∈ Lq (R+ , X) where Z

t

T (t − s)f (s) ds.

u(t; x, f ) = T (t)x + 0

Proposition 1.5. If (Lp (R+ , X), L∞ (R+ , X)), p ∈ [1, ∞] is admissible to T = {T (t)}t≥0 then for every f ∈ Lp (R+ , X) there is an unique x2 ∈ X2 such that u(·; x2 , f ) ∈ L∞ (R+ , X). Proof. Consider f ∈ Lp (R+ , X), x given by Definition 1.4 and x1 ∈ X1 , x2 ∈ X2 with x = x1 + x2 . Then u(·, x2 , f ) = u(·, x, f ) − T (·)x1 . It can be seen that u(·, x2 , f ) is bounded since it is continuous and by hypothesis it belongs to L∞ (R+ , X). Thus the existence part is proved. If we assume that y2 , z2 ∈ X2 and u(·; y2 , f ), u(·, z2 , f ) ∈ C(R+ , X) it is clear that T (·)(y2 − z2 ) = u(·; y2 , f ) − u(·; z2 , f ) ∈ C(R+ , X) which implies that y2 − z2 ∈ X1 ∩ X2 and hence y2 = z2 . So the statement is now proved.

The unique element x2 ∈ X2 for f ∈ Lp (R+ , X) will be denoted in what follows by xf . Proposition 1.6. If the C0 -semigroup S = {S(t)}t≥0 has the property that there is α > 0 such that supt≥0 tα kS(t)k < ∞ then ω(S) < 0. Proof. It is obvious that from the hypothesis we have that there exists a > 0 with kS(a)k < 1. It follows that eaω(S) = r(S(a)) ≤ kS(a)k < 1 and so we obtain ω(S) < 0.

2. The main result Lemma 2.1. Let f, g : R+ → R+ , g continuous with g(t) > 0, for each t ≥ 0. If (i) f (t) ≥ g(t − τ )f (τ ), for each t ≥ τ ≥ 0; (ii) there exists δ > 0 with g(δ) > 1; then there exist N, ν > 0 such that f (t) ≥ N eν(t−t0 ) f (t0 ), for each t ≥ t0 ≥ 0. 0 Proof. Let t ≥ t0 ≥ 0. We denote by n = [ t−t δ ], where [s] denotes the greatest integer less than or equal to s. It follows that t0 + nδ ≤ t < t0 + (n + 1)δ and applying (i) for τ = t0 + nδ we get that

f (t) ≥ g(t − nδ − t0 )f (t0 + nδ) ≥ inf g(s)f (t0 + nδ). s∈[0,δ]

254

C. Preda and P. Preda

IEOT

Denoting β = inf s∈[0,δ] g(s) and applying repeatedly (i) for τ = t0 + (n − 1)δ, τ = t0 + (n − 2)δ, . . . , τ = t0 we get that f (t) ≥ βg(δ)f (t0 + (n − 1)δ) ≥ βg 2 (δ)f (t0 + (n − 2)δ) ≥ · · · ≥ βg n (δ)f (t0 ). Denoting g(δ) = eνδ we get that νδ = ln g(δ) and ν = that

1 δ

ln g(δ) > 0. Thus we have

f (t) ≥ βeνnδ f (t0 ) ≥ βeν(n+1)δ e−νδ f (t0 ) ≥ βe−νδ eν(t−t0 ) f (t0 )

for each t ≥ t0 ≥ 0

and where N = βe−νδ

f (t) ≥ N eν(t−t0 ) f (t0 ), for each t ≥ t0 ≥ 0, 1 = inf s∈[0,δ] g(s) g(δ) , ν = 1δ ln g(δ).

Lemma 2.2. If h is a positive function from Lq (R+ , R), such that the following statement holds, h(r) ≤ ah(t) + b, for all r ≥ t ≥ 0 with r − t ≤ 1, ∞

then h ∈ L (R+ , R). Proof. See Lemma 3.3 from [16].

Proposition 2.3. If (Lp (R+ , X), L∞ (R+ , X)), p ∈ [1, ∞] is admissible to T = {T (t)}t≥0 then there exists K > 0 such that ku(·; xf , f )k∞ ≤ Kkf kp , for all f ∈ Lp (R+ , X). Proof. Define U : Lp (R+ , X) → L∞ (R+ , X), U f = u(·; xf , f ). We note that U is a linear operator. In order to prove that in addition U is also bounded, consider (fn ) a sequence of elements belonging to Lp (R+ , X) and f ∈ Lp (R+ , X), g ∈ L∞ (R+ , X) such that k·kp k·k∞ fn −−−→ f and U fn −−−→ g. Since xfn = (U fn )(0), for all n ∈ N it follows that xfn → g(0)

and so

g(0) ∈ X2

We observe that Z t Z t Z t T (t − s)fn (s) ds − T (t − s)f (s) ds ≤ kT (t − s)(fn (s) − f (s))k ds 0

0

0 1

≤ M eωt t q kfn − f kp , for all t ≥ 0, n ∈ N and q=

 

p p−1 ,

1,  ∞,

p ∈ (1, ∞) p=∞ p=1

Vol. 63 (2009) Admissibility and Exponential Dichotomy for Semigroups

255

Thus we obtain that u(·; g(0), f ) = g ∈ L∞ (R+ , X) which implies that xf = g(0) and hence U f = g. It is now clear that ku(·; xf , f )k∞ = kU f k∞ ≤ kU kkf kp ,

for all f ∈ Lp (R+ , X).

Now we can state the main result of this paper. Theorem 2.4. If (Lp (R+ , X), L∞ (R+ , X)) with p ∈ [1, ∞] is admissible to T = {T (t)}t≥0 then T is exponentially dichotomic. Proof. Consider x ∈ X with Qx 6= 0 and f (τ ) = χ[t0 ,t1 ] (τ )

T (τ )Qx . kT (τ )Qxk

1

Then kf kp = (t1 − t0 ) p and Z ∞ dτ χ[t0 ,t1 ] (τ ) T (t)Qx kT (τ )Qxk 0 Z t Z ∞ dτ dτ = χ[t0 ,t1 ] (τ ) T (t)Qx + χ[t0 ,t1 ] (τ ) T (t)Qx kT (τ )Qxk kT (τ )Qxk 0 t which implies that Z ∞ − χ[t0 ,t1 ] (τ )

dτ T (t)Qx kT (τ )Qxk t Z ∞ Z t = T (t − s)f (s) ds − T (t) χ[t0 ,t1 ] (τ ) 0

0

dτ Qx kT (τ )Qxk

and we know that Z

∞

t −→ −

χ[t0 ,t1 ] (τ ) t

dτ T (t)Qx ∈ L∞ (R+ , X). kT (τ )Qxk

By using Proposition 2.3 we have that there is a K > 0 such that Z ∞ 1 dτ kT (t)Qxk χ[t0 ,t1 ] (τ ) ≤ K(t1 − t0 ) p kT (τ )Qxk t which implies that Z kT (t)Qxk t

t1

1 dτ ≤ K(t1 − t0 ) p kT (τ )Qxk

and setting t = t0 we get Z t1 1 dτ 1 ≤ K(t1 − t) p kT (τ )Qxk kT (t)Qxk t R t1 dτ Denoting φ(t) = t kT (τ )Qxk we can write 1

(t1 − t)− p ≤ −K

φ0 (t) φ(t)

for each t0 ≤ t ≤ t1

for each t ≤ t1 .

for each t < t1

256

C. Preda and P. Preda

which implies that Z t1 −1 1 φ(t1 − 1) (t1 − τ )− p dτ ≤ −K ln φ(s) s

IEOT

for each s ≤ t1 − 1.

It follows that q

φ(t1 − 1)e− K (1−(t1 −s)

1 q

)

≤ φ(s),

for each s ≤ t1 − 1.

Taking t1 = t + 1 we have that Z t+1 Z t+1 1 q 1 dτ dτ 1 q e− K (1−(t+1−s) ) ≤ ≤ K(t + 1 − s) p . kT (τ )Qxk kT (τ )Qxk kT (s)Qxk t s But kT (τ )Qxk ≤ M eω kT (t)Qxk for each τ ∈ [t, t + 1]. It follows that Z t+1 1 1 dτ ≤ M eω kT (t)Qxk kT (τ )Qxk t which implies that 1 q 1 1 1 1 −K (1−(t+1−s) q ) e ≤ K(t + 1 − s) p M eω kT (t)Qxk kT (s)Qxk and so q

1 e− K (1−(t+1−s) kT (s)Qxk KM eω (t + 1 − s) p1

1 q

)

≤ kT (t)Qxk.

Denoting by q

1 q

((u+1) ) q eK 1 g(u) = e− K , g : [0, ∞) → (0, ∞) 1 ω KM e (u + 1) p

we have that limu→∞ g(u) = ∞ and thus supu≥0 g(u) > 1. Thus kT (s)Qxkg(t − s) ≤ kT (t)Qxk for each t ≥ s ≥ 0. So we can find N2 , ν2 > 0 such that kT (t)Qxk ≥ N2 eν2 (t−t0 ) kT (t0 )Qxk while t ≥ t0 ≥ 0. Take now δ > 0 and gδ (t) = χ[0,δ] (t)T (t)P x. Then gδ ∈ Lp (R+ , X) and 1 kgδ kp ≤ δ p Lkxk, where kT (t)P xk ≤ Lkxk (by using the Uniform Boundedness Principle and the definition of X1 ), for each t ≥ 0. Denoting Z t y(t) = χ[0,δ] (τ )T (t − τ )T (τ )P x dτ 0 Z t = T (t)P x χ[0,δ] (τ ) 0 tT (t)P x, t ∈ [0, δ] = δT (t)P x, t ≥ δ

Vol. 63 (2009) Admissibility and Exponential Dichotomy for Semigroups

257

we get that y(·) ∈ L∞ (R+ , X) and y(0) = 0 ∈ X2 , which implies that kyk∞ ≤ Kkgδ kp . But δ2 kT (δ)P xk = 2

δ

Z

tkT (δ)P xk dt 0 δ

Z

tkT (δ − t)P T (t)P xk dt

= 0 δ

Z ≤

tLkT (t)P xk dt 0

Z

δ

ky(t)k dt ≤ Lδkyk∞

=L 0

≤ LKδkgδ kp 1

= L2 Kδ 1+ p kxk which implies that 1

kT (δ)P xk ≤ 2KL2 δ p −1 kxk,

for each δ > 0, x ∈ X.

By Proposition 1.6 we can get N1 , ν1 > 0 such that kT (t)P xk ≤ N1 e−ν1 t kP xk, while t ≥ 0 and x ∈ X. Denoting by ν = min{ν1 , ν2 } we complete the proof and conclude that T = {T (t)}t≥0 is exponentially dichotomic. Theorem 2.5. If (Lp (R+ , X), Lq (R+ , X)), (p, q) 6= (1, ∞) is admissible to the C0 semigroup T = {T (t)}t≥0 then T is exponentially dichotomic. Proof. Taking f ∈ Lp (R+ , X) randomly, we get that there is x ∈ X such that u(·; x, f ) ∈ Lq (R+ , X) with Z r u(r; x, f ) = T (r − t)u(t; x, f ) + T (r − τ )f (τ ) dτ t

for each r ∈ [t, t + 1]. It follows that ku(r; x, f )k ≤ M eω ku(t; x, f )k + M eω kf kp

for each r ∈ [t, t + 1].

By using Lemma 2.2 we get that u(·; x, f ) ∈ L∞ (R+ , X) and from Theorem 2.4 we get that T = {T (t)}t≥0 is exponentially dichotomic. Theorem 2.6. If the C0 -semigroup T = {T (t)}t≥0 is hyperbolic then the pair (Lp (R+ , X), Lq (R+ , X)) is admissible to T, 1 ≤ p ≤ q ≤ ∞. Proof. We take f ∈ Lp (R+ , X) randomly and write Z t Z ∞ v(t) = T (t − s)P f (s) ds − T −1 (s − t)Qf (s) ds 0 t Z t Z t Z ∞ = T (t − s)f (s) ds − T (t − s)Qf (s) ds − T −1 (s − t)Qf (s) ds. 0

0

t

258

C. Preda and P. Preda

IEOT

But Z

∞

T (t)

T

−1

Z (τ )Qf (τ ) dτ = T (t)

0

t

T

−1

Z (τ )Qf (τ ) dτ + T (t)

0

∞

T −1 (τ )Qf (τ ) dτ

t

Z ∞ T (t − τ )T (τ )T −1 (τ )Qf (τ ) dτ + T (t)(T (τ − t)T (t))−1 Qf (τ ) dτ 0 t Z t Z ∞ = T (t − τ )Qf (τ ) dτ + T −1 (τ − t)Qf (τ ) dτ. Z

t

=

0

t

Thus for f ∈ Lp (R+ , X) we get that Z ∞ Z t v(t) = −T (t) T −1 (τ )Qf (τ ) dτ + T (t − s)f (s) ds 0

0

is actually t

Z

T (t − τ )f (τ ) dτ,

u(·; x, f )(t) = T (t)x + 0

R∞ where x = − 0 T −1 (τ )Qf (τ ) dτ ∈ X2 . From the H¨older inequality (see for instance [5, Theorem 6.4, page 477]) it follows that u(·; x, f ) ∈ Lq (R+ , X). Theorem 2.7. If (L1 (R+ , X), L∞ (R+ , X)) is admissible to T = {T (t)}t≥0 then there exist the constants N1 , N2 > 0 such that • kT (t)xk ≤ N1 kT (t0 )xk, for all t ≥ t0 ≥ 0 and all x ∈ X1 ; • kT (t)xk ≥ N2 kT (t0 )xk, for all t ≥ t0 ≥ 0 and all x ∈ X2 . Proof. Let t0 ≥ 0 and f (t) = χ[t0 ,t0 +1] (t) Then kf k1 = 1 and Z ∞ χ[t0 ,t0 +1] (τ ) 0

Z =

T (t)Qx . kT (t)Qxk

dτ T (t)Qx kT (τ )Qxk

t

χ[t0 ,t0 +1] (τ ) 0

dτ T (t)Qx + kT (τ )Qxk

Z

∞

χ[t0 ,t0 +1] (τ ) t

dτ T (t)Qx kT (τ )Qxk

which implies that Z ∞ − χ[t0 ,t0 +1] (τ )

dτ T (t)Qx kT (τ )Qxk t Z ∞ Z t = T (t − s)f (s) ds − T (t) χ[t0 ,t0 +1] (τ ) 0

0

and we know that Z − t

∞

χ[t0 ,t0 +1] (τ )

dτ kT (τ )Qxk

dτ T (·)Qx ∈ L∞ (R+ , X). kT (τ )Qxk

Qx

Vol. 63 (2009) Admissibility and Exponential Dichotomy for Semigroups

Thus there is a K > 0 such that Z ∞ kT (t)Qxk χ[t0 ,t0 +1] (τ ) t

259

dτ ≤K kT (τ )Qxk

which implies that Z

t0 +1

kT (t)Qxk t0

dτ ≤K kT (τ )Qxk

for each t ≤ t0 .

It follows Z

t0 +1

t0

1 dτ ≤ kT (τ )Qxk kT (t)Qxk

for each t ≤ t0 .

But kT (τ )Qxk ≤ M eω kT (t0 )Qxk

for each τ ∈ [t0 , t0 + 1]

implies that 1 ≤ M eω kT (t0 )Qxk

Z

t0 +1

t0

dτ K ≤ . kT (τ )Qxk kT (t)Qxk

Thus 1 ≤ kT (t0 )Qxk KM eω kT (t0 )Qxk

for each t0 ≥ t ≥ 0 and x ∈ X.

Now observe that (by using the Uniform Boundedness Principle) there is N > 0 such that kT (t)P xk ≤ N kxk for each t ≥ 0 and x ∈ X. Then kT (t)P xk = kT (t − t0 )P T (t0 )P xk ≤ N kT (t0 )P xk, for each t ≥ t0 ≥ 0 and x ∈ X. Remark 2.8. Let T = {T (t)}t≥0 be a C0 -semigroup with T (t)P = P T (t) and T (t)|X2 being an isomorphism. If there exist the constants N1 , N2 > 0 such that • kT (t)xk ≤ N1 kT (t0 )xk, for all t ≥ t0 ≥ 0 and all x ∈ X1 ; • kT (t)xk ≥ N2 kT (t0 )xk, for all t ≥ t0 ≥ 0 and all x ∈ X2 ; then (L1 , L∞ ) is admissible to T = {T (t)}t≥0 . Proof. For f ∈ L1 (R+ , X) we denote by Z t Z v(t) = T (t − s)P f (s) ds − 0

∞

T −1 (s − t)Qf (s) ds.

t

Following a similar argument as in Theorem 2.6 we get that Z ∞ Z t v(t) = −T (t) T −1 (τ )Qf (τ ) dτ + T (t − s)f (s) ds 0

0

From here we get that Z t Z kv(t)k = N1 kP f (s)k ds − 0

t

∞

1 kQf (s)k ds ≤ N2

for each t ≥ 0. Thus the proof is complete.

1 N1 kP k + kQk kf k1 , N2

260

C. Preda and P. Preda

IEOT

Acknowledgment The authors would like to thank to the referee for thoroughly reading of the paper and intercepting an error. Also the authors are indebted to the referee for the very helpful comments and suggestions.

References [1] H. Bart, I. Gohberg, M. A. Kaashoek,Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, Mathematical Surveys and Monographs, J. Funct. Anal. 68 (1986), 1–42. [2] C. Chicone, Y. Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs 70, Providence, RO: American Mathematical Society, 1999. [3] J. L. Daleckij, M. G. Krein, Stability of solutions of Differential Equations in Banach spaces, Transl. Math. Monographs 43, Amer. Math. Soc., Providence, RI, 1974. [4] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. 3 (1972), 428-445. [5] K. J. Engel , R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, Berlin, 2000. [6] P. Hartman, Ordinary differential equations, Classics in Applied Mathematics 38, SIAM, Philadelphia, 2002. [7] M. A. Kaashoek, S. M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Diff. Eqs. 112 (1994), 374–406. [8] J. L. Massera J. J. Sch¨ affer, Linear differential equations and function spaces, Academic Press, New York, 1966. [9] C. van der Mee, Exponentially dichotomous operators and applications, Birkh¨ auser OT 182, Basel and Boston, 2008. [10] N. van Minh, On the proof of characterizations of the exponential dichotomy, Proc. Am. Math. Society 127 (1999), 779–782. [11] N. van Minh, F. R¨ abiger, R Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332–353. [12] J.M.A.M. van Neerven, The Asymptotic Behaviour of Semigroups of linear operators, Operator Theory Advances and Applications 88, Birkh¨ auser, Basel, 1996. [13] A Pazy, Semigroups of operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. [14] O. Perron, Die Stabilit¨ atsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703– 728. [15] P. Preda, A. Pogan, C. Preda, On the Perron Problem for the Exponential Dichotomy of C0 -semigroups, Acta Mathematicae Universitae Comenianae 72, no.2 (2003), 207212. [16] P. Preda, A. Pogan, C. Preda, (Lp , Lq )-Admissibility and Exponential Dichotomy of Evolutionay Processes on the Half-Line, Integral Equations and Operator Theory, vol. 49, no. 3 (2004), 405–418.

Vol. 63 (2009) Admissibility and Exponential Dichotomy for Semigroups

261

[17] P. Preda, A. Pogan, C. Preda, Sch¨ affer spaces and exponential dichotomy for evolutionay processes, J. Diff. Eqs, vol. 230, no. 1 (2006), 378–391. [18] J. Pr¨ uss, On the spectrum of C0 -semigroups, Trans. Am. Math. Society 284 (1984), 847–857. Ciprian Preda Department of Mathematics University of California Los Angeles, CA 90095 U.S.A. Current address: West University of Timisoara Bd. V. Pˆ arvan, No. 4 300223 – Timi¸soara Romania e-mail: [email protected] Petre Preda Department of Mathematics West University of Timi¸soara Bd. V. Pˆ arvan, No. 4 300223 – Timi¸soara Romania e-mail: [email protected] Submitted: July 18, 2008. Revised: September 26, 2008.

Integr. equ. oper. theory 63 (2009), 263–280 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020263-18, published online February 2, 2009 DOI 10.1007/s00020-009-1654-5

Integral Equations and Operator Theory

Analysis of Spectral Points of the Operators [∗] [∗] T T and T T in a Krein Space Andr´e Ran and Michal Wojtylak Abstract. Spectra and sets of regular and singular critical points of definitiz[∗] [∗] able operators of the form T T and T T in a Krein space are compared. The relation between the Jordan chains of the above operators (corresponding to the same eigenvalue) is shown. Mathematics Subject Classification (2000). 47B50. Keywords. Regular critical point, singular critical point, Jordan chain, Krein space.

1. Introduction [∗]

[∗]

In this paper we will deal with a pair of operators T T and T T , where T is a possibly unbounded operator in a Krein space. If T is bounded, then by the classical result in [8] the nonzero spectra of the above operators coincide [8]. Apparently, we cannot say much on the unbounded case without assuming anything about the [∗] operator T . Our main additional condition will be that both operators T T and [∗] T T are selfadjoint and definitizable. The reasons for this (nontrivial) requirement can be found in Section 3. In this setting the extended real line R = R∪{∞} divides [∗] (with respect to T T ) into four parts. A point λ either belongs to the resolvent [∗] [∗] [∗] ρ(T T ), or it belongs to the definite spectrum σ+ (T T ) ∪ σ− (T T ), or it is a regular critical point, or it is a singular critical point (see Section 2 for definitions). [∗] The same decomposition can be done with respect to T T . Our motivation for the research presented in this paper was to compare these two divisions of R. A more general problem was studied in the finite dimensional case in the paper [6] by Flanders, already more than half a century ago. The result shows a relation between the Jordan structures of two matrices AB and BA. The Jordan structures corresponding to the nonzero spectral points are the same. The situa∞ tion at the zero eigenvalue is, however, more complicated. If (nj )∞ j=0 and (mj )j=0

264

Ran and Wojtylak

IEOT

are decreasing sequences of sizes of Jordan blocks of AB and BA, respectively, corresponding to the zero eigenvalue, extended by an infinite number of zeros, then |nj − mj | ≤ 1, j = 0, 1, . . . . (1.1) Our second aim was to prove an analogue of the aforementioned for operators of [∗] [∗] the form T T and T T acting in an infinite dimensional Pontryagin space. A (not complete) analysis of Jordan structures and canonical forms [7] of [∗] [∗] the pair of operators T T and T T in a finite dimensional space was done in [12]. Several special cases where treated there. In [16, Theorem 4.2], a complete [∗] [∗] [∗] description of the relations between T , T , T T and T T is given for the finite dimensional case. The approach taken there is to consider analogues of singular value decompositions of matrices in indefinite inner product spaces (see also [3]). Also the real case is treated there, as well as several cases with other symmetries. In [17] the relation to polar decomposition of this problem was discussed: in the finite dimensional case it turns out that an operator T admits polar decomposition [∗] [∗] if and only if T T and T T have the same canonical form with respect to the given indefinite inner product. Polar decomposition in Pontryagin spaces was discussed in [17] (for normal operators in the indefinite inner product) and [15]. The latter [∗] paper also contains some results on operators of the form T T , which we shall use later on. In the present paper we focus attention mostly on those aspects that really belong to the infinite dimensional situation, opposed to the finite dimensional case. The outcome of this paper can be summarized as follows. In Section 4 we prove [∗] [∗] that for nonzero λ ∈ R the properties of λ as a spectral point of T T and T T are strongly related. It is also shown that the nonreal spectra of the operators coincide. The spectral point zero is considered in Sections 4, 6. It appears that the four possibilities mentioned above can occur in almost any combination for [∗] [∗] T T and T T , as can be seen from the following table. The table is obviously TT

[∗]

[∗]

\T T

0∈ρ 0 ∈ σ+ ∪ σ− 0 is reg. crit.

0∈ρ

0 ∈ σ+ ∪ σ−

possible

possible possible

0 is reg. crit. possible Ex. 6.1 possible Ex. 6.2 possible

0 is sing. crit.

0 is sing. crit. impossible Prop. 4.4 possible Ex. 6.3 possible Ex. 6.4 possible [15, Ex. 3.8], [11]

Table 1. Zero as a spectral point

symmetric with respect to the diagonal and the part under the diagonal was left empty for the sake of clarity. The fact that zero cannot be a resolvent point of

[∗]

Vol. 63 (2009)

Analysis of Spectral Points of T T and T T

[∗]

265

one of the operators and at the same time a singular critical point of the other operator is proved in Section 4. The examples showing that all other cases are possible even in the class of bounded operators in a separable Π1 -space are given in Section 6. The behavior of infinity as a spectral point (with the usual understanding of the four notions of resolvent point, critical point etc.) is also rather peculiar, and is investigated in Section 5. The results are presented in the next table. [∗]

[∗]

TT \ T T ∞∈ρ ∞ ∈ σ+ ∪ σ− ∞ is reg. crit.

∞∈ρ possible

∞ ∈ σ+ ∪ σ− impossible possible

∞ is reg. crit. impossible possible Ex. 5.1 impossible Prop. 5.1

∞ is sing. crit.

∞ is sing. crit. impossible possible Ex. 5.1 impossible Prop. 5.1 impossible Prop. 5.1

Table 2. Infinity as a spectral point

In the last section we work in a Pontryagin space. This assumption assures us that at each eigenvalue there is only a finite number of Jordan chains longer than one for each of the operators. Hence, we are able to compare the Jordan structures of the operators at each eigenvalue. A reduction argument allows us to apply the theorem of Flanders and to obtain a similar result (Theorem 7.2).

2. Preliminaries In the whole paper (K, [·, ·]) stands for a Krein space (in Sections 6 and 7 it will be a Pontryagin space). At this point we fix one of the complete norms on K such that the inner product is continuous, and denote it by k·k. Note that all such norms are equivalent (see [1, 14]), and none of the arguments below depend on a choice of an equivalent norm. If A is an operator in K then by D(A), N (A) and R(A) we understand the domain, the kernel and the range of A, respectively. The sum and product of unbounded operators is understood in a standard way, see e.g. [5, 19]. [∗] We write B(K) for the space of all bounded operators with domain equal K. A denotes the adjoint of a densely defined operator A in a Krein space. As usual, σ(A) (ρ(A)) stand for the spectrum (the resolvent set) of a closed, densely defined operator A in K. Let A be a selfadjoint operator in K. We call A definitizable if ρ(A) 6= ∅ and there exists a (real or complex) polynomial p such that [p(A)f, f ] ≥ 0 for f ∈ D(p(A)). Any polynomial p satisfying the last inequality is called a definitizing polynomial for A. Note that even the degree of a definitizing polynomial is usually not unique (see [14, 10] for all results mentioned in this paragraph). We define the

266

Ran and Wojtylak

IEOT

set of critical points of a definitizable operator A as c(A) := c0 (A) ∩ σ(A) ∩ R, where \ c0 (A) := p−1 (0). p definitizing for A

It is well known that c(A) = c0 (A) ∩ R. The Jordan chains corresponding to an eigenvalue λ of a definitizable operator are not longer than k(λ) + 1 (k(λ) for nonreal λ), where k(λ) is the multiplicity of λ as a zero of (any) definitizing polynomial p. This is the same as to say that the algebraic root space Sλ (A) := {f ∈ D(A) : ∃n ∈ N \ {0} : (A − λ)n f = 0} equals N ((A − λ)k(λ)+1 ). By R(A) we understand the semiring generated by finite intervals and their complements with endpoints not in c(A). On this semiring we define the spectral mapping E of the operator A, see [14] for the definition and properties. The definition which we use involves contour integrals (as in [14]), although similar results could be obtained also with the usage of functional calculus from [10]. By σ+ (A) (σ− (A)) we denote the set of all λ ∈ σ(A) ∩ R for which there exists an interval τ ∈ R(A), λ ∈ τ , such that R(E(τ )) is a positive (negative) subspace of K. Critical points are those points λ of the real spectrum for which the space R(E(τ )) is indefinite for any neighborhood τ of λ. We call a critical point λ regular if the limits limx↑λ E([λ0 , x]) and limx↓λ E([x, λ1 ]) exist in the strong operator topology for any (some) not critical λ0 ≤ λ, λ1 ≥ λ. This is equivalent to saying that for every neighborhood τ of λ such that τ ∩ c(A) = {λ} and the spectral function E is bounded on subsets of τ [14, Theorem 5.7], or, again equivalently, that there exists a neighborhood τ with τ ∩ c(A) = {λ} such that the spectral function E is bounded on subsets of τ . We call a critical point singular, if it is not regular. An isolated point of the real spectrum is either in the definite part σ+ (A) ∪ σ− (A) of the spectrum or is a regular critical point. In each case the spectral projection E({λ}) (understood as a limit) equals the Riesz’s projection [5] onto the algebraic root subspace corresponding to the eigenvalue λ.

[∗]

3. Operators of the form T T in a Krein space [∗]

Let us say some words on the operators of the form T T in the unbounded, Krein [∗] space case. Such an operator is naturally symmetric in the sense that [T T f, f ] ∈ R [∗] for f ∈ D(T T ), but it does not have to be even densely defined. Consider the Hilbert space (H, h·, ·i) and introduce on the space K = H × H the Krein space inner product: [(f, g), (h, k)] := hf, ki + hg, hi, f, g, h, k ∈ H. Now let us take T = A ⊕ B, where A and B are both closed, densely defined operators in H (i.e. T (f, g) := (Af, Bg), (f, g) ∈ D(T ) = D(A) × D(B)). It is clear that [∗]

T T = (B ∗ A) ⊕ (A∗ B),

[∗]

T T = (AB ∗ ) ⊕ (BA∗ ).

[∗]

Vol. 63 (2009)

Analysis of Spectral Points of T T and T T

[∗]

267

If we choose as A any unbounded selfadjoint (in H) operator and set B = h·, ei f [∗] [∗] with f ∈ D(A), e ∈ / D(A) then T T is densely defined while T T is not. Consider now the following conditions, for a densely defined and closed operator T in a Krein space K: [∗]

[∗]

(t1) T T and T T are (densely defined and) selfadjoint operators in K. [∗] [∗] (t2) T T and T T have nonempty resolvent sets. [∗] (t3) T T is definitizable. [∗]

Theorem 3.1. Under the conditions (t1) and (t2) the operator T T is definitizable [∗] if and only if T T is definitizable. Moreover, if p(t) is a definitizing polynomial [∗] [∗] for T T then tp(t) is a definitizing polynomial for T T . Consequently, [∗]

[∗]

c0 (T T ) ∪ {0} = c0 (T T ) ∪ {0} . [∗]

(3.1)

[∗]

Proof. For f ∈ D((T T )p(T T )) we have [∗]

[∗]

[∗]

[∗]

[∗]

[∗]

[∗]

[∗]

[(T T )p(T T )f, f ] = [T p(T T )f, T f ] = [p(T T )T f, T f ] ≥ 0. [∗]

This proves the first two sentences of the theorem. In consequence, c0 (T T )∪{0} ⊆ [∗] c0 (T T ) ∪ {0}. We obtain the converse inclusion by interchanging the roles of T [∗] and T . From now on we assume that T satisfies (t1)–(t3). The last example and theorem show our reasons for this assumption. Another motivation is that in a Pontryagin space (t1)–(t3) are always satisfied for a densely defined, closed T . This comes from the fact, that by [13] the second power of a selfadjoint operator is selfadjoint and we can apply Nelson’s trick [18, top of page 143]. To be precise, let T be a densely defined operator in a Πκ -space (K, [·, ·]). Consider the space K × K with the Π2κ -inner product [(f, g), (h, k)] = [f, h] + [g, k] (f, g, h, k ∈ K) and the operator [∗]

Q(f, g) = (T g, T f ),

[∗]

(f, g) ∈ D(Q) = D(T ) × D(T ).

Since Q is selfadjoint in K × K so is Q2 , by the result of Langer [13]. Hence, both [∗] [∗] T T and T T are selfadjoint (and thus definitizable).

4. Definitizable operators in Krein spaces. Positive results on types of spectral point [∗]

Now let (t1)–(t3) hold an let E and E∗ denote the spectral function of T T and [∗] T T respectively. By R0 we denote the semiring generated by finite intervals and [∗] their complements with endpoints not in c(T T ) ∪ {0}. We put R00 for the family of all its bounded elements.

268

Ran and Wojtylak

IEOT

Theorem 4.1. The following inclusions hold [∗]

E∗ (τ )T ⊆ T E(τ ),

[∗]

E(τ )T ⊆ T E∗ (τ ),

τ ∈ R0 .

(4.1)

Moreover, for τ ∈ R00 we have [∗]

E(τ )T [∗] = T E∗ (τ ) ∈ B(K).

E∗ (τ )T = T E(τ ) ∈ B(K),

(4.2)

Proof. We prove only the first inclusion in (4.1), the proof of the second one is similar. First let τ ∈ R00 . By definition, Z 1 [∗] E∗ (τ )h = − lim lim (T T − z)−1 h dz, h ∈ K, 2π i ε→0 δ→0 Cτδ ε

Cσδ

where the contour and the set τε are defined as in [14] (see also [19]). The integral is understood in the strong sense as a limit of Riemann sums N (P )

lim

X

∆P →0

[∗]

(tj − tj−1 )φ0δε (sj )(T T − φδε (sj ))−1 h,

j=1

where φδε : [0, 1] → C is a parametrization of Cτδε , P = (t0 , . . . , tN (p) ) is a partition of [0, 1], ∆P is the mesh of the partition and sj ∈ [tj−1 , tj ]. Now for f ∈ D(T ) we have Z N (P ) X [∗] [∗] (T T − z)−1 T f dz = lim T (tj − tj−1 )φ0δε (sj )(T T − φδε (sj ))−1 f, ∆P →0

Cτδε

[∗]

j=1

[∗]

because (T T − z)−1 T f = T (T T − z)−1 f if only both resolvent operators exist in B(K). Since T is a closed operator, the element Z

N (P ) [∗]

−1

(T T − z)

X

f dz = lim

∆P →0

Cτδε

[∗]

(tj − tj−1 )φ0δε (sj )(T T − φδε (sj ))−1 f

j=1

is in the domain of T and Z N (P ) X [∗] [∗] T (T T − z)−1 f dz = lim T (tj − tj−1 )φ0δε (sj )(T T − φδε (sj ))−1 f ∆P →0

Cτδε

Z =

j=1 [∗]

(T T − z)−1 T f dz.

Cτδε

In the same way we can interchange T with the other two limits in the definition of E∗ (τ ) and get that E∗ (τ )T f = T E(τ )f for f ∈ D(T ) = D(E∗ (τ )T ). This proves the first of the inclusions (4.1) in the case τ ∈ R00 . Note that in this case we also have (see [14, Theorem 3.1(6)]) [∗]

R(E(τ )) ⊆ D(T T ) ⊆ D(T ). Since T is a closed operator we get also the first part of (4.2).

(4.3)

[∗]

Vol. 63 (2009)

Analysis of Spectral Points of T T and T T

[∗]

269

By definition we have E(R \ τ ) = I − E(τ ) (τ ∈ R00 ). Note that E(R \ τ )(D(T )) ⊆ D(T ) and so D(E∗ (R \ τ )T ) = D(T ) ⊆ D(T E(R \ τ )). Moreover, for f ∈ D(T ) we have E∗ (R \ τ )T f = (I − E∗ (τ ))T f = T f − E∗ (τ )T f = T f − T E(τ )f = T E(R \ τ )f. Theorem 4.2. Let λ ∈ R \ {0}. Then [∗]

[∗]

(i) λ ∈ ρ(T T ) if and only if λ ∈ ρ(T T ), [∗] [∗] (ii) λ is a critical point of T T if and only if it is a critical point of T T , [∗] (iii) λ is a regular critical point of T T if and only if it is a regular critical point [∗] of T T . [∗]

Proof. (i) Let λ ∈ τ ⊆ ρ(T T ) for some open τ ∈ R00 . Since E(τ ) = 0, we [∗] [∗] [∗] have E(τ )T f = 0 for f ∈ D(T ). From (4.2) we get T E∗ (τ ) = 0 and conse[∗] [∗] quently T T |R(E∗ (τ )) = 0. But on the other hand 0 ∈ ρ(T T |R(E∗ (τ )) ), because [∗] [∗] σ(T T |R(E∗ (τ )) ) ⊆ τ , [14, Theorem 3.1(7)]. Therefore, E∗ (τ ) = 0 and λ ∈ ρ(T T ). Point (ii) follows from Theorem 3.1 and the fact that c(A) = c0 (A) ∩ R for definitizable A. Let us turn to the proof of (iii). Suppose that λ is a regular critical [∗] point of T T and let us take a bounded closed neighborhood τ of λ such that [∗] [∗] τ ∩ (c(T T ) ∪ {0}) = ∅. Since λ is a regular critical point of T T the spectral function E is bounded on subsets of τ , i.e. there exists a constant c ≥ 0 such that kE(σ)k ≤ c,

σ ⊆ τ,

To prove that λ is a regular critical point for T T bounded on the subsets of τ , [14]. First set R1 := R(E(τ )),

σ ∈ R0 . [∗]

(4.4)

it is enough to show that E∗ is

R2 := R(E∗ (τ )),

and note that [14, p. 30(6)] [∗]

T T|R1 ∈ B(R1 ). [∗]

[∗]

Since 0 ∈ / τ we have that 0 ∈ ρ(T T|R1 ) [14, p. 30(7)] which means that T T is a [∗] bijection from R1 onto itself. Similarly, T T is a bijection from R2 onto itself. On the other hand Theorem 4.1 shows that [∗]

T (R2 ) ⊆ R1 , [∗]

T (R1 ) ⊆ R2 .

(4.5)

Hence R2 = T (T (R2 )) ⊆ T (R1 ) ⊆ R2 and consequently T (R1 ) = R2 . Similarly [∗] [∗] T (R2 ) = R1 . Therefore, T is a bijection from R1 onto R2 (and T is a bijection from R2 onto R1 ). Now for σ ⊆ τ such that λ ∈ σ ∈ R0 we get, since R(E∗ (σ)) ⊆ R2 ,

kE∗ (σ)k = kE∗ (σ)E∗ (τ )k = (T|R2 )−1 T|R2 E∗ (σ)E∗ (τ )

≤ (T|R2 )−1 kE(σ)T|R2 E∗ (τ )k

≤ (T|R2 )−1 c kT|R2 E∗ (τ )k .

270

Ran and Wojtylak

IEOT

To have a complete picture let us deal in this moment with the nonreal spectra. Proposition 4.3. Let τ = {λ} with λ ∈ / R. Then the formulas in Theorem 4.1 hold as well, with the interpretation that E({λ}) and E∗ ({λ}) are the Riesz’s projections [∗] [∗] onto the algebraic root spaces Sλ (T T ) and Sλ (T T ), respectively. Consequently, [∗] [∗] the nonreal spectra of T T and T T coincide. Proof. The proofs of the mentioned formulas are the same as the proof of The[∗] orem 4.1, only the limits in δ and ε are not necessary. Now let λ ∈ (ρ(T T ) ∩ [∗] σ(T T )) \ R. Since each nonreal point of spectrum is necessarily a common zero [∗] of all definitizing polynomials [14, p. 28], λ must be in c0 (T T ). By the first part [∗] [∗] of this proposition we have T E∗ ({λ}) = 0, and consequently T T |R(E∗ ({λ})) = 0. Since λ 6= 0, we have R(E({λ})) = {0}. The only result about the zero eigenvalue we can prove is the following. [∗]

Proposition 4.4. If 0 is in the resolvent of T T then it is not a singular critical [∗] point of T T . [∗]

Proof. Since the resolvent set is open and the nonzero spectrum of T T is equal [∗] [∗] to the nonzero spectrum of T T , zero is an isolated point of spectrum of T T and thus it can not be a singular critical point.

5. Analysis of infinity as a spectral point Let A be a definitizable operator. We write ∞ ∈ ρ(A) if and only if A is bounded (equivalently, if σ(A) is bounded, see [14, 10]). We say that infinity is in the positive (negative) spectrum if there exists a real neighborhood of infinity τ such that E(τ ) is positive (negative). We call infinity a critical point of a definitizable operator A if in each real neighborhood of infinity there exist points from both σ+ (S) and σ− (S). If infinity is a critical point we call it regular if the limits limx↑+∞ E([λ, x]) and limx↓−∞ E([x, λ]) exist in the strong operator topology for any (some) not critical λ ∈ R, otherwise we call it singular . Let us assume now (t1)–(t3) and look at Table 2. The reader can surely find examples for ∞ ∈ σ+ (T 2 ) and ∞ ∈ ρ(T 2 ), with T selfadjoint in a Hilbert space. By virtue of the results of the previous section (Theorem 4.2), we get ∞ ∈ [∗] [∗] ρ(T T ) ⇔ ∞ ∈ ρ(T T ), which completes the first row of the table. A direct consequence of Theorem 3.1 and Theorem 4.2 (and [14, Theorem 3.1(4)]) is the following proposition; we use the notation R± := {x ∈ R : ±x > 0}. Proposition 5.1. We have [∗]

[∗]

σ± (T T ) ∩ R+ = σ± (T T ) ∩ R+ ,

[∗]

[∗]

σ± (T T ) ∩ R− = σ∓ (T T ) ∩ R− .

(5.1) [∗]

Consequently, infinity can be a critical point of at most one of the operators T T [∗] and T T .

Vol. 63 (2009)

[∗]

Analysis of Spectral Points of T T and T T

[∗]

271

What remains to complete Table 2, is to show that infinity indeed can be a [∗] singular or regular critical point of an operator of the form T T . Example 5.1. Let us take a positive operator A with a singular (regular) critical point at infinity and 0 ∈ ρ(A) in a separable Krein space K with an infinite dimensional, uniformly positive [1] subspace K+ (e.g. take the one from [4] for the singular critical point, the regular case is left to reader as a simple exercise). [∗] We will show now that A = T T for some closed, densely defined T (cf. [1, Theorem VII.3.1] for the bounded case). Since A is positive and invertible, the space (D(A), [A·, ·]) is a unitary space. Indeed, if f ∈ D(A) is such that [Af, f ] = 0 then, by the Schwartz inequality, [Af, g] = 0 for g ∈ D(A). Consequently, Af = 0 and so f = 0. Note that the graph Γ(A) = {(f, Af ) : f ∈ D(A)}, with the topology inherited from K×K, is separable, as a closed subspace of a separable Hilbert (Krein) space. The linear mapping Γ(A) 3 (f, Af ) 7→ f ∈ D(A) is onto and continuous with respect to the [A·, ·]2 2 inner product topology on D(A), since [Af, f ] ≤ c kAf k kf k ≤ c(kAf k + kf k ) (f ∈ D(A)) for some c ≥ 0. Hence, (D(A), [A·, ·]) is separable as well. Let us take (H, h·, ·i) as the completion of the unitary space (D(A), [A·, ·]) to a Hilbert space. Since D(A) is dense in H, H is a separable Hilbert space. Therefore, there exists an isometric mapping U from the separable Hilbert space (H, h·, ·i) into the separable Hilbert space (K+ , [·, ·]). We define the operator Te in K as Tef = U f (f ∈ D(A)) and we denote by T its closure. Observe that [T f, T g] = [Af, g],

f, g ∈ D(A). [∗]

[∗]

If we fix f ∈ D(A) in the above, we get T f ∈ D(T ) and T T f = Af . This shows [∗] that T T ⊇ A. The operator on the left hand side is symmetric and the one on [∗] the right hand side is selfadjoint, hence T T = A.

6. Zero as a spectral point. Counterexamples in Π1 -spaces The last two sections concern Pontryagin spaces. In this section we discuss the results indicated in Table 1, insofar as they have not already been proved in Section 4. More information on the dimensions of the algebraic root spaces will be given in the next section. Recall that for T closed and densely defined in a Πκ -space the assumptions (t1)–(t3) are fulfilled. Let us now start completing Table 1. Proposition 4.4 has already been proved. Observe that there are obvious examples (in the class of bounded operators in a [∗] [∗] [∗] [∗] [∗] Hilbert space) for 0 ∈ ρ(T T ) ∩ ρ(T T ), 0 ∈ σ+ (T T ) ∩ σ+ (T T ), 0 ∈ σ+ (T T ) ∩ [∗] ρ(T T ). If T is a zero operator in a 2-dimensional Π1 -space then zero is a regular [∗] [∗] critical point for T T = T T . Now let us turn to more complicated examples.

272

Ran and Wojtylak

IEOT

[∗]

[∗]

Example 6.1. Zero is in the resolvent of T T and is a regular critical point of T T . Let us consider the Hilbert space `2 (Z \ {0}) and the fundamental symmetry J ∈ B(`2 (Z \ {0})), which is uniquely determined below by its action on the canonical basis (ej )j∈Z\{0} ,   ej : |j| > 1 e−1 : j = 1 J(ej ) = j ∈ Z \ {0} .  e1 : j = −1 It is clear, that `2 (Z \ {0}) with the inner product [·, ·] = hJ·, ·i is a Π1 space. We define the operator T ∈ B(`2 (Z \ {0})) by ej+1 : j > 0 T (ej ) = j ∈ Z \ {0} . ej−1 : j < 0 [∗]

It is easy to verify that T = JT ∗ J  0      e−1 [∗] e1 T (ej ) =   e  j−1   ej+1

acts in the following way: : : : : :

j j j j j

= 1, −1 =2 = −2 >2 < −2

j ∈ Z \ {0} .

Hence

[∗]

T T (ej ) = and [∗]

T T (ej ) =

e−j ej

 

0 e−j  ej

[∗]

: |j| = −1, 1 : |j| > 1

j ∈ Z \ {0}

: j = 1, −1 : j = 2, −2 : |j| > 2

j ∈ Z \ {0} .

Observe that T T can be viewed as a block space `2 (Z \ {0}) as follows: [∗] 0 T T =I⊕ 1 [∗]

operator matrix acting on a Hilbert 1 ⊕ I. 0 [∗]

From this it is apparent that σ(T T ) = {−1, 1} and likewise, σ(T T ) = {0, −1, 1}. [∗] Moreover, the algebraic root subspace corresponding to the zero eigenvalue of T T satisfies [∗] [∗] S0 (T T ) = N (T T ) = lin {e1 , e−1 } . Hence, it is a nondegenerate indefinite subspace and zero is a regular critical point [∗] of T T . [∗]

In the example above all Jordan chains for the zero eigenvalue for T T have [∗] length one. In fact, this is the maximal length if zero is in the resolvent of T T , by Theorem 7.2 in the next section. The next example illustrates the situation, [∗] [∗] when zero is in σ+ (T T ) and is a regular critical point of T T . Moreover, there

Vol. 63 (2009)

[∗]

Analysis of Spectral Points of T T and T T

[∗]

273

is a Jordan chain of length two corresponding to the zero eigenvalue of T T the operator T acts on a Π1 -space.

[∗]

and

Example 6.2. Let us consider the space `2 and the Π1 -inner product on `2 given by the fundamental symmetry J ∈ B(`2 ), which is defined on the canonical basis (ej )∞ j=0 as   e2 : j = 0 e0 : j = 2 J(ej ) = j ∈ N.  ej : j 6= 0, 2 We define the operator T by T (ej ) = [∗] T = JT ∗ J satisfies  ej+1    [∗] 0 T (ej ) = e0    ej−1

ej+1 (j ∈ N). It is easy to compute that : : : :

j j j j

= 0, 1 =2 =3 >3

j∈N

and hence  e2    [∗] 0 T T (ej ) = e  0   ej

: : : :

j j j j

=0 =1 =2 >2

 ej+2    [∗] 0 T T (ej ) = e  1   ej

: : : :

j j j j

= 0, 1 =2 =3 > 3.

j∈N

and

[∗]

j ∈ N.

[∗]

It is apparent that σ(T T ) = σ(T T ) = {−1, 0, 1}. However, 0 is in the positive [∗] part of spectrum of T T , since [∗]

[∗]

S0 (T T ) = N (T T ) = lin {e1 } , which is a positive space. On the other hand, (e0 , e2 ) is a Jordan chain for the [∗] eigenvalue 0 of T T and the space [∗]

[∗]

S0 (T T ) = N ((T T )2 ) = lin {e0 , e2 } [∗]

is indefinite and nondegenerate. Hence, 0 is a regular critical point for T T . The example [15, Example 3.8] (see also [11]) shows a bounded, selfadjoint operator T in a Π1 -space such that zero is a singular critical point of T 2 . This is the lower-right corner of Table 1. A modification of that example, which is shown [∗] below, leads to a situation where zero is in the positive part of spectrum of T T [∗] and is a singular critical point of T T .

274

Ran and Wojtylak

IEOT

Example 6.3. Let K be the Hilbert space L2 [0, 1] ⊕ C2 ⊕ `2 with the natural scalar product h·, ·i. We define the fundamental symmetry J(f, x, y, l) = (f, y, x, l) for all f ∈ L2 [0, 1], x, y ∈ C, l ∈ `2 . Obviously, (K, hJ·, ·i) is a Π1 -space. Consider the operator   M√ t 0 π(1) 0 h·, 1i 0 0 0  T :=   0 0 0 0 0 π(e1 ) 0 S where Mφ ∈ B(L2 [0, 1]) denotes the multiplication operator by a bounded function φ, S is the shift operator in `2 (Sej = ej+1 , j ∈ N), π(g) (where g is an element of some Hilbert space) maps x ∈ C to xg and 1 ∈ L2 [0, 1] is a function constantly equal one. It is not hard to compute that   M√t π(1) 0 0  0 [∗] 0 0 h·, e1 i J T =J h·, 1i 0 0 0  0 0 0 S∗   M√t 0 π(1) 0 h·, 1i 0 0 0  . =  0 0 0 h·, e1 i 0 0 0 S∗ Next we compute

√   √t 0 π( t) 0

M  ·, t 0 [∗] 1 0 . T T =  0 1 0 0 0 0 0 I`2

(The zero in position (3,4) is because hSl, e1 i = 0 for all l ∈ `2 ; the zero at (4,2) [∗] is because S ∗ e1 = 0.) Let us note that 0 is not an eigenvalue of T T . Indeed, if [∗] T T (f, x, y, l) = 0 for some (f, x, y, l) ∈ K then in particular √ tf (t) + y t = 0 a.e. in t on [0, 1], x = 0, l = 0. If y 6= 0 then the first equation does not have any solution in f ∈ L2 [0, 1]. Hence y = 0 and consequently f = 0. And so we proved that zero is not in the point [∗] [∗] spectrum of T T . Since this T T is selfadjoint operator in a Pontryagin space, we [∗] know that zero is either in the positive spectrum or in the resolvent of T T . The latter option is not possible, since we will now prove that zero is a singular critical [∗] [∗] point of T T and hence, by Proposition 4.4, it cannot be in ρ(T T ).

[∗]

Vol. 63 (2009)

Analysis of Spectral Points of T T and T T

[∗]

275

A simple computation shows that √   √t 0 π( t) h·, e1 i 1

M  ·, t [∗] 0 1 0  . TT =   0 0 0 0  h·, 1i e1 0 0 SS ∗ [∗]

An element (f, x, y, l) of the kernel of T T satisfies √ tf (t) + y t + hl, e1 i = 0 a.e. in t on [0, 1], D √E f, t + y = 0,

(6.1)

hf, 1i e1 + SS ∗ l = 0. The first equation shows that y = 0 and hl, e1 i = 0,

(6.2)

2

otherwise it has no solutions in f ∈ L ([0, 1]). Hence, f = 0. Therefore, the last equation gives l = ce1 with some c ∈ C. By (6.2), c = 0. Therefore, the space [∗] {(0, x, 0, 0) : x ∈ C} is the kernel of T T . Observe that it is also the algebraic root [∗] space. Indeed, if T T (f, x, y, l) = (0, 1, 0, 0) then (6.1) holds and consequently f = 0, y = 0 and l = 0. Note that {(0, x, 0, 0) : x ∈ C} is a degenerate subspace. [∗] [∗] Resuming, zero is in σ+ (T T ) and it is a singular critical point of T T . Example 6.4. Consider the space K = L2 [0, 1]⊕C2 with the fundamental symmetry J(f, x, y) = (f, y, x) for all f ∈ L2 [0, 1], x, y ∈ C, which makes it a Π1 -space. The operator   M√ t 0 0 T := h·, 1i 0 0 0 0 0 [∗]

has the property that zero is a regular critical point for T T and singular critical [∗] point for T T . The details are similar to the ones in the previous example and therefore they are left to the reader. [∗]

[∗]

7. Operators T T and T T in Pontryagin spaces. Comparing the Jordan chains The reasoning in this section is independent on the type of spectral point λ. We use only linear algebra combined with the information below concerning the lengths of the Jordan chains. The difference between non-critical, singular and regular critical points lies in the fact that the spectral projection E({λ}) may or may not exist, but this does not influence the Jordan structure at all. If A is a selfadjoint operator in a Πκ space then all the Jordan chains are not longer than 2κ+1, i.e. Sλ (A) = N ((A−λ)2κ+1 ), [1, 14, 9]. For each nonreal spectral

276

Ran and Wojtylak

IEOT

˙ λ¯ (A) point λ the algebraic root subspace Sλ (A) is finite dimensional and Sλ (A)+S is nondegenerate (hence, it is a Pontryagin space). The notion of a Jordan chain corresponding to an eigenvalue of A makes sense as well. Namely, for each eigenvalue λ of A there exists a decomposition Sλ (A) = K0 u K1 (formally, we should write Kiλ , i = 0, 1), such that both spaces K0 , K1 are invariant for A, K0 is finite dimensional, and A |K1 has no Jordan chains longer than one. The construction given in [9, Theorem 7.2] is not unique and for our purposes we need to proceed in a slightly different way, although the aforementioned result guarantees that our reasoning makes sense. In particular we can define the Segre characteristic (nj )∞ j=0 for the operator A and a point λ ∈ C. Namely, as n0 , . . . , nk we set the lengths of Jordan chains of A |K0 in decreasing order. If the space K0 is trivial, we set k = −1. We put nj = 1 for j = k + 1, . . . , k + dim K1 and nj = 0 for j > k + dim K1 , if dim K1 < ∞. Obviously, in the finite dimensional case this definition agrees with the standard one (decreasing sequence of sizes of Jordan blocks extended by an infinite number of zeros). Note that the linear space Sλ (A)/N (A − λ) is finite dimensional. Moreover, the operator [A]λ : Sλ (A)/N (A − λ) 3 [f ]λ 7→ [Af ]λ ∈ Sλ (A)/N (A − λ) is well defined. Later on we will omit the subscripts λ and we will not distinguish (in notation) between the λI operators in the original and quotient space. Lemma 7.1. Let A be a selfadjoint operator in a Pontryagin space, let λ be an eigenvalue of A. There exists a one-to-one correspondence between the Jordan chains longer than one of A |K0 and the oJordan chains of [A], in the following n sense: if

(j)

fi

(j)

: i = 0, . . . , nj , j = 0, . . . , k (j)

(A − λ)fi = fi−1 (i = 1, . . . , nj ), n (j) then fi : i = 1, . . . , nj , j = 0, . . . , k, and (j) (j) ([A] − λ) fi = fi−1 , (j) (j) ([A] − λ) f1 = f0 = 0,

is a basis for K0 such that (j)

(A − λ)f0 = 0, j = 0, . . . , k, (7.1) o nj ≥ 1 is a basis for Sλ (A)/N (A − λ) i = 1, . . . , nj ,

j = 0, . . . , k,

j = 0, . . . , k.

(7.2) (7.3)

Consequently, if (nj )∞ j=0 is the Segre characteristic for A, then n ˜ j = max {nj − 1, 0} ,

j = 0, 1, . . . ,

is the Segre characteristic for [A]. Proof. It is apparent, that (7.1) implies (7.2) and (7.3). For simplicity set J := {(i, j) : i = 1, . . . , nj , j = 0, . . . , l, nj ≥ 1} .

(7.4)

[∗]

Vol. 63 (2009)

Analysis of Spectral Points of T T and T T

[∗]

277

n o (j) Now we prove that the vectors fi : (i, j) ∈ J are linearly independent. Indeed, if X (j) αij fi =0 (i,j)∈J

for some complex numbers αij with (i, j) ∈ J then X (j) αij fi ∈ N (A − λ). (i,j)∈J

n o (j) But since fi : i = 0, . . . , nj , j = 0, . . . , l is a Jordan basis for the operator A|K0 , we get αij = 0 for (i, j) ∈ J. The mapping K0 3 f 7→ [f ] ∈ Sλ (A)/N (A − λ) n o (j) is onto, so fi : (i, j) ∈ J is a basis for Sλ (A)/N (A − λ) . The claim on the relation between Segre characteristics follows directly from the forms of basis. Now we return to the case where the operator A under consideration is of the [∗] [∗] [∗] form T T or T T . First we recall that the negative part of the spectrum of T T is finite, thus there are no singular critical points on the negative part of the real axis. Moreover, all the algebraic root spaces corresponding to negative eigenvalues are finite dimensional (see, e.g., [15]). Next let T be a closed densely defined operator in a Pontryagin space and let λ ∈ C. Note that for j = 1, . . . , 2κ + 1 we have [∗] [∗] [∗] [∗] [∗] T N ((T T − λ)j ) ⊆ N ((T T − λ)j ), T N ((T T − λ)j ) ⊆ N ((T T − λ)j ). (7.5) In particular, the following operators are well defined: [∗]

[∗]

[∗]

[∗]

[∗]

[∗]

[T ] : Sλ (T T )/N (T T − λ) 3 [f ] 7→ [T f ] ∈ Sλ (T T )/N (T T − λ) [∗]

[∗]

[∗]

[∗]

[T ] : Sλ (T T )/N (T T − λ) 3 [f ] 7→ [T f ] ∈ Sλ (T T )/N (T T − λ) [∗]

[∗]

([g] stands for the equivalence class of an element g both in Sλ (T T )/N (T T − λ) [∗] [∗] and Sλ (T T )/N (T T −λ), the subscript λ has been omitted as before). Moreover, [∗]

[∗]

[T T ] = [T ][T ],

[∗]

[∗]

[T T ] = [T ][T ].

(7.6)

∞ Theorem 7.2. Let λ be a complex number, and denote by (nj )∞ j=1 and (mj )j=1 the [∗]

[∗]

Segre characteristics for T T and T T respectively, corresponding to λ. If λ 6= 0 then nj = mj for all j ∈ N. If λ = 0 then |nj − mj | ≤ 1 for j ∈ N. Proof. First let λ = 0. The result of Flanders [6, Theorem 2], applied to the [∗] [∗] operators [T ][T ] and [T ][T ], together with (7.6) and Lemma 7.1 give |˜ nj − m ˜ j | ≤ 1,

j ∈ N,

(7.7)

278

Ran and Wojtylak

IEOT [∗]

[∗]

where (˜ nj )∞ ˜ j )∞ j=0 and (m j=0 are Segre characteristics for [T T ] and [T T ], respectively. The formulas (7.7) and (7.4) show that |nl − ml | ≤ 1 for all l such that nl ≥ 3 or ml ≥ 3. Consequently, |nl − ml | ≤ 1 also if nl = 1 or if ml = 1. Hence, the only case we have to exclude is nl = 2, ml = 0 (or conversely) for some l ∈ N. (0) (l) Let then nl = 2. Take the vectors f1 , . . . , f1 from Lemma 7.1, which is possible (0) (l) because nj ≥ nl ≥ 2 for j = 0, . . . , l. Note that the vectors T f1 , . . . , T f1 are linearly independent. Indeed, if l X

(j)

αj T f1

=0

l X

(j)

for some α1 , . . . , αl ∈ C,

j=0

then αj f1

[∗]

∈ N (T ) ⊆ N (T T ).

j=0

(j) Since the vectors fi , j = 1, . . . , l are linearly independent (Lemma 7.1), we get αj = 0 (j = 0, . . . , l). Hence, there are l linearly independent vectors in [∗] [∗] T (N (T T )) ⊆ N (T T ). Consequently ml ≥ 1 (otherwise there are at most l − 1 [∗] linearly independent vectors in N (T T )). The case λ 6= 0 is similar. Indeed, in this case we can follow the same argument applying Theorem 1 in [6] instead of Theorem 2, to obtain that n ˜j = m ˜ j for j ∈ N. As a result, nj = mj whenever either one is larger than one. It remains to exclude the case nl = 1 and ml = 0 (or conversely) for some l ∈ N. It should be noted [∗] [∗] [∗] that T maps N (T T − λ) in a one-to-one way onto N (T T − λ) (λ−1 T is the [∗] inverse). Now suppose that ml = 0, then it follows that N (T T − λ) is finite dimensional and by the observation in the previous sentence it follows that also [∗] N (T T − λ) is finite dimensional and has the same dimension. Hence, ml must be zero as well.

8. Final remarks The condition given by Flanders is necessary and sufficient. Namely, given two ∞ sequences (nj )∞ j=0 and (mj )j=0 satisfying (1.1) we can always construct matrices A and B such that AB and BA have only zero in the spectrum and (nj )∞ j=0 and (mj )∞ j=0 are the Segre characteristics for AB and BA. This solves the finite [∗]

[∗]

dimensional problem completely. For the pair of operators T T and T T in a finite dimensional Pontryagin space this result is not true. For example it is easy to show [∗] [∗] that it is not possible that both operators T T and T T have only one Jordan chain of the same length bigger than one. Some parts of the analysis were done in [12], while a different perspective is taken in [16]. The latter paper solves the finite dimensional problem completely in different terms than ours. The reduction of the Pontryagin space case to the finite dimensional case involves, among other

Vol. 63 (2009)

[∗]

Analysis of Spectral Points of T T and T T

[∗]

279

things, the procedure described in the last section. We shall return to this issue in a subsequent paper.

References [1] J. Bogn´ ar, Indefinite Inner Product Spaces, Springer-Verlag, New York, Heidelberg, 1974. [2] Y. Bolshakov, C.V.M. van der Mee, A.C.M. Ran, B. Reichstein, L. Rodman, Polar Decompositions in Finite Dimensional Indefinite Scalar Product Spaces: General Theory, Linear Algebra and Its Applications, 261 (1997), 91–147. [3] Y. Bolshakov, B. Reichstein. Unitary equivalence in an indefinite scalar product: An analogue of singular value decomposition, Linear Algebra and its Applications, 222 (1995), 155–226. ´ [4] B. Curgus, A. Gheondea, H. Langer, On singular critical points of positive operators in Krein spaces, Proc. AMS, 128 (2000), 2621–2626. [5] N. Dunford, J.T. Schwartz, Linear Operators Part II: Spectral theory. Self adjoint operators in Hilbert space. Interscience Publishers John Wiley & Sons, New York, London, 1963. [6] H. Flanders, Elementary divisors of AB and BA, Proc. AMS, 2 (1951), 871–874. [7] I. Gohberg, P. Lancaster, L. Rodman, Indefinite Linear Algebra and Applications, Birkh¨ auser, Basel, Boston, Berlin, 2005. [8] P.R. Halmos, A Hilbert Space Problem Book, Van Nostrand, New York, 1967. [9] I.S. Iohvidov, M.G. Krein, H. Langer, Introduction to spectral theory of operators in spaces with indefinite metric, Mathematical Research, vol. 9. Akademie-Verlag, Berlin, 1982. [10] P. Jonas, On the functional calculus and the spectral function for definitizable operator in Krein space, Beitr¨ age Anal., 16 (1981), 121–135. [11] P. Jonas, H. Langer, B. Textorius, Models and unitary equivalence of cyclic selfadjoint operators in Pontrjagin spaces, Operator Theory: Advances and Applications, 59 (1992), 252–284. [12] J.S. Kes, A.C.M. Ran, On the relation between XX [∗] and X [∗] X in an indefinite inner product space, Operators and Matrices, 1, No. 2 (2007), 181–197. ¨ [13] M.G. Krein, H. Langer, Uber die Q-Funktion eines π-hermiteschen Operators im Raume Πκ , Acta. Sci. Math. (Szeged), 34 (1973), 191–230. [14] H. Langer, Spectral functions of definitizable operators in Krein spaces, Proc. Graduate School “Functional Analysis”, Dubrovnik 1981. Lecture Notes in Math. 948, Springer Verlag, Berlin, 1982, pp. 1–46. [15] C.V.M. van der Mee, A.C.M. Ran, L. Rodman, Polar decompositions and related classes of Operators in spaces Πκ , Integral Equations and Operator Theory, 44 (2002), 50–70. [16] C. Mehl, V. Mehrmann, H. Xu, Structured decompositions for matrix triples: SVDlike concepts for structured matrices. Submitted for publication. See also Technical Report No. 514, DFG Research Center Matheon, Berlin, 2008.

280

Ran and Wojtylak

IEOT

[17] C. Mehl, A.C.M. Ran, L.Rodman, Polar decompositions of normal operators in indefinite inner product spaces, Proceedings of 3rd Workshop on Indefinite Inner Products, Operator Theory: Advances and Applications 162, (2006), 277–292. [18] B. Thaller, The Dirac Equation, Springer-Verlag, Berlin, Heidelberg, 1992. [19] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, New York, Berlin, 1980. Andr´e Ran VU University Department of Mathematics Faculty of Exact Sciences De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: [email protected] Michal Wojtylak VU University Department of Mathematics Faculty of Exact Sciences De Boelelaan 1081a 1081 HV Amsterdam The Netherlands Permanent address: Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Krak´ ow Poland e-mail: [email protected] Submitted: June 18, 2008. Revised: September 11, 2008.

Integr. equ. oper. theory 63 (2009), 281–296 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020281-16, published online January 27, 2009 DOI 10.1007/s00020-008-1649-7

Integral Equations and Operator Theory

Integral Representations for Generalized Difference Kernels Having a Finite Number of Negative Squares J. Rovnyak and L. A. Sakhnovich Abstract. An integral representation is derived for matrix-valued generalized difference kernels which have a finite number of negative squares. The representation is used to extend such kernels to the real line with a bound on the number of negative squares. The main results are obtained by means of an operator interpolation theorem. The nondegenerate case is assumed. Mathematics Subject Classification (2000). Primary 47A57; Secondary 47A56, 30E05. Keywords. Difference kernel, negative squares, extension problem, generalized Nevanlinna function, interpolation, operator identity.

1. Introduction If s(x) is a measurable m × m matrix-valued function on a finite interval (−`, `), then the formula Z ` d (Sf )(x) = s(x − t)f (t) dt (1.1) dx 0 defines a bounded operator on the Lebesgue space L2m (0, `) of m-dimensional vector-valued functions on (0, `) if (1) for every g in Cm , s(x)g belongs to L2m (−`, `),R and ` (2) for every f in L2m (0, `), the function F (x) = 0 s(x − t)f (t) dt is absolutely 2 continuous and has derivative in Lm (0, `). In this case we call s(x − t) a generalized difference kernel. The operator S is selfadjoint if s(x) = −s(−x)∗ a.e. on (−`, `). The class of bounded selfadjoint operators of the form (1.1) coincides with the set of solutions of an operator identity involving the classical Volterra operator (see Theorem 2.1). The class includes many integral operators with ordinary difference kernels. For example, if C is a

282

Rovnyak and Sakhnovich

IEOT

selfadjoint m × m matrix and k(x) = k(−x)∗ is an integrable m × m matrix-valued function on (−`, `), the operator Z ` (Sf )(x) = Cf (x) + k(x − t)f (t) dt (1.2) 0

Rx on L2m (0, `) has the form (1.1) with s(x) = 12 sgn(x) C + 0 k(u) du on (−`, `). Bounded selfadjoint operators of the form (1.1) which satisfy the additional condition S ≥ 0 play an important role in many areas including interpolation problems and the spectral theory of canonical differential systems [17, 18, 19]. The condition S ≥ 0 in (1.1) implies that s(x) has a representation Z ∞ d itx itx dτ (t) s(x) = 1+ − e + iC0 , (1.3) dx −∞ 1 + t2 t2 where τ (t) is a nondecreasing m×m matrix-valued function such that dτ (t)/(1+t2 ) is integrable over the real line and C0 is a constant selfadjoint m × m matrix (see [18, p. 22] and [20, p. 503]). The formula (1.3) is derived from the Nevanlinna representation of a certain Nevanlinna function which solves a related abstract interpolation problem. The condition S ≥ 0 is referred to as the definite case. In this paper we adapt similar ideas to the indefinite theory. The condition S ≥ 0 is replaced by the requirement that κS < ∞,

(1.4)

that is, the negative spectrum of S consists of finitely many eigenvalues having finite total multiplicity κS . When (1.4) is satisfied, we say that the generalized difference kernel s(x − t) has a finite number of negative squares. In the indefinite case, we use generalized Nevanlinna functions and recent extensions of the theory of operator identities which appear in [13, 14]. A substitute for the Nevanlinna representation is available in the Kre˘ın-Langer integral representation [2, 9, 12] of a generalized Nevanlinna function. Our main result, Theorem 3.1, is a generalization of (1.3) to generalized difference kernels s(x−t) having a finite number of negative squares. For technical reasons, we also assume that S is invertible (nondegenerate case). An immediate consequence of the representation is that s(x − t) can be extended to arbitrarily large intervals with a bound on the number of negative squares (see Theorem 3.3). Our results extend those of A. L. Sakhnovich [15, Section 3]. The study of difference kernels on finite intervals for both scalar- and matrixvalued functions has a long history, which we only partially give. In the case of positive kernels, the study was initiated by Kre˘ın [7], who showed that if k(x) is a continuous scalar-valued function on a finite interval [−`, `] such that the kernel k(x − t) is positive definite, then Z ∞ k(x) = eixt dτ (t) −∞

Vol. 63 (2009)

Integral Representations for Generalized Difference Kernels

283

for some bounded nondecreasing function τ (t). Analogous problems for operators of the form (1.2) are said to be of accelerant type. See Arov and Dym [1] for a comprehensive account and literature references in the positive case. In the indefinite case, an extension theorem and integral representation for continuous Hermitian kernels was given by Kre˘ın [8]. Grossmann and Langer [3, p. 314] and Kaltenb¨ack, Winkler, and Woracek [5, p. 270] obtain more precise extension theorems for such kernels. Connections with generalized Nevanlinna functions, interpolation theory, and canonical differential systems are detailed in Kre˘ın and Langer [9, 10]. Related questions in the indefinite theory are investigated in Kaltenb¨ack and Woracek [6] and Langer, Langer, and Sasv´ ari [11]. Section 2 is devoted to preliminaries on generalized difference kernels, operator identities, and generalized Nevanlinna functions. Our main results are presented in Section 3. A technical detail is deferred to an appendix.

2. Operator identities and generalized Nevanlinna functions We use the theory of operator identities AS − SA∗ = i Φ1 Φ∗2 + Φ2 Φ∗1 ,

(2.1)

where A, S ∈ L(H) and Φ1 , Φ2 ∈ L(G, H) for some Hilbert spaces H and G with dim G < ∞. In our applications, H = L2m (0, `), G = Cm , where ` is a positive number and m is a positive integer (which are fixed throughout), and Z x (Af )(x) = i f (t) dt, (2.2) 0

Φ2 g = g,

(2.3)

for all f in H and g in G. When A is given by (2.2), an operator S which satisfies an identity of the form (2.1) is necessarily selfadjoint. In fact, by (2.1) the operator X = S ∗ − S satisfies AX − XA∗ = 0, and hence X = 0, that is, S = S ∗ . Our first result identifies the class of selfadjoint operators satisfying an operator identity (2.1) as the class of selfadjoint operators of the form (1.1). Theorem 2.1. Let H = L2m (0, `), G = Cm , and define A ∈ L(H) and Φ2 ∈ L(G, H) by (2.2) and (2.3). (i) If S ∈ L(H) and Φ1 ∈ L(G, H) satisfy (2.1), there exists a measurable m × m matrix-valued function s(x) on (−`, `) such that s(x) = −s(−x)∗ a.e. and Z ` d (Sf )(x) = s(x − t)f (t) dt, (2.4) dx 0 (Φ1 g)(x) = ϕ1 (x)g,

ϕ1 (x) = s(x),

0 < x < `,

(2.5)

for each f in H and g in G. (ii) Conversely, if S ∈ L(H) and Φ1 ∈ L(G, H) have the form (2.4) and (2.5) and s(x) = −s(−x)∗ a.e. on (−`, `), then S and Φ1 satisfy (2.1).

284

Rovnyak and Sakhnovich

IEOT

In the scalar case, this result is given in [17, pp. 10,19] The matrix case can be proved similarly using [16, Lemma 2]. For completeness, we outline a direct proof. Proof. (i) Since Φ1 ∈ L(G, H), there is a measurable m×m matrix-valued function ϕ1 (x) on (0, `) such that (Φ1 g)(x) = ϕ1 (x)g for all g in G. Define s(x) = ϕ1 (x) on (0, `), and extend the definition so that s(x) = −s(−x)∗ a.e. on (−`, `). Then (2.5) holds by construction. To prove (2.4), define an operator S1 ∈ L(H) by Z `Z t S1 f = s(x − u) − s(−u) du f (t) dt. 0

0

A straightforward calculation shows that AS1 − S1 A∗ = i A Φ1 Φ∗2 + Φ2 Φ∗1 A∗ . The same identity is satisfied with S1 replaced by ASA∗ , and therefore the operator X = S1 −ASA∗ satisfies AX −XA∗ = 0. It follows that X = 0, that is, S1 = ASA∗ . Hence for any h in H, Z `Z t ∗ ASA h = s(x − u) − s(−u) du h(t) dt 0

0

Z

`

s(x − u) − s(−u) (−i)

=i 0

Z =i

Z

`

h(t) dt du u

`

s(x − u) − s(−u) (A∗ h)(u) du.

0

Therefore for a dense set of functions f in H, Z ` ASf = i s(x − t) − s(−t) f (t) dt.

(2.6)

0

By approximation, the same equation holds for all f in H. In other words, for each f in H and x in (0, `), Z x Z ` Z ` (Sf )(t) dt = s(x − t)f (t) dt − s(−t)f (t) dt. (2.7) 0

0

0

This proves (2.4), and (i) follows. (ii) Let S and Φ1 have the forms (2.4) and (2.5). From the definition of S we deduce (2.7). Hence AS is given by (2.6). The condition s(x) = −s(−x)∗ implies that S = S ∗ , and therefore a formula for SA∗ = (AS)∗ can obtained from (2.6). Then (2.1) follows by a routine calculation. By the generalized Nevanlinna class Nκ we mean the set of m × m matrixvalued meromorphic functions v(z) on the union C+ ∪ C− of the upper and lower ¯ has κ negahalf-planes such that v(z) = v(¯ z )∗ and the kernel [v(z)−v(ζ)∗ ]/(z − ζ) tive squares. The generalized Nevanlinna functions which occur in our applications

Vol. 63 (2009)

Integral Representations for Generalized Difference Kernels

285

satisfy lim

|y|→∞

v(iy) = 0. y

(2.8)

Every generalized Nevanlinna function v(z) which satisfies (2.8) has a Kre˘ın-Langer integral representation r Z X 1 v(z) = − Sj (t, z) dτ (t) + R(z), (2.9) t−z j=0 ∆j where ∆1 , . . . , ∆r are bounded open intervals having disjoint closures, ∆0 is the complement of their union in the real line, and (1◦ ) there are points αj ∈ ∆j , j = 1, . . . , r, and positive integers ρ1 , . . . , ρr such that 2ρj 1 1 t − αj − Sj (t, z) = on ∆j , j = 1, . . . , r , t−z t − z z − αj 1 1 + tz 1 − S0 (t, z) = on ∆0 ; t−z t − z 1 + t2 (2◦ ) τ (t) is an m × m matrix-valued function which is nondecreasing on each of the r + 1 open intervals determined by α1 , . . . , αr such that the integral Z ∞ (t − α1 )2ρ1 · · · (t − αr )2ρr dτ (t) (1 + t2 )ρ1 +···+ρr 1 + t2 −∞ is convergent; (3◦ ) R(z) is an m × m matrix-valued rational function which is analytic at infinity and satisfies R(z) = R(¯ z )∗ ; equivalently, s 1 1 ∗ X R(z) = C0 − Rk + Rk , (2.10) z − λk z¯ − λk k=1

where λ1 , . . . , λs are distinct points in the closed upper half-plane, the functions R1 (z), . . . , Rs (z) are polynomials of the form Rk (z) = Rk1 z + · · · + Rk,σk z σk ,

k = 1, . . . , s,

and C0 = R(∞) is a selfadjoint m × m matrix. Conversely, every function of the form (2.9) is a generalized Nevanlinna function which satisfies (2.8). A Stieltjes inversion formula recovers increments of τ (t) in the open intervals of the real line determined by the points α1 , . . . , αr . For details, see [2, 9, 12]. We recall some constructions from [14]. Consider arbitrary Hilbert spaces H and G, dim G < ∞, and operators A ∈ L(H) and Φ2 ∈ L(G, H). Let v(z) be a generalized Nevanlinna function satisfying (2.8) and having Kre˘ın-Langer

286

Rovnyak and Sakhnovich

IEOT

representation (2.9). Assume that the spectrum of A contains only the point at the origin and that Z

dτ (t) Φ∗2 (I − A∗ t)−1 h, Φ∗2 (I − A∗ t)−1 h < ∞, h ∈ H. (2.11) ∆0

Then the formulas r Z n o X Sv = (I − At)−1 Φ2 [dτ (t)]Φ∗2 (I − A∗ t)−1 − dτj (t; A, Φ2 ) j=0

(2.12)

∆j

Z 1 (I − λA)−1 Φ2 R(λ)Φ∗2 (I − λA∗ )−1 dλ , 2πi Γ r Z n o X = A(I − At)−1 − Sj (t; A) Φ2 [dτ (t)] −

i Φ1,v

j=0

(2.13)

∆j

Z 1 − A(I − λA)−1 Φ2 R(λ) dλ − Φ2 R(∞) . 2πi Γ define operators Sv ∈ L(H) and Φ1,v ∈ L(G, H). Here Γ is any closed contour that winds once counterclockwise about each pole of R(λ). The convergence terms dτj (t; A, Φ2 ) and Sj (t; A), j = 1, . . . , r, are defined to be the Taylor polynomials about t = αj of order 2ρj − 1 of the main terms F (t) = (I − At)−1 Φ2 [dτ (t)]Φ∗2 (I − A∗ t)−1

G(t) = A(I − At)−1 .

and

For j = 0, dτ0 (t; A, Φ2 ) = 0 and S0 (t; A) = −tI/(1 + t2 ). With these choices, the integrals in (2.12) and (2.13) converge weakly by the conditions (1◦ ) and (2◦ ) and the assumption (2.11). By Theorems 3.4 and 3.5 of [14], the definitions of Sv and Φ1,v are independent of the choice of Kre˘ın-Langer representation, Sv is selfadjoint, and ASv − Sv A∗ = i Φ1,v Φ∗2 + Φ2 Φ∗1,v . (2.14) In an appendix, we prove that r s X X κSv ≤ m ρj + σk . j=1

(2.15)

k=1

We return to the case when H = L2m (0, `), G = Cm , and A and Φ2 are given by (2.2) and (2.3). Then the preceding formulas take a more concrete form. In particular, the hypothesis (2.11) is expressed as an integrability property of certain Fourier integrals. Let v(z) be a generalized Nevanlinna function satisfying (2.8) and having Kre˘ın-Langer representation (2.9). Since Z ` Φ∗2 (I − A∗ z)−1 f = e−izt f (t) dt, (2.16) 0

the condition (2.11) asserts that for all f ∈ L2m (0, `), Z Z ` F (t)∗ [dτ (t)] F (t) < ∞ where F (z) = e−izt f (t) dt. ∆0

0

(2.17)

Vol. 63 (2009)

Integral Representations for Generalized Difference Kernels

287

In view of the identity Z 2` Z ` Z ` −izt −izt −iz` e f (t) dt = e f (t) dt + e e−izt f (t + `) dt, 0

0

0

this property is independent of the choice of `. Definition 2.2. A generalized Nevanlinna function v(z) with representation (2.9) is called admissible if it satisfies (2.17) for some and hence every ` > 0. If v(z) is admissible, then operators Sv and Φ1,v are defined, and they have the forms (2.4) and (2.5) by Theorem 2.1. Theorem 2.3. Let H = L2m (0, `) and G = Cm , and let A and Φ2 be given by (2.2) and (2.3). Let S ∈ L(H) and Φ1 ∈ L(G, H) satisfy (2.1). If κS < ∞ and S is invertible, there is an admissible generalized Nevanlinna function v(z) such that S = Sv

and

Φ1 = Φ1,v .

(2.18)

Theorem 2.3 is proved in [13] by an explicit construction of a family of admissible generalized Nevanlinna functions v(z) satisfying (2.18).

3. Main results Given a nonnegative integer κ, let S`,κ be the set of measurable m × m matrixvalued functions s(x) on (−`, `) such that s(x) = −s(−x)∗ a.e. and such that (1.1) defines a bounded operator S with κS = κ. Theorem 3.1. Let s(x) ∈ S`,κ , and assume that the associated operator S defined by (1.1) is invertible. Then there is an admissible generalized Nevanlinna function v(z) such that S = Sv . If v(z) is represented in the form (2.9), then s(ξ) =

r X

sj (ξ) + sd (ξ) ,

(3.1)

j=0

where d s0 (ξ) = dξ

Z

d sj (ξ) = dξ

Z

1+

∆0

"

∆j

itξ itξ dτ (t) −e , 1 + t2 t2

(3.2)

# 2ρj −1 X 1 + itξ − eitξ ν − Qν (αj , ξ)(t − αj ) dτ (t) , t2 ν=0

(3.3)

j = 1, . . . , r, and sd (ξ) =

σk h s X X k=1 ν=1

a.e. on (−`, `).

¯ k , ξ)R∗ Pν−1 (λk , ξ)Rkν + Pν−1 (λ kν

i

+ iC0 ,

(3.4)

288

Rovnyak and Sakhnovich

IEOT

The formulas (3.1)–(3.4) use integrated forms of the Fourier kernel, which we write as ∞

eitx − 1 X = Pν (λ, x)(t − λ)ν , it ν=0 ∞

X 1 + itx − eitx = Qν (λ, x)(t − λ)ν . t2 ν=0 For each ν = 0, 1, 2, . . . , 1 ∂ ν eitx − 1 Pν (λ, x) = ν! ∂t it t=λ " # ν 1 (−1)ν iλx X (−iλx)k e = −1 , i λν+1 k! k=0 1 ∂ ν 1 + itx − eitx Qν (λ, x) = ν! ∂t t2 t=λ " # ν ν+1 k X (−1) (−iλx) = eiλx (ν − k + 1) − ν − 1 − iλx . λν+2 k! k=0

When λ = 0, these expressions are interpreted by continuity: Pν (0, x) =

iν xν+1 , (ν + 1)!

Qν (0, x) =

iν xν+2 . (ν + 2)!

The equations (3.2) and (3.3) will be proved in the forms Z ξ Z 1 + itξ − eitξ itξ s0 (u) du = − dτ (t), t2 1 + t2 0 ∆0 # Z ξ Z " 2ρj −1 X 1 + itξ − eitξ ν sj (u) du = − Qν (αj , ξ)(t − αj ) dτ (t) , t2 0 ∆j ν=0

(3.5) (3.6)

where −` < ξ < `. The integrals on the right sides of (3.5) and (3.6) converge absolutely for all real ξ by the condition (2◦ ) in the Kre˘ın-Langer representation of v(z). Proof. Let H = L2m (0, `), G = Cm , and define A, S ∈ L(H) and Φ1 , Φ2 ∈ L(G, H) by (2.2)–(2.5). By Theorem 2.1, these operators satisfy (2.1). Since s(x) ∈ S`,κ , κS < ∞. Hence by Theorem 2.3, S = Sv and Φ1 = Φ1,v for some admissible generalized Nevanlinna function v(z). Let v(z) have the representation (2.9). Write Pr (j) (d) (2.13) in the form Φ1,v = j=0 Φ1,v + Φ1,v , where Z 1 (j) Φ1,v = A(I − At)−1 − Sj (t; A) Φ2 [dτ (t)] , j = 0, . . . , r, (3.7) i ∆j

Vol. 63 (2009) (d)

Φ1,v = −

Integral Representations for Generalized Difference Kernels

289

Z 1 1 A(I − λA)−1 Φ2 R(λ) dλ + Φ2 R(∞) . i 2πi Γ

(3.8)

Then (j)

Φ1,v g = sj (x)g,

j = 0, . . . , r,

and

(d)

Φ1,v g = sd (x)g,

(3.9)

for some m × m matrix-valued functions sj (x), j = 0, . . . , r, and sd (x) on (0, `). Since Φ1 = Φ1,v , (3.1) holds on (0, `). If we extend the functions in (3.9) so that sj (x) = −sj (−x)∗ , j = 0, . . . , r, and sd (x) = −sd (−x)∗ , then (3.1) holds on (−`, `). It remains to prove (3.5), (3.6), and (3.4). It is sufficient to take 0 < ξ < `. In fact, the integrals on the right sides of (3.5) and (3.6) are unchanged when they are conjugated and ξ is replaced by −ξ, and from this property we easily see that the identities (3.5) and (3.6) hold on (−`, `) if they hold on (0, `). Similarly, (3.4) holds on (−`, `) if it holds on (0, `). Proofs of (3.5) and (3.6). Fix ξ in (0, `). Then Z ξ D E (j) g2∗ sj (u)g1 du = Φ1,v g1 , hξ ,

j = 0, . . . , r,

(3.10)

0

where g1 , g2 are arbitrary vectors in G and ( g2 , 0 < x < ξ, hξ (x) = 0, ξ < x < `. Case 1: j = 0. As in the definite case [18, p. 2], we use the identity A(I − At)−1 − S0 (t; A) = A(I − At)−1 +

tI (A + tI)(I − tA)−1 = 1 + t2 1 + t2

and (3.7) to write D E 1Z g1 (0) −1 Φ1,v g1 , hξ = (A + tI)(I − tA) Φ2 dτ (t) , hξ i ∆0 1 + t2 1 = i

g1 ∗ ∗ −1 ∗ , Φ (I − tA ) A hξ 1 + t2 2 ∆0 Z 1 tg1 ∗ ∗ −1 + dτ (t) , Φ (I − tA ) hξ . i ∆0 1 + t2 2

Z

dτ (t)

Short calculations yield Φ∗2 (I − tA∗ )−1 hξ = i Φ∗2 (I − tA∗ )−1 A∗ hξ = i

e−itξ − 1 g2 , t e−itξ − 1 + itξ g2 . t2

290

Rovnyak and Sakhnovich

IEOT

Thus by (3.10), Z ξ D E (0) g2∗ s0 (u)g1 du = Φ1,v g1 , hξ 0

g1 e−itξ − 1 + itξ , i g 2 1 + t2 t2 ∆0 Z 1 tg1 e−itξ − 1 + dτ (t) ,i g2 i ∆0 1 + t2 t

1 i

Z

=

1 i

Z

=

dτ (t)

eitξ − 1 − itξ eitξ − 1 −i 2 − i g2∗ dτ (t)g1 2) 2 t (1 + t 1 + t ∆0

Z = ∆0

1 + itξ − eitξ itξ − t2 1 + t2

g2∗ dτ (t)g1 .

This proves (3.5). Case 2: j = 1, . . . , r. By (3.7), D E (j) Φ1,v g1 , hξ 1 = i

(3.11)

Φ∗2 A∗ (I − tA∗ )−1 − Sj (t; A)∗ hξ ρj dτ (t)(t − αj ) g1 , . (t − αj )ρj ∆j

Z

By [14, Theorem 3.2], the convergence term Sj (t; A) is given by 2ρj −1

Sj (t; A) =

X

Aν+1 (I − αj A)−ν−1 (t − αj )ν .

ν=0

By induction, for any g ∈ G, Aν+1 (I − λA)−ν−1 Φ2 g = iPν (λ, x)g ,

ν = 0, 1, 2, . . . .

(3.12)

Hence A(I − tA)−1 − Sj (t; A) Φ2 g 2ρj −1

= A(I − tA)−1 Φ2 g −

X

Aν+1 (I − αj A)−ν−1 Φ2 g (t − αj )ν

ν=0

=

itx

e

−1 − t

2ρj −1

X ν=0

and so Φ∗2 A∗ (I − tA∗ )−1 − Sj (t; A)∗ hξ

iPν (αj , x)(t − αj ) g , ν

Vol. 63 (2009)

Integral Representations for Generalized Difference Kernels ξ

Z

= 0

291

2ρj −1 X e−itx − 1 − ν − [iPν (αj , x)] (t − αj ) dx g2 . t ν=0

Thus (3.11) yields D

(j) Φ1,v g1 , hξ

E

2ρj −1 X eitx − 1 ν − iPν (αj , x)(t − αj ) dx g2∗ dτ (t) g1 t ∆j 0 ν=0 Z 2ρj −1 itξ X 1 + itξ − e ν = − Qν (αj , ξ)(t − αj ) g2∗ dτ (t) g1 , t2 ∆j ν=0 1 = i

Z

Z

ξ

where in the last equality we used the identity Z ξ Qν (λ, ξ) = Pν (λ, x) dx,

ν = 0, 1, 2, . . . .

0

By (3.10), Z ξ D E (j) g2∗ sj (u)g1 du = Φ1,v g1 , hξ 0

Z = ∆j

2ρj −1 X 1 + itξ − eitξ ν − Qν (αj , ξ)(t − αj ) g2∗ dτ (t) g1 , t2 ν=0

and (3.6) follows. Proof of (3.4). The integral in (3.8) can be evaluated as a sum of residues: s X 1 (d) Φ1,v = i Res A(I − λA)−1 Φ2 Rk λ − λk k=1 λ=λk ∗ s X 1 −1 +i Res A(I − λA) Φ2 Rk + i Φ2 C0 . ¯ − λk ¯ λ k=1 λ=λk For each k = 1, . . . , s, by (3.12), σk X 1 A(I − λA)−1 Φ2 Rkν Res A(I − λA)−1 Φ2 Rk g= Res g λ − λk (λ − λk )ν λ=λk ν=1 λ=λk =

σk X ν=1

Aν (I − λk A)ν Φ2 Rkν g =

σk X

iPν−1 (λk , x)Rkν g.

ν=1

The residue is calculated with the aid of the formula (I − λA)−1 Res = Aν−1 (I − λ0 A)−ν . λ=λ0 (λ − λ0 )ν ¯ k and Rkν replaced by R∗ . This A similar equation holds with λk replaced by λ kν yields (3.4).

292

Rovnyak and Sakhnovich

IEOT

We prove a converse result. Theorem 3.2. Let v(z) be an admissible generalized Nevanlinna function having the representation (2.9). Then (3.1)–(3.4) define a measurable m × m matrix-valued function s(x) on (−∞, ∞) such that for every of s(x) to Pr` > 0, the Ps restriction (−`, `) belongs to a class S`,κ` with κ` ≤ m j=1 ρj + k=1 σk . Proof. Fix `, and let H = L2m (0, `) and G = Cm . By the definition of an admissible function, operators Sv ∈ L(H) and Φ1,v ∈ L(G, H) are defined for the given function v(z). These operators satisfy (2.14) with A and Φ2 given by (2.2) and (2.3). Hence by Theorem 2.1, Z ` d (Sv f )(x) = sv (x − t)f (t) dt, dx 0 (Φ1,v g)(x) = ϕ1,v (x)g,

ϕ1,v (x) = sv (x),

0 < x < `,

for some m×m matrix-valued function sv (x) = −sv (−x)∗ on (−`, `). The function sv (x) is calculated from v(z) as in proof of Theorem 3.1 (the invertibility hypothesis in Theorem 3.1 is not used in the calculation). The calculation shows that sv (x) agrees with the restriction of the function s(x) defined by (3.1)–(3.4) to (−`, `). Thus the P restriction P of s(x) to (−`, `) belongs to S`,κ` , where κ` = κSv . By (2.15), r s κ` ≤ m ρ + j j=1 k=1 σk . Theorems 3.1 and 3.2 together provide an extension theorem. Theorem 3.3. Let s(x) ∈ S`,κ , and assume that the corresponding operator (1.1) ˜ is invertible. Then s(x) has an extension to real line Pthe Ps such that for every ` > `, r s(x) belongs to a class S`,e e≤m ˜ κ with κ j=1 ρj + k=1 σk . Proof. Theorem 3.1 represents s(x) in the form (3.1)–(3.4). Then the required extensions are provided by Theorem 3.2.

4. Appendix: An estimate of negative squares We sketch a proof of the inequality (2.15) for any generalized Nevanlinna function v(z) represented in the form (2.9). As a preliminary, note the identities n X X

Xp∗ Mν Xq = X ∗ M X,

(4.1)

ν=0 p+q=ν p,q≥0 n X X h ν=0 p+q=ν p,q≥0

Xp∗ Mν Yq

+

Yp∗ Mν∗ Xq

i

∗ X 0 = Y M∗

M 0

X , Y

(4.2)

Vol. 63 (2009)

Integral Representations for Generalized Difference Kernels

which hold for any block operator matrices      X0 Y0 M0  X1   Y1   M1      X =  . , Y =  . , M =  ..   ..   Mn Xn Yn

M2 M3 0

··· ··· ··· ···

Mn−1 Mn 0

293

 Mn 0   .  0

Lemma 4.1. Let M be a selfadjoint 2k × 2k matrix of the form A B M= , B∗ 0 where each block has size k × k. Then κM ≤ k. Proof. Define Mε by the same formula with A and B replaced by Aε = A + εI and Bε = B + εI for any ε > 0. Choose δ > 0 such that Aε and Bε are invertible for 0 < ε < δ. Since I 0 Aε 0 I A−1 ε Bε Mε = , Bε∗ A−1 I 0 −Bε∗ A−1 0 I ε ε Bε two applications of Sylvester’s law of inertia [4, p. 223] show that κMε = k for 0 < ε < δ. In the limit, we get κM ≤ k by [4, p. 540]. Proof of (2.15). It is sufficient to prove (2.15) when v(z) is a single term in (2.9). Z h i 1 Case 1: v(z) = − Sj (t, z) dτ (t) for some j = 1, . . . , r ∆j t − z We show that κSv ≤ mρj in this case. By definition, Z n o Sv = (I − At)−1 Φ2 [dτ (t)]Φ∗2 (I − A∗ t)−1 − dτj (t; A, Φ2 ) . ∆j

By [14, Theorem 3.2], 2ρj −1

dτj (t; A, Φ2 ) =

X

X

o Ap (αj )Φ2 (t − αj )ν [dτ (t)] Φ∗2 Aq (αj )∗ ,

ν=0 p+q=ν p,q≥0

where Ap (λ) = Ap (I − λA)−p−1 for each p ≥ 0. Thus Z n Sv = (I − At)−1 Φ2 [dτ (t)]Φ∗2 (I − A∗ t)−1 ∆j 2ρj −1

−

X

X

o Ap (αj )Φ2 (t − αj )ν [dτ (t)] Φ∗2 Aq (αj )∗ .

ν=0 p+q=ν p,q≥0

By approximation, we may assume that τ (t) is constant in small intervals (αj − ε, αj ) and (αj , αj + ε). In this case, Sv = T1 + T2 , where Z T1 = (I − At)−1 Φ2 [dτ (t)]Φ∗2 (I − A∗ t)−1 , ∆j

294

Rovnyak and Sakhnovich

IEOT

2ρj −1

T2 =

X

X

Ap (αj )Φ2 Hν Φ∗2 Aq (αj )∗ ,

ν=0 p+q=ν p,q≥0

and

Z Hν = −

(t − αj )ν dτ (t),

ν = 0, 1, 2, . . . .

∆j

Clearly, T1 ≥ 0, so κT1 = 0. By (4.1) and Lemma 4.1, κT2 ≤ mρj , and hence κSv ≤ κT1 + κT2 ≤ mρj . Z h i 1 Case 2: v(z) = − S0 (t, z) dτ (t) ∆0 t − z In this case, the operator Z Sv = (I − At)−1 Φ2 [dτ (t)]Φ∗2 (I − A∗ t)−1 ∆0

is nonnegative, and so κSv = 0. Case 3: v(z) is one of the terms in (2.10) The constant C0 makes no contribution, so suppose that 1 1 ∗ v(z) = −Rk − Rk , z − λk z¯ − λk where Rk (z) = Rk1 z + · · · + Rk,σk z σk for some k = 1, . . . , s. Then Z 1 1 Sv = (I − λA)−1 Φ2 Rk Φ∗2 (I − λA∗ )−1 dλ 2πi Γ λ − λk Z 1 ∗ 1 + (I − λA)−1 Φ2 Rk Φ∗2 (I − λA∗ )−1 dλ ¯ 2πi Γ λ − λk σk X (I − λA)−1 Φ2 Rkν Φ∗2 (I − λA∗ )−1 = Res λ=λk (λ − λk )ν ν=1 +

σk X ν=1

Res

¯k λ=λ

∗ (I − λA)−1 Φ2 Rkν Φ∗2 (I − λA∗ )−1 . ¯ k )ν (λ − λ

To evaluate the residues, we use the formula Res

λ=λ0

X (I − λA)−1 C(I − λA∗ )−1 ¯ 0 )∗ , = Ap (λ0 )CAq (λ (λ − λ0 )ν p+q=ν−1 p,q≥0

where Ap (λ) = Ap (I − λA)−p−1 for each p ≥ 0. This yields σk i X X h ¯ k )∗ + Ap (λ ¯ k )Φ2 R∗ Φ∗ Aq (λk )∗ . Sv = Ap (λk )Φ2 Rkν Φ∗2 Aq (λ kν 2 ν=1 p+q=ν−1 p,q≥0

Then by (4.2) and Lemma 4.1, κSv ≤ mσk .

Vol. 63 (2009)

Integral Representations for Generalized Difference Kernels

295

All terms in the formula (2.12) have been considered, and the proof of (2.15) is complete. Added in proof. In Theorems 2.3 and 3.1, v(z) can be chosen such that κv ≤ κS , ¯ We omit the where κv is the number of negative squares of [v(z) − v(ζ)∗ ]/(z − ζ). details but remark that this follows from the proof of [13, Theorem 3.1], where it can be seen that v(z) may be chosen with this property.

References [1] D. Z. Arov and H. Dym, On three Krein extension problems and some generalizations, Integral Equations Operator Theory 31 (1998), no. 1, 1–91. [2] K. Daho and H. Langer, Matrix functions of the class Nκ , Math. Nachr. 120 (1985), 275–294. ¨ [3] M. Grossmann and H. Langer, Uber indexerhaltende Erweiterungen eines hermiteschen Operators im Pontrjaginraum, Math. Nachr. 64 (1974), 289–317. [4] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. [5] M. Kaltenb¨ ack, H. Winkler, and H. Woracek, Almost Pontryagin spaces, Recent advances in operator theory and its applications, Oper. Theory Adv. Appl., vol. 160, Birkh¨ auser, Basel, 2005, pp. 253–271. [6] M. Kaltenb¨ ack and H. Woracek, On extensions of Hermitian functions with a finite number of negative squares, J. Operator Theory 40 (1998), no. 1, 147–183. [7] M. G. Kre˘ın, Sur le probl`eme du prolongement des fonctions hermitiennes positives et continues, C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940), 17–22; Selected Works. I, Akad. Nauk Ukrainy Inst. Mat., Kiev, 1993, pp. 102–110. [8] , Integral representation of a continuous Hermitian-indefinite function with a finite number of negative squares, Dokl. Akad. Nauk SSSR 125 (1959), 31–34. ¨ [9] M. G. Kre˘ın and H. Langer, Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ angen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236. [10] , On some continuation problems which are closely related to the theory of operators in spaces Πκ . IV. Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), no. 2, 299–417. [11] H. Langer, M. Langer, and Z. Sasv´ ari, Continuations of Hermitian indefinite functions and corresponding canonical systems: an example, Methods Funct. Anal. Topology 10 (2004), no. 1, 39–53. [12] J. Rovnyak and L. A. Sakhnovich, On the Kre˘ın-Langer integral representation of generalized Nevanlinna functions, Electron. J. Linear Algebra 11 (2004), 1–15 (electronic). [13] , Interpolation problems for matrix integro-differential operators with difference kernels and with a finite number of negative squares, Operator theory, structured matrices, and dilations: Tiberiu Constantinescu memorial volume, Theta Foundation, Bucharest, 2007, pp. 325–340.

296 [14]

[15]

[16] [17] [18] [19] [20]

Rovnyak and Sakhnovich

IEOT

, On indefinite cases of operator identities which arise in interpolation theory, The extended field of operator theory, Oper. Theory Adv. Appl., vol. 171, Birkh¨ auser, Basel, 2007, pp. 281–322. A. L. Sakhnovich, Modification of V. P. Potapov’s scheme in the indefinite case, Matrix and operator valued functions, Oper. Theory Adv. Appl., vol. 72, Birkh¨ auser, Basel, 1994, pp. 185–201. ˇ 13 (1972), 868–883, Engl. L. A. Sakhnovich, Similarity of operators, Sibirsk. Mat. Z. transl., Siberian Math. J. 13(4) (1973), 604–615. , Integral equations with difference kernels on finite intervals, Oper. Theory Adv. Appl., vol. 84, Birkh¨ auser Verlag, Basel, 1996. , Interpolation theory and its applications, Kluwer, Dordrecht, 1997. , Spectral theory of canonical differential systems. Method of operator identities, Oper. Theory Adv. Appl., vol. 107, Birkh¨ auser Verlag, Basel, 1999. , On reducing the canonical system to two dual differential systems, J. Math. Anal. Appl. 255 (2001), no. 2, 499–509.

J. Rovnyak University of Virginia Department of Mathematics P. O. Box 400137 Charlottesville, VA 22904–4137 USA e-mail: [email protected] L. A. Sakhnovich 735 Crawford Avenue Brooklyn, NY 11223 USA e-mail: [email protected] Submitted: January 7, 2008. Revised: December 18, 2008.

Integr. equ. oper. theory 63 (2009), 297–320 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030297-24, published online February 24, 2009 DOI 10.1007/s00020-009-1668-z

Integral Equations and Operator Theory

The Abstract Titchmarsh-Weyl M -function for Adjoint Operator Pairs and its Relation to the Spectrum Malcolm Brown, James Hinchcliffe, Marco Marletta, Serguei Naboko and Ian Wood Abstract. In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M -function see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and S˜ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M -function is analytic. We present three examples – one involving a Hain-L¨ ust type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible. Mathematics Subject Classification (2000). Primary 47A10; Secondary 35J25, 35P05, 47A11. Keywords. M -function, boundary triples, adjoint pairs, spectrum.

1. Introduction In the theory of inverse problems for Schr¨odinger operators on a half line, −y 00 + q(x)y = λy,

x ∈ (0, ∞),

(1.1)

it has been well known since the work of Borg [4], of Marchenko [23] and of Gelfand and Levitan [9] that the function q is uniquely determined by the Titchmarsh-Weyl James Hinchcliffe and Serguei Naboko wish to thank the British EPSRC for financial support under grant EP/C008324/1 “Spectral Problems on Families of Domains and Operator M -functions”. Serguei Naboko wishes to thank the Russian RFBR for grant 06-01-00219. All authors wish to thank INTAS for financial support under INTAS Project No. 051000008-7883. The authors wish to thank the referee for many helpful comments.

298

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

function for the problem. Here q is assumed to be real valued and integrable over any finite sub-interval of [0, ∞) and to give rise to a so-called limit point case at infinity: that is, one requires only a boundary condition at the origin, and no boundary condition at infinity, in order to obtain a selfadjoint operator associated with the expression on the left hand side of (1.1). The Titchmarsh Weyl function M (λ) for this problem can be regarded as a Dirichlet to Neumann map for the problem. Suppose that we define a ‘maximal’ operator A∗ by D(A∗ ) = {y ∈ L2 (0, ∞) | − y 00 + qy ∈ L2 (0, ∞)}, A∗ y = −y 00 + qy, where y is to be understood in the sense of weak derivatives; also define some ‘boundary’ operators Γ1 and Γ2 on D(A∗ ) by 00

Γ1 y = y(0), Γ2 y = −y 0 (0). Then the Titchmarsh Weyl function may be defined by the expression −1 M (λ) = Γ2 Γ1 |ker(A∗ −λI) , or equivalently M (λ)y(0) = −y 0 (0) when −y 00 + qy = λy and y ∈ L2 (0, ∞). If we let AD denote the ‘Dirichlet restriction’ of A∗ , that is the restriction of A∗ to D(AD ) = D(A∗ ) ∩ ker(Γ1 ), then the M -function is easily seen to be well defined for λ 6∈ σ(AD ). One may show that (AD − λ)−1 has the same poles as M (λ), and the famous Weyl Kodaira formula relates the spectral measure ρ of AD to M : 1 dρ(k) = w − lim&0 =M (k + i)dk. π In short, complete information about the original operator is encoded in M . For PDEs, similar inverse results are also available. For Schr¨odinger operators on smooth domains with smooth potentials, for instance, the Dirichlet-to-Neumann map M (λ) determines the potential uniquely. Moreover in this PDE case it is not necessary to know M (λ) as a function of λ: it suffices to know it for one value of λ for which it is well defined. For more general classes of PDEs there are many results guaranteeing that the coefficients can be recovered up to some explicit transformations. See Isakov [14] for a review of inverse problems for elliptic PDEs. In this paper we consider similar questions in the totally abstract setting of boundary triples (cf. Section 2 for the definition). As shown in the papers by Kre˘ın, Langer and Textorius [16, 17, 18] on extensions of symmetric operators, under an assumption of complete nonselfadjointness of the underlying symmetric minimal operator, the maximal operator is determined up to unitary equivalence by the M -function. Moreover, recently Ryzhov [27] has shown that under the same assumptions and an additional invertibility condition imposed on the Dirichlet

Vol. 63 (2009)

Abstract M -functions

299

restriction AD , the operators AD and Γ2 A−1 D are determined by the difference M (z) − M (0) up to unitary equivalence. For the non-symmetric case, the authors considered in [7] the question of behaviour of the abstract M -function(s) near the boundary of the essential spectrum and asked: to what extent does the M -function see the same singularities as the resolvent of a certain restriction AB of the maximal operator? In this paper we obtain results showing that it is possible to describe explicitly certain spaces S and S˜ such that the bordered resolvent PS˜(AB − λI)−1 PS , in which the P are orthogonal projections onto the spaces indicated, is analytic everywhere that M (λ) is analytic. The spaces S and S˜ are, in general, not closed. However we present three examples – one involving a Hain-L¨ ust type operator, one involving a perturbed Friedrichs operator and one involving simple ordinary differential operators on a half line – which together indicate that the abstract results in Section 3 are probably best possible. As a result we conclude that the abstract approach to inverse problems may yield rather limited results unless further hypotheses are introduced which reflect properties of problems involving concrete ordinary and partial differential expressions. We should mention that since their introduction by Vishik [28] for second order elliptic operators and Lyantze and Storozh [19] for adjoint pairs of abstract operators, boundary triplets have been widely used to characterise extensions of operators and investigate spectral properties using Weyl M -functions. An extension of the theory to relations can be found in the work of Malamud and Mogilevskii [21, 22]. For related recent results, particularly in the context of PDEs, we refer to the works of Alpay and Behrndt [1], Behrndt and Langer [3], Brown, Grubb, Wood [6], Gesztesy and Mitrea [10, 11, 12] and also to Posilicano [24, 25] and Post [26].

2. Background theory of boundary triples and Weyl functions Throughout this article we will make the following assumptions: 1. A, A˜ are closed, densely defined operators in a Hilbert space H. 2. A and A˜ are an adjoint pair, i.e. A∗ ⊇ A˜ and A˜∗ ⊇ A. Proposition 2.1. [19, (Lyantze, Storozh ’83)]. For each adjoint pair of closed densely defined operators on H, there exist “boundary spaces” H, K and “boundary operators” ˜ 1 : D(A∗ ) → K and Γ ˜ 2 : D(A∗ ) → H Γ1 : D(A˜∗ ) → H, Γ2 : D(A˜∗ ) → K, Γ such that for u ∈ D(A˜∗ ) and v ∈ D(A∗ ) we have an abstract Green formula ˜ 2 v)H − (Γ2 u, Γ ˜ 1 v)K . (A˜∗ u, v)H − (u, A∗ v)H = (Γ1 u, Γ (2.1) ˜ ˜ The boundary operators Γ1 , Γ2 , Γ1 and Γ2 are bounded with respect to the graph ˜1, Γ ˜ 2 ) is surjective onto norm. The pair (Γ1 , Γ2 ) is surjective onto H × K and (Γ K × H. Moreover, we have ˜ = D(A∗ ) ∩ ker Γ ˜ 1 ∩ ker Γ ˜ 2 . (2.2) D(A) = D(A˜∗ ) ∩ ker Γ1 ∩ ker Γ2 and D(A)

300

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

˜1, Γ ˜ 2 )} is called a boundary triplet for the adjoint The collection {H⊕K, (Γ1 , Γ2 ), (Γ ˜ pair A, A. Malamud and Mogilevskii [21, 22] use this setting to define Weyl M -functions and γ-fields associated with boundary triplets and to obtain Kre˘ın formulae for the resolvents. We now summarize some results, using however a slightly different setting taken from [7] in which the boundary conditions and Weyl function contain an additional operator B ∈ L(K, H). ˜ ∈ L(H, K). We define extensions of A and Definition 2.2. Let B ∈ L(K, H) and B ˜ A (respectively) by AB := A˜∗ |ker(Γ −BΓ ) and A˜ ˜ := A∗ | ˜ ˜ ˜ . 1

2

B

ker(Γ1 −B Γ2 )

In the following, we assume ρ(AB ) 6= ∅, in particular AB is a closed operator. For λ ∈ ρ(AB ), we define the M -function via MB (λ) : Ran(Γ1 − BΓ2 ) → K, MB (λ)(Γ1 − BΓ2 )u = Γ2 u for all u ∈ ker(A˜∗ − λ) and for λ ∈ ρ(A˜B˜ ), we define ˜ ˜ (λ) : Ran(Γ ˜1 − B ˜Γ ˜ 2 ) → H, M ˜ ˜ (λ)(Γ ˜1 − B ˜Γ ˜ 2 )v = Γ ˜ 2 v for all v ∈ ker(A∗ − λ). M B B ˜ ˜ (λ) are well defined for λ ∈ ρ(AB ) It is easy to prove that MB (λ) and M B and λ ∈ ρ(A˜B˜ ) respectively. Definition 2.3. (Solution Operator) For λ ∈ ρ(AB ), we define the linear operator Sλ,B : Ran(Γ1 − BΓ2 ) → ker(A˜∗ − λ) by

i.e. Sλ,B

(A˜∗ − λ)Sλ,B f = 0, (Γ1 − BΓ2 )Sλ,B f = f, −1 . = (Γ1 − BΓ2 )|ker(A˜∗ −λ)

(2.3)

Since we shall use solution operators quite extensively in the sequel, we include the proof of the following lemma, for completeness. Lemma 2.4. Sλ,B is well-defined for λ ∈ ρ(AB ). Moreover for each f ∈ Ran(Γ1 − BΓ2 ) the map from ρ(AB ) → H given by λ 7→ Sλ,B f is analytic. Proof. For f ∈ Ran(Γ1 −BΓ2 ), choose any w ∈ D(A˜∗ ) such that (Γ1 −BΓ2 )w = f . Let v = −(AB −λ)−1 (A˜∗ −λ)w. Then v +w ∈ ker(A˜∗ −λ) and (Γ1 −BΓ2 )(v +w) = (Γ1 − BΓ2 )w = f , so a solution to (2.3) exists and is given by Sλ,B f = I − (AB − λ)−1 (A˜∗ − λ) w (2.4) for any w ∈ D(A˜∗ ) such that (Γ1 − BΓ2 )w = f . Moreover Sλ,B f is well defined because the solution to (2.3) is unique. For suppose u1 and u2 are two solutions. Then (u1 −u2 ) ∈ ker(A˜∗ −λ)∩ker(Γ1 −BΓ2 ), so u1 −u2 ∈ D(AB ) and (AB − λ)(u1 − u2 ) = 0. As λ ∈ ρ(AB ), u1 = u2 . The analyticity of Sλ,B as a function of λ is immediate from (2.4) using the fact that the choice of w does not depend on λ.

Vol. 63 (2009)

Abstract M -functions

301

Corollary 2.5. Under the hypotheses of Lemma 2.4, Sλ,B = Sλ0 ,B + (λ − λ0 )(AB − λ)−1 Sλ0 ,B .

(2.5)

Proof. Fix λ0 ∈ ρ(AB ) and choose w = Sλ0 ,B f . Then Sλ,B f = Sλ0 ,B − (AB − λ)−1 (A˜∗ − λ)Sλ0 ,B f = Sλ0 ,B f + (λ − λ0 )(AB − λ)−1 Sλ0 ,B f.

Note that the identity (2.5) may be regarded as a Hilbert identity for the difference of resolvents corresponding to different boundary conditions. To be able to study spectral properties of the operator AB via the M -function, we need to relate the M -function to the resolvent. This can be done in the following way: Theorem 2.6.

1. Let λ, λ0 ∈ ρ(AB ). Then on Ran(Γ1 − BΓ2 ) MB (λ) = Γ2 I + (λ − λ0 )(AB − λ)−1 Sλ0 ,B = Γ2 (AB − λ0 )(AB − λ)−1 Sλ0 ,B .

2. Let B, C ∈ L(K, H), λ ∈ ρ(AB ) ∩ ρ(AC ). Then (AB − λ)−1 = (AC − λ)−1 − Sλ,C (I + (B − C)MB (λ))(Γ1 − BΓ2 )(AC − λ)−1 = (AC − λ)−1 − Sλ,C (I + (B − C)MB (λ))(C − B)Γ2 (AC − λ)−1 . Proof. Part (1) is just Proposition 4.6 from [7], while part (2) is a slight improvement to Theorem 4.7 of the same paper. We include the proof of (2) for completeness: Let u ∈ H. Set v := (AB − λ)−1 − (AC − λ)−1 u. Since v ∈ ker(A˜∗ − λ), we have MB (λ)(Γ1 − BΓ2 )v = Γ2 v. Then (Γ1 − CΓ2 ) v = [Γ1 − BΓ2 + (B − C)MB (λ)(Γ1 − BΓ2 )] v = (I + (B − C)MB (λ))(Γ1 − BΓ2 )v = −(I + (B − C)MB (λ))(Γ1 − BΓ2 )(AC − λ)−1 u. Set f := −(I + (B − C)MB (λ))(Γ1 − BΓ2 )(AC − λ)−1 u. Then by the above calculation, f ∈ Ran(Γ1 − CΓ2 ) and Sλ,C f = v = (AB − λ)−1 − (AC − λ)−1 u. Therefore, (AB − λ)−1 = (AC − λ)−1 − Sλ,C (I + (B − C)MB (λ))(Γ1 − BΓ2 )(AC − λ)−1 .

3. How much of an operator can its Weyl function determine? In this section we wish to know how much of the spectrum of an operator can be seen by its Weyl function. In the symmetric case, complete non-selfadjointness of the minimal operator A is required to recover the operator (up to unitary

302

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

equivalence) from the Weyl function (see e.g. [27]). Motivated by this, we fix µ0 6∈ σ(AB ) and define the spaces S = Spanδ6∈σ(AB ) (AB − δI)−1 Ran(Sµ0 ,B ),

(3.1)

T = Spanµ6∈σ(AB ) Ran(Sµ,B ), (3.2) −1 where Sµ,B = (Γ1 − BΓ2 )|ker(A˜∗ −µI) is the solution operator. Here Span denotes the set of finite linear combinations of vectors from the sets indicated. The spaces S depend on the choice of µ0 , but this dependence will not be indicated explicitly. Moreover the closures of S do not depend on µ0 , as the following lemma shows.

Lemma 3.1. Suppose that there exists a sequence (zn ) in C which tends to infinity and is such that the family of operators (zn (AB − zn I)−1 )n∈N is bounded. Then S =T. In particular, S does not depend on µ0 . Proof. From the hypothesis that the operators zn (AB − zn I)−1 are bounded it follows that the operators AB (AB − zn I)−1 = I + zn (AB − zn I)−1 are uniformly bounded. Let φ ∈ H be arbitrary. Given > 0 exploit the density of D(AB ) in H to choose ψ ∈ D(AB ) such that kφ − ψk < . Now because ψ ∈ D(AB ), it follows that AB (AB − zn I)−1 ψ = (AB − zn I)−1 AB ψ and so kAB (AB − zn I)−1 ψk ≤ kAB ψk

kzn (AB − zn I)−1 k → 0 (n → ∞). |zn |

Hence for all sufficiently large n, kAB (AB − zn I)−1 ψk < . But we know that the operators AB (AB − zn I)−1 are uniformly bounded, so for all sufficiently large n kAB (AB − zn I)−1 φk ≤ kAB (AB − zn I)−1 kkφ − ψk + kAB (AB − zn I)−1 ψk < C + for some C > 0. Hence kAB (AB − zn I)−1 φk → 0 (n → ∞) for each fixed φ ∈ H. Similar arguments may be found in, e.g., [8, Lemma II.3.4]. Let µ0 be as in the definition of S and let φ = Sµ0 ,B f for some f in the boundary space. Evidently kAB (AB − zn I)−1 φk → 0 and so −zn (AB − zn I)−1 Sµ0 ,B f → Sµ0 ,B f. It follows from the definition of S that Sµ0 ,B f ∈ S. Now if µ is another point in the resolvent set of AB then the identity Sµ,B = Sµ0 ,B + (µ − µ0 )(AB − µ)−1 Sµ0 ,B from Corollary 2.5 immediately shows that Sµ,B f lies in S also. It follows that T ⊆ S and hence T ⊆ S.

Vol. 63 (2009)

Abstract M -functions

303

Next we show that if f lies in the boundary space and µ, δ do not lie in σ(AB ) then (AB − δ)−1 Sµ,B f lies in T . For this we again use the formula (2.5) which gives, for δ 6= µ, (AB − δ)−1 Sµ,B f =

1 (Sδ,B f − Sµ,B f ); δ−µ

the right hand side of this expression obviously lies in T . Taking the limit as µ → δ it follows that (AB − δ)−1 Sδ,B f lies in T . Thus S ⊆ T and S ⊆ T . Remark 3.2. In fact with some mild additional assumptions one may show that S is generically independent of B (as well as of µ0 ), using the identity Sµ0 ,C (I − (C − B)Γ2 Sµ0 ,B ) = Sµ0 ,B from Proposition 4.5 of [7]. Remark 3.3. The hypothesis that one can choose (zn ) tending to infinity such that (zn (AB − zn I)−1 )n∈N is bounded holds in the case when the numerical range ω(AB ) is contained in a half plane, for in this case the zn can be chosen so that zn dist(zn , ω(AB )) is uniformly bounded in n. Lemma 3.4. The space S is a regular invariant space of the resolvent of the operator AB : that is, (AB − µI)−1 S = S for all µ ∈ ρ(AB ). Proof. We start by showing that (AB − µI)−1 S ⊆ S for all µ ∈ ρ(AB ). Choose f of the form N X (AB − δj I)−1 Sµ0 ,B fj (3.3) f= j=1

for some functions fj in H, and note that such f are dense in S. It follows from the resolvent identity (AB − µI)−1 (AB − δI)−1 =

1 {(AB − µI)−1 − (AB − δI)−1 } µ−δ

(3.4)

that (AB −µI)−1 f also admits a representation of the form (3.3); thus (AB −µ)−1 f also lies in S, giving (AB − µI)−1 S ⊆ S. Now suppose that f lies in S. We can write f = limN →∞ fN where fN has the form N X fN = (AB − δj,N I)−1 Sµ0 ,B fj,N j=1

304

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

and so fN = (AB − µI)−1

N X

(AB − µI)(AB − δj,N I)−1 Sµ0 ,B fj,N

j=1

= (AB − µI)−1

N X

Sµ0 ,B fj,N + (δj,N − µ)(AB − δj,N I)−1 Sµ0 ,B fj,N .

j=1

PN

Now the term j=1 Sµ0 ,B fj,N lies in the space T of (3.2) which is contained in S by Lemma 3.1. Thus fN has the form (AB − µI)−1 hN for some hN ∈ S. Hence f lies in (AB − µI)−1 S, in other words S ⊆ (AB − µI)−1 S. This completes the proof. Corresponding to the spaces S and T we define, from the formally adjoint operators, the spaces S˜ = Spanδ6∈σ(A˜B∗ ) (A˜B ∗ − δI)−1 Ran(S˜µ˜,B ∗ ),

where S˜µ,B ∗

1

(3.5)

T˜ = Spanµ6∈σ(A˜B∗ ) Ran(S˜µ,B ∗ ), (3.6) −1 ˜ 2 ) ˜1 − B∗Γ is the corresponding solution opera= (Γ ker(A∗ −µI)

tor. Once again, it may be shown that S˜ = T and so S˜ does not depend on µ ˜. We have so far defined the Weyl function MB (·) on ρ(AB ) where it is an analytic function. In what follows we will call a point λ0 ∈ C a point of analyticity of MB if all analytic continuations of MB coincide in a neighbourhood of λ0 . Theorem 3.5. Suppose that a point λ0 is a point of analyticity of MB and is also a limit point of points of analyticity of λ 7→ (AB − λI)−1 – that is, λ ∈ ρ(AB ). Let S be as in (3.1) and, for positive integers N and M , let PN,S and PM,S˜ denote projections onto any N and M -dimensional subspaces of S and S˜ respectively. Then PM,S˜(AB − λI)−1 PN,S is analytic at λ = λ0 . A similar result holds when one uses projections PN,T and PM,T˜ onto finite-dimensional subspaces of T and T˜ . Proof. Let f ∈ Ran(Γ1 − BΓ2 ) and let F = Sµ,B f for µ ∈ ρ(AB ). Then for each λ ∈ C, (A˜∗ − λI)F = (µ − λ)F = (µ − λ)Sµ,B f. (3.7) From the resolvent identity (3.4) it follows that for λ, δ ∈ ρ(AB ), (AB − λI)−1 (AB − δI)−1 (A˜∗ − λI)F µ−λ (AB − λI)−1 Sµ,B f − (AB − δI)−1 Sµ,B f = λ−δ 1 In

˜B ∗ is the adjoint of AB . fact we showed in [7] that A

Vol. 63 (2009)

Abstract M -functions

305

and hence, replacing (A˜∗ − λI)F on the left hand side by (µ − λ)Sµ,B f and the first copy of Sµ,B f on the right hand side by (µ − λ)−1 (A˜∗ − λI)F , (AB − λI)−1 [(AB − δI)−1 Sµ,B f ] =

1 (AB − δI)−1 (AB − λ)−1 (A˜∗ − λI)F − Sµ,B f. (µ − λ)(λ − δ) λ−δ

Let v ∈ D(A∗ ) and recall that (Γ1 − BΓ2 )F = f . The remainder of our proof relies heavily on the identity F − (AB − λ)−1 (A˜∗ − λ)F, (A∗ − λI)v (3.8) ˜ 2 v)H + (MB (λ)f, (Γ ˜1 − B∗Γ ˜ 2 )v)K = −(f, Γ which is eqn. (5.1) in [7]. Note that on the right hand side of this equation, the only λ-dependent term is MB (λ). Using this identity yields (AB − λI)−1 [(AB − δI)−1 Sµ,B f ], (A∗ − λI)v o n 1 ˜ 2 v)H − (MB (λ)f, (Γ ˜1 − B∗Γ ˜ 2 )v)K = F, (A∗ − λI)v + (f, Γ (µ − λ)(λ − δ) 1 − (AB − δI)−1 Sµ,B f, (A∗ − λI)v (3.9) λ−δ If we now select N points δj in the resolvent set of AB and N functions fj in Ran(Γ1 − BΓ2 ), and define Φ :=

N X

(AB − δj I)−1 Sµ,B fj ∈ S, Ψ :=

j=1

Θ :=

N X Sµ,B fj j=1

N X j=1

(AB − δj I)−1

Sµ,B fj ∈ S, λ − δj

λ − δj

φ :=

N X

∈T,

fj ,

j=1

then we obtain, upon summing the identities (3.9) with δ 7→ δj and f 7→ fj , (AB − λI)−1 Φ, (A∗ − λI)v = − Θ, (A∗ − λI)v o (3.10) 1 n ˜ 2 v)H − (MB (λ)φ, (Γ ˜1 − B∗Γ ˜ 2 )v)K + Ψ, (A∗ − λI)v + (φ, Γ µ−λ We have thus developed from (3.8) an expression in which (A∗ − λ)F has been replaced by the arbitrary element Φ of any finite-dimensional subspace of S. From the right hand side of the expression (3.10), since MB (λ) is analytic at λ0 and since none of the δj is equal to λ0 , it follows that ((AB − λI)−1 Φ, (A∗ − λI)v) is analytic at λ0 . Now the term (A∗ − λ)v may also be turned into an arbitrary ˜ of any finite-dimensional subspace of S˜ by similar reasoning, and so element Φ ˜ is analytic at λ0 . ((AB − λI)−1 Φ, Φ) The reasoning is similar but slightly simpler when working with elements of T .

306

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

In the case of isolated spectral points this theorem can be generalized as follows. Theorem 3.6. Suppose that a point λ0 is a point of analyticity of MB and that λ0 is at worst an isolated singularity of (AB −λI)−1 and suppose that the resolvent set ρ(A˜B ∗ ) has finitely many connected components. Let PS and PS˜ denote orthogonal projections onto the closures of S and S˜ respectively. Then PS˜(AB − λI)−1 PS is analytic at λ = λ0 . Proof. Assume that λ 6∈ ρ(AB ), otherwise the statement is trivial. In eqn. (3.10) ˜1 − B∗Γ ˜ 2 ) and any µ take v = (S˜µ˜,B ∗ )g for any g ∈ Ran(Γ ˜ not in the spectrum of ∗ ∗ ˜ ˜ µ − λ)(Sµ˜,B ∗ )g and so from (3.10), AB . Then (A − λI)v = (˜ 1 (3.11) Θ, (A∗ − λ)v (AB − λI)−1 Φ, (S˜µ˜,B ∗ )g = − µ ˜−λ n o 1 ˜ 2 v)H − (MB (λ)φ, (Γ ˜1 − B∗Γ ˜ 2 )v)K + Ψ, (A∗ − λ)v + (φ, Γ (˜ µ − λ)(µ − λ) Since µ ˜ lies in the resolvent set of A˜B ∗ = (AB )∗ we know that µ ˜ 6= λ0 . Let Γ be any smooth closed contour surrounding λ0 , not enclosing µ or µ ˜ and bounded away from the spectrum of AB . Integrating (3.11) around Γ yields Z (AB − λI)−1 Φ, (S˜µ˜,B ∗ )g dλ = 0. Γ

ˆ having a representation of the form It follows that for any Φ ˆ= Φ

M X

(S˜µ˜j ,B ∗ )gj

(3.12)

j=1

in which the points µ ˜j lie outside Γ, we have Z ˆ dλ = 0. (AB − λI)−1 Φ, Φ

(3.13)

Γ

˜ in S = T . Given > 0, such a Φ ˜ can be approximated Consider now a general Φ ˜ to accuracy by Φ of the form ˜ = Φ

M X

(S˜µ˜j, ,B ∗ )gj,

(3.14)

j=1

in which the points µ ˜j, could, however, lie inside Γ. However the solution operator ˜ Sµ˜,B ∗ is analytic for µ ˜ in the resolvent set ρ(A˜B ∗ ). If the curve Γ is chosen in a sufficiently small neighbourhood of λ0 then its image under complex conjugation, denoted Γ, lies in a single connected component of the resolvent set ρ(A˜B ∗ ). Denote this connected component by U and choose any open set O in U outside Γ. The values of the analytic function µ ˜ 7→ S˜µ˜,B ∗ at any point in U (and hence, in particular, at any points µ ˜j, inside Γ) are uniquely determined by the values of

Vol. 63 (2009)

Abstract M -functions

307

˜ of the form (3.12) to this function in O, so it must be possible to approximate Φ accuracy by approximations of the form ˆ = Φ

K X

(S˜ζj, ,B ∗ )hj,

(3.15)

j=1

in which the points ζj, either lie in O or in a completely different component of ˜ −Φ ˆ k < 2 and we also have, from (3.13), the resolvent set ρ(A˜B ∗ ). We have kΦ Z ˆ dλ = 0. (AB − λI)−1 Φ, Φ (3.16) Γ

Since the vectors (AB − λI)−1 Φ are uniformly bounded on Γ, which does not intersect the spectrum of AB , we can take limits in and obtain Z ˜ dλ = 0 (AB − λI)−1 Φ, Φ (3.17) Γ

˜ ∈ S. ˜ The result is now immediate from Morera’s theorem. for all Φ ∈ S, Φ

4. A first-order example An obvious question arising from the previous section is whether or not the result of Theorem 3.5 remains true if one omits projections onto finite dimensional subspaces: if MB (λ) is analytic at some point which is a non-isolated spectral point of AB , is PS˜ (AB − λI)−1 PS also analytic at this point? A simple example shows that this result is false. Consider in L2 (0, ∞) the operator A = A˜ given by D(A) = H01 (0, ∞) with Af = i

df . dx

(4.1)

The operator A is maximal symmetric and D(A∗ ) = H 1 (0, ∞). Define the bound˜1, Γ ˜ 2 by ary spaces H = C, K = {0}, and boundary value operators Γ1 , Γ2 , Γ ˜ 2 f = f (0). Γ1 f = if (0), Γ

(4.2)

˜ 1 f = 0, Γ2 f = 0. Γ (4.3) ˜1, Γ ˜ 2 ) are surjective and a simple It is easy to see that the pairs (Γ1 , Γ2 ) and (Γ integration shows that ˜ 2 g − Γ2 f Γ ˜ 1 g. (A∗ f, g) − (f, A∗ g) = if (0)g(0) = Γ1 f Γ ˜ 1 and Γ2 are trivial it follows immediately from the definitions that Because Γ ˜ ˜ (λ) = −1/B. ˜ MB (λ) = 0; M B Moreover, σ(AB ) = C+ .

308

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

Now we consider the space T , for simplicity in the case B = 0. For =(µ) < 0 a typical element of T has the form yµ = Sµ,0 f and therefore satisfies iyµ0 = µyµ with yµ (0) = f ; in other words, for some complex number f , yµ (x) = f exp(−iµx). ⊥

Now suppose that u ∈ T . Then (u, yµ ) = 0 for all =(µ) < 0. This means Z ∞ u(x) exp(iµx)dx = 0 0

for all =(µ) < 0. Setting µ = ω − ir, r > 0, we deduce that for all ω ∈ R Z ∞ u(x) exp(−rx) exp(iωx)dx = 0. 0

From inverse Fourier transformation this implies that u(x) exp(−rx) = 0 for all x and hence u(x) ≡ 0. Thus we have proved that for this example, T = T˜ = L2 (0, ∞) and so (AB − λI)−1 is not reduced by the bordering projection operators PT and PT˜ . It follows that for this example, the set of singular points of the bordered resolvent is strictly greater than the set of singular points of MB (λ).

5. A Hain-L¨ ust type example In this section we consider a block operator matrix example in which PS˜(AB − λI)−1 PS has exactly the same singularities as MB (λ), even though some of these singularities are not isolated. In other words, for the example which we present here, a stronger result holds than those available in Theorems 3.5 and 3.6. It is not yet clear to us what special properties of this example mean that, unlike for the example of Section 4, better results hold here than those in Theorems 3.5 and 3.6. Let ! 2

A˜∗ =

d − dx w(x) 2 + q(x) w(x) u(x)

,

(5.1)

where q, u and w are complex-valued L∞ -functions, and the domain of the operator is given by D(A˜∗ ) = H 2 (0, 1) × L2 (0, 1). (5.2) Also let 2

∗

A =

d − dx w(x) 2 + q(x)

w(x)

u(x)

! , with D(A∗ ) = D(A˜∗ ).

(5.3)

Vol. 63 (2009)

Abstract M -functions

309

It is then easy to see that y f y f ∗ ∗ ˜ A , − ,A z g z g y f y f = Γ1 , Γ2 − Γ2 , Γ1 , z g z g where

y Γ1 z Consider the operator

=

−y 0 (1) y 0 (0)

, Γ2

Aαβ := A˜∗

y z

=

ker(Γ1 −BΓ2 )

y(1) y(0)

,

(5.4)

.

(5.5)

where, for simplicity, B=

cot β 0

0 − cot α

.

(5.6)

It is known [2] that σess (Aαβ ) = essran(u) := {z ∈ C |∀ > 0, meas ({x ∈ [0, 1] | |u(x) − z| < }) > 0} . This result is independent of the choice of boundary conditions. The measure used is Lebesgue. Note also that σ(Aαβ ) is not the whole of C for essentially bounded q, u and w. For future use we also define the set W = {x ∈ [0, 1] | w(x) 6= 0}. The function w is defined only almost everywhere, but this is sufficient to define W up to a set of measure zero, which can be neglected. m11 (λ) m12 (λ) We now calculate the function M (λ) = such that m21 (λ) m22 (λ) y y M (λ)(Γ1 − BΓ2 ) = Γ2 z z y for ∈ ker(A˜∗ − λ). In our calculation we assume that λ 6∈ σess (Aαβ ). The z y condition ∈ ker(A˜∗ − λ) yields the equations z −y 00 + (q − λ)y + wz = 0;

wy + (u − λ)z = 0

which, in particular, give −y 00 + (q − λ)y +

w2 y = 0. λ−u

(5.7)

y1 The linear space ker(A˜∗ − λ) is thus spanned by the functions wy /(λ − u) 1 y2 and where y1 and y2 are solutions of the initial value problems wy2 /(λ − u)

310

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

consisting of the differential equation (5.7) equipped with initial conditions y1 (0) = cos α, y10 (0) = sin α, y2 (0) = − sin α,

y20 (0)

= cos α,

(5.8) (5.9)

where α is as in (5.6). A straightforward calculation shows that y(1) m11 (λ) m12 (λ) −y 0 (1) − cos β y(1)/ sin β = . y(0) m21 (λ) m22 (λ) y 0 (0) + cos α y(0)/ sin α Note that the yj depend on x and λ but that the λ-dependence is suppressed in the notation, except when necessary. Another elementary calculation now shows that y2 (1, λ) , y20 (1, λ) + cot β y2 (1, λ) sin α m21 (λ) = m12 (λ) = 0 , y2 (1, λ) + cot β y2 (1, λ) 0 y1 (1, λ) + cot β y1 (1, λ) m22 (λ) = sin α cos α + sin2 α . y20 (1, λ) + cot β y2 (1, λ) m11 (λ) = −

(5.10) (5.11) (5.12)

As an aside, notice that all these expressions contain a denominator y20 (1, λ)+ cot β y2 (1, λ) and that λ 6∈ essran(u|W ) is an eigenvalue precisely when this denominator is zero. Remark 5.1. For λ ∈ C \ essran(u|W ), the coefficient w(x)2 /(u(x) − λ) in (5.7) is analytic as a function of λ. Therefore, the solutions y1 and y2 are analytic in λ. The M -function may have an isolated pole at some point λ if y20 (1, λ) + cot β y2 (1, λ) happens to be zero; such a pole will be an eigenvalue of the operator Aαβ and may or may not be embedded in the essential spectrum of the operator. As a consequence of this remark, the M -function can be analytic at points in the essential range of u, as long as those points are outside the essential range of u|W : Lemma 5.2. Apart from poles at eigenvalues of Aαβ , the M -function M (λ) is analytic in the set C \ essran ( u|W ). We now turn our attention to the behaviour of the resolvent (Aαβ − λI)−1 on the spaces T and T . Theorem 5.3. S=T ⊆

L2 (0, 1) L2 (W)

.

Moreover if MB (λ) is analytic at a point λ not in essran ( u|W ) then y f := (Aαβ − λI)−1 z g admits analytic continuation for any f ∈ L2 (0, 1) and g ∈ L2 (W).

(5.13)

(5.14)

Vol. 63 (2009)

Abstract M -functions

311

Proof. Suppose that (f1 , f2 ) ∈ C2 and that µ ∈ ρ(Aαβ ). Since µ does not lie in the essential spectrum, it does not lie in the essential range of u, so 1/(u − µ) is essentially bounded. Consider the functions yµ , zµ defined by yµ f1 = Sµ,B ; zµ f2 eliminating zµ from these equations using wyµ zµ = u−µ we find that yµ satisfies the ODE

(5.15)

w2 yµ = 0 (5.16) µ−u with boundary conditions yµ0 (1)+cot(β)yµ (1) = −f1 and yµ0 (0)+cot(α)yµ (0) = f2 . The boundary value problem for yµ is uniquely solvable because µ ∈ ρ(Aαβ ) and so yµ ∈ L2 (0, 1). It follows from (5.15) that zµ ∈ L2 (W). This proves the inclusion (5.13). We decompose the space 2 2 L (0, 1) L (0, 1) M 0 = (5.17) L2 (0, 1) L2 (W) L2 (W c ) 2 L (0, 1) 0 where W c = [0, 1] \ W. Denote H1 = and H = . 2 L2 (W) L2 (W c ) We shall now show Aαβ . It that these are reducing subspaces for the operator h h is clear that if ∈ D(Aαβ ) then the projections of onto H1 and g g H2 will also lie in the domain ofthe operator as H2 ⊆ D(Aαβ ). The conditions h h Aαβ PHi ∈ Hi when ∈ D(Aαβ ) for i = 1, 2 are a simple calculation. g g Here Pi denotes the orthogonal projection onto Hi . We have σess (Aαβ |H1 ) = essran(u|W ). By Remark 5.1, any eigenvalue of the operator Aαβ |H1 will be a pole of MB (λ). Hence, if MB (λ) is analytic ata point f λ not in essran ( u|W ), we have that λ ∈ ρ(Aαβ |H1 ) and for any ∈ H1 , g f (Aαβ − λI)−1 admits analytic continuation. g −yµ00 + (q − µ)yµ +

As an immediate corollary of this theorem we have Corollary 5.4. For λ 6∈ essran ( u|W ) the bordered resolvent PS˜(Aαβ − λI)−1 PS is analytic precisely where MB (λ) is analytic. Proof. Since (Aαβ − λI)−1 |H1 is analytic on the space H1 which is larger than S by Theorem 5.3, it is immediate that the bordered resolvent is analytic wherever MB (·) is analytic. The fact that MB (·) is analytic whenever the bordered resolvent is analytic follows from (3.10).

312

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

Remark 5.5. Generically one expects that MB (·) will not be analytic at points in essran ( u|W ). The analyticity or otherwise depends on the analyticity or otherwise of solutions of the ODE (5.7). It is worth mentioning also the following result. Proposition 5.6. Let λ be any fixed point in the resolvent set ρ(Aαβ ). Then 2 2 L (0, 1) L (0, 1) (Aαβ − λI)−1 = . (5.18) L2 (W) L2 (W) Proof. The domain of the differential expression −

d2 w2 +q−λ− 2 dx u−λ

equipped with boundary conditions y(1)+cot(β)y(1) = 0 and y 0 (0)+cot(α)y(0) = 0, is dense in L2 (0, 1). Thus any function in L2 (0, 1) can be approximated to arbitrary accuracy by a solution y of a boundary value problem (−

w2 d2 +q −λ− )y = h ∈ L2 (0, 1) y(1)+cot(β)y(1) = 0 = y 0 (0)+cot(α)y(0) dx2 u−λ

for a suitably chosen h. Having fixed such y and h, then for any z ∈ L2 (W) we may define g to satisfy 1 z= (g − wy) u−λ and clearly have g ∈ L2 (W). Finally we define f ∈ L2 (0, 1) by f = h + wg/(u − λ) so that f ∈ L2 (0, 1) and h = f − wg/(u − λ). We thus have (−

d2 w2 + q − λ − )y = f − wg/(u − λ), dx2 u−λ

z=

1 (g − wy). u−λ

(5.19)

This is 2equivalent to (5.14). We have therefore approximated 2 an arbitrary element L (0, 1) L (0, 1) −1 of by a function in (Aαβ − λI) . To get the opposite L2 (W) L2 (W) inclusion consider y f −1 = (Aαβ − λI) z g in which g ∈ L2 (W). We need to show that z ∈ L2 (W) also. The expression for z is given in (5.19); evidently wy ∈ L2 (W) and g ∈ L2 (W) so the result is immediate.

6. A perturbed multiplication operator in L2 (R) The results of the foregoing sections show that there are often wide gaps between what may be true at an abstract level about the relationship between resolvents and M -functions, and what may be achievable in concrete examples.

Vol. 63 (2009)

Abstract M -functions

313

In light of these gaps, in this section we consider boundary triplets and Weyl M -functions for a simple Friedrichs model with a singular perturbation. Our purpose is to show even more unexpected and counter-intuitive results. For example, in [7, Section 4] it is shown that isolated eigenvalues of an operator correspond to isolated poles of the associated M -function assuming unique continuation holds, i.e. ker(A˜∗ − λ) ∩ ker(Γ1 ) ∩ ker(Γ2 ) = {0}, while [22, Proposition 5.2] shows this result under the assumption that the point under consideration is in the resolvent set of an extension of the minimal operator. In this section, we shall find that these hypotheses which have seemed reasonable in the development of an abstract theory of boundary triplets are not satisfied by a rather simple example. As a consequence, the relationship between the M -function and the spectrum of the operator becomes more interesting. We consider in L2 (R) the operator A with domain given by ( ) Z R f (x)dx exists and is zero , (6.1) D(A) = f ∈ L2 (R)|xf (x) ∈ L2 (R), lim R→∞

−R

given by the expression (Af )(x) = xf (x) + hf, φiψ(x),

(6.2)

2

where φ, ψ are in L (R). Observe that since the constant function 1 does not lie in L2 (R) the domain of A is dense in L2 (R). Formally, the expression xf (x) + hf, φiψ(x) is equivalent, by Fourier transformation, to a sum of a first order differential operator and an inner R product (integral) term acting on the Fourier transform fˆ. The condition R f = 0 is equivalent to a ‘boundary’ condition fˆ(0) = 0. Lemma 6.1. The adjoint of A is given on the domain D(A∗ ) = f ∈ L2 (R) | ∃cf ∈ C : xf (x) − cf 1 ∈ L2 (R) ,

(6.3)

by the formula A∗ f = xf (x) − cf 1 + hf, ψiφ.

(6.4)

Proof. Suppose that f 7→ hAf, gi is a bounded linear functional on D(A). A direct calculation shows that Z hAf, gi = f (x)(xg(x) + hg, ψiφ(x))dx. R

(Note that the integralRis convergent since xf (x) ∈ L2 (R) and g ∈ L2 (R).) In view of the constraint R f = 0 and the density of D(A) in L2 (R), the L2 (R)boundedness of this functional implies that for some constant cg , xg + hg, ψiφ = cg 1 + h 2

for some h ∈ L (R). Since φ ∈ L2 (R) this implies that xg − cg 1 ∈ L2 (R) and so hAf, gi = hf, xg − cg 1 + hg, ψiφi. The density of D(A) in L2 (R) now gives A∗ g = xg − cg 1 + hg, ψiφ.

314

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

Remark 6.2. For f sufficiently well behaved at infinity, the constant cf appearing in Lemma 6.1 is given by cf = lim xf (x). x→∞

For later reference we can calculate the deficiency indices of A. To this end we may neglect the finite rank term h·, φiψ and calculate the set of u such that xu(x) − cu 1 = ±iu. 1 ; cu x∓i

the factor cu is a normalization. A simple calculation shows This yields u = that c(x∓i)−1 = 1 and so we may choose u(x) = (x ∓ i)−1 as the deficiency elements, showing that A has deficiency indices (1, 1). We now introduce ‘boundary value’ operators Γ1 and Γ2 on D(A∗ ) as follows: Z (6.5) Γ1 u = (u(x) − cu 1sign(x)(x2 + 1)−1/2 )dx, Γ2 u = cu . R

We make the following observations. Lemma 6.3. The operators Γ1 and Γ2 are bounded relative to A∗ . Proof. Observe that cu = −A∗ u + xu + hu, ψiφ. Multiply both sides by the characteristic function χ(0,1) of the interval (0, 1), then take L2 -norms, to obtain |cu | ≤ kA∗ uk + kuk + kukkψkkφk which shows that Γ2 is bounded relative to A∗ . Similarly, an elementary calculation shows that Z p sign(x) Γ1 u = {( x2 + 1sign(x) − x)u(x) + (xu(x) − cu 1)} √ dx; x2 + 1 R √ since ( x2 + 1)sign(x) − x ∈ L2 (R), this shows that Γ1 is bounded relative to A∗ . Lemma 6.4. The following ‘Green’s identity’ holds: hA∗ f, gi − hf, A∗ gi = Γ1 f Γ2 g − Γ2 f Γ1 g + hf, ψihφ, gi − hf, φihψ, gi.

(6.6)

Consequently, in the case when φ = ψ, the operators A∗ |ker(Γ1 −BΓ2 ) are selfadjoint for any real number B. Proof. The identity (6.6) is a simple calculation. In the case when φ = ψ the operator A is symmetric and the selfadjointess of the extensions A∗ |ker(Γ1 −BΓ2 ) is a well known result from theory of boundary value spaces: see, e.g., Gorbachuk and Gorbachuk [13].

Vol. 63 (2009)

Abstract M -functions

315

In the case when φ 6= ψ, the terms hf, ψihφ, gi − hf, φihψ, gi on the right hand side of (6.6) arise from the fact that A∗ is not an extension of A. In order to eliminate these terms we follow the formalism of Lyantze and Storozh [19] and introduce an operator A˜ in which φ and ψ are swapped: ( ) Z R 2 2 ˜ D(A) = f ∈ L (R) | xf (x) ∈ L (R), lim f (x)dx = 0 , (6.7) R→∞

−R

˜ )(x) = xf (x) + hf, ψiφ. (Af In view of Lemma 6.1 we immediately see that D(A˜∗ ) = D(A∗ ) and that A˜∗ f = xf (x) − cf 1 + hf, φiψ. ˜∗

(6.8)

(6.9)

˜ and the following result is Thus A is an extension of A, A is an extension of A, easily proved. Lemma 6.5.

∗

A = A˜∗

ker(Γ1 )∩ker(Γ2 )

; A˜ = A∗ |ker(Γ1 )∩ker(Γ2 ) ;

(6.10)

moreover, the Green’s formula (6.6) can be modified to hA∗ f, gi − hf, A˜∗ gi = Γ1 f Γ2 g − Γ2 f Γ1 g.

(6.11)

This is a slight simplification of the situation in [19] as only two boundary operators are required, rather than four. For any fixed complex number B and suitable λ ∈ C, by the ‘Weyl function MB (λ)’ we shall mean the map −1 MB (λ) := Γ2 (Γ1 − BΓ2 )|ker(A˜∗ −λ) . (6.12) We now calculate MB (λ). Suppose that =λ 6= 0 and that f ∈ ker(A˜∗ − λI). Then xf (x) − cf + hf, φiψ = λf and simple algebra yields cf − hf, φiψ . x−λ Taking inner products with φ and recalling that Γ2 f = cf yields f=

hf, φiD(λ) = Γ2 f h(x − λ)−1 , φi where D(λ) = 1 +

R

(x − λ)−1 ψφdx. Substituting back into (6.13) yields 1 h(x − λ)−1 , φi ψ f = Γ2 f − , x−λ D(λ) x−λ

(6.13)

(6.14)

R

It follows upon calculating the relevant integrals that h(x − λ)−1 , ψih(x − λ)−1 , φi Γ1 f = sign(=λ)πi + Γ2 f, D(λ)

(6.15)

(6.16)

316

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

and so −1 h(x − λ)−1 , ψih(x − λ)−1 , φi MB (λ) = sign(=λ)πi + −B . D(λ)

(6.17)

Remark 6.6. If D(λ) is nonzero then a local unique continuation property holds: f ∈ ker(A˜∗ − λ) ∩ ker(Γ1 ) ∩ ker(Γ2 ) = 0 =⇒ f = 0. (6.18) To see this observe that from (6.15) we see that Γ2 f = 0 implies f = 0, giving unique continuation a fortiori. Remark 6.7. Generically, the M -function MB (λ) ‘sees’ the whole essential spectrum: the term sign(=(λ)πi) has a discontinuity across the real axis which one cannot expect to be cancelled by the other terms, except possibly on a set of measure zero. 2 Example. If φ and ψ both lie in the Hardy space H+ (see Koosis [15] for definitions and properties of Hardy spaces) then the inner product h(x − λ)−1 , φi is zero for =λ > 0 and the inner product h(x−λ)−1 , ψi is zero for =λ < 0. In this case MB (λ) has no poles and is given by

MB (λ) = (sign(=λ)πi − B)−1 . If B = πi then the entire upper half plane is filled with eigenvalues of the operator A˜∗ |ker(Γ1 −BΓ2 ) ; if B = −πi then it is the lower half plane which is entirely filled with eigenvalues. Example. We construct an example with a particularly interesting property: an eigenvalue which is not a pole of the M -function. 2 ; fix λ0 and, by choice of φ Consider the case where φ and ψ both lie in H+ and ψ, arrange that D(λ0 ) = 0. Avoid the pathological cases where eigenvalues fill the entire upper or lower half planes by choosing B = 0; we have 1 M0 (λ) = sign(=λ). πi Consider the function ψ(x) u(x) = . x − λ0 Since D(λ0 ) = 0 it follows that hu, φi = −1. Moreover it is easy to check that Γ2 u = cu = 0. It is now easy to check that u satisfies λ0 ψ xu(x) + hu, φiψ = x − λ0 ∗ ˜ and so u is an eigenfunction of A |ker(Γ ) with eigenvalue λ0 . However λ0 is not a 2

pole of M0−1 (λ), in apparent contradiction to the results in [22] and [7] mentioned at the beginning of this section. Which hypotheses have failed? If =λ0 < 0 then observe that Γ1 u = h(x − λ0 )−1 , ψi = 0, so the eigenfunction u belongs to the domain of the minimal operator A, and hence to the domain

Vol. 63 (2009)

Abstract M -functions

317

of every extension: thus unique continuation fails, so there is no contradiction to the theorems in [7]. The failure of unique continuation implies that there is no extension of A for which λ0 lies in the resolvent set, and so there is no contradiction to the results in [22] either. If =λ0 > 0 then although λ0 is no longer an eigenvalue for every extension, it nevertheless lies in the spectrum of every extension. To see this, attempt to solve (x − λ)u − Γ1 u + hu, φiψ = f, (Γ1 − CΓ2 )u = 0, with =λ > 0. Taking the inner products of both sides and remembering that h(x − λ)−1 , φi = 0 in the upper half plane we obtain hu, φi = h

ψ f , φi − hu, φih , φi. x−λ x−λ

ψ , φi = −1 since D(λ0 ) = 0 and so we obtain At λ = λ0 we have h x−λ 0

h

f , φi = 0. x − λ0

(6.19)

Thus the problem can only be solved for f satisfying the condition (6.19) and so λ0 lies in the spectrum of every extension of A˜∗ . This gives a further reason why we would not expect λ0 to be a pole of any M -function. 2 the operators A˜∗ |ker(Γ1 −BΓ2 ) are selfadjoint for Example. In the case φ = ψ ∈ H+ real B. The functions MB (λ) still cannot ‘see’ φ and ψ, however, being given by

MB (λ) = (sign(=λ)πi − B)−1 . Any eigenvalues of the operator will obviously be real and will be imbedded in the essential spectrum. If λ0 ∈ R and ψ(λ0 ) = 0 and Z |ψ(x)|2 dx = −1, R x − λ0 which can always be arranged, then λ0 will be an eigenvalue with eigenfunction ψ/(x − λ0 ). The operator will not be unitarily equivalent to the unperturbed operator, which has no eigenvalues. This is not surprising as the eigenfunction here belongs to the minimal operator, which therefore fails to be completely nonselfadjoint. There is therefore no contradiction to the results of Kre˘ın, Langer and Textorius [16, 17, 18] and Ryzhov [27] which state that if the minimal operator is completely non-selfadjoint then the maximal operator is determined up to unitary equivalence by the M -function.

318

Brown, Hinchcliffe, Marletta, Naboko and Wood

IEOT

References [1] Alpay, D. and Behrndt, J.; Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators. Preprint, arXiv:0807.0095. [2] Atkinson, F.V.; Langer, H.; Mennicken, R.; Shkalikov, A.A.; The essential spectrum of some matrix operators. Math. Nachr. 167 (1994), 5–20. [3] Behrndt, J. and Langer, M.; Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal. 243 (2007), 536–565. [4] Borg, G.; Uniqueness theorems in the spectral theory of y 00 +(λ−q(x))y = 0. Den 11th Skandinaviske Matematikerkongress, Trondheim, 1949; pp. 276–287. John Grundt Tanums Forlag, Oslo, 1952. [5] Brown, B.M.; Langer, M.; Marletta, M.; Spectral concentrations and resonances of a second-order block operator matrix and an associated λ-rational Sturm-Liouville problem. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2052, 3403–3420. [6] Brown, B.M.; Grubb, G.; Wood, I.; M -functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. To appear in Math. Nachr. [7] 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. London Math. Soc. (2) 77 (2008), 700–718. [8] Engel, K.-J. and Nagel, R.; One-parameter Semigroups for Linear Evolution Equations. GTM, Springer, 1999. [9] Gelfand, I.M. and Levitan, B.M.; On the determination of a differential equation from its spectral function. Am. Math. Soc. Trans. 1 (1951), 253–304. [10] Gesztesy, F.; Mitrea, M.; Zinchenko, M.; Variations on a theme by Jost and Pais. J. Funct. Anal. 253 (2007), 399–448. [11] Gesztesy, F. and Mitrea, M.; Generalized Robin Boundary Conditions, Robin-toDirichlet Maps, and Krein-Type Resolvent Formulas for Schr¨ odinger Operators on Bounded Lipschitz Domains. Preprint, arXiv:0803.3179. [12] Gesztesy, F. and Mitrea, M.; Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schr¨ odinger Operators on Bounded Lipschitz Domains. Preprint, arXiv:0803.3072. [13] Gorbachuk, V.I. and Gorbachuk, M.L.; Boundary value problems for operator differential equations. Kluwer, Dordrecht (1991). [14] Isakov, V.; Inverse problems for partial differential equations. Applied Mathematical Sciences, 127. Springer, New York, 1998. [15] Koosis, P.; Introduction to Hp spaces. Second edition. Cambridge Tracts in Mathematics, 115. Cambridge University Press, Cambridge, 1998. ¨ [16] Kre˘ın, M.G. and Langer, H.; Uber die Q-Funktion eines π-hermiteschen Operators im Raume Πκ . Acta Sci. Math. (Szeged) 34 (1973), 191–230. ¨ [17] Kre˘ın, M.G. and Langer, H.; Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ angen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77 (1977), 187–236.

Vol. 63 (2009)

Abstract M -functions

319

[18] Langer, H. and Textorius, B.; On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pacific J. Math. 72, 1 (1977), 135–165. [19] Lyantze, V.E. and Storozh, O.G.; Methods of the Theory of Unbounded Operators, (Russian) (Naukova Dumka, Kiev, 1983). [20] Malamud, M.M. and Mogilevskii, V.I.; On extensions of dual pairs of operators. Dopovidi Nation. Akad. Nauk Ukraine 1 (1997), 30–37. [21] Malamud, M.M. and Mogilevskii, V.I.; On Weyl functions and Q-function of dual pairs of linear relations. Dopovidi Nation. Akad. Nauk Ukraine 4 (1999), 32–37. [22] Malamud, M.M. and Mogilevskii, V.I.; Kre˘ın type formula for canonical resolvents of dual pairs of linear relations. Methods Funct. Anal. Topology (4) 8 (2002), 72–100. [23] Marchenko, V.A.; Concerning the theory of a differential operator of the second order. (Russian) Doklady Akad. Nauk SSSR. (N.S.) 72 (1950) 457–460. [24] Posilicano, A.; Self-adjoint extensions of restrictions. To appear in Operators and Matrices. [25] Posilicano, A. and Raimondi, L.; Krein’s Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators. To appear in J. Math. Phys.A. [26] Post, O.; First order operators and boundary triples. Russian J. Math. Phys. 14 (2007), 482–492. [27] Ryzhov, V.; A general boundary value problem and its Weyl function. Opuscula Math. (2) 27 (2007), 305–331. [28] Vishik, V.I.; On general boundary value problems for elliptic differential operators. Trudy Mosc. Mat. Obsv. 1 (1952) 187–246; Amer. Math. Soc. Transl. (2) 24 (1963), 107–172.

Malcolm Brown School of Computer Science Cardiff University Queen’s Buildings, 5 The Parade Cardiff CF24 3AA United Kingdom e-mail: [email protected] James Hinchcliffe and Marco Marletta School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG United Kingdom e-mail: [email protected] [email protected]

320

Brown, Hinchcliffe, Marletta, Naboko and Wood

Serguei Naboko Department of Math. Physics Institute of Physics St. Petersburg State University 1 Ulianovskaia, St. Petergoff St. Petersburg, 198504 Russia e-mail: [email protected] Ian Wood Institute of Mathematics and Physics Aberystwyth University Penglais, Aberystwyth Ceredigion SY 23 3BZ UK e-mail: [email protected] Submitted: August 27, 2008. Revised: November 18, 2008.

IEOT

Integr. equ. oper. theory 63 (2009), 321–335 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030321-15, published online February 2, 2009 DOI 10.1007/s00020-009-1653-6

Integral Equations and Operator Theory

Bounded Berezin-Toeplitz Operators on the Segal-Bargmann Space Hiroyuki Chihara Abstract. We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex n-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sj¨ ostrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sj¨ ostrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B32, 47G30. Keywords. Bargmann transform, Segal-Bargmann space, Berezin-Toeplitz operator, pseudodifferential operator.

1. Introduction We study the boundedness and the deformation estimates of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) introduced by Sj¨ostrand in [15]. This space is a reproducing kernel Hilbert space of square-integrable holomorphic functions on the complex n-space, and is characterized by the image of a global Bargmann-type transform. We begin with a review of Sj¨ostrand’s “linear” theory in [15] to introduce the setting of the present paper. Let φ(X, Y ) be a quadratic form of (X, Y ) ∈ Cn × Cn of the form 1 1 hX, AXi + hX, BY i + hY, CY i, 2 2 where A, B and C are complex n × n matrices, t A = A, t C = C and hX, √ Yi = X1 Y1 + · · · + Xn Yn for X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Yn ). Set i = −1, φ(X, Y ) =

Supported by the JSPS Grant-in-Aid for Scientific Research #20540151.

322

Chihara

IEOT

¯ ¯ CR = (C + C)/2 and CI = (C − C)/2i. We denote by In the n×n identity matrix. Assume that det φ00XY = det B 6= 0, Im

φ00Y Y

(1)

= CI > 0.

−1/2

−1/2

(2) −1/2

−1/2

We remark that det C = det CI det(CI CR CI +iIn ) 6= 0 since CI CR CI is a real symmetric matrix. Let h ∈ (0, 1] be a semiclassical parameter, and let S (Rn ) be the Schwartz class on Rn . A global Bargmann-type transformation of u ∈ S (Rn ) is defined by Z −3n/4 eiφ(X,y)/h u(y)dy, T u(X) = Cφ h Rn

where Cφ is a normalizing constant as Cφ = 2−n/2 π −3n/4 |det B|(det CI )−1/4 . The assumption (2) guarantees the existence of a function Φ(X) = maxn {− Im φ(X, y)} y∈R

1 1 hIm(t BX), CI−1 Im (t BX)i − Im hX, AXi 2 2 ¯ + Re hX, Φ00XX Xi, = hX, Φ00X X¯ Xi =

¯ BCI−1 t B BCI−1 t B A > 0, Φ00XX = − − . 4 4 2i p ¯ for X ∈ We denote the Lebesgue measure on Cn by L. Set |X| = hX, Xi n n 2 C . Let LΦ be the set of all square-integrable functions on C with respect to e−2Φ(X)/h L(dX), and let HΦ be the set of all holomorphic functions in L2Φ . We remark that 1 1/2 Re {iφ(X, y)} = Φ(X) − |CI (y + CI−1 Im (t BX))|2 . 2 The Bargmann transform T is well-defined for any tempered distribution u ∈ S 0 (Rn ). Moreover T u satisfies e−Φ(X)/h T u(X) ∈ S 0 (Cn ), and is holomorphic on Cn . In particular, T gives a Hilbert space isomorphism of L2 (Rn ) onto HΦ , where L2 (Rn ) is the set of all Lebesgue square-integrable functions on Rn . We here remark that e−Φ(X)/h T (S (Rn )) ⊂ S (Cn ), and T (S (Rn )) is densely embedded in HΦ and T (S 0 (Rn )) respectively since S (Rn ) is densely embedded in L2 (Rn ) and S 0 (Rn ) respectively. The Bargmann transform T is interpreted as a Fourier integral operator associated with a linear canonical transform Φ00X X¯ =

κT : Cn × Cn 3 (Y, −φ0Y (X, Y )) 7→ (X, φ0X (X, Y )) ∈ Cn × Cn , κT (x, ξ) = (−t B −1 (Cx + ξ), Bx − At B −1 (Cx + ξ)). If we set

( ΛΦ =

2 ∂Φ X, (X) i ∂X

) n , X∈C

Vol. 63 (2009)

Berezin-Toeplitz Operators

323

then ΛΦ = κT (R2n ). This means that the singularities of u ∈ S 0 (Rn ) described in the phase space R2n are translated into those of T u described in the Lagrangian submanifold ΛΦ . Let Ψ(X, Y ) be a holomorphic quadratic function on Cn × Cn defined by the ¯ critical value of −{φ(X, Z) − φ(Y¯ , Z)}/2i for Z ∈ Cn , that is, 1 1 Ψ(X, Y ) = hX, Φ00X X¯ Y i + hX, Φ00XX Xi + hY, Φ00XX Y i. 2 2 ∗ ¯ Note that Ψ(X, X) = Φ(X). T T is an orthogonal projector of L2Φ onto HΦ , and given by Z CΦ ¯ e[2Ψ(X,Y )−2Φ(Y )]/h u(Y )L(dY ), (3) T T ∗ u(X) = n h Cn n 2 CΦ = det(Φ00X X¯ ) = (2π)−n |det B|2 (det CI )−1 . π Here we state the definition of Berezin-Toeplitz operators on HΦ . If we set −1/2 t R = CI B/2, then R∗ R = Φ00XX ¯ . Let T be a class of symbols defined by Z ( ) −2|R(X−Y )|2 /h 2 n T = b(X) e |b(Y )| L(dY ) < ∞ for any X ∈ C . Cn A Berezin-Toeplitz operator T˜b associated with a symbol b ∈ T is defined by T˜b u = T T ∗ (bu) for u∈HΦ . Since Re {2Ψ(X, Y¯ ) − 2Φ(Y )} = Φ(X) − Φ(Y ) − |R(X − Y )|2 , e−Φ(X)/h T˜b u(X) takes a finite value for each X ∈ Cn provided that u∈L2Φ and b ∈ T . Historically, Berezin introduced this type of operators acting on a class of holomorphic functions over some complex spaces or manifolds, and established the foundation of geometric quantization in his celebrated paper [1]. Properties of such operators and related problems on the usual Segal-Bargmann space have been investigated in several papers. See [2], [3], [5], [6], [7], [17] and references therein. Here we give two examples of HΦ . Example 1. If φ(X, Y ) = iβ(X 2 /2 − 2XY + Y 2 ), β > 0 and XY = hX, Y i, then HΦ is the usual Segal-Bargmann space (the Fock space), and β ¯ i i ¯ Ψ(X, Y ) = X Y , κT (x, ξ) = x − ξ, −iβ x + ξ . 2 2β 2β It is remarkable that Φ(X) = β|X|2 /2 is strictly convex and Φ00XX = 0 in this case. The strict convexity justifies the change of quantization parameter. See [15, Proposition 1.3]. These facts are effectively used in the analysis on the usual SegalBargmann space. See e.g., [8] for the detail.

324

Chihara

IEOT

Example 2. If we set φ(X, Y ) = i(X − Y )2 /2, then T is the heat kernel transform, and (X − Y¯ )2 (Im X)2 Ψ(X, Y¯ ) = − , Φ(X) = , κT (x, ξ) = (x − iξ, ξ). 8 2 In this case, the global FBI transform e−Φ(X)/h T and the space HΦ are used as strong tools for microlocal and semiclassical analysis of linear differential operators on Rn . See [12] for the detail. The purpose of the present paper is to study the boundedness and the deformation estimates of Berezin-Toeplitz operators on the generalized Segal-Bargmann space HΦ . To state our results, we introduce notation and review pseudodifferential calculus on HΦ developed in [15]. We denote by L (HΦ ) the set of all bounded linear operators of HΦ to HΦ , and set ∂a −1 ∂b , (Φ00XX ) Q(a, b) = ¯ ¯ , {a, b} = iQ(a, b) − iQ(b, a) ∂X ∂X R for a, b∈C 1 (Cn ). Pick up χ ∈ S (Cn ) such that Cn χ(X)L(dX) 6= 0. Sj¨ostrand’s Wiener algebra SW (Cn ) is the set of all tempered distributions on Cn satisfying U (ζ; b) = sup |F [uτZ χ](ζ)| ∈ L1 (Cnζ ),

(4)

Z∈Cn

where F is the usual (not semiclassical) Fourier transform on Cn ' R2n , τZ χ(X) = χ(X − Z), and L1 (Cn ) is the set of all Lebesgue integrable functions on Cn . Set kbkSW = kU (·; b)kL1 (Cn ) . We also denote by L∞ (Cn ) the set of all essentially bounded functions on Cn . The definition of SW (Cn ) is independent of the choice of χ, and SW (Cn ) is invariant under linear transforms on Cn . It is remarkable that B 2n+1 (Cn ) ⊂ SW (Cn ) ⊂ B 0 (Cn ), and the Weyl quantization of any element of SW is a bounded linear operator. Set N0 = {0, 1, 2, . . . } for short. B k (Cn ), k ∈ N0 is the set of all bounded C k -functions on Cn whose derivatives of any order up to k are also bounded on Cn . Next we introduce the Weyl quantization on HΦ . For fixed X ∈ Cn , set ( ) 2 ∂Φ X + Y n , Γ(X) = (Y, θ) Y ∈ C , θ = i ∂X 2 and a volume of Γ(X) is defined by dΩ = dY1 ∧ · · · ∧ dYn ∧ dθ1 ∧ · · · ∧ dθn . For u∈HΦ , the reproducing formula u = T T ∗ u has another expression Z 1 u(X) = eihX−Y,θi/h u(Y )dΩ. (5) (2πh)n Γ(X) The right hand sides of (3) and (5) coincide to each other via the change of variables

Vol. 63 (2009)

Berezin-Toeplitz Operators

325

called the Kuranishi trick. The Weyl quantization of a symbol a(X, θ)∈SW (ΛΦ ) = ∗ 2n (κ−1 T ) SW (R ) is defined by Z 1 X +Y ihX−Y,θi/h e , θ u(Y )dΩ OpW (a)u(X) = a h (2πh)n Γ(X) 2 n for u∈T (S (Rn )). OpW h (a)u is holomorphic in C since ∂ ihX−Y,θi/h ∂ ihX−Y,θi/h X +Y X +Y ,θ = ¯ e ,θ a a ¯e 2 2 ∂X ∂Y

in the sense of distribution. The Weyl quantization of a ◦ κT is defined by Z x+y 1 ihx−y,ξi/h e a ◦ κ , ξ u(y)dydξ OpW (a ◦ κ )u(x) = T h (2πh)n R2n 2 for u ∈ S (Rn ). It is remarkable that OpW h (SW (ΛΦ )) is extended on HΦ and a subalgebra of L (HΦ ), and the exact Egorov theorem W OpW h (a) ◦ T = T ◦ Oph (a ◦ κT )

holds for a∈SW (ΛΦ ). Moreover, Guillemin discovered in [9] that T˜b = ¯ where for b(X) = b(X, X), i 0 00 −1 00 b1/2 (X, θ) = b1/2 X, (ΦX X¯ ) θ − ΦXX X , 2

(6) 0 OpW h (b1/2 )

and {bt }t>0 is the heat flow of b defined by bt (X) = eth∆ b(X) Z 2 CΦ = e−2|R(X−Y )| /th b(Y )L(dY ), (th)n Cn ∂ 1 −1 ∂ , (Φ00XX ) ∆= ¯ ¯ . 2 ∂X ∂X bt makes sense for b ∈ T and t ∈ (0, 2). We use only t ∈ [0, 1] as a quantization parameter. b1 is said to be the Berezin symbol of a Berezin-Toeplitz operator T˜b . These facts show that pseudodifferential calculus (See e.g., [10], [12] and [16]) and the heat flow determined by the phase function play essential roles in the analysis of Berezin-Toeplitz operators. Here we state our results. Theorem 1. Suppose that b ∈ T . (i) If T˜b ∈ L (HΦ ), then for any t ∈ (1/2, 1], kbt kL∞ (Cn ) 6

kT˜b kL (HΦ ) . (2t − 1)n

(ii) If bt ∈L∞ (Cn ) for some t ∈ [0, 1/2), then T˜b ∈ L (HΦ ).

(7)

326

Chihara

IEOT

(iii) Suppose that b ∈ S 0 (Cn ) in addition. Set bλ (X) = ei RehX,λi b(X) for λ ∈ Cn . Then, b1/2 ∈SW (Cn ) if and only if k(bλ )1 (·)kL∞ (Cn ) e−h|

t

R−1 λ|2 /8

∈ L1 (Cnλ ).

(8)

In this case, T˜b ∈ L (HΦ ). α β α β Theorem 2. Suppose that ∂X ∂X¯ a, ∂X ∂X¯ b ∈ SW (Cn ) for any multi-indices satisfying |α + β| 6 3. Then, there exists a positive constant C0 which is independent of a, b and h, such that ) (

ih h ˜ ˜ ˜ ˜ ˜ ˜ ˜

, max

[Ta , Tb ] − 2 T{a,b}

Ta ◦ Tb − Tab + 2 TQ(a,b) L (HΦ ) L (HΦ ) X X µ ν 2 α β 6 C0 h k∂X ∂X¯ akSW k∂X ∂X¯ bkSW . |α+β|63

|µ+ν|63

Here we explain the known results and the detail of our results. Theorem 1-(i) is a refinement and a generalization of the results of Berger and Coburn in [3]. They proved that kbt kL∞ (Cn ) 6 C(t)kT˜b kL (HΦ ) for t ∈ (1/2, 1] with some function C(t) in case that HΦ is the usual Segal-Bargmann space. For a general HΦ , we need some ideas to avoid difficulties coming from Φ00XX 6= 0. Theorem 1-(ii) is obvious by the L2 -boundedness theorem of pseudodifferential operators of order zero with smooth symbols. The condition (8) is a special form of the condition for which b1/2 ∈SW (Cn ). This is given by a special choice of a Schwartz function χ appearing in the definition of SW (Cn ). Theorem 1-(iii) seems to extend the known results by Berger and Coburn in [3, Theorem 13], that is, if b > 0 and b1 ∈L∞ (Cn ), then T˜b ∈ L (HΦ ). Theorem 2 reminds us of the recent interesting results of Lerner and Morimoto in [11] on the Fefferman-Phong inequality. Coburn proved in [5] the deformation estimates on the usual Segal-Bargmann space under the assumption a, b ∈ the set of all trigonometric polynomials + C02n+6 (Cn ), where C02n+6 (Cn ) is the set of all compactly supported C 2n+6 -functions on Cn . Roughly speaking, Theorem 2 asserts that the deformation estimates hold for a, b ∈ B 2n+4 (Cn ). The relationship between Berezin-Toeplitz operators and Weyl pseudodifferential operators on HΦ gives a formal identity T˜a ◦ T˜b = T˜c ,

c = e−h∆/2 (a01/2 #b01/2 ),

where # is the product of SW (ΛΦ ) in the sense of the Weyl calculus introduced later. Unfortunately, however, the backward heat kernel e−h∆/2 can act only on a class of real-analytic symbols, and it is very hard to obtain the symbol c. We apply the forward heat kernel eth∆ to the construction of the asymptotic expansion of the backward heat kernel h e−h∆/2 = 1 − ∆ + O(h2 ), 2

Vol. 63 (2009)

Berezin-Toeplitz Operators

327

and give an elementary proof of Theorem 2. The organization of the present paper is as follows. In Section 2 we prove (i) and (iii) of Theorem 1. In Section 3 we prove Theorem 2.

2. Boundedness of Berezin-Toeplitz operators In this section we prove (i) and (iii) of Theorem 1. On one hand, to prove (i), we express the boundedness of T˜b in terms of a complete orthonormal system of HΦ . We introduce a trace class operator defined by T˜b and the complete orthonornal system, and take its trace which becomes bt (X) for any fixed X ∈ Cn . This idea is basically due to Berger and Coburn in [3]. In our case, however, Φ(X) is not supposed to be strictly convex, nor Φ00XX is not supposed to vanish. We need to be careful about these obstructions. On the other hand, the proof of (iii) is a simple computation. We choose a Schwartz function χ as a heat kernel at the time t = 1/2. Here we give two lemmas used in the proof of (i). For u, v∈HΦ , the inner product h·, ·iHΦ is defined by Z hu, viHΦ = u(X)v(X)e−2Φ(X)/h L(dX), Cn

which is the restriction of h·, ·iL2Φ on HΦ . Set uα (X) =

CΦ 2|α| hn α!h|α|

1/2

00

(RX)α ehX,ΦXX Xi/h

for a multi-index α ∈ Nn0 . The first lemma is concerned with a complete orthonormal system of HΦ which is naturally generated by the Taylor expansion of the ¯ reproducing kernel e2Ψ(X,Y )/h . Lemma 3. {uα }α∈Nn0 is a complete orthonormal system of HΦ . In case that HΦ is the usual Segal-Bargmann space, the proof of Lemma 3 is given in [8, page 40, (1.63) Theorem]. In this case, {T ∗ uα }α∈Nn0 is said to be the family of Hermite functions. The general case can be proved in the same way, and we here omit the proof of Lemma 3. Next lemma is concerned with the family of Weyl operators, which is a family of unitary operators on HΦ and acts on symbols of Berezin-Toeplitz operators as a group of shifts on Cn . The family of Weyl operators {Wλ }λ∈Cn on HΦ is defined by Wλ u(X) = e[2ϕ(X,λ)−ϕ(λ,λ)]/h u(X − λ), where ¯ + hX, Φ00 λi. ϕ(X, λ) = hX, Φ00X X¯ λi XX We remark that ϕ(X, λ) is holomorphic in X, and if u is holomorphic, then Wλ u is also. Properties of Weyl operators are the following.

328

Chihara

IEOT

Lemma 4. We have (i) Wλ∗ = W−λ on HΦ . (ii) Wλ∗ ◦Wλ = I on HΦ . (iii) Wλ∗ ◦ T˜b ◦Wλ = T˜b(·+λ) on HΦ for b ∈ T . Proof. A direct computation shows that 2ϕ(X + λ, λ) − ϕ(λ, λ) − 2Φ(X + λ) = −2ϕ(X + λ, λ) + ϕ(λ, λ) − 2Φ(X)

(9)

= 2ϕ(X, −λ) − ϕ(−λ, −λ) − 2Φ(X). (10) Let u, v∈HΦ . Using a translation X 7→ X + λ and (10), we deduce Z hWλ u, viHΦ = e[2ϕ(X,λ)−ϕ(λ,λ)−2Φ(X)]/h u(X − λ)v(X)L(dX) n ZC = e[2ϕ(X+λ,λ)−ϕ(λ,λ)−2Φ(X+λ)]/h u(X)v(X + λ)L(dX) Cn Z = e[2ϕ(X,−λ)−ϕ(−λ,−λ)−2Φ(X)]/h u(X)v(X + λ)L(dX) Cn

= hu, W−λ viHΦ , which shows that Wλ∗ = W−λ . Wλ∗ ◦Wλ = I is also proved by a direct computation Wλ∗ ◦Wλ u(X) = W−λ ◦Wλ u = e[2ϕ(X,−λ)−ϕ(−λ,−λ)]/h (Wλ u)(X + λ) = e[−2ϕ(X,λ)−ϕ(λ,λ)]/h (Wλ u)(X + λ) = e[−2ϕ(X,λ)−ϕ(λ,λ)+2ϕ(X+λ,λ)−ϕ(λ,λ)] u(X) = u(X), since ϕ(X + λ, λ) = ϕ(X, λ) + ϕ(λ, λ). T T ∗ is self-adjoint on L2Φ and T T ∗ Wλ v = Wλ v for v∈HΦ . Using this and (9), we deduce hWλ∗ ◦ T˜b ◦Wλ u, viHΦ = hT˜b ◦Wλ u, Wλ viH

Φ

= hT T ∗ (bWλ u), Wλ viHΦ = hbWλ u, Wλ viL2Φ Z = b(X)e[2ϕ(X,λ)+2ϕ(X,λ)−ϕ(λ,λ)−ϕ(λ,λ)−2Φ(X)]/h Cn

× u(X − λ)v(X − λ)L(dX) Z

b(X + λ)e[2ϕ(X+λ,λ)+2ϕ(X+λ,λ)−ϕ(λ,λ)−ϕ(λ,λ)−2Φ(X+λ)]/h

= Cn

× u(X)v(X)L(dX)

Vol. 63 (2009)

Berezin-Toeplitz Operators

Z

329

b(X + λ)e−2Φ(X)/h u(X)v(X)L(dX)

= Cn

= hT˜b(·+λ) u, viHΦ , which proves Wλ∗ ◦ T˜b ◦Wλ = T˜b(·+λ) .

Here we prove Theorem 1-(i). Proof of Theorem 1-(i). Suppose T˜b ∈ L (HΦ ), and set M = kT˜b kL (HΦ ) for short. Lemma 4 shows that T˜b(·+X) ∈ L (HΦ ) and M = kT˜b(·+X) kL (HΦ ) for any X ∈ Cn . In terms of the complete orthonormal system given in Lemma 3, T˜b ∈ L (HΦ ) implies that |hT˜b uα , uβ iHΦ | 6 M for any α, β ∈ Nn0 . Since Φ(Y ) = |RY |2 + Re hY, Φ00XX Y i, we deduce that for any X ∈ Cn hT˜b(·+X) uα , uβ iHΦ = hT T ∗ (b(· + X)uα ), uβ iHΦ = hb(· + X)uα , uβ iL2Φ 1/2 Z 1 CΦ b(X + Y ) = n h α!β! Cn )β ( )α ( 1/2 1/2 2 2 2 RY e−2|RX| /h L(dY ). × RY h h In particular, if we take α = β and sum it up for |α| = k, then we have X hT˜b(·+X) uα , uα iHΦ |α|=k

CΦ = n h

Z Cn

1 k!

2|RY |2 h

k

e−2|RY |

2

/h

b(X + Y )L(dY ).

(11)

Fix (t, X) ∈ (1/2, 1] × Cn . When k = 0, (11) shows that hT˜b(·+X) u0 , u0 iHΦ = b1 (X), and |hT˜b(·+X) u0 , u0 iHΦ | 6 M implies that kb1 kL∞ (Cn ) 6 M , which is (7) at t = 1. We consider (t, X) ∈ (1/2, 1) × Cn below, and set s = 1/t − 1 ∈ (0, 1). Here we introduce a trace class operator Hs,X u =

∞ X

(−s)k

k=0

X

hu, uα iHΦ T˜b(·+X) uα

|α|=k

for u∈HΦ . Let Ks,X (Y, Z) be the integral kernel of Hs,X , that is, Ks,X (Y, Z) =

∞ X k=0

(−s)k

X |α|=k

T˜b(·+X) uα (Y )uα (Z).

330

Chihara

IEOT

It is easy to see that Ks,X (Y, Y )∈L1 (Cn ; e−2Φ(Y )/h L(dY )) since ∞ X X Z sk |T˜b(·+X) uα (Y )||uα (Y )|e−2Φ(Y )/h L(dY ) k=0

|α|=k

6M

X

Cn ∞ X

s|α| = M

α∈Nn 0

!n sk

= M (1 − s)−n =

k=0

M tn . (2t − 1)n

Then, the Lebesgue convergence theorem and (11) impliy that Z X N X t−n (−s)k T˜b(·+X) uα (Y )uα (Y )e−2Φ(Y )/h L(dY ) Cn k=0

=

CΦ (th)n

Z Cn

|α|=k N X

(−s)k

k=0

1 k!

2|RY |2 h

k

e−2|RY |

2

/h

b(X + Y )L(dY )

(12)

converges as N → ∞. Thus we have (7) for t ∈ (1/2, 1) since the right hand side of (12) converges to bt (X). Next we prove Theorem 1-(iii). Boulkhemair proved in [4] that (4) is equivalent to sup |F −1 [F [b]τλ χ](X)| ˜ ∈ L1 (Cnλ ) (13) X∈Cn R with some χ ˜ ∈ S (Cn ) satisfying Cn χ(X)L(dX) ˜ 6= 0, where F −1 is the usual n inverse Fourier transform on C . Proof of Theorem 1-(iii). We compute the condition (13). We choose F [χ](X) = 2 C1 e−4|RX| /h which is the heat kernel at the time t = 1/2, and expect a comprehensive expression coming from the parallelogram law. Let X ∗ ∈ Cn be the dual variable under the Fourier transform. We choose a constant C1 > 0 so that t ¯ −1 ∗ 2 χ(X ∗ ) = e−h| R X | /16 . Set χλ = τλ χ for short. The parallelogram law implies that F [b1/2 ](X ∗ )χ2λ¯ (X ∗ ) = e−h|

t

¯ 2 /16 ¯ −1 X ∗ |2 /16−h|t R ¯ −1 (X ∗ −2λ)| R

= e−h|

t

¯ 2 /8 ¯ −1 (X ∗ −λ)| R−1 λ|2 /8−h|t R

F [b](X ∗ )

F [b](X ∗ ).

Taking the inverse Fourier transformation of the above, we deduce F −1 [F [b1/2 χ2λ¯ ]](X) Z 2 −h|t R−1 λ|2 /8 CΦ =e ei Re hX−Y,λi−2|R(X−Y )| /h b(Y )L(dY ) hn C n = ei Re hX,λi−h|

t

R−1 λ|2 /8

(b−λ )1 (X).

Hence, we obtain sup |F −1 [F [b1/2 ]χ−2λ¯ ](X)| = e−h|

X∈Cn

This completes the proof.

t

R−1 λ|2 /8

k(bλ )1 kL∞ (Cn ) .

Vol. 63 (2009)

Berezin-Toeplitz Operators

331

3. Deformation estimates for compositions Finally, we prove Theorem 2. We first review the composition of pseudodifferential operators on HΦ . Let σ be a canonical symplectic form on C2n , that is, n X σ = dΞ∧dX = dΞj ∧dXj j=1 n

n

at (X, Ξ) ∈ C ×C . Split σ into real and imaginary parts, and denote σ = σR +iσI . R2n and ΛΦ are I-Lagrangian and R-symplectic. Indeed, this is obvious for R2n , and a direct computation shows that σI |ΛΦ = 0 and n X 2 ∂Φ ∂2Φ ¯ σR |ΛΦ = 2i ¯ k dXj ∧dXk for θ = i ∂X (X), ∂Xj ∂ X j,k=1

which is nondegenerate. We use this fact as κ∗T σ = σR on R2n . Let a0 , b0 ∈SW (ΛΦ ). It is well-known that W 0 W 0 0 0 OpW h (a ◦ κT ) ◦ Oph (b ◦ κT ) = Oph (a ◦ κT #b ◦ κT ), Z 1 e−2iσR (y,η;z,ζ)/h a0 ◦ κT #b0 ◦ κT (x, ξ) = (2πh)2n R4n × a0 ◦ κT (x + y, ξ + η)b0 ◦ κT (x + z, ξ + ζ)dydηdzdζ.

Set θ(X) = −2iΦ0X (X) for X ∈ Cn . Using the exact Egorov theorem (6) together with the symplectic transform κT or a direct computation, we have W 0 W 0 0 0 OpW h (a ) ◦ Oph (b ) = Oph (a #b ), n 2 Z 2 CΦ a0 #b0 (X, θ(X)) = e−2iσ(Y,θ(Y );Z,θ(Z))/h hn 2n C × a0 (X + Y, θ(X + Y ))b0 (X + Z, θ(X + Z))L(dY )L(dZ). α β α β ∂X¯ b ∈ ∂X¯ a, ∂X Here we begin the proof of Theorem 2. Suppose that ∂X n th∆ SW (C ) for any multi-indices satisfying |α + β| 6 3. Set at = e a, bt = eth∆ b, 2 i a01/2 (X, θ) = a1/2 X, (Φ00X X¯ )−1 θ − Φ00XX X , 2 i i 2 b01/2 (X, θ) = b1/2 X, (Φ00X X¯ )−1 θ − Φ00XX X . 2 i ¯ Then, we have T˜a ◦ T˜b = OpW (a0 #b0 ). Since a0 (X, θ(X)) = a1/2 (X, X), h

1/2

1/2

1/2

¯ and bt (X) = bt (X, X) ¯ simply, then T˜a ◦ T˜b = if we write at (X) = at (X, X) OpW (a #b ), and 1/2 1/2 h n 2 Z 2 CΦ at #bt (X) = e−2iσ(Y,θ(Y );Z,θ(Z))/h hn C2n × at (X + Y )bt (X + Z)L(dY )L(dZ).

332

Chihara

IEOT

To complete the proof of Theorem 2, we have only to show that a1/2 #b1/2 ≡ eh∆/2 (ab) −

h h∆/2 e Q(a, b) 2

mod h2 SW (Cn ).

(14)

Here we remark that ¯ − 4hZ, Φ00 ¯ Y¯ i = 8i ImhY, Φ00 ¯ Zi, ¯ −2iσ(Y, θ(Y ); Z, θ(Z)) = 4hY, Φ00X X¯ Zi XX XX h 00 −1 ∂ −2iσ(Y,θ(Y );Z,θ(Z))/h (Φ ¯ ) e , 4 XX ∂ Z¯ h ∂ −2iσ(Y,θ(Y );Z,θ(Z))/h e Y¯ e−2iσ(Y,θ(Y );Z,θ(Z))/h = − (Φ00X X¯ )−1 . 4 ∂Z From Taylor’s formula and the integration by parts we derive Y e−2iσ(Y,θ(Y );Z,θ(Z))/h =

h h Q(at , bt )(X) + Q(bt , at )(X) + h2 rt (X; h), 4 4 where {rt (X; h)}h∈(0,1] is bounded in B ∞ (Cn ) for fixed t > 0. We approximate the main term of at #bt which is at #bt (X) = at bt (X) −

h h Q(at , bt ) + Q(bt , at ), 4 4 by constructing an approximate solution to the initial value problem for the heat equation satisfied by ct . In other words, we construct an asymptotic solution to the transport equation whose main term is given by the heat operator ∂t − h∆. It is easy to see that ct = at bt −

α β α β ∂X ∂X¯ at , ∂X ∂X¯ bt ∈ C([0, ∞); SW (Cn )) (1)

for |α + β| 6 3. Set pt = eth∆ (ab) + hpt (1)

pt

and

1 1 = − eth∆ Q(a, b) + eth∆ Q(b, a) 4 4 Z 1 t (t−s)h∆ − e {Q(as , bs ) + Q(bs , as )}ds. 2 0

Then, ct and pt solve ∂ h h2 − h∆ ct = − {Q(at , bt ) + Q(bt , at )} + Q1 (at , bt ), ∂t 2 4 h h c0 = ab − Q(a, b) + Q(b, a), 4 4 2 2 00 −1 ∂ a 00 −1 ∂ b Q1 (a, b) = (ΦX X¯ ) , (Φ ) ¯ XX ¯2 , ∂X 2 ∂X ∂ h − h∆ pt = − {Q(at , bt ) + Q(bt , at )}, ∂t 2 h h p0 = ab − Q(a, b) + Q(b, a), 4 4

Vol. 63 (2009)

Berezin-Toeplitz Operators

333

respectively. Hence, h2 ct − pt = 4

t

Z

e(t−s)h∆ Q1 (as , bs )ds ∈ h2 C([0, ∞); SW (Cn )).

0 (1)

We show that the main part of the second term in pt is −teth∆ {Q(a, b) + Q(b, a)}/2, that is, Z t e(t−s)h∆ {Q(as , bs ) + Q(bs , as )}ds = teth∆ {Q(a, b) + Q(b, a)} + O(h). 0

For this purpose, we estimate Z t Z t e(t−s)h∆ Q(as , bs )ds − teth∆ Q(a, b) = {e(t−s)h∆ Q(as , bs ) − Q(as , bs )}ds 0 0 Z t + {Q(as , bs ) − Q(at , bt )}ds 0

+ t{Q(at , bt ) − eth∆ Q(a, b)} = Ft + Gt + tHt . We here remark that the heat kernel eth∆ is an even function in the space variable. Combining this fact and Taylor’s formula, we can obtain the desired estimates of Ft and Gt . This technique has been frequently used for approximating symbols. Changing the variables in the explicit formula of the heat kernel, we have Z t Z 2 CΦ Ft (X) = ds e−2|RY | /(t−s)h n 0 {(t − s)h} Cn × {Q(as , bs )(X + Y ) − Q(as , bs )(X)}L(dy) Z t Z 2 = CΦ ds e−2|RY | 0 Cn p (15) × {Q(as , bs )(X + (t − s)hY ) − Q(as , bs )(X)}L(dy). Substituting Taylor’s formula Q(as , bs )(X + Y ) = Q(as , bs )(X) + hY, ∂X Q(as , bs )(X)i + hY¯ , ∂X¯ Q(as , bs )(X)i + Q2 (as , bs )(X, Y ), X

Q2 (as , bs )(X, Y ) =

|α+β|=2

Y α Y¯ β α!β!

Z

1

(1 − τ )

0

∂ 2 Q(as , bs ) ¯β ∂X α ∂ X

(X + τ Y )dτ

into (15), we have Z Ft (X) = CΦ

t

Z ds

0

2

e−2|RY | Q2 (as , bs )(X,

Cn

which belongs to hC([0, ∞); SW (Cn )).

p (t − s)hY )L(dY ),

334

Chihara

IEOT

We split Gt into two parts Z t Z t Gt = {Q(as , bs ) − Q(at , bt )}ds = {Q(as − at , bs ) + Q(at , bs − bt )}ds. 0

0

Since Z

2

e−2|RY | {a(X +

as (X) − at (X) = CΦ ZC

√

shY ) − a(X +

√ thY )}L(dY )

n

√ √ √ 2 e−2|RY | a ˜(X, ( s − t) hY )L(dY ),

= CΦ Cn

X

a ˜(X, Y ) =

|α+β|=2

Y α Y¯ β α!β!

Z

1

(1 − τ )

0

∂2a ¯β ∂X α ∂ X

we can show that Gt ∈hC([0, ∞); SW (Cn )). It follows that Ht ∈hC([0, ∞); SW (Cn )) since ∂ − h∆ Ht ∈ hC([0, ∞); SW (Cn )), ∂t

(X + τ Y )dτ,

H0 = 0.

Combining the estimates of Ft , Gt and Ht , we have Z t e(t−s)h∆ Q(as , bs )ds − teth∆ Q(a, b) ∈ hC([0, ∞); SW (Cn )).

(16)

0 (1)

Applying (16) to pt , we obtain h 1 h 1 th∆ th∆ ct = e (ab) − + t e Q(a, b) + − t eth∆ Q(b, a) + O(h2 ). 2 2 2 2 If we take t = 1/2, we obtain (14). This completes the proof of Theorem 2. Acknowledgment The author would like to express to the referee his sincere gratitude for valuable comments. In particular, the present paper improved in the presentation following the referee’s suggestion on the logic in the first section.

References [1] F. A. Berezin, Quantization, Math. USSR Izvestija 8 (1974), 1109–1163. [2] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), 813–829. [3] C. A. Berger and L. A. Coburn, Heat flow and Berezin-Toeplitz estimates, Amer. J. Math. 116 (1994), 563–590. [4] A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett. 4 (1997), 53–67. [5] L. A. Coburn, Deformation estimates for the Berezin-Toeplitz quantization, Comm. Math. Phys. 149 (1992), 415–424.

Vol. 63 (2009)

Berezin-Toeplitz Operators

335

[6] L. A. Coburn, On the Berezin-Toeplitz calculus, Proc. Amer. Math. Soc. 129 (2001), 3331–3338. [7] L. A. Coburn, A Lipschitz estimate for Berezin’s operator calculus, Proc. Amer. Math. Soc. 133 (2005), 127–131. [8] G. B. Folland, “Harmonic Analysis in Phase Space”, Princeton University Press, 1989. [9] V. Guillemin, Toeplitz operators in n dimensions, Integral Equations Operator Theory 7 (1984), 145–205. [10] H. Kumano-go, “Pseudodifferential Operators”, MIT Press, 1981. [11] N. Lerner and Y. Morimoto, On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators, Publ. Res. Inst. Math. Sci. 43 (2007), 329–371. [12] A. Martinez, “An Introduction to Semiclassical and Microlocal Analysis”, SpringerVerlag, 2002. [13] J. Sj¨ ostrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994), 185–192. [14] J. Sj¨ ostrand, Wiener type algebras of pseudodifferential operators. S´eminaire sur les ´ ´ Equations aux D´eriv´ees Partielles, 1994–1995, Exp. No. IV, 21 pp., Ecole Polytech., Palaiseau, 1995. [15] J. Sj¨ ostrand, Function spaces associated to global I-Lagrangian manifolds, “Structure of solutions of differential equations (Katata/Kyoto, 1995)”, 369–423, World Sci. Publ., 1996. [16] M. A. Shubin, “Pseudodifferential Operators and Spectral Theory, Second Edition”, Springer-Verlag, 2001. [17] K. Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), 3–16. Hiroyuki Chihara Mathematical Institute Tohoku University Sendai 980-8578 Japan e-mail: [email protected] Submitted: June 2, 2008. Revised: November 17, 2008.

Integr. equ. oper. theory 63 (2009), 337–349 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030337-13, published online February 24, 2009 DOI 10.1007/s00020-009-1662-5

Integral Equations and Operator Theory

On C ∗-Extreme Maps and ∗-Homomorphisms of a Commutative C ∗-Algebra Martha Case Gregg Abstract. The generalized state space of a commutative C ∗ -algebra, denoted SH (C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H. C ∗ -convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C ∗ -extreme point of SH (C(X)) satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C ∗ -extreme maps from C(X) into K+ , the algebra generated by the compact and scalar operators, are multiplicative. This generalizes a result of D. Farenick and P. Morenz. We then determine the structure of these maps. Mathematics Subject Classification (2000). Primary 46L05; Secondary 46L30. Keywords. C ∗ -extreme, C ∗ -convex, non-commutative convexity, generalized state.

Several non-commutative analogs of convexity have appeared in the literature including CP-convexity [4] and matrix convexity [2], as well as C ∗ -convexity [3], [6], which is the topic of this paper. In [6], Hopenwasser, Moore, and Paulsen characterized operators which are C ∗ -extreme in the unit ball of B(H) and obtained results about other C ∗ -convex sets and their extreme points. In [3], Farenick and Morenz extend the idea of C ∗ -convexity to the space of completely positive maps on a C ∗ -algebra. They show that C ∗ -extreme maps with their range in K+ are also extreme (in the classical sense) and obtain a characterization of C ∗ -extreme maps on a commutative C ∗ -algebra which have their range in Mn , the C ∗ -algebra of n × n complex matrices. Subsequently, Zhou [9] gave two necessary and sufficient conditions for a completely positive map to be C ∗ -extreme, and described the structure of C ∗ -extreme maps with range in Mn . The main results presented here are Theorem 5, which gives a necessary condition for a map φ : C(X) → B(H) This paper constitutes a part of the author’s Ph.D. thesis at the University of Nebraska-Lincoln.

338

Gregg

IEOT

to be C ∗ -extreme, and Theorem 10, which then shows that a positive unital map φ : C(X) → K+ is C ∗ -extreme if and only if it is multiplicative. We then determine the structure of such maps. The author wishes to express her gratitude to Professor David Pitts for many lively conversations which led to significant improvements in the paper. Throughout, let X be a compact Hausdorff space, C(X) the C ∗ -algebra of continuous functions on X, and H a Hilbert space. Definition 1. The generalized state space of C(X) is SH (C(X)) = {φ : C(X) → B(H) | φ is positive and φ(1) = I}. Note that in the case of a non-commutative C ∗ -algebra A, the generalized state space SH (A) is the set of completely positive unital maps. However, for a commutative C ∗ -algebra, every positive map is also completely positive [8, Theorem 4]. If H = C, the generalized state space SC (A) coincides with the classical state space of A. Definition 2. We say that φ, ψ ∈ SH (C(X)) are unitarily equivalent, and write φ ∼ ψ, if there is a unitary u ∈ B(H) such that φ(f ) = u∗ ψ(f )u for every f ∈ C(X). Definition 3. If φ, ψ1 , . . . ψn ∈ SH (C(X)) and t1 , . . . tn ∈ B(H) are invertible with t∗1 t1 + . . . + t∗n tn = I, then we say φ(f ) = t∗1 ψ1 (f )t1 + . . . + t∗n ψn (f )tn for every f ∈ C(X), is a proper C ∗ -convex combination. We call a map φ ∈ SH (C(X)) C ∗ -extreme if, whenever φ is written as a proper C ∗ -convex combination of ψ1 , . . . , ψn , then ψj ∼ φ for each j = 1, . . . , n. We begin with a discussion of B(H)-valued measures, which closely follows the development given in Paulsen [7]. These operator valued measures play a key role in the proof of Theorem 5, below. Given a bounded linear map φ : C(X) → B(H) and vectors x, y ∈ H, the bounded linear functional f 7→ hφ(f )x, yi corresponds to a unique regular Borel measure µx,y on X such that Z f dµx,y := hφ(f )x, yi for any f ∈ C(X). X

Denote the σ-algebra of Borel sets of X by S. For a set B ∈ S, the sesquilinear form (x, y) 7→ µx,y (B) then determines an operator µ(B). Thus we obtain an operator-valued measure µ : S −→ B(H) which is:

C ∗ -Extreme Maps

Vol. 63 (2009)

339

1. weakly countably additive, i.e., if {Bi }∞ i=1 ⊆ S are pairwise disjoint, and B = S∞ B , then i=1 i hµ(B)x, yi =

∞ X

hµ(Bi )x, yi for every x, y ∈ H.

i=1

2. bounded, i.e., kµk := sup{kµ(B)k : B ∈ S} < ∞. 3. regular, i.e., for each pair of vectors x and y in H, the complex measure µx,y is regular. Furthermore, this process works in reverse: given a regular bounded operatorvalued measure µ : S −→ B(H), define Borel measures µx,y (B) := hµ(B)x, yi for each x, y ∈ H. Then the operator φ(f ) is uniquely defined by the equations Z hφ(f )x, yi := f dµx,y ; X

the map φ : C(X) −→ B(H) is then seen to be bounded and linear. This construction shows that each operator valued-measure gives rise to a unique bounded linear map, and vice-versa. The following proposition summarizes properties shared by operator valued-measures and their associated linear maps. We will be most concerned with parts (2) and (4) of Proposition 4; part (4) is, of course, the Spectral Theorem. Proposition 4. [7, Proposition 4.5] Given an operator valued measure µ and its associated linear map φ, 1. φ is self-adjoint if and only if µ is self-adjoint, 2. φ is positive if and only if µ is positive, 3. φ is a homomorphism if and only if µ(B1 ∩ B2 ) = µ(B1 )µ(B2 ) for all B1 , B2 ∈ S, 4. φ is a ∗-homomorphism if and only if µ is spectral (i.e., projection-valued). We note the following important features of positive operator valued measures, and their relationship to the associated positive maps. (1) Let F(X) = {f : X → C | f is a bounded Borel measurable function}. If φ : C(X) → B(H) is a positive map, we may use the corresponding positive operator-valued measure to extend φ to a map φ˜ : F(X) → B(H) by defining Z ˜ )= φ(f f dµφ , X

for every f ∈ F(X). The measure µφ may then be viewed as the restriction of φ˜ to the characteristic functions of Borel sets. For simplicity, we will simply write φ, ˜ and use the notations µφ (F ) and φ(χ ) interchangeably. rather than φ, F

340

Gregg

IEOT

(2) A positive unital map φ ∈ SH (C(X)) is C ∗ -extreme if and only if the associated operator-valued measure µφ is C ∗ -extreme. (Here, a positive operatorvalued measure µφ is called C ∗ -extreme if, whenever µφ is written µφ = t∗1 µ1 t1 + ... + t∗n µn tn , Pn

∗ where j=1 tj tj = I and each µj is a positive operator-valued measure, then µj ∼ µφ for each j = 1, ...n.) (3) Finally, if φ : C(X) → B(H) is a positive bounded linear map, and µφ the associated operator-valued measure, then for each Borel set F ⊆ X, µφ (F ) ∈ wot-cl φ(C(X)), the weak operator topology closure of φ(C(X)). The proof of this fact requires some care, because while φ(C(X)) is an operator space, it is not generally an algebra.

Proof. Let G ⊆ X be an open set. Then a basic WOT-open set in B(H) centered at φ(χG ) has the form: O = {T ∈ B(H) : |h(T − φ(χG ))xi , yi i| < ε for i = 1 . . . n}, where xi , yi ∈ H and ε > 0. We wish to show that for any such open set there is a function f ∈ C(X) with φ(f ) ∈ O. For each j, we can write µxj ,yj = µj,1 − µj,2 + i(µj,3 − µj,4 ), where each µj,k is a positive measure. Since each of these measures is regular, we may choose compact sets Kj,k ⊆ G for j = 1, . . . , n and k = 1, . . . , 4 such that ε µj,k (G \ Kj,k ) < . 4 Then, setting n [ 4 [ K= Kj,k j=1 k=1

we have, for j = 1, . . . , n, |µxj ,yj (G \ K)| ≤ |µj,1 (G \ K)| + · · · + |µj,4 (G \ K)| < ε. Urysohn’s Lemma now guarantees the existence of a continuous function f : X → [0, 1] with f |K = 1 and f |GC = 0. Hence, for each j = 1, . . . , n, Z Z (f − χK )dµj,2 |h(φ(f ) − µφ (K))xj , yj i| = (f − χK )dµj,1 − X X Z Z +i (f − χK )dµj,1 − (f − χK )dµj,4 X X Z Z ≤ χG\K dµj,1 + · · · + χG\K dµj,4 X

X

< ε. Therefore φ(f ) ∈ O, as required; hence φ(f ) ∈ wot-cl φ(C(X)).

C ∗ -Extreme Maps

Vol. 63 (2009)

341

Now let F = {F ⊆ X : F is a Borel set and φ(χF ) ⊆ wot-cl φ(C(X))} . We will prove that F is a σ-algebra containing the Borel sets, and hence that F = S. Our discussion above shows that F contains every open set S∞of X. Suppose that {Bi } is a countable family of sets in F and set B = i=1 Bi . Assume without loss of generality that {Bi } are a disjoint family. Then, since µφ is weakly countably additive, ∞ X hµφ (B)x, yi = hµφ (Bi )x, yi i=1

for any x, y ∈ H. That is µφ (B) = wot lim µφ N

[ N

Bi ;

i=1

It follows that B ∈ F. Furthermore, if F ∈ F, then φ(χF C ) = φ(1 − χF ) = I − φ(χF ), so that F C ∈ F also. Therefore F is the σ-algebra of Borel sets of X.

Thus, if the range of φ is contained in a C ∗ -subalgebra A of B(H), then the range of µφ is contained in the weak operator topology closure of A, i.e, A00 . We can now prove the following theorem, which gives a necessary condition for a positive map φ on a commutative C ∗ -algebra (or equivalently its associated positive operator-valued measure ) to be C ∗ -extreme. Theorem 5. Let X be a compact Hausdorff space, and φ : C(X) −→ B(H) a unital, positive map. Denote by µφ the unique positive operator-valued measure associated to φ. If φ is C ∗ -extreme, then for every Borel set F ⊂ X, either (1) µφ (F ) is a projection, in which case µφ (F ) ∈ φ(C(X))0 , or (2) σ(µφ (F )) = [0, 1]. Moreover, if (2) occurs and µφ (F ) has an eigenvalue in (0, 1), then the point spectrum of µφ (F ) must contain (0, 1). Proof. Suppose there is a Borel set F ⊆ X so that µφ (F ) is not a projection and σ(µφ (F )) 6= [0, 1]. We will show that φ is not C ∗ -extreme by constructing a proper C ∗ -convex combination t∗1 ψ1 t1 + t∗2 ψ2 t2 = φ in which ψ1 and ψ2 are not unitarily equivalent to φ. Choose x ∈ (0, 1)\σ(µφ (F )) and let (a, b) be the largest open subinterval of (0, 1) which contains x but does not intersect σ(µφ (F )). To be precise, let [ (a, b) = {(α, β) ⊆ (0, 1) : x ∈ (α, β), (α, β) ∩ σ(µφ (F )) = ∅} Note that this choice of the interval (a, b) insures that at least one of the pair {a, b} is in σ(µφ (F )). In particular, if a > 0, then a ∈ σ(µφ (F )) and if b < 1, then

342

Gregg

IEOT

b ∈ σ(µφ (F )). Choose s1 ∈ 41 , 21 with s1 > 21 a−ab b−ab , and set s2 = 1 − s1 . For k = 1, 2, define 1 1 Qk = µφ (F ) + sk µφ (F C ) = sk I + ( − sk )µφ (F ). 2 2 Note that 0 6∈ σ(Qk ) = sk + ( 12 − sk )σ(µφ (F )), so that both Qk ’s are invertible. Now define new positive operator-valued measures µ1 and µ2 by 1 − 12 −1 C µk (B) = Qk µφ (B ∩ F ) + sk µφ (B ∩ F ) Qk 2 , 2 where B is any Borel set of X. Observe that each of the µk ’s is a positive operator1 valued measure with µk (X) = I. Next, define tk = Qk2 , for k = 1, 2. Then, for any Borel set B of X, 1 µφ (B ∩ F ) + s1 µφ (B ∩ F C ) 2 1 + µφ (B ∩ F ) + s2 µφ (B ∩ F C ) 2 = µφ (B).

t∗1 µ1 (B)t1 + t∗2 µ2 (B)t2 =

Each tk is invertible and t∗1 t1 + t∗2 t2 = Q1 + Q2 = µφ (F ) + µφ (F C ) = I. Thus t∗1 µ1 t1 + t∗2 µ2 t2 is a proper C ∗ -convex combination of µ1 and µ2 . It is still necessary to show that µφ is not unitarily equivalent to at least one 1 of µ1 or µ2 . For k = 1, 2, set gk (t) = [sk + (sk − 21 )t]− 2 . As each gk is continuous − 21

on [0, 1], and Qk we have

− 12

= gk (µφ (F )), Qk

1 −1 µk (F ) = µφ (F ) Qk 2 2 −1 1 1 = µφ (F ) sk I + − sk µφ (F ) . 2 2 −1 Let fk (t) = 12 t sk + 12 − sk t . Observe that each fk is continuous on [0, 1], and that µk (F ) = fk (µφ (F )). Therefore, by the spectral mapping theorem, σ(µk (F )) = fk (σ(µφ (F ))). It is easy to check that for t ∈ (0, 1), t < f1 (t) < 1, while 0< f2 (t) < t, and that both fk ’s are strictly increasing. In addition, since s1 > 12 a−ab b−ab , if a > 0, 1 a 1 a < f1 (a) = a < a(1−b) = b ≤ f1 (b). 1 2 s1 + ( 2 − s1 )a b(1−a) (1 − a) + a −1 Qk 2

commutes with µφ (F ). Thus, for k = 1, 2,

Consider the following two cases:

Vol. 63 (2009)

C ∗ -Extreme Maps

343

Case (i). a 6= 0. In this case a ∈ σ(µφ (F )). Thus f1 (a) ∈ σ(µ1 (F )), but since f1 (a) ∈ (a, b), f1 (a) ∈ / σ(µφ (F )). This shows that σ(µφ (F )) 6= σ(µ1 (F )); therefore µφ and µ1 are not unitarily equivalent. Case (ii). a = 0. In this case, b < 1 and b ∈ σ(µφ (F )). As a = 0 < f2 (b) < b, we have f2 (b) ∈ σ(µ2 (F )) \ σ(µφ (F )). In this case, µ2 is not unitarily equivalent to µφ . Let ψk be the positive map determined by µk . Then φ = t∗1 ψ1 t1 + t∗2 ψ2 t2 ; this is a proper C ∗ -convex combination of ψ1 and ψ2 , where φ is not unitarily equivalent to at least one of the maps ψk . Therefore, φ is not C ∗ -extreme. Now suppose that σ(µφ (F )) = [0, 1] and that σpt (µφ (F )) intersects (0, 1), but does not contain (0, 1). It is not difficult to convince oneself that it is possible to choose a, b ∈ (0, 1) satisfying both 2a , and (i) a < b < a+1 (ii) exactly one of the pair {a, b} is an eigenvalue. and define positive operator-valued measures µ1 and µ2 as Set s1 = 12 a−ab b−ab above. As in the previous computation, µ1 (F ) = f1 (µφ (F )). As a result of our choice of s1 , f1 (a) = b. Application of the Spectral Mapping Theorem then shows that either b ∈ σpt (µ1 (F )) \ σpt (µφ (F )), or b ∈ σpt (µφ (F )) \ σpt (µ1 (F )). Since the point spectrum is also a unitary invariant, and µφ = t∗1 µ1 t1 + t∗2 µ2 t2 , this shows that φ is not C ∗ -extreme. Finally, we wish to show that any projection in the range of µφ must commute with φ(C(X)). Suppose that µφ (F ) is a projection and choose f ∈ C(X) with 0 ≤ f ≤ 1. Write f = χF f + (1 − χF )f. Then φ(χF f ) ≤ µφ (F ), so these operators commute. Similarly, φ((1 − χF )f ) ≤ µφ (X \ F ) = I − µφ (F ), so that φ((1 − χF )f ) also commutes with µφ (F ). Therefore φ(f ) commutes with µφ (F ). If f is an arbitrary continuous function, we can express f as a linear combination of positive functions with ranges in [0, 1]. Thus f will commute with µφ (F ). In their paper of 1997 [3], Farenick and Morenz show that a positive map from a commutative C ∗ -algebra into a matrix algebra Mn is C ∗ -extreme if and only if it is a ∗-homomorphism. In view of the spectral condition given by Theorem 5, a shorter proof is possible. Corollary 6. [3, Proposition 2.2] Let X be a compact Hausdorff space and φ : C(X) −→ Mn a positive map. Then φ is C ∗ -extreme if and only if it is a ∗homomorphism.

344

Gregg

IEOT

Proof. It is already known that if φ is a representation (i.e.,∗-homomorphism), then φ is C ∗ -extreme [3, Proposition 1.2]. On the other hand, if φ is not a representation, then the associated positive operator-valued measure µφ is not a spectral measure. In this case, there is a Borel set F ⊂ X for which µφ (F ) is not a projection. As µφ (F ) is an n × n matrix, σ(µφ (F )) consists of at most n isolated points. We may therefore apply the theorem to conclude that φ is not C ∗ -extreme. −1

1

Note that in the proof of Theorem 5, Qk , Qk 2 and tk = Qk2 are elements of the C ∗ -algebra generated by µφ (F ). As noted in the remark preceding Theorem 5, the range of µφ is contained in the WOT-closure of the range of φ. Thus we have the following corollary: Corollary 7. Let M ⊆ B(H) be a von Neumann algebra, φ : C(X) −→ M a unital positive map, and µφ the positive operator-valued measure associated to φ. If φ fails to meet the spectral condition described in Theorem 5, then φ can be written as a proper C ∗ -convex combination φ = t∗1 ψ1 t1 + t∗2 ψ2 t2 , where each tk ∈ M, each ψk : C(X) −→ M, and, for at least one choice of k, ψk is not unitarily equivalent to φ in B(H). We now consider an example of a C ∗ -extreme map which is not multiplicative. The positive map φ defined below was considered by Arveson [1, p. 164] as an example of an extreme point in the generalized state space. Farenick and Morenz [3, Example 2] subsequently showed that φ is also a C ∗ -extreme point, although not a homomorphism. Consider the Hilbert spaces L2 (T, m), where m is normalized Lebesgue measure on T, and H 2 , the classical Hardy space. Let P be the projection of L2 (T, m) onto H 2 . For a function f ∈ L2 (T, m) denote by Mf multiplication by f and by Tf = P Mf P the Toeplitz operator for f . Example 8. [1], [3] Consider the representation π : C(T) −→ B(L2 (T, m)) given by π(f ) = Mf . The spectral measure associated to π is given by µπ (B) = MχB , where B ⊆ X is a Borel set. Define a unital positive map φ : C(T) −→ B(H 2 ) by φ(f ) = P Mf P. Since µπ (B) = MχB , we have µφ (B) = P MχB P = TχB , a Toeplitz operator. Thus σ(µφ (B)) = σ(TχB ). Since χB is a real-valued L∞ function, σ(TχB ) is the closed convex hull of the essential range of χB [5, p. 868]. Therefore, if µφ (B) 6∈ {0, I}, then σ(µφ (B)) = [0, 1] . Thus, for any Borel set B ⊆ X, either µφ (B) = [0, 1] or µφ (B) is a trivial projection; that is, φ satisfies the conditions of the theorem. Now let us consider the case of a unital positive map φ on a commutative C ∗ -algebra C(X) whose range is in K+ , the C ∗ -algebra generated by the compact operators and the identity operator. In [3, Proposition 1.1] Farenick and Morenz

C ∗ -Extreme Maps

Vol. 63 (2009)

345

show that if such a map φ is C ∗ -extreme, then φ is also extreme. It is possible, however, to say more. Theorem 5 requires the operators in the range of the positive operator-valued measure µφ either to be projections, or to have spectrum equal to [0, 1]. In contrast, the spectrum of a positive operator K + αI ∈ K+ must be a sequence of positive numbers with a single limit point at α. This dichotomy suggests that Theorem 5 may give additional information about these maps. In fact, both the result of Theorem 5 (the spectral condition on the operators in the range of µφ ) and the technique used in its proof, will be used below. The result is Theorem 10, which shows that such maps must be multiplicative, and gives their structure. In the succeeding lemma and theorem, let q be the usual quotient map q : B(H) → B(H)/K(H), and set τ = q ◦ φ. Then τ is a positive linear functional, so there is a unique positive real-valued Borel measure µτ on X so that Z τ (f ) = f dµτ for every f ∈ C(X). X

For any function f ∈ C(X), write φ(f ) = Kf + τ (f )I, where Kf ∈ K is a compact operator. Lemma 9. Let φ : C(X) → K+ be unital, positive, and C ∗ -extreme. Then the map τ is multiplicative. Proof. As in the proof of Theorem 5, we will prove the contrapositive. Assume that τ is not multiplicative; then the support of µτ must contain at least two distinct points, which we will call s1 and s2 . Let N1 be a neighborhood of s1 which does not contain s2 . By Urysohn’s Lemma, there exists a continuous function f : X → [0, 1] such that f (s1 ) = 1 and f |N1C = 0. Choose α and β in (0, 1) with α > β and let Q1 = αφ(f ) + βφ(1 − f ) = (α − β)φ(f ) + βI, and Q2 = (1 − α)φ(f ) + (1 − β)φ(1 − f ) = (β − α)φ(f ) + (1 − β)I. Note that since 0 ≤ f ≤ 1, the spectrum of φ(f ) is contained in the closed unit interval. Thus, σ(Q1 ) ⊆ [β, α], and σ(Q2 ) ⊆ [1 − α, 1 − β]. So both Qj ’s are invertible positive operators. Define maps ψ1 and ψ2 by − 21

ψ1 (g) = Q1

− 21

ψ2 (g) = Q2

−1

[αφ(f g) + βφ((1 − f )g)] Q1 2 , and −1

[(1 − α)φ(f g) + (1 − β)φ((1 − f )g)] Q2 2 .

346

Gregg

IEOT 1

Both ψj ’s are positive, unital maps with ranges in K+ . Setting tj = Qj2 , we have t∗1 ψ1 (g)t1 + t∗2 ψ2 (g)t2 = αφ(f g) + βφ(g − f g) + (1 − α)φ(f g) + (1 − β)φ(g − f g) = φ(f g) + φ(g − f g) = φ(g), for every g ∈ C(X). t∗1 t1

t∗2 t2

Since + = I, the above expression gives φ as a proper C ∗ -convex combination of ψ1 and ψ2 . We now wish to show that ψ1 and ψ2 are not unitarily equivalent. To this end, let N2 be a neighborhood of s2 with N1 ∩ N2 = ∅. Then we may choose a continuous function h : X → [0, 1] with h|N2C = 0 (i.e., supp h ⊆ N2 ) and h(s2 ) = 1; thus f h = 0 and (1 − f )h = h. Since h ∈ C(X), φ(h) = Kh + τ (h)I ∈ K+ . Note that τ (h) > 0, since h > 0 on some neighborhood of s2 , and that the essential spectrum of φ(h) is {τ (h)}. Now compute − 21

ψ1 (h) = Q1

− 12

(αφ(f h) + βφ((1 − f )h)) Q1

−1

− 12

= βQ1 2 φ(h)Q1 −1

− 12

= βQ1 2 Kh Q1

+ βτ (h)Q−1 1 .

The first term in this sum is compact, while the second term can be written −1 βτ (h)Q−1 , 1 = βτ (h)[(α − β)Kf + ((α − β)τ (f ) + β)I]

where φ(f ) = Kf + τ (f )I. Thus (q ◦ ψ1 )(h) =

βτ (h) I + K. (α − β)τ (f ) + β

Similar computations yield −1

− 12

ψ2 (h) = (1 − β)Q2 2 Kh Q2 (q ◦ ψ2 )(h) =

+ (1 − β)τ (h)Q−1 2 , and

(1 − β)τ (h) I + K. (β − α)τ (f ) + (1 − β)

So the essential spectra of ψ1 (h) and ψ2 (h) are βτ (h) (1 − β)τ (h) and , (α − β)τ (f ) + β ((β − α)τ (f ) + (1 − β) respectively. However, if these are equal, then β(β − α)τ (f ) + β(1 − β) = (1 − β)(α − β)τ (f ) + β(1 − β), so that, β = β − 1, which is clearly impossible. This shows that the essential spectra of ψ1 (h) and ψ2 (h) are distinct, so that ψ1 (h) and ψ2 (h) are not unitarily equivalent. Thus φ = t∗1 ψ1 t1 + t∗2 ψ2 t2

C ∗ -Extreme Maps

Vol. 63 (2009)

347

expresses φ as a proper C ∗ -convex combination of positive unital maps ψ1 and ψ2 which are not both unitarily equivalent to φ, demonstrating that φ is not C ∗ extreme. This proves the lemma. We can now prove the following: Theorem 10. Let φ : C(X) → K+ be unital and positive. Then φ is C ∗ -extreme if and only if φ is a homomorphism. Proof. If φ is multiplicative, then φ is C ∗ -extreme [3, Proposition 1.2]. Conversely, if φ is C ∗ -extreme, Lemma 9 shows that the map τ = q ◦ φ is multiplicative, so τ is a point evaluation τ (f ) = f (s0 ) for some point s0 ∈ X. Let N be any neighborhood of s0 . Then there exists a continuous function gN : X → [0, 1] with gN (s0 ) = 0 and gN |N C = 1. In this case τ (gN ) = 0, so φ(gN ) = KgN ∈ K. Note that χN C ≤ gN , so that φ(χN C ) ≤ φ(gN ). Since K is hereditary, it follows that φ(χN C ) is compact. By Theorem 5, either φ(χN C ) is a projection or σ(φ(χN C )) = [0, 1]. As a compact operator cannot have the unit interval as its spectrum, φ(χN C ) must be a projection of finite rank. Thus φ(χN ) is also a projection. Let B be any Borel set of X which does not contain s0 . Set Λ := {K ⊆ B : K closed}, and partially order Λ by inclusion. Then µφ (K) is an increasing net of projections. Thus the SOT-lim µφ (K) =: Q exists, and is a projection, namely the projection K [ onto ran µφ (K). Since the measures µx,x are regular for any choice of x ∈ H, K∈Λ

we have µx,x (B) = sup µx,x (K) K∈Λ

or, equivalently, hµφ (B)x, xi = sup hµφ (K)x, xi K∈Λ

= hQx, xi. As this holds for any x ∈ H, Q = µφ (B). If B is a Borel set in X which does contain s0 , then the preceeding argument shows that µφ (B C ) is a projection. Thus µφ (B) is also a projection. Hence µφ is a projection valued measure, and φ is a homomorphism. Remark 11. When φ : C(X) → K+ , as in Theorem 10, we can obtain more information regarding the support of µφ . We have shown above that for any closed set K with s0 6∈ K, µφ (K) is a finite rank projection, say of rank n. If s1 , s2 are

348

Gregg

IEOT

distinct points of K ∩ supp µφ , let N1 ⊆ K be a neighborhood of s1 which does not contain s2 . Then K \N1 is closed and s0 6∈ K \N1 , so µφ (K \N1 ) is a projection of finite rank and 0 < rank µφ (K \ N1 ) < rank µφ (K) = n. Since µφ (K) = µφ (K \ N1 ) + µφ (N1 ), it follows that µφ (N1 ) is also a projection with 0 < rank µφ (N1 ) < n. Clearly this process can be iterated at most n times; we conclude that any closed set K 63 s0 contains at most finitely many points of supp µφ . Consequently, supp µφ \{s0 } is a discrete set with at most one accumulation point at s0 . If H is a separable Hilbert space, then it is clear from the proof of Theorem 10 and the preceding remark that the support of µφ must be at most countable with a single limit point at s0 . In this case, φ must have the form X φ(f ) = f (s)Ps , s∈supp(µφ )

where Ps = µφ ({s}) is a finite rank projection for each s 6= s0 . The rank of Ps0 , on the other hand, may be finite or infinite. The following example, in which we consider the case of a nonseparable Hilbert space, illustrates the structure of unital positive maps φ : C(X) → K+ . Example 12. Let H be a nonseparable Hilbert space with dimension at least as great as the cardinality of R, and let X = R∪{ω} be the one point compactification of (R, d), the reals equipped with the discrete topology. Choose an orthonormal set {es }s∈R in H indexed by the reals, and write Ps for the projection onto the span of es . Then, for any function f ∈ C(X), the set S(f ) := {s ∈ X : f (s) 6= f (ω)} is at most countable, and lim f (sn ) = f (ω),

n→∞

where {sn } is any enumeration of S(f ). Define a positive map φ on C(X) by X φ(f ) = [f (s) − f (ω)]Ps + f (ω)I. s∈S(f )

Then for each s ∈ R, the function δs = χ{s} is continuous and φ(δs ) = µφ ({s}) = Ps . As in the proof of Theorem 10, if G is any neighborhood of ω, then GC is a closed set not containing ω, and φ(χG ) is a projection. In this case the descending net φ(χG ) of projections converges to the projection φ(χ{ω} ) = 0. Thus µφ is a projection valued measure.

C ∗ -Extreme Maps

Vol. 63 (2009)

349

Note that we could define similar maps φ1 and φ2 by X φ1 (f ) = [f (s) − f (ω)]P1/s + f (ω)I, s∈S(f )

and φ2 (f ) =

X

[f (s) − f (ω)]Parctan s + f (ω)I.

s∈S(f )

For these two maps, we have φ1 (χ{ω} ) = P0 , while φ2 (χ{ω} ) is the projection onto the closed span {ran Ps : s ∈ (−∞, π/2] ∪ [π/2, ∞)}. Thus, the measure of {ω} may be a projection of either finite or infinite rank.

References [1] William B. Arveson. Subalgebras of C*-Algebras. Acta Mathematica, 123:141–224, 1969. [2] Edward G. Effros and Soren Winkler. Matrix Convexity: Operator Analogues of the Bipolar and Hahn-Banach Theorems. J. Funct. Anal., 144(1):117–152, 1997. [3] Douglas R. Farenick and Phillip B. Morenz. C*-extreme Points in the Generalized State Spaces of a C*-Algebra. Transactions of the American Mathematical Society, 349:1725–1748, 1997. [4] Ichiro Fujimoto. A Gelfand-Naimark theorem for C ∗ -algebras. Pacific J. Math., 184(1):95–119, 1998. [5] Phillip Hartman and Aurel Wintner. The Spectra of Toeplitz’s Matrices. American Journal of Mathematics, 76:867–882, 1954. [6] Alan Hopenwasser, Robert L. Moore, and V. I. Paulsen. C ∗ -extreme Points. Trans. Amer. Math. Soc., 266(1):291–307, 1981. [7] Vern Paulsen. Completely Bounded Maps and Operator Algebras, volume 78 of Cambridge Studies in Advanced Mathematics. Press Syndicate of the University of Cambridge, Cambridge, UK, 2002. [8] W. Forrest Stinespring. Positive Functions on C*-Algebras. Proceedings of the American Mathematical Society, 6:211–216, 1955. [9] Hongding Zhou. C*-Extreme Points in Spaces of Completely Bounded Positive Maps. PhD thesis, University of Regina, 1998. Martha Case Gregg Dept. of Mathematics Augustana College 2001 South Summit Avenue Sioux Falls, SD 57197 USA e-mail: [email protected] Submitted: January 2, 2008. Revised: February 5, 2009.

Integr. equ. oper. theory 63 (2009), 351–371 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030351-21, published online January 27, 2009 DOI 10.1007/s00020-008-1652-z

Integral Equations and Operator Theory

Embedding of Semigroups of Lipschitz Maps into Positive Linear Semigroups on Ordered Banach Spaces Generated by Measures Sander C. Hille and Dani¨el T.H. Worm Abstract. Interpretation, derivation and application of a variation of constants formula for measure-valued functions motivate our investigation of properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals. Spaces of measures densely embed into these preduals. The metric space embeds continuously in these preduals, even isometrically in a specific case. Under mild conditions, a semigroup of Lipschitz transformations on the metric space then embeds into a strongly continuous semigroups of positive linear operators on these Banach spaces generated by measures. Mathematics Subject Classification (2000). Primary 46E27; Secondary 47H20, 47D06. Keywords. Nonlinear semigroups, metric spaces, Lipschitz functions, finite measures.

1. Introduction The concept of a continuous-time deterministic or causal dynamical system in a set S can be expressed by the existence of a family of maps Φt : S → S, parametrised by the nonnegative real numbers t ∈ R+ , that satisfy the semigroup properties: Φt ◦ Φs = Φt+s and Φ0 = IdS . The evolution of the system in time from its initial state x0 ∈ S is described by the orbit t 7→ Φt (x0 ). (An interesting essay on the history of this concept can be found in [10]). If Σ is a σ-algebra of subsets of S and each Φt is (Σ, Σ)-measurable, then each Φt induces a linear operator TΦ (t) on the space of signed measures M(S) on Σ by means of TΦ (t)µ := µ ◦ Φ−1 t .

(1)

352

Hille and Worm

IEOT

The family of operators (TΦ (t))t≥0 leaves the cone of positive measures M+ (S) invariant. It constitutes a positive linear semigroup in M(S) and Φt can be recovered from TΦ (t) through the relation TΦ (t)δx = δΦt (x) . In this sense, any semigroup of measurable maps on a measurable space (S, Σ) embeds into a positive linear semigroup on the space of signed measures on S. This paper studies properties of this embedding in detail when S is a metric space with the Borel σ-algebra and the transformations Φt are Lipschitz maps. We are motivated by the study of long-term dynamics in structured population models where deterministic behaviour of an individual is ‘perturbed’ at random discrete time points by a deterministic or random (approximately) instantaneous change in state. Examples include branching random evolution [9], kinetic chemotaxis models concerning the run-and-tumble type of movement of flagellated bacteria like E. coli, B. subtilis or V. cholerae [18, 19, 2] and the extension of these to amoebae like Dictyostelium discoideum [3, 11], and cell cycle models in which a cell divides at random time points coupled to deterministic growth [5]. Our approach to these systems is to consider them as deterministic dynamical systems in the space (or cone of positive) finite Borel measures on the individual’s state space S. The dynamics are then governed by a suitable variation of constants formula Z

t

TΦ (t − s)F (µs )ds

µt = TΦ (t)µ0 +

(2)

0

in a space of measures on S. The interpretation, derivation and application of (2) require a detailed examination of topologies and functional analytic properties of spaces or sets of measures and operators thereon. There are some preliminary issues here, which are the primary concern of this paper. First, the representation (1) of TΦ (t) is practical in the context of (2) only when Φt is invertible, which is rarely the case in applications. For a functional analytic treatment we therefore need a ‘better’ representation of TΦ (t). Second, what topology is ‘natural’ in this setting and allows the application of numerous results on perturbations of linear semigroups in the literature? The total variation norm in M(S) is of little use in our context. The embedding x 7→ δx : S → M(S) is not continuous for k · kTV , nor is (TΦ (t))t≥0 strongly continuous, unless (TΦ (t))t≥0 is constant. Our investigations continue along the line set out by Dudley [7, 8] mainly, based on [21]. Third, we need to have appropriate regularity of the map t 7→ TΦ (t)µ for the existence of the integral in (2) in some sense (weak, Bochner, etc.). Concerning the topologies on spaces of measures we would like to point out that clearly M(S) is a subspace of Cb (S)∗ and can therefore be endowed with the restriction of the weak-star topology on Cb (S)∗ . This topology is often used in probability theory. There is an interesting result by Varadarajan, that the restriction to M+ (S) is metrisable (when S is separable, or when one restricts to separable positive measures), by a complete metric if S is complete ([21, Theorem 13 and Theorem 18]). Later Dudley showed ([7, Theorem 9 and Theorem 18]) that

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

353

the metric given by dBL (µ, ν) = kµ −

νk∗BL

Z f d(µ − ν) : kf kBL ≤ 1 , = sup

may be used. It is this point that we pursue further. Moreover, Peng and Xu [20] provide an embedding of a nonlinear semigroup of Lipschitz transformations into a linear semigroup as well. Their approach involves the use of quotient spaces and their duals however. These are rather inconvenient and the relationship of their results to the targeted semigroup (TΦ (t))t≥0 on measures is not as clear and direct as the approach we advocate. The outline of the paper is as follows: Section 2 and 3 introduce Banach spaces of Lipschitz functions on S, BL(S) and Lipe (S), investigate their dual spaces and introduce preduals for both, SBL and Se respectively. The latter are closed sub∗ spaces of BL(S)∗ and Lipe (S) . While assuming for simplicity of this introductory exposition that S is separable, the space of finite measures M(S) and its subspace of measures with first moment, M1 (S), are densely embedded in SBL and Se respectively (Theorem 3.9 and 3.14). The latter spaces equal these spaces of measures only in the case that S is uniformly discrete (Theorem 3.11). The embeddings yield through the map x 7→ δx , the Dirac measure at x ∈ S, an embedding ∗ of S into BL(S)∗ and Lipe (S) that is continuous in the first case (Lemma 3.5) and an isometric embedding in the latter (Lemma 3.4). Section 4 discusses the relationship between the natural pointwise ordering on Lipschitz functions, positive functionals on BL(S) and Lipe (S) and cones of positive measures. Section 5 presents the main result on the embedding of a semigroup of Lipschitz transformations Φt on S into a positive linear semigroup on SBL and Se . We give a sufficient condition for strong continuity of these semigroups in terms of (Φt )t≥0 . Section 6 concludes with a discussion of some issues concerning topologies on spaces or cones of measures.

2. Banach spaces of Lipschitz functions Let (S, d) be a metric space, consisting of at least two points. Lip(S) denotes the vector space of real-valued Lipschitz functions on S. We only consider real-valued functions, because ordering will play a role. Moreover, it seems that real-valued functions are more ‘natural’ in the theory of spaces of Lipschitz functions (see [22, p. 13]). The Lipschitz seminorm | · |Lip is defined on Lip(S) by means of |f (x) − f (y)| : x, y ∈ S, x 6= y . |f |Lip := sup d(x, y) Clearly, |f |Lip = 0 if and only if f is constant. We start with some basic facts on Lipschitz functions that we will use repeatedly. First, the distance function is a Lipschitz function: Lemma 2.1. Let E be a nonempty subset of S. Then x 7→ d(x, E) is in Lip(S). If E = S, then d(·, E) ≡ 0 and if E is a proper subset of S, then |d(·, E)|Lip = 1.

354

Hille and Worm

IEOT

This follows from the triangle inequality and the fact that d(x, E) = d(x, E). In particular Lemma 2.1 implies that x 7→ d(x, y) ∈ Lip(S) for all y ∈ S. Second, the pointwise minima and maxima of a finite number of Lipschitz functions are again Lipschitz functions: Lemma 2.2. ([7, Lemma 4]) Given f1 , . . . , fn ∈ Lip(S) we define g(x) := min(f1 (x), . . . , fn (x)) and h(x) := max(f1 (x), . . . , fn (x)). Then g, h ∈ Lip(S) and max(|g|Lip , |h|Lip ) ≤ max(|f1 |Lip , . . . , |fn |Lip ). In the sequel two normed spaces of Lipschitz functions on S and their Banach space properties will be the central objects of study. First, for each e ∈ S we introduce the norm k · ke on Lip(S) by kf ke := |f (e)| + |f |Lip , f ∈ Lip(S). If e0 is another element in S, then kf ke

≤ ≤

|f (e0 )| + |f (e) − f (e0 )| + |f |Lip ≤ |f (e0 )| + |f |Lip (d(e, e0 ) + 1) kf ke0 (d(e, e0 ) + 1).

Thus k · ke and k · ke0 are equivalent norms on Lip(S). For the rest of the paper, we fix an element e ∈ S and write Lipe (S) for the normed vector space Lip(S) with norm k · ke . The following property is straightforward: Lemma 2.3. If f ∈ Lipe (S) and x ∈ S, then |f (x)| ≤ max(1, d(x, e))kf ke . Proposition 2.4. Lipe (S) is a Banach space. Proof. Let (fn )n be a Cauchy sequence in Lipe (S). Let x ∈ S. Then Lemma 2.3 implies that (fn (x))n is a Cauchy sequence for every x ∈ S. Put f (x) := limn→∞ fn (x). Let > 0. There is an N ∈ N, such that |fn − fm |Lip ≤ for all n, m ≥ N . Then for x, y ∈ S, m ≥ N , |(f − fm )(x)) − (f − fm )(y)| =

lim |(fn − fm )(x) − (fn − fm )(y)|

n→∞

≤ d(x, y). Hence |f − fm |Lip ≤ for all m ≥ N . This implies that f ∈ Lipe (S) and |f − fn |Lip → 0 as n → ∞. Thus kf −fn ke → 0 as n → ∞, and Lipe (S) is complete. Second, let BL(S) be the vector space of bounded Lipschitz functions from S to R. For f ∈ BL(S) we define: kf kBL := kf k∞ + |f |Lip . Then k · kBL is a norm on BL(S). Proposition 2.5. BL(S) is complete with respect to k · kBL . The proof of this proposition proceeds in a similar way to that of Proposition 2.4. See also [22, Proposition 1.6.2 (a)]. There, completeness is proved for the alternative (but equivalent) norm kf kBL,max = max(kf k∞ , |f |Lip ). If f ∈ BL(S), then f ∈ Lipe (S), so there is a canonical embedding j : BL(S) → Lipe (S), where j(f ) = f . Clearly kj(f )ke ≤ kf kBL . Thus BL(S)

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

355

embeds continuously into Lipe (S). If S has finite diameter, then BL(S) = Lipe (S), and it is easy to see that in this case the norms k · kBL and k · ke are equivalent. Otherwise we can consider the closure of BL(S) in Lipe (S) with respect to k · ke : Proposition 2.6. Let S be a metric space with infinite diameter. Then k·ke

BL(S) ( BL(S) Proof. Define f (x) :=

p

( Lipe (S).

d(x, e) + 1. Then

d(x, y) |d(x, e) − d(y, e)| p p ≤p . |f (x) − f (y)| = p d(x, e) + 1 + d(y, e) + 1 d(x, e) + 1 + d(y, e) + 1 So f is in Lipe (S), but not in BL(S), since S has infinite diameter. We will show k·ke

that f ∈ BL(S) . Let fn (x) := min(f (x), n). Then fn ∈ BL(S) by Lemma 2.2. Let gn := f − fn . Now let x, y ∈ S, x 6= y. Then if f (x) ≤ n and f (y) ≤ n, |gn (x) − gn (y)| = 0. If f (x) > n and f (y) > n, then |gn (x) − gn (y)| = |f (x) − f (y)| ≤ d(x,y) 2n . d(x,y) If f (x) > n and f (y) ≤ n, then |gn (x)−gn (y)| = |f (x)−n| ≤ |f (x)−f (y)| ≤ n+1 . 1 1 So |f − fn |Lip = |gn |Lip ≤ n+1 . Therefore kf − fn ke ≤ n+1 and fn → f in Lipe (S). Now define g(x) = d(x, e). Then g is in Lipe (S), but not in BL(S). Suppose k·ke

that g ∈ BL(S) , then there is a h ∈ BL(S), with kg − hke < Lemma 2.3 yields 1 |g(x) − h(x)| ≤ max(1, d(x, e)). 2 This implies that 1 1 d(x, e) − . 2 2 Because S has infinite diameter, this contradicts that h is bounded.

1 2.

Moreover,

|h(x)| ≥ |g(x)| − |g(x) − h(x)| ≥

∗

Note that the adjoint map j ∗ : Lipe (S) → BL(S)∗ , which restricts a ϕ ∈ ∗ Lipe (S) to BL(S), is continuous, with kj ∗ (ϕ)k∗BL ≤ kϕk∗e . k·ke

( Lipe (S), by Proposition 2.6. Whenever S has infinite diameter, BL(S) From this and the Hahn-Banach Theorem it follows that there exists a non-zero ∗ φ ∈ Lipe (S) such that φ|BL(S) = 0, hence j ∗ is not injective. We will use the term Lipschitz spaces to refer to BL(S) and Lipe (S). Remark. Various authors consider other Banach spaces of Lipschitz functions, such as e.g. Weaver [22], looking at Lip0 (S) consisting of all Lipschitz functions on S that vanish at some distinct point e ∈ S. On this subspace of Lip(S), | · |Lip is a norm for which Lip0 (S) is complete. Peng and Xu [20] for example, perform the standard construction of dividing out the constant functions in Lip(S). Then this space of equivalence classes of Lipschitz functions Lip(S)/R1 is complete with respect to the norm | · |Lip and it is isometrically isomorphic to Lip0 (S). Working with these spaces is somewhat cumbersome for our applications.

356

Hille and Worm

IEOT

3. Dual and predual of Lipschitz spaces Various spaces of Lipschitz functions have been shown to be isometrically isomorphic to the dual of a Banach space. For instance, Lip0 (S) is the dual of the so-called Arens-Eells space (see [1] and [22, Section 2.2] ). It is also known that BL(S) endowed with the norm kf kBL,max := max(kf k∞ , |f |Lip ) is isometrically isomorphic to the dual of a Banach space. For instance in [15, Theorem 4.1] the more general result is proven for BL(S, E ∗ ), where E ∗ is the dual of a Banach space. Our aim in this section is to show that BL(S) with the norm k · kBL can also be viewed as the dual of a Banach space, SBL , and that Lipe (S) is the dual of a Banach space, Se , as well. Furthermore, we will show that natural spaces of measures are densely contained in SBL and Se . 3.1. Embedding of measures in dual of Lipschitz spaces In this section we are concerned with embedding measures into BL(S)∗ and ∗ Lipe (S) . We shall write k · k∗BL to denote the dual norm on BL(S)∗ and k · k∗e to denote the dual norm on Lipe (S)∗ . Let M(S) be the space of all signed finite Borel measures on S and M+ (S) the convex cone of positive measures in M(S). Let k·kTV denote the total variation norm on M(S). It is a standard result that M(S) endowed with k·kTV is a Banach space. The Baire σ-algebra is the smallest σ-algebra on S for which all continuous real-valued functions on S are measurable. Since S is a metric space, the Baire and Borel σ-algebras coincide, because for any closed C ⊂ S, fC : x 7→ d(x, C) is Lipschitz continuous by Lemma 2.1. Therefore we can apply some of the results from Dudley [7] on Baire measures. Each µ ∈ M(S) defines a linear functional Iµ on BL(S), by means of Iµ (f ) := R f dµ. Then S Z kIµ k∗BL = sup f dµ : kf kBL ≤ 1 Z ≤ sup |f |d|µ| : kf kBL ≤ 1 ≤ |µ|(S) = kµkTV , (3) thus Iµ ∈ BL(S)∗ . Moreover, one has Lemma 3.1. Let µ ∈ M+ (S). Then kIµ k∗BL = kµkTV . Proof. Suppose µ ∈ M+ (S). From (3) it follows that kIµ k∗BL ≤ kµkTV . Clearly the R constant function 1 is in BL(S), with k1k∗BL = 1. Then kµkTV = µ(S) = 1dµ ≤ kIµ k∗BL . Hence kIµ k∗BL = kµkTV . Lemma 3.2. ([7, Lemma 6]) The linear map µ 7→ Iµ : M(S) → BL(S)∗ is injective. Thus we can continuously embed M(S) into BL(S)∗ and identify µ ∈ M(S) with Iµ ∈ BL(S)∗ . When a functional ϕ ∈ BL(S)∗ can be represented by a measure, we shall write ϕ ∈ M(S).

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

357

We define the subspace of M(S) of measures with finite first moment as follows: Z M1 (S) := µ ∈ M(S) : d(x, e)d|µ|(x) < ∞ . + And we put M+ 1 (S) := M1 (S) ∩ M (S). For µ ∈ M1 (S) we define kµk1 := R max(1, Rd(x, e))d|µ|(x). Then k · k1 is a norm on M1 (S). Let µ ∈ M1 (S). Then Iµ (f ) := f dµ is well defined for every f ∈ Lipe (S), and Iµ is a linear functional on Lipe (S). ∗

Lemma 3.3. Let µ ∈ M1 (S). Then Iµ ∈ Lipe (S) and kIµ k∗e ≤ kµk1 . ∗

Moreover, the linear map µ 7→ Iµ : M1 (S) → Lipe (S) is injective. Proof. Let µ ∈ M1 (S) and f ∈ Lipe (S). Using Lemma 2.3 we obtain Z Z Z | f dµ| ≤ |f |d|µ| ≤ kf ke max(1, d(x, e))d|µ| ≤ kf ke kµk1 . M1 (S) is a subspace of M(S) and thus embeds into BL(S)∗ . The image of µ ∈ M1 (S) in BL(S)∗ coincides with the one obtained by mapping M1 (S) into ∗ Lipe (S) and then restricting to BL(S). Therefore µ 7→ Iµ is injective. ∗

Thus we can identify µ ∈ M1 (S) with Iµ ∈ Lipe (S) , and embed M1 (S) ∗ ∗ into Lipe (S) . When a functional ϕ ∈ Lipe (S) can be represented by a measure in M1 (S), we shall write ϕ ∈ M1 (S). We can embed S into M(S) or M1 (S), by sending x to the Dirac measure δx . This embedding is not continuous in general with respect to the total variation norm, since kδx − δy kTV = 2 whenever x 6= y. However, we do have an isometric ∗ embedding into Lipe (S) : ∗

Lemma 3.4. Let x ∈ S, then δx is in Lipe (S) with kδx k∗e = max(1, d(x, e)). The ∗ map x 7→ δx is an isometric embedding from S into Lipe (S) . Proof. Let f ∈ Lipe (S) and x ∈ S. Then Lemma 2.3 implies that kδx k∗e max(1, d(x, e)). For the reverse estimate, consider f (x) := d(x, e). Then f Lipe (S) and |f |Lip = 1, according to Lemma 2.1. Hence kf ke = 1, and |δx (f )| d(x, e) for every x ∈ S. Also, the constant function 1 ∈ Lipe (S) and k1ke 1. Furthermore, |δx (1)| = 1. Hence kδx k∗e ≥ max(1, d(x, e)) and thus kδx k∗e max(1, d(x, e)). Now, let x, y ∈ S, x 6= y and f ∈ Lipe (S). Then

≤ ∈ = = =

|(δx − δy )(f )| = |f (x) − f (y)| ≤ |f |Lip d(x, y) ≤ kf ke d(x, y). Let f (z) := d(x, z) − d(x, e). Then |f |Lip = |d(x, ·)|Lip = 1, kf ke = 1 and |δx (f ) − δy (f )| = d(x, y). Hence kδx − δy k∗e = d(x, y) and x 7→ δx is an isometric embedding ∗ from S into Lipe (S) . The situation for the embedding of S into BL(S)∗ is similar, though slightly different: the embedding is not isometric in general.

358

Hille and Worm

IEOT

Lemma 3.5. For every x ∈ S, δx is in BL(S)∗ , and kδx k∗BL = 1. Furthermore for every x, y ∈ S, kδx − δy k∗BL =

2d(x, y) ≤ min(2, d(x, y)). 2 + d(x, y)

(4)

Proof. Let x ∈ S and f ∈ BL(S). Then |δx (f )| = |f (x)| ≤ kf kBL , hence kδx k∗BL ≤ 1. The constant function 1 is in BL(S) and |δx (1)| = 1 = k1kBL , so kδx k∗BL = 1. If x = y, then (4) is satisfied. Suppose x 6= y. Let f ∈ BL(S). Then |f (x) − f (y)| ≤ min(|f |Lip d(x, y), 2kf k∞ ). Hence (2 + d(x, y))|f (x) − f (y)| ≤ 2d(x, y)kf kBL , so kδx − δy k∗BL = Define f (z) :=

sup

d(z,y)−d(z,x) 2+d(x,y) .

|f |Lip ≤

|f (x) − f (y)| ≤

kf kBL ≤1

2d(x, y) . 2 + d(x, y)

Then

1 2 |d(·, y) − d(·, x)|Lip ≤ , 2 + d(x, y) 2 + d(x, y)

where we use that |d(·, x)|Lip = 1, by Lemma 2.1. Since |d(z, y) − d(z, x)| ≤ d(x, y) d(x,y) for all z ∈ S, we can conclude that kf k∞ ≤ 2+d(x,y) . Hence kf kBL ≤ 1. Furthermore 2d(x, y) . |δx (f ) − δy (f )| = |f (x) − f (y)| = 2 + d(x, y) Hence kδx − δy k∗BL =

2d(x,y) 2+d(x,y) .

Remark. Instead of the norms k·kBL and k·ke , we could also consider the equivalent norms k · kBL,max and kf ke,max := max(|f (e)|, |f |Lip ). Then Lemma 3.4 holds with k · k∗e replaced by k · k∗e,max . The corresponding statement to (4) in Lemma 3.5 for k · k∗BL,max norm is that kδx − δy k∗BL,max = min(2, d(x, y)), which can be shown using the function f (z) := min(−1 + d(x, z), 1) if d(x, y) < 2 and f (z) := min(−1 + 2d(x,z) d(x,y) , 1) if d(x, y) ≥ 2. 3.2. Predual of Lipe (S) and BL(S) Let ( D := span{δx |x ∈ S} =

n X

) αk δxk : n ∈ N, αk ∈ R, xk ∈ S

.

k=1 ∗

We define Se to be the closure of the linear subspace D in Lipe (S) with respect to k · k∗e , and SBL to be the closure of D in BL(S)∗ with respect to k · k∗BL . Theorem 3.6. Se∗ is isometrically isomorphic to Lipe (S) under the map ψ 7→ T ψ, where T ψ(x) := ψ(δx ).

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

359

∗

Proof. Since Se ⊂ Lipe (S) , we can define R : Lipe (S) → Se∗ such that Rf (ϕ) := ϕ(f ) for all ϕ ∈ Se . Clearly |Rf (ϕ)| ≤ kϕk∗e kf ke , hence kRf kSe∗ ≤ kf ke . Now define T : Se∗ → Lipe (S) such that T ψ(x) := ψ(δx ) for all x ∈ S. It can easily be verified that T ψ is indeed in Lipe (S), and that T is linear. Now we want to show that kT ψke ≤ kψkSe∗ . Let x, y ∈ S, x 6= y. Note that for real numbers a and b, |a| + |b| = max(|a − b|, |a + b|). Therefore, using the fact that we only consider real-valued Lipschitz functions and hence real Banach spaces, ψ(δx − δy ) ψ(δx − δy ) |ψ(δx − δy )| = max |ψ(δe ) − |, |ψ(δe ) + | |ψ(δe )| + d(x, y) d(x, y) d(x, y) δx − δy δx − δy )|, |ψ(δe + )| = max |ψ(δe − d(x, y) d(x, y) δx − δy ∗ δx − δy ∗ ≤ kψkSe∗ max kδe − k , kδe + k d(x, y) e d(x, y) e Now for all f ∈ Lipe (S), with kf ke ≤ 1, we have |(δe −

f (x) − f (y) δx − δy )(f )| = |f (e) − | d(x, y) d(x, y) |f (x) − f (y)| ≤ |f (e)| + ≤ 1. d(x, y)

δx −δy d(x,y) )(1)| = 1, kδe δx −δy ∗ d(x,y) ke = 1 Thus for

δx −δy ∗ d(x,y) ke

And since |(δe −

−

also get kδe +

all x, y ∈ S, x 6= y,

|ψ(δe )| + sup x,y∈S x6=y

= 1. By interchanging x and y, we

|ψ(δx − δy )| ≤ kψkSe∗ . d(x, y)

Consequently, kT ψke ≤ kψkSe∗ , for all ψ ∈ Se∗ . Now we need to show R and T are each other’s inverses. Let f ∈ Lipe (S), then T (Rf )(x) = Rf (δx ) = f (x), for all x ∈ S. Hence T ◦ R = IdLip (S) . Now let ψ ∈ Se∗ , and let d ∈ D, then d = e for certain αk ∈ R and xk ∈ S. Then n n X X R(T ψ)(d) = αk T ψ(xk ) = αk ψ(δxk ) = ψ(d). k=1

Pn

k=1

αk δxk ,

k=1

Hence R(T ψ) = ψ on a dense subset of Se , so R(T ψ) = ψ on Se . Hence R ◦ T = IdSe∗ . Consequently we get that for all f ∈ Lipe (S) : kRf k∗e ≤ kf ke = kT (Rf )ke ≤ kRf k∗e , hence R is an isometric isomorphism from Lipe (S) to Se∗ , with T as its inverse. A similar result holds for BL(S):

360

Hille and Worm

IEOT

∗ Theorem 3.7. SBL is isometrically isomorphic to BL(S) under the map ψ 7→ T ψ, where T ψ(x) := ψ(δx ). ∗ Proof. We define R : BL(S) → SBL such that Rf (ϕ) := ϕ(f ) for all ϕ ∈ SBL ⊂ ∗ ∗ BL(S) . And we define T : SBL → BL(S) such that T ψ(x) := ψ(δx ) for all x ∈ S. ∗ Then analogous to the proof of Theorem 3.6 we can show that kRf kSBL ≤ kf ke , ∗ that kT ψkBL ≤ kψkSBL and that R and T are each other’s inverses. Hence R is an ∗ isometric isomorphism from BL(S) to SBL , with T as its inverse.

3.3. Identification of SBL A Borel measure µ ∈ M(S) is called separable if there is a separable Borel measurable subset E of S, such that µ is concentrated on E, i.e. |µ|(S\E) = 0. Let Ms (S) be the separable Borel measures on S, and M+ s (S) the set of positive, finite and separable Borel measures on S. If S is separable, Ms (S) = M(S). It is easy to see that Ms (S) is a closed subspace of M(S) with respect to k · kTV . Let ( n ) X + αi δxi : n ∈ N, αi ∈ R+ , xi ∈ S . D := i=1 + + ⊂ We define SBL to be the closure of D+ with respect to k · k∗BL . Notice that SBL + SBL and all ϕ ∈ SBL are positive: ϕ(f ) ≥ 0 for all 0 ≤ f ∈ BL(S). We will need the following theorem, which is based on a result from [7]: ∗ Theorem 3.8. M+ s (S) is norm closed in BL(S) if and only if S is complete. ∗ Proof. If S is complete, then M+ s (S) is norm closed in BL(S) by [7, Theorem 9]. Suppose S is not complete. Then there exists a Cauchy sequence (xn )n in S that does not converge to an element in S. Then (xn )n cannot have a convergent subsequence. This implies that for every x ∈ S there must be an > 0 and an M ∈ N, such that d(x, xm ) ≥ for all m ∈ N, m ≥ M , otherwise (xn )n∈N has a subsequence that converges to x. ∗ We will show that M+ s (S) cannot be norm closed in BL(S) . By Lemma 3.5 ∗ δxn is a Cauchy sequence in BL(S) . Now assume there is a µ ∈ M+ s (S), such that kδxn − µk∗BL → 0. Then

kµk∗BL = lim kδxn k∗BL = 1. n→∞

We will show that µ must be zero, which gives a contradiction. We can assume, by taking a subsequence, that kδxn − µk∗BL < n12 . Now define fn (x) := min(nd(x, xn ), 1). Then fn ∈ BL(S), with |fn |Lip ≤ n and kfn k∞ ≤ 1. Hence Z Z n+1 | fn dµ| = |δxn (fn ) − fn dµ| < → 0 as n → ∞. n2 Now let x ∈ S. Then there exists an > 0 and an M ∈ N, such that d(x, xm ) ≥ for all m ∈ N, m ≥ M . This implies that fn (x) → 1 as n → ∞. Hence, by the Lebesgue Dominated Convergence Theorem, Z Z | 1 dµ| = | lim fn dµ| = 0, n→∞

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

which implies that µ is zero.

361

Our main result in this section is the following theorem: + + + Theorem 3.9. M+ s (S) ⊂ SBL . Furthermore, SBL = Ms (S) if and only if S is complete. + + Proof. First we show that M+ s (S) ⊂ SBL . Let µ ∈ Ms (S), and let E be a measurable separable subset of S on which µ is concentrated. We want to show that there R + is an element ϕ ∈ SBL such that ϕ(f ) = f dµ for all f ∈ BL(S). If µ(S) = 0 this is clear, so we assume µ(S) > 0. We define the map δ : S → SBL , sending x to δx . Then δ is Lipschitz continuous by Lemma 3.5. Also, since E is separable and δ is continuous, δ(E) is a separable subset of SBL . Because µ(S\E) = 0, δ is µ-essentially separably val∗ ued. For any f ∈ BL(S) ∼ the function x 7→ hδx , f i = f (x) is measurable, = SBL so x 7→ δx is weakly measurable. By the Pettis Measurability Theorem (e.g. [6, Theorem 2]), δ is strongly µ-measurable. Furthermore, Z Z ∗ kδx kBL dµ(x) = dµ < ∞,

R therefore δ : S → SBL is µ-Bochner integrable and δx dµ(x) defines an element in SBL . By [6, Corollary 8] we get that Z 1 + δx dµ(x) ∈ conv{δx : x ∈ E} ⊂ SBL . µ(S) R + Hence δx dµ(x) ∈ SBL . Furthermore, by [6, RTheorem 6] we obtain R R R for all f ∈ BL(S) that h δx dµ(x), f i = hδx , f idµ(x) = f dµ. This implies δx dµ(x) is a + + functional in SBL represented by µ. Thus M+ s (S) ⊂ SBL . Now assume S is complete. It is clear that for all x ∈ S, δx ∈ M+ s (S). + (S) is norm closed in (S). From Theorem 3.8 we obtain that M Hence D+ ⊂ M+ s s + + (S) (S). If S is not complete, then by Theorem 3.8, M BL(S)∗ , hence SBL ⊂ M+ s s + is not norm closed in BL(S)∗ , which implies that M+ s (S) ( SBL . The crucial observation towards identification of SBL is the following: Corollary 3.10. Ms (S) is a k · k∗BL -dense subspace of SBL . One might ask when SBL = Ms (S). To answer this question we need the notion of a uniformly discrete metric space. S is uniformly discrete if there is an > 0 such that d(x, y) > for all x, y ∈ S, x 6= y. The following theorem settles our question: Theorem 3.11. Ms (S) is norm closed in BL(S)∗ if and only if S is uniformly discrete. k·k∗ BL

= Ms (S). Then (Ms (S), k · k∗BL ) is a Banach space. Proof. Suppose Ms (S) Let I be the identity map from (Ms (S), k · kTV ) to (Ms (S), k · k∗BL ). Then, since

362

Hille and Worm

IEOT

kµk∗BL ≤ kµkTV , I is a bounded linear map. Clearly, I is bijective, hence by the Inverse Mapping Theorem the inverse of I is a bounded linear map. Assume S is not uniformly discrete, then there are xn , yn ∈ S, such that 0 < d(xn , yn ) < n1 . Let µn = δxn − δyn . Then kµn kTV = 2 , while kµn k∗BL ≤ d(xn , yn ) < n1 , for all n ∈ N. This implies I −1 cannot be bounded, which gives us a contradiction. Hence S must be uniformly discrete. Now suppose S is uniformly discrete. Then there is an > 0 such that d(x, y) > for all x, y ∈ S, x 6= y. Let µ ∈ Ms (S). Let S = P ∪ N be the Hahn decomposition of S corresponding to µ, then µ+ = µ|P and µ− = µ|N . Define min(/4, 1/2) if x ∈ P ; f (x) := − min(/4, 1/2) if x ∈ N . Then kf k∞ ≤ 1/2 and |f |Lip = sup x6=y

|f (x) − f (y)| /2 1 ≤ = . d(x, y) 2

Hence kf kBL ≤ 1. Furthermore, Z Z Z | f dµ| = | min(/4, 1/2)d µ − min(/4, 1/2)dµ| P

N

= |µ+ (S) + µ− (S)| min(/4, 1/2) = kµkTV min(/4, 1/2). Hence kµkTV ≤ kµk∗BL

1 , min(/4, 1/2)

for all µ ∈ Ms (S). Also, kµk∗BL ≤ kµkTV for all µ ∈ Ms (S), hence the norms k · k∗BL and k · kTV are equivalent on Ms (S). This implies that k·k∗ BL

Ms (S)

k·kT V

= Ms (S)

= Ms (S).

Remark. Note that all the arguments in the proof of Theorem 3.11 hold when we k·k∗ BL

replace Ms (S) by M(S). Hence M(S) discrete.

= M(S) if and only if S is uniformly

Corollary 3.12. If S is not uniformly discrete, there are elements in SBL , hence in BL(S)∗ , that cannot be represented by a measure in M(S). 3.4. Identification of Se We start with the observation that each ϕ ∈ Se is completely determined by its restriction to BL(S); more precise: Lemma 3.13. Let ϕ ∈ Se , f ∈ Lipe (S). Define fn (x) := max(min(f (x), n), −n). Then limn→∞ ϕ(fn ) = ϕ(f ).

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

363

Proof. Obviously, kfn ke ≤ kf ke for all n ∈ N. Let > 0. Then there is a d ∈ D such that kϕ − dk∗e < 2(kf ke +1) . Let Nd be such that d(f − fn ) = 0 for all n ≥ Nd . Then for n ≥ Nd we have |ϕ(f ) − ϕ(fn )|

≤ |ϕ(f ) − d(f )| + |d(f ) − d(fn )| + |ϕ(fn ) − d(fn )| ≤ 2kϕ − dk∗e kf ke < .

Hence limn→∞ ϕ(fn ) = ϕ(f ).

Just as before, we restrict to the separable Borel measures: Let Ms,1 (S) := + 1 + M1 (S) ∩ Ms (S), and M+ s,1 (S) := Ms (S) ∩ M+ (S). Similar to SBL , we define Se + ∗ to be the closure of D with respect to k · ke . Now we can prove the analogue to Theorem 3.9: + + + Theorem 3.14. M+ s,1 (S) ⊂ Se . Furthermore, Se = Ms,1 (S) if and only if S is complete. + + Proof. First we will show that M+ s,1 (S) ⊂ Se . Let µ ∈ Ms,1 (S) and define δ : ∗ S → Lipe (S) , x 7→ δx . Then we can prove, using similar Rtechniques as in the proof Rof Theorem 3.9,Rthat δ is µ-Bochner integrable, that δx dµ(x) ∈ Se+ and + that h δx dµ(x), f i = f dµ for all f ∈ Lipe (S). This implies that M+ s,1 (S) ⊂ Se . + + Now suppose that S is complete. It is clear that D ⊂ Ms,1 (S). Let ϕ ∈ Se+ , then there are dn ∈ D+ such that kϕ − dn k∗e → 0. Because

kj ∗ (ϕ) − dn k∗BL = kj ∗ (ϕ − dn )k∗BL ≤ kϕ − dn k∗e → 0, ∗ there is a µ ∈ M+ s (S), according to Theorem 3.9, such that j (ϕ(f )) = ϕ(f R R )= + f d µ for all f ∈ BL(S). We need to show that µ ∈ Ms,1 (S) and ϕ(f ) = f d µ for all f ∈ Lipe (S). Let f ∈ Lipe (S), f ≥ 0. Then Z Z fn dµ = f dµ < ∞ ϕ(f ) = lim ϕ(fn ) = lim n→∞

n→∞

by Theorem. In particular, R Lemma 3.13 and the Monotone Convergence d(x, e)dµ = ϕ(d(·, e)) < ∞, hence µ ∈ M+ (S). s,1 Using f = f + − f − for general f ∈ Lipe (S), Rwhere f + = max(f, 0) and f − = − min(f, 0), we find that f ∈ L1 (µ) and ϕ(f ) = f dµ for every f ∈ Lipe (S). Hence Se+ ⊂ M+ s,1 (S). Now suppose S is not complete. Then there is a Cauchy sequence (xn )n in S that does not converge to an element in S. This implies by Lemma 3.4 that ∗ (δxn )n is a Cauchy sequence in Lipe (S) . Suppose that µ ∈ M+ s,1 (S) is such that kδxn − µk∗e → 0. Then kδxn − µk∗BL → 0, but from the proof of Theorem 3.8 it follows that this is not possible. Hence M+ s,1 (S) is not norm closed in Lipe (S), and + + + since M+ (S) ⊂ S , this implies that M e s,1 s,1 (S) ( Se . The following corollaries follows easily from Theorem 3.14: Corollary 3.15. Ms,1 (S) is a k · k∗e -dense subspace of Se .

364

Hille and Worm

IEOT

Corollary 3.16. M+ s,1 (S) is norm closed in Se if and only if S is complete. Remark. In [16, Theorem 4.2] it is shown that the metric space S is complete if and only if the set of separable probability measures of finite first moment, Ps,1 (S), is complete with respect to the metric H, where Z Z H(µ, ν) = sup | f dµ − f dν|. f ∈Lip(S) |f |Lip ≤1

From Corollary 3.16 we can also conclude this theorem: it follows that when S is complete, the subset of separable probability measures of finite first moment P 1,s (S) is also a closed set of Se , hence complete with respect to k · k∗e . Let µ, ν ∈ P 1,s (S), then kµ − νk∗e is equal to H(µ, ν), since for f ∈ Lip(S) with |f |Lip ≤ 1 we have Z Z Z Z | f dµ − f dν| = | f − f (e)d µ − f − f (e)d µ|, and g(x) := f (x) − f (e) satisfies: kgke = |f |Lip ≤ 1. Furthermore, when S is not ∗ complete, M+ s,1 (S) is not complete with respect to k · ke . Then it is not difficult to see that then P 1,s (S) also cannot be complete with respect to k · k∗e , hence it is not complete with respect to H. Recall the natural embedding j : BL(S) → Lipe (S), and the adjoint j ∗ : ∗ Lipe (S) → BL(S)∗ . Then, as a consequence of Proposition 2.6, j ∗ is not injective whenever S has infinite diameter. Consider however the restriction je∗ of j ∗ to Se . Lemma 3.17. je∗ maps Se injectively and densely into SBL . Proof. Let φ ∈ Se be such that je∗ (φ) = 0. Then φ(f ) = 0 for all f ∈ BL(S). Hence Lemma 3.13 implies that φ(f ) = 0 for all f ∈ Lipe (S), hence φ = 0. So je∗ is injective. By continuity of je∗ , je∗ (Se ) = je∗ (D

k·k∗ e

k·k∗ BL

) ⊂ je∗ (D)

=D

k·k∗ BL

= SBL .

So we can continuously embed Se into SBL and je∗ (Se ) is dense in SBL , since je∗ (D) = D is dense in SBL .

4. Positivity We can endow BL(S) and Lipe (S) with pointwise ordering, so f ≥ g if f (x) ≥ g(x) for all x ∈ S. From Lemma 2.2 it follows that BL(S) and Lipe (S) are Riesz spaces with respect to this ordering. However, k · kBL and k · ke are not Riesz norms, since |f | ≤ |g| need not imply that |f |Lip ≤ |g|Lip . We are interested in the question whether all the positive functionals BL(S)∗+ := {φ ∈ BL(S)∗ : φ(f ) ≥ 0 for all f ∈ BL(S), f ≥ 0} can be represented by measures on S.

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

365

Let Cub (S) denote the Banach space of bounded uniformly continuous realvalued functions on S, with the supremum norm k · k∞ . Then BL(S) ⊂ Cub (S) is dense [7, Lemma 8]. Let φ ∈ BL(S)∗+ . Then |φ(f )| = |φ(f + ) − φ(f − )| ≤ φ(|f |) ≤ φ(kf k∞ · 1) = kf k∞ φ(1)

(5)

by positivity of φ. This φ can be uniquely extended to a positive continuous linear functional on Cub (S). Let S be complete. If S is compact, then Cub (S) = C(S), and by the Riesz representation theorem, every φ ∈ Cub (S)∗ can be represented by a measure. If S is not compact, then C0 (S) ( Cub (S), and it is possible to show the existence of a non-zero functional φ ∈ Cub (S)∗+ , such that φ|C0 (S) = 0, which implies that φ cannot be represented by a measure. However, when we also demand that φ is in SBL , it can be represented by a measure, by a corollary of the following theorem: + Theorem 4.1. SBL ∩ BL(S)∗+ = SBL . + Proof. Clearly, SBL ⊂ SBL ∩BL(S)∗+ . Suppose that there exists a φ ∈ SBL ∩BL(S)∗+ + such that φ 6∈ SBL . If φ(1) = 0, then φ(f ) = 0 for every f ∈ BL(S), by positivity + of φ and (5), hence φ ∈ SBL . So φ(1) > 0. Let ( n ) X M := αi δxi : n ∈ N, 0 ≤ αi ≤ φ(1), xi ∈ S, for i = 1, . . . , n , i=1 + with respect to k · k∗BL . By then M ⊂ Let M be the closure of M in SBL assumption, φ is not in M . Since M is convex, M is a closed convex subset of SBL . Thus φ is strictly separated from M by [4, Corollary IV.3.10]: there is an ∗ = BL(S) and an α ∈ R, such that hm, f i < α for all m ∈ M , and f ∈ SBL hφ, f i = φ(f ) > α. Clearly φ(1)δx ∈ M for all x ∈ S, hence + SBL .

hφ(1)δx , f i = φ(1)f (x) < α for all x ∈ S. So f <

α φ(1)

and by positivity of φ, φ(f ) < φ(

α1 ) = α, φ(1)

+ which is a contradiction. So SBL ∩ BL(S)∗+ = SBL .

From Theorem 3.9 and Theorem 4.1 we get the following result: ∗ ∗ + Corollary 4.2. M+ s (S) ⊂ SBL ∩ BL(S)+ , and SBL ∩ BL(S)+ = Ms (S) if and only if S is complete.

The following theorem can be proved similarly to Theorem 4.1: ∗

Theorem 4.3. Se ∩ Lipe (S)+ = Se+ . And the following corollary follows from Theorem 3.14 and Theorem 4.3: ∗

∗

+ Corollary 4.4. M+ s,1 (S) ⊂ Se ∩ Lipe (S)+ , and Se ∩ Lipe (S)+ = Ms,1 (S) if and only if S is complete.

366

Hille and Worm

IEOT

We have seen in Lemma 3.17 that Se can be considered as a dense subspace + of SBL . The closed convex cones Se+ and SBL in both spaces relate as follows: + Proposition 4.5. SBL ∩ Se = Se+ .

Proof. Using Theorem 4.1, we obtain + SBL ∩ Se

= BL(S)∗+ ∩ Se = {φ ∈ Se : φ(f ) ≥ 0, for all 0 ≤ f ∈ BL(S)} =: P.

Now, if φ ∈ Se is such that φ(f ) ≥ 0 for all positive f ∈ BL(S), then, by Lemma 3.13, φ(g) ≥ 0 for all positive g ∈ Lipe (S). Hence φ ∈ Se+ and P ⊂ Se+ . Clearly Se+ ⊂ P . + The closed convex cone SBL defines a partial ordering ‘ ≥0 on SBL by means ∗ of φ ≥ ψ if and only if φ − ψ ∈ SBL . Then (SBL , ≥) is an ordered Banach space. In a similar fashion, Se+ introduces a partial ordering in Se . Proposition 4.5 implies that both orderings are compatible and obtained from the ordering in BL(S)∗+ and ∗ Lipe (S)+ according to Theorem 4.1 and Theorem 4.3 respectively. + Note that SBL is not a generating cone in SBL , unless S is uniformly discrete (Theorem 3.11).

5. Embedding into positive linear semigroups on dual Lipschitz spaces Let Lip(S, S) be the space of Lipschitz maps on S. For T ∈ Lip(S, S), we define d(T (x), T (y)) : x, y ∈ S, x 6= y . |T |Lip := sup d(x, y) Lemma 5.1. Let T ∈ Lip(S, S). For any f ∈ Lipe (S), kf ◦ T ke ≤ max(1, d(e, T (e)) + |T |Lip )kf ke , and for g ∈ BL(S), kg ◦ T kBL ≤ max(1, |T |Lip )kgkBL . Proof. It is easy to check that for f ∈ Lipe (S), |f ◦ T |Lip ≤ |f |Lip |T |Lip , hence we have kf ◦ T ke

≤ |f (T (e))| + |f |Lip |T |Lip ≤ |f (e)| + |f |Lip d(e, T (e)) + |f |Lip |T |Lip ≤ max(1, d(e, T (e)) + |T |Lip )kf ke .

And for g ∈ BL(S), we have kg ◦ T kBL .

≤ kg ◦ T k∞ + |g|Lip |T |Lip ≤ kgk∞ + |g|Lip |T |Lip ≤ max(1, |T |Lip )kgkBL .

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

367

Definition 5.2. A family of maps (Φt )t≥0 from S into S is a Lipschitz semigroup on S if (i) for all t ≥ 0, Φt ∈ Lip(S, S), (ii) for all s, t ≥ 0, Φt ◦ Φs = Φt+s and Φ0 = IdS . A Lipschitz semigroup (Φt )t≥0 on S is called strongly continuous if t 7→ Φt (x) is continuous at t = 0 for all x ∈ S. From property (ii) it then follows that t 7→ Φt (x) is continuous on R+ for all x ∈ S. Let (Φt )t≥0 be a Lipschitz semigroup on S. Then we define a semigroup of operators on Lipe (S). Let f ∈ Lipe (S) and t ≥ 0, and let SΦ (t)f := f ◦ Φt . Then SΦ (t) is a bounded linear operator on Lipe (S), by Lemma 5.1, and kSΦ (t)kL(Lip (S)) ≤ max(1, d(e, Φt (e))+|Φt |Lip ). Hence (SΦ (t))t≥0 is a semigroup e

of bounded linear operators on Lipe (S). ∗ So the dual operators (SΦ (t))t≥0 form a semigroup of bounded linear opera∗ tors on Lipe (S) . ∗ Lemma 5.3. SΦ (t)(Se ) ⊂ Se .

Proof. Let f ∈ Lipe (S). Then ∗ (SΦ (t)δx )(f ) = δx (SΦ (t)f ) = δx (f ◦ Φt ) = f (Φt (x)) = δΦt (f ),

for all x ∈ S, t ≥ 0. Thus Se .

∗ SΦ (t)(D)

⊂ D. Hence, by continuity of

∗ SΦ (t),

(6)

∗ SΦ (t)(Se )

⊂

Thus we can define a semigroup (TˆΦ (t))t≥0 of bounded linear operators on Se by setting ∗ TˆΦ (t)ϕ := SΦ (t)ϕ, for all ϕ ∈ Se , t ≥ 0. Theorem 5.4. For all x, y ∈ S and s, t ≥ 0, d(Φs (x), Φt (y)) = kTˆΦ (s)δx − TˆΦ (t)δy k∗ . e

(7)

Furthermore, the following are equivalent: (i) (TˆΦ (t))t≥0 is a strongly continuous semigroup on Se . (ii) (Φt )t≥0 is strongly continuous and lim supt↓0 |Φt |Lip < ∞. (iii) (Φt )t≥0 is strongly continuous and there exist M ≥ 1 and ω ∈ R such that |Φt |Lip ≤ M eωt for all t ≥ 0. Proof. From Lemma 3.4 and (6) we get that for every x, y ∈ S and t, s ≥ 0 kTˆΦ (s)δx − TˆΦ (t)δy k∗e

= kδΦs (x) − δΦt (y) k∗e = d(Φs (x), Φt (y)).

(i)⇒(iii): There exist M ≥ 1 and ω ∈ R such that kTˆΦ (t))kL(Se ) ≤ M eωt for all t ≥ 0. Hence it follows from (7) that for x, y ∈ S and t ≥ 0, d(Φt (x), Φt (y))

= kTˆΦ (t)δx − TˆΦ (t)δy k∗e ≤ M eωt kδx − δy k∗e = M eωt d(x, y).

368

Hille and Worm

IEOT

Hence |Φt |Lip ≤ M eωt for all t ≥ 0. From (7) and strong continuity of (TˆΦ (t))t≥0 it follows that (Φt )t≥0 is strongly continuous. (iii)⇒(ii): This is trivial. (ii)⇒(i): We want to show that there is a δ > 0 and an M ≥ 1 such that sup0≤t≤δ kTˆΦ (t)kL(Se ) ≤ M , and that (TˆΦ (t))t≥0 is strongly continuous on D. Then we can conclude by [10, Proposition 5.3] that (TˆΦ (t))t≥0 is strongly continuous on Se , since D is dense in Se by definition. Since lim supt↓0 |Φt |Lip < ∞, there exist M1 , δ > 0 such that |Φt |Lip ≤ M1 for all 0 ≤ t ≤ δ. We know that kTˆΦ (t)kL(Se )

≤

∗ kSΦ (t)kL(Lipe (S)∗ )

= kSΦ (t)kL(Lipe (S)) ≤ max(1, d(e, Φt (e)) + |Φt |Lip ). Now, since [0, δ] is compact, Φ[0,δ] (e) is compact, hence bounded, in S, so there is an M2 > 0 such that d(e, Φt (e)) ≤ M2 for all 0 ≤ t ≤ δ. Hence sup kTˆΦ (t)kL(Se ) ≤ max(1, M1 + M2 ) =: M < ∞.

0≤t≤δ

By (7) and strong continuity of (Φt )t≥0 we have for every x ∈ S that kTˆΦ (t)δx − δx k∗e = d(Φt (x), x) → 0 as t ↓ 0. Hence by linearity limt↓0 kTˆΦ (t)d − dk∗e = 0 for all d ∈ D. Remarks.

1) Notice that for all ϕ ∈ Se , f ∈ Lipe (S) and t ≥ 0, we have

∗ f (TˆΦ (t)ϕ) = (TˆΦ (t)ϕ)(f ) = (SΦ (t)ϕ)(f ) = ϕ(SΦ (t)(f )) = (SΦ (t)f )(ϕ).

Therefore TˆΦ∗ (t)f = SΦ (t)f for all f ∈ Lipe (S) and under the equivalent conditions of Theorem 5.4, (SΦ (t))t≥0 is the dual semigroup of a strongly continuous semigroup. As Se is not reflexive in general, (SΦ (t))t≥0 cannot be expected to be strongly continuous. It is on the smaller space Se by definition. It would be interesting to be able to identify the latter space. 2) In [20, Corollary 3 and Remark 4] a result similar to Theorem 5.4 is proven, but in less generality, since there S is taken to be a closed subset of a Banach space. In [20] the duality of spaces of Lipschitz functions is also exploited to show this result, but there the Banach space Lip0 (S) is used, consisting of the Lipschitz functions vanishing at some distinct point e in S. Since the semigroup TΦ (t) will in general not map Lip0 (S) into itself, unless e is a fixed point of (Φt )t≥0 , the proof in [20] needs to make use of the Banach space Lip(S)/R1. By making use of the space Lipe (S), we have no such difficulties. Notice that the semigroup (SΦ (t))t≥0 defined above is also a semigroup of ∗ bounded linear operators on BL(S), by Lemma 5.1. Then (SΦ (t))t≥0 is a semigroup ∗ of bounded linear operators on BL(S) . Using very similar techniques as above, we ∗ can show that SΦ (t)(SBL ) ⊂ SBL for all t ≥ 0. Hence we can define a semigroup ∗ (TΦ (t))t≥0 on SBL by restricting SΦ (t) to SBL . Under the equivalent conditions of Theorem 5.4 this semigroup is strongly continuous:

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

369

Theorem 5.5. For all x, y ∈ S,s, t ≥ 0, kTΦ (s)(δx ) − TΦ (t)(δy )k∗BL

2d(Φs (x), Φt (y)) 2 + d(Φs (x), Φt (y)) ≤ min(2, d(Φs (x), Φt (y))). =

If lim supt↓0 |Φt |Lip < ∞ and (Φt )t≥0 is strongly continuous, then (TΦ (t))t≥0 is a strongly continuous semigroup on SBL . The proof is similar to the proof of Theorem 5.4, but here the equality follows from Lemma 3.5. Let t ≥ 0. Then TˆΦ (t)(D+ ) ⊂ D+ and TΦ (t)(D+ ) ⊂ D+ , hence by the conti+ nuity of TˆΦ (t) and TΦ (t) we can conclude that TˆΦ (t)(Se+ ) ⊂ Se+ and TΦ (t)(SBL )⊂ + ˆ SBL . Thus (TΦ (t))t≥0 and (TΦ (t))t≥0 are positive semigroups. Thus, if S is complete, TˆΦ (t)(M+ (S)) ⊂ M+ (S) s,1

s,1

and + TΦ (t)(M+ s (S)) ⊂ Ms (S). In the following proposition we will show that this also holds if S is not complete.

Proposition 5.6. Let t ≥ 0. Then TΦ (t) and TˆΦ (t) leave Ms (S) and Ms,1 (S) invariant, respectively. Moreover, they are given by (1). Proof. Let µ ∈ Ms (S). Then for all f ∈ BL(S) and t ≥ 0 we have: Z Z TΦ (t)(µ)(f ) = µ(SΦ (t)f ) = f ◦ Φt dµ = f d(µ ◦ Φ−1 t ), where µ◦Φ−1 t is again a Borel measure, since Φt is continuous on S. Hence TΦ (t)(µ) −1 is represented by the measure µ ◦ Φ−1 is a t . We now want to show that µ ◦ Φt separable measure. Since µ is separable, there is a separable Borel measurable subset E of S, such that |µ|(S\E) = 0. By continuity of Φt , Φt (E) is separable, and so is Φt (E). For any Borel measurable A ⊂ S\Φt (E), µ◦Φ−1 t (A) = 0. Therefore −1 |µ ◦ Φ−1 |(S\Φ (E)) = 0, so µ ◦ Φ is separable. t t t Similarly we get that for µ ∈ Ms,1 (S) and t ≥ 0, TˆΦ (t)(µ) is represented by −1 the separable Borel measure µ ◦ Φ−1 ∈ M1 (S), t . Then, by Lemma 3.3, µ ◦ Φt ˆ hence in Ms,1 (S). So TΦ (t)(Ms,1 (S)) ⊂ Ms,1 (S). + Corollary 5.7. Let t ≥ 0. Then TΦ (t) and TˆΦ (t) leave M+ s (S) and Ms (S) invariant, respectively.

So we see that the strongly continuous semigroup (TΦ (t))t≥0 on SBL , when restricted to Ms (S), is the semigroup defined by (1). This gives us the proper functional analytic framework that will enable us to study (2).

370

Hille and Worm

IEOT

References [1] R.F. Arens and J. Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397–403 . [2] F.A.C.C. Chalub, P.A. Markowich, B. Perthame and Ch. Schmeiser, Kinetic Models for Chemotaxis and their Drift-Diffusion Limits, Monatsh. Math. 142 (2004), 123– 141. [3] F.A.C.C. Chalub, Y. Dolak-Struss, P.A. Markovich, D. Oelz, Ch. Schmeiser and A. Soreff, Model Hierarchies for Cell Aggregation by Chemotaxis, Math. Models Methods Appl. Sci. 16 (2006), 1173–1197. [4] J.B. Conway, A Course in Functional Analysis, 2nd edition, Springer, 1990. [5] O. Diekmann, M. Gyllenberg, H.R. Thieme and S.M. Verduyn Lunel, A cell-cycle model revisited, Amsterdam: CWI report AM-R9305 (1993). [6] J. Diestel and J.J. Uhl Jr., Vector Measures, Mathematical Surveys, no. 15, Providence: American Mathematical Society, 1977. [7] R.M. Dudley, Convergence of Baire measures, Stud. Math. 27 (1966), 251–268. [8] R.M. Dudley, Correction to: “Convergence of Baire measures”, Stud. Math. 51 (1974), 275. [9] S.R. Dunbar , A branching random evolution and a nonlinear hyperbolic equation, SIAM J. Appl. Math. 48(6) (1988), 1510–1526. [10] K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, New York: Springer-Verlag, 2000. [11] R. Erban and H.G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol. 54 (2007) , 847–885. [12] G.B. Folland, Real analysis: modern techniques and their applications, New York: John Wiley & Sons Inc, 1999. [13] L.G. Hanin, An extension of the Kantorovich norm, in: Monge–Amp´ere Equation: Applications to Geometry and Optimization, Contemp. Math. 226, Providence: American Mathematical Society, 1999, 113–130. [14] E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Providence: American Mathematical Society, 1957. [15] J.A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147–169. [16] A.S. Kravchenko, Completeness of the space of separable measures in the Kantorovich-Rubinstein metric, Sib. Math. Journal 47(1) (2006), 68–76. [17] E.J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40(12) (1934), 837–842. [18] H.G. Othmer, S.R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol 26 (1988), 263–298. [19] H.G. Othmer and T. Hillen, The diffusion limit of transport equations II: chemotaxis equations, SIAM J. Appl. Math. 62(4) (2002), 1222–1250. [20] J. Peng and Z. Xu, A novel dual approach to nonlinear semigroups of Lipschitz operators, Trans. Amer. Math. Soc. 357(1) (2005), 409–424.

Vol. 63 (2009)

Embedding of Semigroups of Lipschitz Maps

371

[21] V.S. Varadarajan, Measures on topological spaces, Mat. Sb. (N.S.) 55(97) (1961), 35-100; English translation: Amer. Math. Soc. Transl. 2(48) (1965), 161-228. [22] N. Weaver, Lipschitz Algebras, World Scientific Publishing Co. Pte. Ltd., 1999. Sander C. Hille and Dani¨el T.H. Worm Mathematical Institute Leiden University P.O. Box 9512 2300 RA Leiden The Netherlands e-mail: [email protected] [email protected] Submitted: July 7, 2008. Revised: October 9, 2008.

Integr. equ. oper. theory 63 (2009), 373–402 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030373-30, published online February 24, 2009 DOI 10.1007/s00020-009-1663-4

Integral Equations and Operator Theory

R-Boundedness of Smooth Operator-Valued Functions Tuomas Hyt¨onen and Mark Veraar Abstract. In this paper we study R-boundedness of operator families T ⊂ B(X, Y ), where X and Y are Banach spaces. Under cotype and type assumptions on X and Y we give sufficient conditions for R-boundedness. In the first part we show that certain integral operator are R-bounded. This will be used to obtain R-boundedness in the case that T is the range of an operator-valued function T : Rd → B(X, Y ) which is in a certain Besov space d/r Br,1 (Rd ; B(X, Y )). The results will be applied to obtain R-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. Mathematics Subject Classification (2000). Primary 47B99; Secondary 46B09, 46E35, 46E40, 60B05. Keywords. R-boundedness, operator theory, type and cotype, Besov space, semigroup theory, evolution family, stochastic Cauchy problem.

1. Introduction The notion of R-boundedness (see Section 2.3 for definition) appeared implicitly in the work of Bourgain [6] and was formalized by Berkson and Gillespie [5]. Cl´ement, de Pagter, Suckochev and Witvliet [7] studied it in more detail in relation to vector-valued Schauder decompositions, and shortly after Weis [40] found a characterization of maximal regularity for the Cauchy problem u0 = Au + f , u(0) = 0, in terms of R-boundedness of the resolvent of A or the associated semigroup. After this, many authors have used R-boundedness techniques in the theory of Fourier multipliers and Cauchy problems (cf. [8, 16, 21] and references therein). Tuomas Hyt¨ onen is supported by the Academy of Finland (grant 114374). Mark Veraar is supported by the Alexander von Humboldt foundation. His visit to Helsinki, which started this project, was funded by the Finnish Centre of Excellence in Analysis and Dynamics Research.

374

Hyt¨ onen and Veraar

IEOT

For Hilbert space operators, R-boundedness is equivalent to uniform boundedness. The basic philosophy underlying much of the work cited above is that many results for Hilbert spaces remain true in certain Banach spaces if one replaces boundedness by R-boundedness. Thus it is useful to be able to recognize R-bounded sets of operators. Let X and Y be Banach spaces. In this paper we will study R-boundedness of some subsets of B(X, Y ) under type and cotype assumptions. Although the definition of R-boundedness suggests connections with type and cotype, there are only few results on this in the literature. Arendt and Bu [3, Proposition 1.13] pointed out that uniform boundedness already implies R-boundedness if (and only if) X has cotype 2 and Y has type 2. Recently, van Gaans [12] showed that a countable union of R-bounded sets remains R-bounded if the individual R-bounds are `r summable for an appropriate r depending on the type and cotype assumptions, improving on the trivial result with r = 1 (the triangle inequality!) valid for any Banach spaces. Implicitly, one can find similar ideas already in Figiel [11]. In [13, Theorem 5.1], Girardi and Weis have found criteria for R-boundedness of the range of operator-valued functions T : Rd → B(X, Y ) in terms of their smoothness and the Fourier type of the Banach space Y . Their result states that if d p (Rd ; B(X, Y )), then Y has Fourier type p ∈ [1, 2] and T is in the Besov space Bp,1 d {T (t) : t ∈ R } is R-bounded.

We will prove a similar result as [13, Theorem 5.1] under assumptions on the cotype of X and the type of Y . More precisely, if X has cotype q and Y has type d r p and if T ∈ Br,1 (Rd ; B(X, Y )) for some r ∈ [1, ∞] such that 1r = p1 − 1q , then d {T (t) : t ∈ R } is R-bounded (see Theorem 5.1 below). Our result improves [13, Theorem 5.1]. This follows from the fact that every space with Fourier type p has type p. Furthermore, we note that the only spaces which have Fourier type 2 are spaces which are isomorphic to a Hilbert space. However, there are many Banach spaces with type 2, e.g., all Lp spaces with p ∈ [2, ∞). In the limit case that X has 0 cotype 2 and Y has type 2 our assumption on T becomes T ∈ B∞,1 (Rd ; B(X, Y )). This condition is quite close to uniform boundedness of {T (t) : t ∈ Rd } which under these assumption on X and Y is equivalent to R-boundedness. Following [13, Section 5], we apply the sufficient condition for R-boundedness to strongly continuous semigroups. Furthermore, we show that our results are sharp in the case of the translation semigroup on Lp (R). The R-boundedness result for semigroups leads to existence, uniqueness and regularity results for stochastic equations with additive noise. As a second application we present an Rboundedness result for evolution families, assuming the conditions of Acquistapace and Terreni [1]. We will write a . b if there exists a universal constant C > 0 such that a ≤ Cb, and a h b if a . b . a. If the constant C is allowed to depend on some parameter t, we write a .t b and a ht b instead.

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

375

2. Preliminaries Throughout this paper (Ω, A, P) denotes a probability space, and E is the expectation. Let X and Y be Banach spaces. Let (rn )n≥1 be a Rademacher sequence on Ω, i.e. an independent sequence with P(rn = 1) = P(rn = −1) =

1 . 2

For N ≥ 1 and x1 , . . . , xN ∈ E, recall the Kahane-Khinchine inequalities (cf. [10, Section 11.1] or [23, Proposition 3.4.1]): for all p, q ∈ [1, ∞), we have N N

q q1

p p1 X X

hp,q E rn xn . E rn xn

(2.1)

n=1

n=1

These inequalities will be often applied without referring to it explicitly. For each integer N , the space RadN (X) ⊂ L2 (Ω; X) is defined as all elements PN of the form n=1 rn xn , where (xn )N n=1 are in X. 2.1. Type and cotype Let p ∈ [1, 2] and q ∈ [2, ∞]. The space X is said to have type p if there exists a constant C ≥ 0 such that for all (xn )N n=1 in X we have N N

2 21 X p1 X

kxn kp . rn xn ≤C E n=1

n=1

The space X is said to have cotype q if there exists a constant C ≥ 0 such that for all (xn )N n=1 in X we have N X n=1

kxn kq

q1

N

2 21 X

rn xn ≤ C E , n=1

with the obvious modification in the case q = ∞. For a detailed study of type and cotype we refer to [10]. Every Banach space has type 1 and cotype ∞ with constant 1. Therefore, we say that X has non-trivial type (non-trivial cotype) if X has type p for some p ∈ (1, 2] (cotype q for some 2 ≤ q < ∞). If the space X has non-trivial type, it has non-trivial cotype. Hilbert spaces have type 2 and cotype 2 with constants 1. For p ∈ [1, ∞) the Lp -spaces have type p ∧ 2 and cotype p ∨ 2. Recall the following duality result for RadN (X) (cf. [34] or [10, Chapter 13]). If X has non-trivial type then RadN (X)∗ = RadN (X ∗ ) isomorphically with constants independent of N .

(2.2)

376

Hyt¨ onen and Veraar

IEOT

2.2. Fourier type The Fourier transform fb = Ff of a function f ∈ L1 (Rd ; X) will be normalized as Z 1 fb(ξ) = f (x)e−ix·ξ dx, ξ ∈ Rd . (2π)d/2 Rd Let p ∈ [1, 2] and p0 be the conjugate exponent, p1 + p10 = 1. The space X has Fourier type p, if F defines a bounded linear operator for some (and then for all) 0 d = 1, 2, . . . from Lp (Rd ; X) to Lp (Rd ; X). If X has Fourier type p, then it has both type p and cotype p0 . In particular, spaces isomorphic to a Hilbert space are the only ones with Fourier type 2 (see [22]). The Lp -spaces have Fourier type p ∧ p0 (see [30]), while every Banach space has Fourier type 1. The notion becomes more restrictive with increasing p. 2.3. R-boundedness A collection T ⊂ B(X, Y ) is said to be R-bounded if there exists a constant M ≥ 0 such that N N

2 12

2 21 X X

≤ M E rn xn , rn Tn xn E Y

n=1

n=1

X

N for all N ≥ 1 and all sequences (Tn )N n=1 in T and (xn )n=1 in X. The least constant M for which this estimate holds is called the R-bound of T , notation R(T ). By (2.1), the role of the exponent 2 may be replaced by any exponent 1 ≤ p < ∞ (at the expense of a possibly different constant). The notion of R-boundedness has played an important role in recent progress in Fourier multiplier theory and this has applications to regularity theory of parabolic evolution equations. For details on the subject we refer to [8, 21] and references therein. A property which we will need later on is the following. If T ⊂ B(X, Y ) is R-bounded and X has non-trivial type, then it follows from (2.2) that the set of adjoint operators T ∗ = {T ∗ ∈ B(Y ∗ , X ∗ ) : T ∈ T } is R-bounded as well.

2.4. Lorentz spaces We recall the definition of the Lorentz space (cf. [14, 38]). Let (S, Σ, µ) be a σ-finite measure space. For f ∈ L1 (S) + L∞ (S) define the non-increasing rearrangement of f as f ∗ (s) = inf{t > 0 : µ(|f | > t) ≤ s}, s > 0. For p, q ∈ [1, ∞] define Lp,q (S) = {f ∈ L1 (S) + L∞ (S) : kf kLp,q (S) < ∞}, where kf kLp,q (S) =

  R∞ 0

tq/p f ∗ (t)q dt t

sup 1/p ∗ f (t) t>0 t

1/q

if q ∈ [1, ∞), if q = ∞.

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

377

For p ∈ [1, ∞] and q1 < q2 one has kf kLp,p (S) = kf kLp (S) , kf kLp,q2 (S) ≤ cp,q1 ,q2 kf kLp,q1 (S) . Also recall (e.g. [35, pp. 331–2]) that for p, q ∈ [1, ∞) Z ∞ q/p dt q1 ||f kLp,q (S) = p tq µ(|f | > t) ; t 0

(2.3)

indeed, just compare the two iterated integrals of sq−1 tq/p−1 over the subset {f ∗ (t) > s} = {µ(|f | > s) > t} of (0, ∞)2 . 2.5. Besov spaces We recall the definition of Besov spaces using the so-called Littlewood-Paley decomposition (cf. [4, 38]). Let φ ∈ S (Rd ) be a fixed Schwartz function whose Fourier transform φb is nonnegative and has support in {ξ ∈ Rd : 21 ≤ |ξ| ≤ 2} and which satisfies X b −k ξ) = 1 for ξ ∈ Rd \ {0}. φ(2 k∈Z

Such a function can easily be constructed (cf. [4, Lemma 6.1.7]). Define the sequence (ϕk )k≥0 in S (Rd ) by X b −k ξ) for k = 1, 2, . . . and ϕ ϕ ck (ξ) = φ(2 c0 (ξ) = 1 − ϕ ck (ξ), ξ ∈ Rd . k≥1

Similar as in the real case one can define S (Rd ; X) as the usual Schwartz space of rapidly decreasing X-valued smooth functions on Rd . As in the real case this is a Fr´echet space. Let the space of X-valued tempered distributions S 0 (Rd ; X) be defined as the continuous linear operators from S (Rd ) into X. s For 1 ≤ p, q ≤ ∞ and s ∈ R the Besov space Bp,q (Rd ; X) is defined as the 0 d space of all X-valued tempered distributions f ∈ S (R ; X) for which

ks

kf kBp,q ϕk ∗ f k≥0 q p d s (Rd ;X) := 2 l (L (R ;X))

s (Rd ; X) Bp,q

is a Banach space, and up to an is finite. Endowed with this norm, equivalent norm this space is independent of the choice of the initial function φ. The sequence (ϕk ∗ f )k≥0 is called the Littlewood-Paley decomposition of f associated with the function φ. s If 1 ≤ p, q < ∞, then Bp,q (Rd ; X) contains the Schwartz space S (Rd ; X) as a dense subspace. For 1 ≤ p1 ≤ p2 ≤ ∞, q ∈ [1, ∞] and s1 , s2 ∈ R with s1 − pd1 = s2 − pd2 the following continuous inclusion holds (cf. [38, Theorem 2.8.1(a)] Bps11 ,q (Rd ; X) ,→ Bps22 ,q (Rd ; X). Next we give an alternative definition of Besov spaces. Let I = (a, b) with −∞ ≤ a < b ≤ ∞. For h ∈ R and a function f : I → X we define the function

378

Hyt¨ onen and Veraar

IEOT

T (h)f : R → X as the translate of f by h, i.e. ( φ(t + h) if t + h ∈ I, (T (h)φ)(t) := 0 otherwise. For h ∈ R put n o I[h] := r ∈ I : r + h ∈ I . For a strongly measurable function f ∈ Lp (I; X) and t > 0 let Z p1 %p (f, t) := sup kT (h)f (r) − f (r)kp dr . |h|≤t

I

We use the obvious modification if p = ∞. For p, q ∈ [1, ∞] and s ∈ (0, 1) define Λsp,q (I; X) := {f ∈ Lp (I; X) : kf kΛsp,q (I;X) < ∞}, where kf kΛsp,q (I;X) =

Z

kf (r)kp dr

p1

I

+

1

Z 0

q dt q1 t−s %p (f, t) t

(2.4)

with the obvious modification if p = ∞ or q = ∞. Endowed with the norm k · kΛsp,q (I;X) , Λsp,q (I; X) is a Banach space. Moreover, if I = R, then Λsp,q (R; X) = s (R; X) with equivalent norms (cf. [31, Proposition 3.1] and [36, Theorem Bp,q 4.3.3]). Similarly, if I 6= R, then for every f ∈ Λsp,q (I; X) there exists a function g ∈ Λsp,q (R; X) such that g|I = f and there exists a constant C > 0 independent of f and g such that C −1 kgkΛsp,q (R;X) ≤ kf kΛsp,q (I;X) ≤ kgkΛsp,q (R;X) .

(2.5)

3. Tensor products We start with a basic lemma, which can be viewed as a generalization of the Kahane-contraction principle. Lemma 3.1. Let X be a Banach space and let (S, Σ, µ) be a σ-finite measure space and let q ∈ [2, ∞). The following assertions hold: 1. If X has cotype q, then there exists a constant C such that for all (fn )N n=1 in in X Lq,1 (S) and (xn )N n=1 N

X

rn fn xn

Lq (S;L2 (Ω;X))

n=1

Z ≤C 0

∞

N

X

max µ(|fn | > t)1/q dt rn xn

1≤n≤N

n=1

L2 (Ω;X)

(3.1) .

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

379

2. If X has cotype q, then for all q˜ ∈ (q, ∞] there exists a constant C such that q˜ N for all (fn )N n=1 in L (S) and (xn )n=1 in X N

X

rn fn xn

Lq˜(S;L2 (Ω;X))

n=1

N

X

≤ C sup kfn kLq˜(S) rn xn 1≤n≤N

L2 (Ω;X)

n=1

.

(3.2)

Moreover, if q ∈ {2, ∞}, then (3.2) holds with q˜ = q. 3. Assume (S, Σ, µ) contains N disjoint sets of equal finite positive measure for every N ∈ Z+ . If there exists a constant C such that (3.1) holds for all q,1 (fn )N (S) and (xn )N n=1 in L n=1 in X, then X has cotype q. Remark 3.2. Note that by (2.3) for q ∈ [1, ∞) it holds that Z ∞ Z ∞

1/q 1 max µ(|fn | > t)1/q dt ≤ λ max (fn∗ ) > t dt = max (fn∗ ) Lq,1 , 1≤n≤N 1≤n≤N q 1≤n≤N 0 0 where λ denotes the Lebesgue measure on (0, ∞). Moreover, if the f1 , . . . , fN are identically distributed, then one has Z ∞ 1 max µ(|fn | > t)1/q dt = kf1 kLq,1 (S) . 1≤n≤N q 0 With this in mind, one can also view (3.1) as an extension of [24, Proposition 9.14] and [33, Proposition 3.2(ii)]. There it is shown that (3.1) holds for the case that µ(S) = 1, (fn )n≥1 are i.i.d. and symmetric. Remark 3.3. By (2.1), one could rephrase (3.2) as follows when q˜ < ∞. In the natural embedding Lq˜(S) ,→ B(X, Lq˜(S; X)), f 7→ f ⊗ (·), the unit ball BLq˜(S) becomes an R-bounded subset of B(X, Lq˜(S; X)). 2 Proof of Lemma 3.1. (1) and (2): Define the operator T : `∞ N → L (Ω; X) by PN T (a) = n=1 rn an xn . By the Kahane contraction principle there holds

kT k

2 B(`∞ N ,L (Ω;X))

N

X

rn xn ≤ 2

L2 (Ω;X)

n=1

.

Since L2 (Ω; X) has cotype q it follows from [10, Theorem 11.14] that T is (q, 1)summing with πq,1 (T ) ≤ CX,q kT k. Then [32, Theorem 2.1] (also see [10, Theorem 10.9]) implies that there is a probability measure ν on {1, 2, . . . , N } such that q,1 q,1 2 ˜ T = T˜j, where j : `∞ N → `N (ν) is the embedding and T ∈ B(`N (ν), L (Ω; X)) sat1 −1+ N qπ isfies kT˜kB(`q,1 (ν),L2 (Ω;X)) ≤ q q,1 (T ). It follows that for all scalars (an )n=1 , N

N

X

rn an xn

n=1

L2 (Ω;X))

N

X

q,1 ≤ CX,q k(an )N k r x

n n n=1 ` (ν) N

n=1

L2 (Ω;X)

N

X

≤ CX,q,˜q k(an )N rn xn n=1 k`q˜ (ν) N

n=1

L2 (Ω;X)

,

380

Hyt¨ onen and Veraar

IEOT

q,1 where `q,1 defined on {1, . . . , N } with measure ν, N (ν) denotes the Lorentz space L q˜ and the second step follows from the embedding of `q,1 ˜ ∈ (q, ∞]. N (ν) into `N (ν) for q If we apply this with an = fn (s) and take the Lq (µ)-norms, then it follows from (2.3) and Minkowski’s inequality that N N

X

−1

X

× r x r f x

2

n n n n n q 2 L (S;L (Ω;X))

n=1

≤ CX,q

∞

Z Z S

Z ≤ CX,q

0

= CX,q ≤ CX,q

1/q

dt

q

q1 dµ(s)

1/q ν(n)1{|fn (s)|>t} dµ(s) dt

S n=1 N ∞X

0

Z

ν(n)1{|fn (s)|>t}

n=1

N ∞Z X

0

Z

N X

L (Ω;X)

n=1

1/q ν(n)µ(|fn | > t) dt

n=1 ∞

1/q max µ(|fn | > t) dt.

1≤n≤N

0 q˜

Similarly with L (µ)-norms, it follows that N N

−1

X

X

rn xn r f x ×

n n n q˜ 2 2 L (S;L (Ω;X))

n=1

n=1

L (Ω;X)

N q˜/˜q 1/˜q Z X ν(n)|fn (s)|q˜ ≤ CX,q,˜q dµ(s) S N X

= CX,q,˜q

n=1

Z ν(n) S

n=1

≤ CX,q,˜q

Z max

1≤n≤N

1/˜q |fn (s)|q˜ dµ(s)

1/˜q |fn (s)|q˜ dµ(s) .

S

(3): Let x1 , . . . , xN ∈ X. Let (Sn )N n=1 be disjoint sets in Σ with µ(Sn ) = µ(S1 ) ∈ (0, ∞) for all n. Letting fn = µ(S1 )−1/q 1Sn for n = 1, 2, . . . , N , we obtain N N

q

X X

rn fn xn = kxn kq .

q 2 n=1

L (S;L (Ω;X))

n=1

On the other hand, µ(|fn | > t) = µ(S1 ) · 1[0,µ(S1 )−1/q ) (t) for all n = 1, . . . , N . Therefore, (3.1) implies that N N

X

q X

kxn kq ≤ C q rn xn n=1

which shows that X has cotype q.

n=1

L2 (Ω;X)

,

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

381

We do not know, whether we can take q˜ = q in Lemma 3.1(2) if q 6= 2. However, if X is an Lq -space with q ≥ 2, then one may take q˜ = q. This follows e = R. from the next remark in the case that X Remark 3.4. Let (A, A, ν) be a σ-finite measure space. Let 2 ≤ q1 < q < ∞. Let e be a Banach space with cotype q1 . If X = Lq (A; X), e then (3.2) of Lemma 3.1(2) X holds with q˜ = q. e and (2.1) we obtain that Proof. By Fubini’s theorem, Lemma 3.1(2) applied to X N N

X

X

h r f x rn fn xn q

q n n n q q 2 2 e n=1

L (S;L (Ω;X))

n=1

L (A;L (S;L (Ω;X)))

N

X

rn xn ≤ C sup kfn kLq (S) 1≤n≤N

n=1

e Lq (A;L2 (Ω;X))

N

X

rn xn hq C sup kfn kLq (S) 1≤n≤N

n=1

L2 (Ω;X)

.

.

Remark 3.5. Notice that a version of Lemma 3.1 also holds for quasi-Banach spaces. This can be proved in a similar way as above. Instead of [32] one has to use the factorization result of [18, Theorem 4.1]. Note that in [18] the role of the Lorentz space Lq,1 (S) is replaced by Lq,r (S), where r is some number in (0, 1] which depends on X. One can see from the above proof that this number r will also appear in the quasi-Banach space version of (3.2). The details are left to the interested reader. The following dual version of Lemma 3.1 holds: Lemma 3.6. Let X be a Banach space, let (S, Σ, µ) be a σ-finite measure space and let p ∈ (1, 2]. The following assertions hold: 1. If X has type p, then there exists a constant C such that for all (fn )N n=1 in Lp,∞ (S) which are identically distributed and (xn )N in X n=1 N N

X

X

(3.3) kf1 kLp,∞ (S) rn xn 2 ≤ C rn fn xn p . 2 n=1

L (Ω;X)

n=1

L (S;L (Ω;X))

2. If X has type p, then for all p˜ ∈ [1, p) there exists a constant C such that for p˜ N all (fn )N n=1 in L (S) and (xn )n=1 in X N N

X

X

inf kfn kLp˜(S) rn xn 2 ≤ C rn fn xn p˜ . (3.4) 2 1≤n≤N

n=1

L (Ω;X)

n=1

L (S;L (Ω;X))

Moreover, if p ∈ {1, 2}, then (3.2) holds with p˜ = p. 3. Assume (S, Σ, µ) contains N disjoint sets of equal finite positive measure for every N ∈ Z+ . If there exists a constant C such that (3.3) holds for all p,1 (S) which are identically distributed and (xn )N (fn )N n=1 in L n=1 in X, then X has type p.

382

Hyt¨ onen and Veraar

IEOT

A similar statement as in Remark 3.4 also holds. Proof. (1) : Without loss of generality, kf1 kLp,∞ (S) = supt>0 t1/p f1∗ (t) = 1. Choose t0 so that f1∗ (t0 ) > (2t0 )−1/p , or equivalently t0 < µ |f1 | > (2t0 )−1/p . Let An := {|fn | > (2t0 )−1/p }, so by equidistribution, µ(An ) = µ(A1 ) > t0 . It follows that Z DX N N N X E X 1 hxn , x∗n i = rn 1An xn , rm 1Am x∗m dµ(s) E µ(A ) 1 S n=1 n=1 m=1 ≤

N N

X

1

X rm 1Am x∗m 0 . rn 1An xn

p 2 µ(A1 ) n=1 Lp (S;L2 (Ω;X ∗ )) L (S;L (Ω;X)) m=1

Now X ∗ has cotype p0 , hence by Lemma 3.1(1) there holds N

X

rm 1Am x∗m

Lp0 (S;L2 (Ω;X ∗ ))

m=1

N

X

rm x∗m

1/p0

≤ Cµ(A1 )

L2 (Ω;X ∗ )

m=1

Since X has non-trivial type, taking the supremum over all RadN (X ∗ ) with norm one, it follows that N

X

rn xn

L2 (Ω;X)

n=1

N

X

rn . n=1

PN

.

∗ n=1 rn xn

∈

1An

x

p n L (S;L2 (Ω;X)) µ(A1 )1/p

N

X

rn fn xn .

Lp (S;L2 (Ω;X))

n=1

,

where the last estimate used the contraction principle and the fact that |fn | > (2t0 )−1/p 1An & µ(A1 )−1/p 1An by the definition of An . (2) : The case p = p˜ = 1 follows from Z N

X

rn |fn (s)|dµ(s)xn 2 LHS(3.4) ≤ ≤

L (Ω;X)

S

n=1

Z X N

rn |fn (s)|xn

S

L2 (Ω;X)

n=1

dµ(s) = RHS(3.4),

where the first estimate was the contraction principle. For p > 1, we argue by duality in a similar spirit as in (1): assuming min1≤n≤N kfn kLp˜(S) = 1, choose R 0 gn ∈ Lp˜ (S) of at most unit norm so that S fn · gn dµ = 1 and write N X n=1

hxn , x∗n i = E

Z DX N S

n=1

rn fn xn ,

N X

E rm gm x∗m dµ.

m=1

Then proceed as in (1), only using Lemma 3.1(2) instead of Lemma 3.1(1). (3): This claim follows in a similar way as the corresponding claim in Lemma 3.1.

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

383

4. Integral operators An operator-valued function T : S → B(X, Y ) will be called X-strongly measurable if for all x ∈ X, the Y -valued function s 7→ T (s)x is strongly measurable. Let r ∈ [1, ∞]. For an X-strongly measurable mapping T : S → B(X, Y ) 0 with kT (s)xkLr (S;Y ) ≤ M kxk and f ∈ Lr (S) we will define Tf ∈ B(X, Y ) as Z Tf x = T (s)x f (s) dµ(s). S

By H¨ older’s inequality, we have kTf kB(X,Y ) ≤ M kf kLr0 (S) . If r = 1, then by [21, Corollary 2.17] R {Tf : kf kL∞ (S) ≤ 1} ≤ 2M. (4.1) In the next result we will obtain R-boundedness of {Tf : kf kLr0 (S) ≤ 1} for different exponents r under assumptions on the cotype of X and type of Y . Proposition 4.1. Let X and Y be Banach spaces and let (S, Σ, µ) be a σ-finite measure space. Let p0 ∈ [1, 2] and q0 ∈ [2, ∞]. Assume that X has cotype q0 and Y has type p0 . The following assertions hold: 1. If r ∈ [1, ∞) is such that 1r > p10 − q10 . Then there exists a constant C = C(r, p0 , q0 , X, Y ) such that for all T ∈ Lr (S; B(X, Y )), R {Tf ∈ B(X, Y ) : kf kLr0 (S) ≤ 1} ≤ CkT kLr (S;B(X,Y )) . (4.2) 2. Assume the pair (p0 , q0 ) is an element in {(1, ∞), (2, ∞), (2, 2), (1, 2)}. If r ∈ [1, ∞] is such that 1r = p10 − q10 , then there exists a constant C = C(X, Y ) such that for all T ∈ Lr (S; B(X, Y )) (4.2) holds. Remark 4.2. Since B(X, Y ) is usually non-separable, it could happen that T : S → B(X, Y ) is not strongly measurable and therefore not in Lr (S; B(X, Y )). However, one can replace the assumption that T ∈ Lr (S; B(X, Y )) by the condition that T is X-strongly measurable and s 7→ kT (s)k is in Lr (S) or is dominated by a function in Lr (S). This does not affect the assertion in Proposition 4.1 and the proof is the same. The following will be clear from the proof of Proposition 4.1 and Remark 3.4. Remark 4.3. Let (Ai , Ai , νi ), i = 1, 2 be a σ-finite measure space. Let 1 < p0 < e be a Banach space with cotype q1 and Ye be p1 ≤ 2 and 2 ≤ q1 < q0 < ∞. Let X e and Y = Lp0 (A2 ; Ye ) , then (4.2) a Banach space with type p1 . If X = Lq0 (A1 ; X) 1 1 1 of Proposition 4.1 (1) holds with r = p0 − q0 . 0

r Proof of Proposition 4.1. (1): Let (fn )N n=1 in L (S) be so that supn kfn kLr0 (S) ≤ 1 and x1 , . . . , xN ∈ X. First assume p0 > 1 and q0 < ∞. Let p ∈ (1, p0 ) and q ∈ (q0 , ∞) be such that 0 0 1 1 1 1 1 1 r 0 /q := and hn = sign(fn )|fn |r /p r p − q , and hence r 0 = p0 + q . Let gn = |fn | for n = 1, . . . , N . Then kgn kLq (S) ≤ 1, khn kLp0 (S) ≤ 1 and fn = gn hn for all n.

384

Hyt¨ onen and Veraar

IEOT

P

N ∗ ∗ Let (yn∗ )N n=1 in Y be such that n=1 rn yn

Lp0 (Ω;Y ∗ )

H¨older’s inequality and

1 p

1 r

=

1 q

+

≤ 1. Then it follows from

that

N N N DX E X X E rn Tfn xn , rn yn∗ = hTfn xn , yn∗ i n=1

n=1

=

Z X N

n=1

hgn (s)T (s)xn , hn (s)yn∗ i dµ(s)

S n=1

Z =

N N D E X X E T (s) rn gn (s)xn , rn hn (s)yn∗ dµ(s)

S

n=1

n=1

N

X

≤ T rn gn xn

Lp (S×Ω;Y )

n=1

≤ kT k

Lr (S;B(X,Y

N

X

rn hn yn∗

Lp0 (S×Ω;Y ∗ )

n=1

N

X

rn gn xn ))

Lq (S×Ω;X)

n=1

N

X

rn hn yn∗

Lp0 (S×Ω;Y ∗ )

n=1

.

Since X has cotype q0 < q it follows from Lemma 3.1(2) that N

X

rn gn xn

Lq (S×Ω;X)

n=1

N

X

≤ C1 rn xn

Lq (Ω;X)

n=1

Since Y has type p0 it follows that Y ∗ has cotype p00 < p0 (cf. [10, Proposition 11.10]) and therefore it follows from Lemma 3.1(2) that N

X

rn hn yn∗

Lp0 (S×Ω;Y ∗ )

n=1

N

X

rn yn∗ ≤ C2 n=1

Lp0 (Ω;Y ∗ )

≤ C2 .

We may conclude that N N N

X E DX X

rn yn∗ ≤ C1 C2 kT kLr (S;B(X,Y )) rn xn rn Tfn xn , E n=1

n=1

n=1

Lq0 (Ω;X)

.

By assumption Y has non-trivial type, hence RadN (Y )∗ = RadN (Y ∗ ) isomor∗ phically (see (2.2)). Taking the supremum over all y1∗ , . . . , yN ∈ Y ∗ as above, we obtain that N N

X

X

rn Tfn xn p . kT kLr (S;B(X,Y )) rn xn q .

n=1

L (Ω;Y )

n=1

L (Ω;X)

The result now follows from (2.1). If p0 > 1 and q0 = ∞ one can easily adjust the above argument to obtain the result. In particular, gn = 1 for n = 1, . . . , N in this case. If p0 = 1 and q0 < ∞, then the duality argument does not hold since Y only has the trivial type 1. However, one can argue more directly in this case.

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

385

Now r0 > q0 . By the triangle inequality, H¨older’s inequality and Lemma 3.1, we obtain that Z X N N

X

dµ(s) ≤ rn fn xn 2 rn Tfn xn 2

T

n=1

L (Ω;Y )

S

L (Ω;Y )

n=1

N

X

≤ kT kLr (S;B(X,Y ))

rn fn xn

Lr0 (S;L2 (Ω;Y ))

n=1 N

X

≤ CkT kLr (S;B(X,Y )) rn xn

L2 (Ω;Y )

n=1

.

(2): The case p0 = 1 and q0 = ∞ follows from (4.1). The cases (p0 = 2 and q0 = ∞) and (p0 = q0 = 2) follow from Lemma 3.1 in the same way as in as (1). If p0 = 1 and q0 = 2 then r = 2 and by the Cauchy-Schwartz inequality and Lemma 3.1 we obtain that Z N N

X X

rn fn (s)xn 2 dµ(s) rn Tfn xn 2 ≤

T (s)

n=1

L (Ω;Y )

S

L (Ω;X)

n=1

N

X

rn fn xn ≤ kT kL2 (S;B(X,Y ))

L2 (S×Ω;X)

n=1 N

X

rn xn ≤ CkT kL2 (S;B(X,Y )) n=1

L2 (S×Ω;X)

.

With a certain price to pay, it is possible to relax the norm integrability condition of Proposition 4.1 to the uniform Lr -integrability of the orbits s 7→ T (s)x. This is reached at the expense of not being able to exploit the information about the cotype of X but only the type of Y , as shown in the following remark. In the example further below, it is shown that in general the Lr -integrability of the orbits is not sufficient for the full conclusion of Proposition 4.1. Remark 4.4. Let X and Y be Banach spaces and let (S, Σ, µ) be a σ-finite measure space. Let p0 ∈ [1, 2]. Assume that Y has type p0 . The following assertions hold: 1. Assume p0 ∈ (1, 2). If r ∈ (1, p0 ) and T : S → B(X, Y ) is such that kT xkLr (S;Y ) ≤ CT kxk, x ∈ X. Then there exists a constant C = C(r, p0 , Y ) such that R {Tf ∈ B(X, Y ) : kf kLr0 (S) ≤ 1} ≤ CCT .

(4.3)

(4.4)

2. Assume that p0 = 1 or p0 = 2 and that (4.3) holds for r = p0 . Then there exists a constant C = C(Y ) such that (4.4) holds. If Y is as in Remark 4.3, then (4.4) holds for r = p0 .

386

Hyt¨ onen and Veraar

IEOT

Proof. (1): One can argue as in the proof of Proposition 4.1 with p = r, gn = 1 and hn = fn . Indeed, we have N N N

DX E X X

E rn Tfn xn , rn xn rn yn∗ ≤ T n=1

Lr (S×Ω;Y )

n=1

n=1

N

X

rn hn yn∗

Lr0 (S×Ω;Y ∗ )

n=1

.

By the assumption 4.3 one can estimate N

X

rn xn

T

Lp (S×Ω;Y )

n=1

P

N The term n=1 rn hn yn∗

Lr0 (S×Ω;Y ∗ )

N

X

≤ CT rn xn n=1

Lp (S×Ω;X)

.

can be treated in the same way as in the

proof of Proposition 4.1. (2): The case p0 = 1 follows from (4.1). The case p0 = 2 can be proved as above. In the next example, we will show that even if X has cotype q for some q ∈ (2, ∞) the result in Remark 4.4 cannot be improved. Example 4.5. Consider the spaces X = `q , q ∈ (2, ∞), and Y = R, so that X has cotype q and Y has type 2. Let S = Z+ with the counting measure, and define the B(X, Y )-valued function T on S by T (s)x := t(s)x(s) for some t : S → R. Then we can make the following observations: T ∈ Lr (S; B(X, Y )) if and only if t ∈ Lr (S) = `r . The condition (4.3) means u ktxkr . kxkq (where tx is the pointwise product), which holds if and R only if t ∈ ` , 1/u = (1/r − 1/q) ∨ 0. Under this condition, the operators x 7→ S T (s)f (s)x ds 0 0 are well-defined and bounded from X to Y for f ∈ Lr (S) = `r . Let us then derive a necessary condition for the R-boundedness of the mentioned operators. Let xi ∈ X = `q be α(i)ei , where ei is the ith standard unitvector and α(i) ∈ R. Let fi = ei . The defining inequality of R-boundedness for these functions reduces to N N

X

X

ri t(i)α(i) . E ri α(i)ei kq h kαkq . ktαk2 h E i=1

i=1

This holds if and only if t ∈ `v , 1/v = 1/2−1/q, which is stronger than (4.3) unless r ≤ 2. Conversely, the condition (4.3) suffices for R-boundedness in this range, as shown in Remark 4.4. As a final result in this section we will show how one can use Lemma 3.1 to obtain a version of Proposition 4.1 with sharp exponents. It may seem artificial at first sight, but it enables us to obtain a sharp version of Theorem 5.1 below. For f ∈ L1 (S) + L∞ (S) let Lf (S) = {g ∈ L1 (S) + L∞ (S) : µ(|g| > t) = µ(|f | > t) for all t > 0}.

(4.5)

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

387

Proposition 4.6. Let X and Y be Banach spaces and let (S, Σ, µ) be a σ-finite measure space. Let p ∈ [1, 2] and q ∈ [2, ∞]. Assume that X has cotype q and Y has type p. If r ∈ [1, ∞] is such that 1r = p1 − 1q . Then there exists a constant 0

C = C(p, q, X, Y ) such that for all f0 ∈ Lr ,1 (S) and T ∈ Lr (S; B(X, Y )), R {Tf ∈ B(X, Y ) : f ∈ Lf0 (S)} ≤ CkT kLr (S;B(X,Y )) kf0 kLr0 ,1 (S) .

(4.6)

0

Since each f ∈ Lf0 (S) is also in Lr (S), Tf ∈ B(X, Y ) is well-defined. Note that Remark 4.2 applies also here. In the limit cases p ∈ {1, 2} and q ∈ {2, ∞}, Proposition 4.1 (2) yields a stronger result. Proof. Without loss of generality we may assume that kf0 kLr,1 (S) = 1. Let (fn )N n=1 in Lf0 (S) and x1 , . . . , xN ∈ X. 0 0 0 First assume p > 1 and q < ∞. Let gn = |fn |r /q and hn = sign(fn )|fn |r /p 0 for n = 1, . . . , N . Then the |gn | have the same distribution as |f0 |r /q and the |hn | 0 0 ∗ have the same distribution as |f0 |r /p , and fn = gn hn for all n. Let (yn∗ )N n=1 in Y

PN ∗ ≤ 1. Then it follows from H¨older’s inequality be such that n=1 rn yn 0 Lp (Ω;Y ∗ )

and

1 p

=

1 r

1 q

+

that

N N N E X DX X rn yn∗ = hTfn xn , yn∗ i rn Tfn xn , E n=1

=

Z X N

n=1

n=1

hgn (s)T (s)xn , hn (s)yn∗ i dµ(s)

S n=1

Z =

N N D E X X E T (s) rn gn (s)xn , rn hn (s)yn∗ dµ(s)

S

n=1

n=1

N

X

rn gn xn ≤ T n=1

Lp (S×Ω;Y )

N

X

rn hn yn∗

n=1

N

X

rn gn xn ≤ kT kLr (S;B(X,Y ))

Lq (S×Ω;X)

n=1

Lp0 (S×Ω;Y ∗ ) N

X

rn hn yn∗

Lp0 (S×Ω;Y ∗ )

n=1

Since X has cotype q it follows from Lemma 3.1 (1) that Z ∞ N N

X

X 0

rn gn xn ≤ C1 µ(|f0 | > tq/r )1/q dt rn xn

n=1

Lq (S×Ω;X)

0

0

n=1 0 r0

.

Lq (Ω;X)

It follows from (2.3) and Lr ,1 ,→ Lr , q that Z ∞ Z 0 r0 ∞ dt q/r 0 1/q µ(|f0 | > t ) dt = µ(|f0 | > t)1/q tr /q q t 0 0 0 0 0 r r /q = (r0 )−r /q kf0 k r0 , r0 ≤ Cq,r . q q (S) L

.

388

Hyt¨ onen and Veraar

IEOT

Since Y has type p it follows that Y ∗ has cotype p0 (cf. [10, Proposition 11.10]) and therefore it follows from Lemma 3.1 (1) that N

X

rn hn yn∗

n=1

Z

Lp0 (S×Ω;Y ∗ )

≤ C2 0

∞

N

X 0 0

µ(|f0 | > tp /r )1/q dt rn yn∗

Lp0 (Ω;Y ∗ )

n=1

.

As before it holds that Z

∞

0

0

µ(|f0 | > tp /r )1/q dt ≤ Cp,r .

0

The result now follows with the same duality argument for RadN (Y ) as in Proposition 4.1. If p > 1 and q = ∞ one can easily adjust the above argument to obtain the result. In particular, gn = 1 for n = 1, . . . , N in this case. If p = 1 and q < ∞, then one can argue as in Proposition 4.1, but instead of Lemma 3.1 (2) one has to apply Lemma 3.1 (1). If p = 1 and q = ∞ the result follows from Proposition 4.1. Similar as in Remark 4.4 the following strong version of Proposition 4.6 holds. Remark 4.7. Let X and Y be Banach spaces and let (S, Σ, µ) be a σ-finite measure 0 space. Let p ∈ (1, 2]. Assume that Y has type p. If f0 ∈ Lp ,1 (S) and T : S → B(X, Y ) is such that (4.3) holds, then there exists a constant C = C(p, Y ) such that R {Tf ∈ B(X, Y ) : f ∈ Lf0 (S)} ≤ CCT kf0 kLp0 ,1 (S) . (4.7) Proof. Similar as in Remark 4.4 one can argue as in the proof of Proposition 4.6 with p = r, gn = 1 and hn = fn .

5. Besov spaces and R-boundedness Recall from [40] that for an interval I = (a, b) and T ∈ W 1,1 (I; B(X, Y )), R(T (t) ∈ B(X, Y ) : t ∈ (a, b)) ≤ kT (a)k + kT 0 kL1 (I;B(X,Y ))

(5.1)

In [13, Theorem 5.1] this result has been improved under the assumption that Y has Fourier type p. In the next result we obtain R-boundedness for the range of smooth operator-valued functions under (co)type assumptions on the Banach spaces X and Y . The result below improves [13, Theorem 5.1]. Theorem 5.1. Let X and Y be Banach spaces. Let p ∈ [1, 2] and q ∈ [2, ∞]. Assume that X has cotype q and Y has type p. If r ∈ [1, ∞] is such that 1r = p1 − 1q , then d

r (Rd ; B(X, Y )), there exists a constant C = C(p, q, X, Y ) such that for all T ∈ Br,1 R {T (t) ∈ B(X, Y ) : t ∈ Rd } ≤ CkT k dr . (5.2)

Br,1 (Rd ;B(X,Y ))

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

389

d

r (Rd ; B(X, Y )) ,→ BU C(Rd ; B(X, Y )) (the space of bounded, Note that Br,1 uniformly continuous functions) for all r ∈ [1, ∞] (cf. [38, Theorem 2.8.1(c)]). If p = q = 2 in (2), then the uniform boundedness of {T (t) : t ∈ Rd } ⊂ B(X, Y ) already implies R-boundedness (see [3, Proposition 1.13]). P P P Proof. (1): Let Z = B(X, Y ). We may write T = k≥0 Tk = k≥0 n≥0 ϕn ∗ Tk , where Tk = ϕk ∗ T and the series converges in Z uniformly in Rd . Let ϕ−1 = 0. By [40, Lemma 2.4] we obtain that XX R T (t) ∈ Z : t ∈ Rd ≤ R ϕn ∗ Tk (t) ∈ Z : t ∈ Rd

k≥0 n≥0

=

X k+1 X

R ϕn ∗ Tk (t) ∈ Z : t ∈ Rd .

k≥0 n=k−1

Fix n ∈ {0, 1, 2, . . . , } and t ∈ R and define ϕn,t ∈ S (Rd ) by ϕn,t (s) = ϕn (t − s). Then for all t ∈ Rd , ϕn,t ∈ Lϕn (Rd ), where Lϕn (Rd ) is as in (4.5) and dn kϕn kLr0 ,1 (Rd ) ≤ c2 r . Indeed, it is elementary to check that ϕ∗n (t) = 2dn ϕ∗ (2nd t). Therefore, Z ∞ Z ∞ 1 dn 1 dn dt dt kϕn kLr0 ,1 (Rd ) = 2dn t r0 ϕ∗ (2nd t) =2r t r0 ϕ∗ (t) = 2 r kϕkLr0 ,1 (Rd ) . t t 0 0 d

Letting Tk,ϕn,t ∈ Z be the integral operator from Propositions 4.1 and 4.6, it follows that for all k ≥ 0, dn R ϕn ∗ Tk (t) ∈ Z : t ∈ Rd = R Tk,ϕn,t ∈ Z : t ∈ Rd ≤ C1 2 r kTk kLr (Rd ;Z) . We may conclude that X k+1 X dn R T (t) ∈ Z : t ∈ Rd ≤ C1 2 r kTk kLr (Rd ;Z) k≥0 n=k−1

≤ C2

X

2

dk r

kTk kLr (Rd ;Z) = C2 kT k

k≥0

d r (Rd ;Z) Br,1

.

In [13, Remark 5.2] the following result is presented for operator families which are in a Besov space in the strong sense. If Y has Fourier type p and T : Rd → B(X, Y ) is such that for all x ∈ X, kT xkB d/p (Rd ;Y ) ≤ CT kxk, p,1

d

then {T (t) : t ∈ R } is R-bounded. We will obtain the same conclusion assuming only type p. Notice that many Banach spaces have type 2, whereas all Banach spaces with Fourier 2 are isomorphic to a Hilbert space. We first need an analogue of Proposition 4.1 involving the space γ(H, Y ) of γ-radonifying operators from H to Y . See [29] for information on this space. We note that a version of the following Lemma is also in [15, Proposition 3.19], where it is instead assumed that Y has so-called property (α). Moreover, in

390

Hyt¨ onen and Veraar

IEOT

[15, Remark 5.3 and Proof of Proposition 3.19] it is claimed that this assumption can be relaxed to non-trivial cotype, which is weaker than our assumption below. However, there seems to be a small confusion there: in [15, Remark 5.3] it is observed that non-trivial cotype suffices, thanks to a result in [17], if (a certain Hilbert space) H1 = C, whereas in [15, Proposition 3.19] and the following lemma one has the dual situation: H1 = H is a general Hilbert space and H2 = C. Indeed one could deduce the following lemma by a standard duality argument from the result in [17], but since a self-contained argument is only slightly longer, we provide it for completeness: Lemma 5.2. Let Y be a Banach space with non-trivial type and let H be a Hilbert space. Then there exists a constant C such that for all Ψn ∈ γ(H, Y ) and fn ∈ H, N

X

rn Ψn fn

L2 (Ω;Y )

n=1

N

X

≤ C sup kfn kH rn Ψn 1≤n≤N

L2 (Ω;γ(H,Y ))

n=1

.

Proof. Let (hk )K be an orthonormal basis for the span of (fn )N n=1 in H, so PK k=1 that fn = k=1 (hk , fn )hk . Let further (rn )N be a Rademacher sequence on a n=1 0 a Gaussian sequence on Ω . Then probability space Ω, and (γk )K k=1 K D N X N X X E Ψn hk , (hk , fn )yn∗ hΨn fn , yn∗ i = n=1 k=1

n=1

K N N K DX X E X X ∗ γk rn Ψn hk , = EE0 rm γ` (h` , fm )ym n=1

k=1

m=1

`=1

K N

X X

γk rn Ψn hk ≤

L2 (Ω;L2 (Ω0 ;Y ))

n=1

k=1

N K

X

X

∗ rm γ` (h` , fm )ym ×

m=1

Lq (Ω0 ;L2 (Ω;Y ∗ ))

`=1

q ∈ [2, ∞).

,

The first factor is bounded by N

X

rn Ψn

L2 (Ω;γ(H,Y ))

n=1

by the definition of the norm in γ(H, Y ). As for the second, the non-trivial type of Y implies some non-trivial cotype q0 ∈ [2, ∞) for Y ∗ , and then, taking q ∈ (q0 , ∞) and applying Lemma 3.1(2), N K

X

X

∗ rm γ` (h` , fm )ym

m=1

≤C

Lq (Ω0 ;L2 (Ω;Y ∗ ))

`=1

K

X

sup γ` (h` , fm )

1≤m≤N

`=1

Lq (Ω0 )

N

X

∗ rm y m

m=1

L2 (Ω;Y ∗ )

.

Vol. 63 (2009)

Here PK

R-Boundedness of Smooth Operator-Valued Functions

391

PK

γ` (h` , fm ) is a centered Gaussian variable with variance equal to 2 2 q `=1 (h` , fm ) = kfm kH , hence its L norm is cq kfm kH for a constant cq . PN ∗ The assertion follows by taking the supremum over all m=1 rm ym ∈ ∗ RadN (Y ) of norm 1, using the non-trivial type of Y . `=1

Proposition 5.3. Let X and Y be Banach spaces. Let p ∈ [1, 2], and assume that Y has type p. If T : Rd → B(X, Y ) satisfies kT xkB d/p (Rd ;Y ) ≤ CT kxk, x ∈ X,

(5.3)

p,1

then there exists a constant C = C(p, Y ) such that R {T (t) ∈ B(X, Y ) : t ∈ Rd } ≤ CCT .

(5.4)

Proof. If p = 1, the result follows from [13, Remark 5.2]. Let then p ∈ (1, 2]. Fix for the moment x ∈ X and k ≥ 0. Let fk : Rd → Y be defined as fk (t) = Tk (t)x = ϕk ∗ T (t)x. Then by [19, Theorem 1.1], 1

kfk kγ(L2 (Rd ),Y ) ≤ Ckfk k

d( 1 − 1 ) Bp,pp 2 (Rd ;Y

)

1

≤ C2dk( p − 2 ) kfk kLp (Rd ;Y ) .

(5.5)

d M Choose (tm )M m=1 in R and (xm )m=1 in X arbitrarily. Since Y has type p > 1 it follows from Lemma 5.2 that M

X

rm T (tm )xm

L2 (Ω;Y )

m=1

Z M X X k+1

X = rm

k≥0 n=k−1

.

X k+1 X k≥0 n=k−1

h

X

Rd

m=1

sup 1≤m≤M

M

X X k+1

X rm ϕn ∗ Tk (tm )xm ≤

k≥0 n=k−1

Tk (u)xm ϕn (tm − u)du

L2 (Ω;Y )

M

X

kϕn (tm − ·)kL2 (Rd ) rm Tk xm

m=1

L2 (Ω;γ(L2 (Rd ),Y ))

m=1

M

X

rm xm 2kd/2 Tk

k≥0

L2 (Ω;Y )

m=1

L2 (Ω;γ(L2 (Rd ),Y ))

.

Applying (5.5) pointwise in Ω yields that M

X

rm xm

Tk m=1

γ(L2 (Rd ),Y )

M

X 1 1

. 2kd( p − 2 ) Tk rm xm m=1

Lp (Rd ;Y )

.

(5.6)

392

Hyt¨ onen and Veraar

IEOT

Therefore, we obtain from (5.6) and (2.1) that X

M

X

2kd/2 Tk rm xm

.

L2 (Ω;γ(L2 (Rd ),Y ))

m=1

k≥0

X

M

X 1 1

2kd/2 2kd( p − 2 ) Tk rm xm

h

Z X Ω k≥0

=

M

X

rm xm 2kd/p Tk

Z ≤ Ω

Lp (Rd ;Y )

m=1

Z X M

rm xm

T Ω

L2 (Ω;Lp (Rd ;Y ))

m=1

k≥0

d/p

Bp,1 (Rd ;Y )

m=1

dP

dP

M M

X

X

rm xm rm xm dP ≤ CT CT X

m=1

m=1

L2 (Ω;X)

.

Putting things together yields the required R-boundedness estimate.

As a consequence of Theorem 5.1 we have the following two results. One can similarly derive strong type results from Proposition 5.3. Corollary 5.4. Let X and Y be Banach spaces. Let p ∈ [1, 2] and q ∈ [2, ∞]. Assume that X has cotype q and Y has type p. Let r ∈ [1, ∞] be such that 1r = p1 − 1q . If there exists an M such that Z r1 ≤M (5.7) kDα T krB(X,Y ) Rd

for every α ∈ {0, 1, . . . , d} with |α| ≤ b dr c + 1, then {T (t) ∈ B(X, Y ) : t ∈ Rd } is R-bounded. d

Corollary 5.5. Let X and Y be Banach spaces. Let p ∈ [1, 2] and q ∈ [2, ∞]. Assume that X has cotype q and let Y have type p. Let I = (a, b) with −∞ ≤ a < b ≤ ∞. Let r ∈ (1, ∞] be such that 1r ≥ p1 − 1q . Let α ∈ ( 1r , 1). If T ∈ Lr (R; B(X, Y )) and there exists an A such that kT (s + h) − T (s)k ≤ A|h|α (1 + |s|)−α , s, s + h ∈ I, h ∈ I,

(5.8)

then {T (t) ∈ B(X, Y ) : t ∈ I} is R-bounded by a constant times A. Note that in the case that I is bounded, the factor (1+|s|)−α can be omitted. Proof. By taking a worse p or q it suffices to consider the case that 1r = p1 − 1q . First consider the case that I = R. As in [13, Corollary 5.4] one may check that 1

r T ∈ Λr,1 (R; B(X, Y )), where the latter is defined in Section 2.5, and therefore the result follows from Theorem 5.1. If I 6= R, then one can reduce to the above case by (2.5).

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

393

6. Applications 6.1. R-boundedness of semigroups In the next result we will give a sufficient condition for R-boundedness of strongly continuous semigroups restricted to fractional domain spaces. Theorem 6.1. Let (T (t))t∈R+ be a strongly continuous semigroup on a Banach space X with kT (t)k ≤ M e−ωt for some ω > 0. Assume X has type p ∈ [1, 2] and cotype q ∈ [2, ∞]. Let α > 1r = p1 − 1q and let iα : D((−A)α ) → X be the inclusion mapping. Then {T (t)iα : t ∈ R+ } ⊂ B(D((−A)α ), X) is R-bounded. Proof. For θ ∈ (0, 1) let Xθ = (X, D(A))θ,∞ . Then x ∈ Xθ if and only if kxkXθ := kxk + sup t−θ k(T (t)x − x)k t∈R+

is finite, and this expression defines an equivalent norm on Xθ (cf. [25, Proposition 3.2.1]). If we fix θ ∈ ( 1r , α), then we obtain that sup t−α kT (t)iα − iα kB(D((−A)α ),X) = t∈R+

sup

sup t−α kT (t)x − xkX

kxkD((−A)α ) ≤1 t∈R+

≤

sup

kxkXθ

kxkD((−A)α ) ≤1

.

sup

kxkD((−A)α ) = 1.

kxkD((−A)α ) ≤1

Therefore, kT (s + h)iα − T (s)iα kB(D((−A)α ),X) . M e−ωs hα and the result follows from Corollary 5.5.

The result in Theorem 6.1 is quite sharp as follows from the next example. An application of Theorem 6.1 will be given in Theorem 6.3. For α ∈ R and p ∈ [1, ∞], let H α,p (R) be the Bessel-potential spaces (cf. [38, 2.3.3]). Example 6.2. Let p ∈ [1, ∞). Let (T (t))t∈R be the left-translation group on X = d Lp (R) with generator A = dx . Then for all α ∈ (| p1 − 12 |, 1) and M ∈ R+ , {T (t)iα : t ∈ [−M, M ]} ⊂ B(H α,p (R), Lp (R)), α,p

(6.1)

p

is R-bounded, where iα : H (R) → L (R) denotes the embedding. On the other hand, for α ∈ (0, | p1 − 12 |) and M = 1, the family (6.1) is not R-bounded. Proof. Note that Lp (R) and H α,p have type p ∧ 2 and cotype p ∨ 2. Therefore, for α > | p1 − 12 | the R-boundedness of {e−t T (t)iα : t ∈ R+ } ⊂ B(H α,p (R), Lp (R))

394

Hyt¨ onen and Veraar

IEOT

follows from Theorem 6.1. Therefore, we obtain from the Kahane-contraction principle that {T (t)iα : t ∈ [0, M ]} ⊂ B(H α,p (R), Lp (R)) is R-bounded. Since a similar argument works for T (−t), the R-boundedness of (6.1) follows from the fact that the union of two R-bounded sets is again Rbounded. For the converse, let ψ ∈ C ∞ (R) \ {0} be such that supp(ψ) ⊂ (0, 1). For c ∈ (0, ∞) let ψc (t) = ψ(ct). Then (−A)α ψc = cα [(−A)α ψ]c . Fix an integer N and let fn = f0 := ψN for all n. Then f0 has support in (0, 1/N ) and kf0 kpLp (R) = N −1 kψkpLp (R) . There holds, on the one hand, N N

p

X

p

X

rn f0 (· + n/N ) = rn T (n/N )iα fn

2 2 n=1

L (Ω;X)

L (Ω;X)

n=1

=

N X

kf0 (· + n/N )kpLp (R) = N kf0 kpLp (R) = kψkpLp (R) ,

n=1

and on the other hand, N

p

X

rn fn

2 α n=1

L (Ω;D((−A) ))

p

p

= N 2 kf0 kpD((−A)α ) = N 2 kf0 kpLp (R) + k(−A)α f0 kpLp (R) p

= N 2 kψN kpLp (R) + kN α [(−A)α ψ]N kpLp (R)

p = N −1+ 2 kψkpLp (R) + N αp k(−A)α ψ]kpLp (R) . Therefore, if τ := {T (t)iα : t ∈ [−1, 1]} is R-bounded, then it follows that there exists a constant C such that 1

1

1 ≤ CN − p + 2 +α . Letting N tend to infinity, this implies that α ≥ p1 − 12 , i.e., τ can only be Rbounded in this range. We still have to prove that the R-boundedness also implies α ≥ 12 − p1 . This can be proved by duality. If {T (t) ∈ B(H α,p (R), Lp (R)) : t ∈ [−1, 1]} is R-bounded, 0 0 then {T ∗ (t) ∈ B(Lp (R), H −α,p (R)) : t ∈ [−1, 1]} is R-bounded as well. It follows 0 −α ∗ p0 that {(1−A) T (t) ∈ B(L (R), Lp (R)) : t ∈ [−1, 1]} is R-bounded. This implies 0 0 that {T ∗ (t) ∈ B(H α,p (R), Lp (R)) : t ∈ [−1, 1]} is R-bounded. According to the first part of the proof this implies that α ≥ p10 − 12 = 12 − p1 . 6.2. Stochastic Cauchy problems We apply Theorem 6.1 to stochastic equations with additive Brownian noise. We refer the reader to [29] for details on stochastic Cauchy problems, stochastic integration and γ-radonifying operators. Let (Ω, F, P) be a probability space. Let H be a separable Hilbert space and let WH be a cylindrical Wiener process. Recall from [29] that for an operator-valued function Φ : [0, t] → B(H, E) which belongs

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

395

to γ(L2 (0, t; H), X) (the space of γ-radonifying operators from L2 (0, t; H) to X) we have

Z t

= kΦkγ(L2 (0,t;H),X) . Φ(s) dWH (s) 2

L (Ω;X)

0

On a real Banach space X we consider the following equation. ( dU (t) = AU (t) dt + B(t)dWH (t), t ∈ R+ , U (0) = x.

(SE)

Here A is the generator of a strongly continuous semigroup (T (t))t∈R+ , B : R+ → B(H, E) and x ∈ X. We say that a strongly measurable process U : R+ × Ω → X is a mild solution of (SE) if for all t ∈ R+ , almost surely we have Z t U (t) = T (t)x + T (t − s)B(s) dWH (s). 0

In general (SE) does not have a solution (cf. [29, Example 7.3]). In the case when B(t) = B ∈ γ(H, X) is constant, there are some sufficient conditions for existence. Indeed, if X has type 2 or (T (t))t∈R+ is an analytic semigroup, then (SE) always has a unique mild solution and it has a version with continuous paths (see [39, Corollary 3.4] and [9] respectively). In the next result we prove such an existence and regularity result under assumptions on the noise in terms of the type and cotype of X. Theorem 6.3. Assume X has type p ∈ [1, 2] and cotype q ∈ [2, ∞]. Let w ∈ R be such that limt→∞ ewt T (t) = 0. Let α > p1 − 1q and B ∈ γ(L2 (R+ ; H), D((w−A)α )). Then (SE) has a unique mild solution U . Moreover, if there exists an ε > 0 such that for all M ∈ R+ , sup ks 7→ (t − s)−ε B(s)kγ(L2 (0,t;H),D((w−A)α ) < ∞

(6.2)

t∈[0,M ]

then U has a version with continuous paths. In particular we note that if B(t) = B ∈ γ(H, D((w − A)α )) is constant then for all ε ∈ (0, 21 ) 1

ks 7→ (t − s)−ε B(s)kγ(L2 (0,t;H),D((w−A)α )) = (1 − 2ε)−1 t 2 −ε kBkγ(H,D((w−A)α )) . Remark 6.4. Here is a sufficient condition for (6.2): there is an s ∈ (2, ∞) such that for all M ∈ R+ , 1

−1

p 2 B ∈ Bs,p (0, M ; D((w − A)α )).

Indeed, it follows from [28, Lemma 3.3] that (6.2) holds for all ε ∈ (0, 12 − 1s ). Proof. Assume that (6.2) holds for some ε ∈ [0, 12 ). In the case ε = 0 we will show existence of a solution, and in the other case we show that the solution has a version with continuous paths.

396

Hyt¨ onen and Veraar

IEOT

By Theorem 6.1, {ewt T (t)iα ∈ B(D((w − A)α ), X) : t ≥ 0} is R-bounded. It follows that for fixed M > 0, {T (t)iα ∈ B(D((w − A)α ), X) : t ∈ [0, M ]} is R-bounded by some constant C. Therefore, by [20] (see also [27, Theorem 9.14]), the function s 7→ T (s)iα acts as a multiplier between the spaces γ(L2 (0, t; H), D((w − A)α )) and γ(L2 (0, t; H), X), and we conclude that sup ks 7→ (t − s)−ε T (t − s)B(s)kγ(L2 (0,t;H),X) t∈[0,M ]

≤ C sup ks 7→ (t − s)−ε B(s)kγ(L2 (0,t;H),D((w−A)α )) < ∞. t∈[0,M ]

Now the result follows from [39, Proposition 3.1 and Theorem 3.3].

6.3. R-boundedness of evolution families In the next application we obtain R-boundedness of an evolution family generated by a family (A(t))t∈[0,T ] of unbounded operators which satisfy the conditions (AT) of Acquistapace and Terreni (see [1]). For φ ∈ (0, π], we define the sector Σ(φ) := {0} ∪ {λ ∈ C \ {0} : | arg(λ)| < φ}. The condition (AT) is said to be satisfied if the following two requirements hold: (AT1) The A(t) are linear operators on a Banach space E and there are constants K ≥ 0, and φ ∈ ( π2 , π) such that Σ(φ) ⊂ %(A(t)) and for all λ ∈ Σ(φ) and t ∈ [0, T ], K kR(λ, A(t))k ≤ . 1 + |λ| (AT2) There are constants L ≥ 0 and µ, ν ∈ (0, 1] with µ + ν > 1 such that for all λ ∈ Σ(φ, 0) and s, t ∈ [0, T ], kA(t)R(λ, A(t))(A(t)−1 − A(s)−1 )k ≤ L|t − s|µ (|λ| + 1)−ν . Under these assumptions there exists a unique strongly continuous evolution family (t,s) (P (t, s))0≤s≤t≤T in B(X) such that ∂P∂t = A(t)P (t, s) for 0 ≤ s < t ≤ T . −1 Moreover, kA(t)P (t, s)k ≤ C(t − s) . For analytic semigroup generators one has that for all ε > 0 and T ∈ [0, ∞), {tε S(t) ∈ B(X) : t ∈ [0, T ]} is R-bounded. This easily follows from (5.1). This may be generalized to evolution families (P (t, s))0≤s≤t≤T , where (A(t))t∈[0,T ] satisfies the (AT) conditions. Indeed, then by the same reasoning we obtain that for all α > 0, sup R {(t − s)α P (t, s) ∈ B(X) : t ∈ [s, T ]} < ∞. s∈[0,T ]

This argument does not hold if one considers the R-bound with respect to s ∈ [0, t] instead of t ∈ [s, T ]. This is due to the fact that

∂P (t, s)

(6.3)

≤ C(t − s)−1 ∂s

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

397

might not be true. The R-boundedness with respect to s ∈ [0, t] has applications for instance in the study of non-autonomous stochastic Cauchy problems (see [39]). We also note that (6.3) does hold if (A(t)∗ )t∈[0,T ] satisfies the (AT)-conditions (see [2]). Recall from [42, Theorem 2.3] that for all θ ∈ (0, µ), kP (t, s)(−A(s))θ k ≤ C(t − s)−θ , 0 ≤ s < t ≤ T.

(6.4)

Due to this inequality one might expect that under assumptions on µ, one can still obtain a fractional version of (6.3). This is indeed the case and in the next theorem we will give conditions under which the R-boundedness with respect to s ∈ [0, t] holds. The authors are grateful to Roland Schnaubelt for showing them the following result. Proposition 6.5. Assume (AT). Then for all θ ∈ (0, µ) there exists a constant C such that for all 0 ≤ s ≤ t ≤ T , k(−A(t))−θ (P (t, s) − I)k ≤ C(t − s)θ .

(6.5)

Proof. First let θ ∈ (1 − ν, µ). By [26, equation (A.5)] we can write Z t −θ (−A(t)) (P (t, s) − I) = g(t, s) + (−A(t))−θ P (t, τ )(−A(τ ))θ h(τ, s)dτ, s

where g(t, s) = (−A(t))−θ (e(t−s)A(s) − I), h(t, s) = (−A(t))1−θ (−A(s))−1 − (−A(t))−1 A(s)e(t−s)A(s) . We may write kg(t, s)k ≤k((−A(t))−θ − (−A(s))−θ )(e(t−s)A(s) − I)k + k(−A(s))−θ (e(t−s)A(s) − I)k. By [37, equation (2.10)] k((−A(t))−θ − (−A(s))−θ )(e(t−s)A(s) − I)k . |t − s|µ . (t − s)θ . For the other term it is clear that −θ

k(−A(s))

(t−s)A(s)

(e

Z − I)k ≤

t−s

k(−A(s))1−θ eτ A(s) k dτ . (t − s)θ .

0

This shows that kg(t, s)k . (t − s)θ . By [41, equation (2.2)] we obtain that kh(t, s)k . (t − s)µ−1 . Since by [26, Lemma A.1] V (t, s) = (−A(t))−θ P (t, τ )(−A(τ ))θ is uniformly bounded, it follows that Z t kV (t, τ )h(τ, s)kdτ . (t − s)µ . (t − s)θ . s

We may conclude (6.5) for the special choice of θ.

398

Hyt¨ onen and Veraar

IEOT

For general θ ∈ (0, µ) choose ε > 0 so small that µ − ε > 1 − ν. Then by interpolation with θ/(µ − ε) it follows that k(−A(t))−θ (P (t, s) − I)k . k(−A(t))−(µ−ε) (P (t, s) − I)kθ/(µ−ε) kP (t, s) − Ik1−θ/(µ−ε) . (t − s)θ .

Theorem 6.6. Let X be a Banach space with type p ∈ [1, 2] and cotype q ∈ [2, ∞]. Assume (AT) with 1 1 − . p q

µ> Then for all ε > 0,

sup R {(t − s)ε P (t, s) ∈ B(X) : s ∈ [0, t]} < ∞. t∈[0,T ]

As a consequence one obtains a version of [39, Corollary 4.5] without assuming (t,s) k ≤ C(t − s)−1 . k ∂P∂s Proof. Choose θ ∈ ( p1 − 1q , µ). Let r ∈ (1, ∞) be such that θ > 1r ≥ p1 − 1q and ε > θ − 1r . Fix t ∈ [0, T ]. We will apply Theorem 5.1, with the equivalent norm explained in Section 2.5, to the function f : [0, t] → B(X) defined by f (s) = (t − s)ε P (t, s). Let h ∈ (0, t). By the triangle inequality we can write k(t − s − h)ε P (t, s + h) − (t − s)ε P (t, s)k ≤ k(t − s − h)ε (P (t, s + h) − P (t, s))k + |(t − s − h)ε − (t − s)ε |kP (t, s)k. Since the main point is dealing with small ε, we may assume that (ε − 1)r < −1. Then Z t−h Z 2h Z t |(t − s)ε − (t − s − h)ε |r ds = + |uε − (u − h)ε |r du 0

h

Z

2h

≤

εr

Z

u du + h

ε−1 r

|hε(u − h)

2h r 1+(ε−1)r

.ε,r h1+εr + h h

2h

t εr

| du ≤ h(2h)

r

Z

∞

+ (hε)

u(ε−1)r du

h

h h1+εr .

For the other part it follows from (6.4) and Proposition 6.5 that for all θ < µ kP (t, s + h)−P (t, s)k = kP (t, s+h)(−A(s+h))θ (−A(s+h))−θ (I − P (s+h, s))k ≤ C(t − s − h)−θ k(−A(s + h))−θ (P (s + h, s) − I)k ≤ C(t − s − h)−θ hθ .

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

399

We conclude that Z t−h r1 k(t − s − h)ε P (t, s + h) − (t − s)ε P (t, s)kr ds 0 1

. hε+ r + hθ

Z

t−h

(t − s − h)(ε−θ)r ds

r1

0 1

h hε+ r + hθ . hθ where we used ε > θ − 1r . Similar results hold for h < 0. It follows that %r (f, τ ) . τ θ , for τ ∈ (0, 1), where %r is defined as in Section 2.5. Since θ >

1 r

1

r we can conclude that f ∈ Λr,1 (0, t; B(X)). Now the result 1

1

r r (R; B(X)) and Br,1 (R; B(X)), and follows from (2.5), the norm equivalence of Λr,1 Theorem 5.1.

Acknowledgment The authors thank S. Kwapie´ n for the helpful comments on Lemma 3.1 which eventually led us to parts (1) and (3) of that Lemma. The authors thank R. Schnaubelt for showing them Proposition 6.5.

References [1] P. Acquistapace and B. Terreni. A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova, 78:47–107, 1987. [2] P. Acquistapace and B. Terreni. Regularity properties of the evolution operator for abstract linear parabolic equations. Differential Integral Equations, 5(5):1151–1184, 1992. [3] W. Arendt and S. Bu. The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z., 240(2):311–343, 2002. [4] J. Bergh and J. L¨ ofstr¨ om. Interpolation spaces. An introduction. Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. [5] E. Berkson and T. A. Gillespie. Spectral decompositions and harmonic analysis on UMD spaces. Studia Math., 112(1):13–49, 1994. [6] J. Bourgain. Vector-valued singular integrals and the H 1 -BMO duality. In Probability theory and harmonic analysis (Cleveland, Ohio, 1983), volume 98 of Monogr. Textbooks Pure Appl. Math., pages 1–19. Dekker, New York, 1986. [7] Ph. Cl´ement, B. de Pagter, F. A. Sukochev, and H. Witvliet. Schauder decomposition and multiplier theorems. Studia Math., 138(2):135–163, 2000. [8] R. Denk, M. Hieber, and J. Pr¨ uss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., 166(788), 2003. [9] J. Dettweiler, J. M. A. M. van Neerven, and L. W. Weis. Space-time regularity of solutions of parabolic stochastic evolution equations. Stoch. Anal. Appl., 24:843–869, 2006.

400

Hyt¨ onen and Veraar

IEOT

[10] J. Diestel, H. Jarchow, and A. Tonge. Absolutely summing operators, volume 43 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. [11] T. Figiel. On equivalence of some bases to the Haar system in spaces of vector-valued functions. Bull. Polish Acad. Sci. Math., 36(3-4):119–131 (1989). [12] O. van Gaans. On R-boundedness of unions of sets of operators. In Partial differential equations and functional analysis, volume 168 of Oper. Theory Adv. Appl., pages 97– 111. Birkh¨ auser, Basel, 2006. [13] M. Girardi and L. W. Weis. Criteria for R-boundedness of operator families. In Evolution equations, volume 234 of Lecture Notes in Pure and Appl. Math., pages 203–221. Dekker, New York, 2003. [14] L. Grafakos. Classical and Modern Fourier Analysis. Pearson/Prentice Hall, New Jersey, 2004. [15] B. H. Haak and P. C. Kunstmann. Admissibility of unbounded operators and wellposedness of linear systems in Banach spaces. Integr. Equ. Oper. Theory, 55(4):497– 533, 2006. [16] T. Hyt¨ onen. Fourier embeddings and Mihlin-type multiplier theorems. Math. Nachr., 274/275:74–103, 2004. [17] C. Kaiser and L. W. Weis. Wavelet transform for functions with values in UMD spaces. Studia Math., 186(2):101–126, 2008. [18] N. J. Kalton and S. J. Montgomery-Smith. Set-functions and factorization. Arch. Math. (Basel), 61(2):183–200, 1993. [19] N. J. Kalton, J. M. A. M. van Neerven, M. C. Veraar, and L. W. Weis. Embedding vector-valued Besov spaces into spaces of γ-radonifying operators. Math. Nachr., 281(2):238–252, 2008. [20] N. J. Kalton and L. W. Weis. The H ∞ -calculus and square function estimates. Preprint, 2004. [21] P. C. Kunstmann and L. W. Weis. Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus. In Functional analytic methods for evolution equations, volume 1855 of Lecture Notes in Math., pages 65– 311. Springer, Berlin, 2004. [22] S. Kwapie´ n. Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Studia Math., 44:583–595, 1972. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. [23] S. Kwapie´ n and W. A. Woyczy´ nski. Random series and stochastic integrals: single and multiple. Probability and its Applications. Birkh¨ auser, Boston, MA, 1992. [24] M. Ledoux and M. Talagrand. Probability in Banach spaces, volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1991. Isoperimetry and processes. [25] A. Lunardi. Interpolation Theory. Appunti. Scuola Normale Superiore Pisa, 1999. [26] L. Maniar and R. Schnaubelt. The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions. J. Differential Equations, 235(1):308–339, 2007.

Vol. 63 (2009)

R-Boundedness of Smooth Operator-Valued Functions

401

[27] J. M. A. M. van Neerven. Internet Seminar 2007/2008 Stochastic Evolution Equations. http://fa.its.tudelft.nl/isemwiki. [28] J. M. A. M. van Neerven, M. C. Veraar, and L. W. Weis. Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal., 255(4):940–993, 2008. [29] J. M. A. M. van Neerven and L. W. Weis. Stochastic integration of functions with values in a Banach space. Studia Math., 166(2):131–170, 2005. [30] J. Peetre. Sur la transformation de Fourier des fonctions ` a valeurs vectorielles. Rend. Sem. Mat. Univ. Padova, 42:15–26, 1969. [31] A. Pelczy´ nski and M. Wojciechowski. Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm. Studia Math., 107(1):61– 100, 1993. [32] G. Pisier. Factorization of operators through Lp∞ or Lp1 and noncommutative generalizations. Math. Ann., 276(1):105–136, 1986. [33] G. Pisier. Probabilistic methods in the geometry of Banach spaces. In Probability and analysis (Varenna, 1985), volume 1206 of Lecture Notes in Math., pages 167–241. Springer, Berlin, 1986. [34] G. Pisier. The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1989. [35] E. Sawyer. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans. Amer. Math. Soc., 281(1):329–337, 1984. [36] H.-J. Schmeisser. Vector-valued Sobolev and Besov spaces. In Seminar analysis of the Karl-Weierstraß-Institute of Mathematics 1985/86 (Berlin, 1985/86), volume 96 of Teubner-Texte Math., pages 4–44. Teubner, Leipzig, 1987. [37] R. Schnaubelt. Asymptotic behaviour of parabolic nonautonomous evolution equations. In Functional analytic methods for evolution equations, volume 1855 of Lecture Notes in Math., pages 401–472. Springer, Berlin, 2004. [38] H. Triebel. Interpolation theory, function spaces, differential operators. Johann Ambrosius Barth, Heidelberg, second edition, 1995. [39] M. C. Veraar and J. Zimmerschied. Non-autonomous stochastic Cauchy problems in Banach spaces. Stud. Math., 185(1):1–34, 2008. [40] L. W. Weis. Operator-valued Fourier multiplier theorems and maximal Lp -regularity. Math. Ann., 319(4):735–758, 2001. [41] A. Yagi. Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. II. Funkcial. Ekvac., 33(1):139–150, 1990. [42] A. Yagi. Abstract quasilinear evolution equations of parabolic type in Banach spaces. Boll. Un. Mat. Ital. B (7), 5(2):341–368, 1991. Tuomas Hyt¨ onen Department of Mathematics and Statistics University of Helsinki Gustaf H¨ allstr¨ omin katu 2B FI-00014 Helsinki Finland e-mail: [email protected]

402

Hyt¨ onen and Veraar

Mark Veraar Delft Institute of Applied Mathematics Delft University of Technology P.O. Box 5031 2600 GA Delft The Netherlands e-mail: [email protected], [email protected] Submitted: April 21, 2008. Revised: December 8, 2008.

IEOT

Integr. equ. oper. theory 63 (2009), 403–425 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030403-23, published online February 24, 2009 DOI 10.1007/s00020-009-1665-2

Integral Equations and Operator Theory

Crossed Product of a C ∗-Algebra by a Semigroup of Endomorphisms Generated by Partial Isometries B.K. Kwa´sniewski and A.V. Lebedev Abstract. The paper presents a construction of the crossed product of a C ∗ algebra by a semigroup of endomorphisms generated by partial isometries. Mathematics Subject Classification (2000). 46L05, 46L55. Keywords. C ∗ -algebra, endomorphism, partial isometry, crossed product, finely representable action, transfer operator.

Contents 1. Introduction 2. The dynamical system (A, Γ+ , α) and transfer actions 3. Finely representable systems 4. The Banach ∗ -algebra l1 (Γ, α, A) 5. The crossed product A oα Γ 6. Faithful and regular representations of the crossed product References

403 405 408 412 416 420 424

1. Introduction Given a C ∗ -algebra A and an endomorphism α there is a number of ways to construct a new C ∗ -algebra (an extension of A) called the crossed product (or a covariance algebra, or a C ∗ -algebra associated with a C ∗ -dynamical system). These algebras are now recognized as being among the most important structures in operator algebras as well as in their applications. In quantum theory the term This work was in part supported by Polish Ministry of Science and High Education grant number N N201 382634.

404

Kwa´sniewski and Lebedev

IEOT

covariance algebra means an algebra generated by an algebra of observables and by operators which determine the time evolution of a quantum system (a C ∗ dynamical system), thereby the covariance algebra is an object which carries all the information about the quantum system, see, for example, [8], [24], [18] (and the sources cited there) for this and other connections with mathematical physics. In pure mathematics C ∗ -algebras associated with C ∗ -dynamical systems proved to be useful in different fields: classification of operator algebras [25], [27], [8], [22]; K-theory for C ∗ -algebras [7], [11], [20]; functional and functional differential equations [1], [2] [3], [4]; or even in number theory [16]. This multitude of applications and the complexity of the matter attracted many authors and at present the construction and the theory of the crossed products associated with automorphisms has mostly attained its final shape (recall again in this connection [25], [27]). On the other hand in the endomorphism case even the construction of the corresponding crossed products causes an abundance of various approaches, and this is mainly due to the differences in the nature of irreversibility of endomorphisms. Among the successful constructions of the algebras of this type one would mention, for example, the constructions developed by J. Cuntz and W. Krieger [9, 10], W.L. Paschke [23], P.J. Stacey [26], G.J. Murphy [21], R. Exel [12], [13] and B.K. Kwa´sniewski [14]. Recently A.B. Antonevich, V.I Bakhtin and A.V. Lebedev have developed in [6, 5] the construction of the crossed product that unifies all the previous structures in the situation when the endomorphism is generated by a partial isometry. In addition the crossed product elaborated possesses ’almost all’ the fine properties of the crossed products associated with automorphisms: starting with the semigroup Z+ generated by α it can be represented by means of Fourier type series over the ’group’ Z (associated with α and the corresponding transfer operator), it has a reasonable regular representation and its faithful representations are described by the Isomorphism Theorem (Theorem 3.6, [5]) which is a quite appropriate analogy to the corresponding results for the automorphisms case (cf. Remark 3.8 [5]). This situation calls natural reminiscences of say partial automorphisms situation where R.Exel [11] invented the crossed product by a single partial automorphism and then K. McClanahan [20] developed this construction further up to partial actions of an arbitrary group. His results along with those obtained by A.V. Lebedev [17] show that the crossed product associated with partial action in fact behaves ideally — almost like in the situations of the action by automorphisms. This reminiscence is one of the main impulses that caused the appearance of the present article — once the construction of the crossed product generated by a single endomorphism appeared in [5] it is a natural desire to develop it from the situation of a single endomorphism to a semigroup of endomorphisms. Recalling also that in [5] there sprang out a natural passage from the semigroup Z+ to the ’group’ Z one feels a spontaneous wish to associate with a given semigroup of endomorphisms a certain ’group’ generated by endomorphisms and the corresponding transfer operators. The fulfillment of this wish is the main theme of the article.

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

405

It should be noted at once that the situation with endomorphisms is not quite the same as the situation with partial automorphisms since the endomorphism situation is ’heavily’ irreversible. The naturally arising here in the crossed product construction notion of a finely representable pair (see 2.4, [5]) and the notion of a finely representable C ∗ -dynamical system (see 3.1 of the present article) ’feels’ what is ’left’ and what is ’right’ (see (3.2) and Proposition 5.3). Involving the physical associations one can also say that it feels the ’past’ and the ’future’. In other words it feels the order. Therefore it seems that the natural development of the group Z case here is the case of a totally ordered group. And the material of the paper shows that this case can be worked out perfectly. We also note that a special case of the crossed product elaborated in the present article was considered by J. Lindiarni and I. Raeburn in [19]. The paper is organized as follows. In Section 2 we introduce the necessary notion of an action α of the positive cone Γ+ of a totally ordered abelian group Γ by endomorphisms of a C ∗ -algebra A thus defining a C ∗ -dynamical system (A, Γ+ , α). By using the ideas and methods that originated in essence in [11, 6] we deduce the existence of a complete transfer action for (A, Γ+ , α) (Theorem 2.4). In Section 3 we discuss the principal constructive element of the crossed product defined further — the finely representable C ∗ -dynamical systems. Here the results form a natural development of the corresponding results of [6]. Section 4 is devoted to the explicit presentation of the ’base’ of the crossed product — the Banach ∗ -algebra l1 (Γ, α, A). The crossed product A oα Γ itself is then defined in Section 5 and the main structural result here is Theorem 5.4. Finally in Section 6 we give a criterion for a representation of A oα Γ to be faithful and present the regular representation of A oα Γ. The article is based on [15].

2. The dynamical system (A, Γ+ , α) and transfer actions The results of this section form in essence a natural generalization of the corresponding results from [6, 11] to the totally ordered abelian group situation. Let A be a C ∗ -algebra with an identity 1 and let Γ+ be the positive cone of a totally ordered abelian group Γ with an identity 0: Γ+ = {x ∈ Γ : 0 ≤ x},

Γ = Γ + − Γ+ .

We fix a semigroup homomorphism α : Γ+ → End(A), that is α0 = Id,

αx ◦ αy = αx+y ,

αx , αy ∈ End(A), x, y ∈ Γ+ .

Depending on inclinations one may say that we fix an action α of Γ+ by -endomorphisms of A, or a C ∗ -dynamical system (A, Γ+ , α). In the sequel we shall often make use of the simple fact that {αx (1)}x∈Γ+ is a nonincreasing family of projections. Indeed, the αx (1) are self-adjoint idempotents and if x ≤ y, that is y − x ∈ Γ+ , then ∗

αx (1)αy (1) = αx (1)αx (αy−x (1)) = αx (1αy−x (1)) = αx+y−x (1) = αy (1),

406

Kwa´sniewski and Lebedev

IEOT

that is x ≤ y ⇒ αx (1) ≥ αy (1). Throughout the article as a rule we shall denote by x, y the elements of Γ+ and by g an element of Γ. 2.1. Let L be an action of Γ+ by continuous, linear, positive maps Lx : A → A, that is Lx+y = Ly ◦ Lx , x, y ∈ Γ+ . We shall say that L is a transfer action or an action by transfer operators for (A, Γ+ , α), if the following identity is satisfied Lx (αx (a)b) = aLx (b)

(2.1)

for all a, b ∈ A, x ∈ Γ+ . Note that then Lx (bαx (a)) = Lx (b)a as well. These relations have the following consequences (see [12]): Lx (A) is a two-sided ideal, Lx (1) is a positive central element in A (and therefore Lx (1)A is a two-sided ideal), and the following formula holds Lx ( · ) = Lx (αx (1) · ) = Lx ( · αx (1)).

(2.2)

2.2. The transfer action L will be called non-degenerate if for each x ∈ Γ+ one of the equivalent conditions holds (see [12], Proposition 2.3): (i) the composition Ex = αx ◦ Lx is a conditional expectation onto αx (A), (ii) αx ◦ Lx ◦ αx = αx , (iii) αx (Lx (1)) = αx (1). In particular, as α0 = Id (ii) implies L0 = Id. The non-degeneracy of the transfer action implies (see [6], Propositions 2.5, 2.6) that: Lx (A) = Lx (1)A, the element Lx (1) is a central orthogonal projection in A, and Lx : αx (A) → Lx (A) is a ∗ -isomorphism with inverse αx : Lx (A) → αx (A). Moreover we note that {Lx (1)}x∈Γ+ is a nonincreasing family of projections. Indeed, if x ≤ y, that is y − x ∈ Γ+ , then Lx (1)Ly (1) = Lx (1)Lx Ly−x (1) = Lx αx (Lx (1))Ly−x (1) = Lx αx (1)Ly−x (1) = Lx (Ly−x (1)) = Ly (1), that is x ≤ y ⇒ Lx (1) ≥ Ly (1). 2.3. The transfer action L will be called complete, if αx (Lx (a)) = αx (1)aαx (1),

x ∈ Γ+ ,

a ∈ A.

(2.3)

The completeness of the transfer action implies that for each x ∈ Γ+ we have αx ◦Lx ◦αx = αx and recalling (2.2) we also obtain Lx ◦αx ◦Lx = Lx . In particular, a complete transfer action is non-degenerate, in addition if L is complete then for each x ∈ Γ+ , we have αx (A) = αx (1)Aαx (1) that is αx (A) is a hereditary subalgebra of A (see [12], Proposition 4.1).

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

407

The existence and uniqueness of a transfer action for an arbitrary dynamical system (A, Γ+ , α) is quite a problematic matter. However, thanks to [6] one can write down conditions on (A, Γ+ , α) under which a unique complete transfer action for (A, Γ+ , α) does exist. The next result is a generalization of Theorem 2.8, [6] to the situation under investigation. Theorem 2.4. Let (A, Γ+ , α) be a dynamical system. The following are equivalent: 1) there exists a complete transfer action L for (A, Γ+ , α), 2) (i) there exists a non-degenerate transfer action L for (A, Γ+ , α), (ii) αx (A) is a hereditary subalgebra of A for each x ∈ Γ+ , 3) (i) there exists a family {Px }x∈Γ+ of central orthogonal projections in A such that a) αx (Px+y ) = αx (1)Py , for all x, y ∈ Γ+ , b) the mappings αx : Px A → αx (A) are ∗ -isomorphisms, and (ii) αx (A) = αx (1)Aαx (1) for each x ∈ Γ+ . Moreover the objects in 1) – 3) are defined in a unique way (i. e. the transfer action L in 1) and 2) is unique and the family of projections {Px }x∈Γ+ in 3) is unique as well) and Px = Lx (1), x ∈ Γ+ , (2.4) and Lx (a) = αx−1 (αx (1)aαx (1)), a ∈ A, (2.5) −1 where αx : αx (A) → Px A is the inverse mapping to αx : Px A → α(A), x ∈ Γ+ . Proof. 1) ⇒ 2). Follows from the definition of a complete transfer action, see 2.3. 2) ⇒ 3). It is known that 2) (ii) and 3) (ii) are equivalent, cf. [12], Proposition 4.1. Set Px = Lx (1), x ∈ Γ+ . By 2.2, {Px }x∈Γ+ is a family of central orthogonal projections and the mappings αx : Px A → αx (A) are ∗ -isomorphisms. So 3) (i) b) is true. Recalling (2.2) and using the fact that Lx : αx (A) → Px A is the inverse of αx : Px A → αx (A) we obtain αx (Px+y ) = αx (Lx+y (1)) = αx (Lx (Ly (1))) = αx Lx (αx (1)Py αx (1)) = αx (1)Py αx (1) = αx (1)Py , which proves 3) (i) a). 3) ⇒ 1). Fix x ∈ Γ+ . Let αx−1 : αx (A) → Px A be the inverse mapping to αx : Px A → αx (A). Define the operator Lx by the formula Lx (a) = αx−1 (αx (1)aαx (1)). Clearly Lx is linear and positive, and (2.3) is fulfilled. Note that αx Lx (αx (a)b) = αx (1)αx (a)bαx (1) = αx (a)αx (1)bαx (1) = αx (aLx (b)), and as the elements Lx (αx (a)b) and aLx (b) belong to the ideal Px A where the endomorphism αx is injective, they coincide. Therefore (2.1) holds, and the only thing left to prove is that L is an action of the semigroup Γ+ . For that purpose let us observe that the family {Px }x∈Γ+ is nonincreasing. Indeed, if x, y ∈ Γ+ are such that x ≤ y, then A can be written as the direct sum of ideals in two ways, namely A = ker αx ⊕ (Px A) = ker αy ⊕ (Py A), and ker αx ⊂ ker αy ,

408

Kwa´sniewski and Lebedev

IEOT

whence Py A ⊂ Px A and hence Py ≤ Px . Using 3) (i) a), we have αy (Px+y A) = αy (Px+y )αy (A) = Px αy (A) and as Px+y A ⊂ Py A we obtain that αy : Px+y A → Px αy (A) is a ∗ -isomorphism and the inverse is given by Ly . Thus we have Ly (Lx (A)) = Ly (Px A) = Ly (αy (1)Px Aαy (1)) = Ly (Px αy (A)) = Px+y A. Hence Ly (Lx (a) and Lx+y (a) belong to the ideal Px+y A where the endomorphism αx+y is injective, and as αx+y (Ly (Lx (a)) = αx αy (Ly (Lx (a)) = αx αy (1)Lx (a)αy (1) = αx+y (1)αx (Lx (a))αx+y (1) = αx+y (1)aαx+y (1) = αx+y (Lx+y (a)) we have Lx+y = Ly ◦Lx . The uniqueness of the objects in 1) - 3) can be established exactly in the same way as it is done in the proof of Theorem 2.8 in [6].

3. Finely representable systems Here we discuss one of the main notions of the article — the finely representable systems. They play a principal role in the construction of the crossed product in Section 5 as well as in the construction of its regular representation in Section 6. 3.1. Let A be a C ∗ -algebra containing an identity 1 and α : Γ+ → End(A) be a semigroup homomorphism. We say that the triple (A, Γ+ , α) is finely representable if if there exists a triple (H, π, U ) consisting of a Hilbert space H, a faithful nondegenerate representation π : A → L(H) and a semigroup homomorphism U : Γ+ → L(H) such that for every a ∈ A, x ∈ Γ+ , the following conditions are satisfied π(αx (a)) = Ux π(a)Ux∗ ,

Ux∗ π(a)Ux ∈ π(A)

(3.1)

a ∈ A.

(3.2)

and Ux π(a) = π(αx (a))Ux ,

In this case we also say that A is a coefficient algebra associated with α. Remark 3.2. The notion of a coefficient algebra associated with a single endomorphism was introduced by A.V. Lebedev and A. Odzijewicz in [18] and proved to be of principle importance in the investigation of extensions of C ∗ -algebras by partial isometries. We shall also use certain ideas and methods from [18] while uncovering the internal structure of the crossed product in Sections 5 and 6. It is an easy exercise that in the above definition one can replace condition (3.2) by the condition Ux∗ Ux ∈ Z(π(A)), x ∈ Γ+ (3.3) where Z(π(A)) is the center of π(A), or the condition Ux∗ Ux π(a)Ux∗ π(b)Ux = π(a)Ux∗ π(b)Ux , a, b ∈ A,

x ∈ Γ+ .

(3.4)

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

409

In particular it is clear that a finely representable C ∗ -dynamical system can also be defined as a triple (A, Γ+ , α) such that there exists a triple (H, π, U ) where π : A → L(H) is a faithful non-degenerate representation, Ux ∈ L(H), x ∈ Γ+ and the mappings Ux · Ux∗ , x ∈ Γ+ coincide with the endomorphisms αx on π(A) while the mappings Ux∗ · Ux are transfer operators for αx . The next theorem presents a criterion for the fine representability. The construction given in its proof is in fact a development of the corresponding construction from Theorem 3.1, [6] onto the situation under consideration. Theorem 3.3. (A, Γ+ , α) is finely representable iff there exists a complete transfer action L for (A, Γ+ , α) (that is if either of the equivalent conditions of Theorem 2.4 holds). Proof. Necessity. If conditions (3.1) and (3.2) are satisfied then (identifying A with π(A)) one can set Lx (·) = Ux∗ (·)Ux ,

x ∈ Γ+

and it is easy to verify that L is a complete transfer action. Sufficiency. Let L be a complete transfer action. We shall construct the desired Hilbert space H by means of the elements of the initial algebra A in the following way. Let h · , · i be a certain non-negative inner product on A (differing from a common inner product only in such a way that for certain non-zero elements v ∈ A the expression hv, vi may be equal to zero). For example this inner product may have the form hv, ui = f (u∗ v) where f is some positive linear functional on A. If one factorizes A by all the elements v such that hv, vi = 0 then one obtains a linear space with a strictly positive inner product. We shall call the completion p of this space with respect to the norm kvk = hv, vi the Hilbert space generated by the inner product h · , · i. Let F be the L set of all positive linear functionals on A. The space H will f be the direct sum Hilbert spaces H f . Every H f will in turn f ∈F H of someL f f be the direct sum of Hilbert spaces g∈Γ Hg . The spaces Hg are generated by non-negative inner products h · , · ig on the initial algebra A that are given by the following formulae hv, ui0 = f (u∗ v); ∗

hv, uix = f Lx (u v) , hv, ui−x = f u∗ αx (1)v ,

(3.5) +

x∈Γ ;

(3.6)

+

(3.7)

x∈Γ .

The properties of these inner products are described in the next

410

Kwa´sniewski and Lebedev

IEOT

Lemma 3.4. For any v, u ∈ A, any x ∈ Γ+ and any g ∈ Γ the following equalities are true hαx (v), uig = hv, Lx (u)ig−x , hαx (1)αg (v), uig = hv, Lg (u)ig−x ,

x ≤ g;

(3.8)

0 ≤ g ≤ x;

(3.9)

hαx−g (1)v, uig = hv, αx−g (1)uig−x , g ≤ 0.

(3.10)

Proof. Let x ≤ g. The proof of (3.8) reduces to the verification of the equalities Lg (u∗ αx (v)) = Lg−x Lx (u∗ αx (v) = Lg−x Lx (u)∗ v which follow from the definition of the transfer action (recall x ≤ g iff g − x ∈ Γ+ ). Let 0 ≤ g ≤ x. Formula (3.9) follows from Lg u∗ αx (1)αg (v) = Lg u∗ αg (αx−g (1))αg (v) = Lg u∗ αg (αx−g (1)v) = Ly (u)∗ αx−y (1)v Let g ≤ 0. The proof of (3.10) reduces to the verification of the equality u∗ α−g (1)αx−g (1)v = u∗ αx−g (1)v which follows from α−g (1) ≥ αx−g (1).

The proof of Theorem 3.3 is now resumed. Let us define the semigroup homomorphism U : Γ+ → L(H). For an arbitrary fixed x ∈ Γ+ the operators Ux and all the subspaces H f ⊂ H. The action of these operators Ux∗ will leave invariant L f f on every H = g∈Γ Hg is similar and its scheme is presented in the the next diagrams. αx−g (1) ·

αx−g (1) ·

αx (1)αg (·)

αx (·)

αx (·)

U −−−→ L

α−g (1) ·

α−g (1) ·

H0f

Hxf

Lg+x (·)

Lx (·)

g∈Γ

Lx (·)

U∗ ←−−−− L

g∈Γ

H0f

f H−x

Formally this action is defined in the following way. For any finite sum M h= hg ∈ H f , hg ∈ Hgf , g

we set Ux h =

M g

(Ux h)g

and

Ux∗ h =

M g

Hgf

(Ux∗ h)g

Hgf

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

where

411

  if x ≤ g, αx (hg−x ), (Ux h)g = αx (1)αg (hg−x ), if 0 ≤ g ≤ x,   αx−g (1)hg−x , if g ≤ 0,   if 0 ≤ g, Lx (hg+x ), (Ux∗ h)g = Lg+x (hg+x ), if − x ≤ g ≤ 0,   α−g (1)hg+x , if g < 0.

Lemma 3.4 guarantees that the operators Ux and Ux∗ are well defined (i. e. they preserve factorization and completion by means of which the spaces Hgf were built from the algebra A) and Ux and Ux∗ are mutually adjoint. Let us show that U is a semigroup homomorphism. Take any x, y ∈ Γ+ and hg ∈ Hgf . For x + y ≤ g we have y ≤ g and x ≤ g − y, whence Uy (Ux h) g = αy (Ux h)g−y = αy (αx (hg−y−x ) = αx+y (hg−(x+y) ) = (Ux+y h)g . For 0 ≤ g ≤ x + y two cases are possible: If y ≤ g then we have 0 ≤ g − y ≤ x, whence Uy (Ux h) g = αy (Ux h)g−y = αy αx (1)αg−y (hg−y−x ) = αx+y (1)αg (hg−(x+y) ) = (Ux+y h)g . If g ≤ y then we have g − y ≤ 0, and thus Uy (Ux h) g = αy (1)αg (Ux h)g−y = αy (1)αx+y (1)αg (hg−(x+y) ) = (Ux+y h)g , where in the final equality we used the fact that αx+y (1) ≤ αy (1). For g ≤ 0 we have Uy (Ux h) g = αy−g (1)(Ux h)g−y = αy−g (1)αx+y−g (1)hg−(x+y) = (Ux+y h)g . where in the final equality we used the inequality αy−g (1) ≥ αx+y−g (1). Thus we have proved that U : Γ+ → L(H) is a semigroup homomorphism. Now let us define the representation π : A → L(H). For any a ∈ A the operator π(a) : H → H will leave invariant all the subspaces H f ⊂ H and also all the subspaces Hgf ⊂ H f . If hg ∈ Hgf then we set ( ahg , g ≥ 0, π(a)hg = (3.11) α−g (a)hg , g ≤ 0. The scheme of the action of the operator π(a) is presented in the following diagram. ...

α2x (a)·

αx (a)·

a·

a·

H0f

Hxf

a·

π(a)

... L

f H−2x

f H−x

f H2x

g∈Γ

Hgf

412

Kwa´sniewski and Lebedev

IEOT

If x ≤ g then π(αx (a))hg = αx (a)hg , Ux π(a)Ux∗ hg = αx (aLx (hg )) = αx (a)hg αx (1), and moreover (3.6), inequality x ≤ g and the definition of a transfer operator imply that the element αx (a)hg αx (1) coincides with αx (a)hg in the space Hgf . For 0 ≤ g ≤ x we have π(αx (a))hg = αx (a)hg , Ux π(a)Ux∗ hg

= αx (1)αg (αx−g (a)Lg (hg )) = αx (a)αg (1)hg αg (1) = αx (a)hg αg (1),

where we used the inequality αx (1) ≤ αg (1). The same argument as above shows that αx (a)hg αg (1) coincides with αx (a)hg in the space Hgf . For g ≤ 0 we have π(αx (a))hg = αx−g (a)hg , Ux π(a)Ux∗ hg = αx−g (1)αx−g (a)αx−g (1)hg = αx−g (a)hg . Thus we have proved that Ux π(a)Ux∗ = π(αx (a)) for any a ∈ A. It can be shown similarly that Ux∗ π(a)Ux = π(Lx (a)). Indeed, if 0 ≤ g one then has π(Lx (a))hg = Lx (a)hg , Ux∗ π(a)Ux hg = Lx (aαx (hg )) = Lx (a)hg . For −x ≤ g ≤ 0 one has π(Lx (a))hg = α−g (Lx (a))hg = α−g (L−g (Lx+g (a)))hg = α−g (1)Lx+g (a)α−g (1)hg , Ux∗ π(a)Ux hg = Lg+x (aαx (1)αg+x (hg )) = Lg+x (aαx+g−g (1))(hg ) = Lx+g (a)α−g (1)hg , and moreover (3.7) implies that α−g (1)Lx+g (a)α−g (1)hg coincides with an element Lx+g (a)α−g (1)hg in the space Hgf . For g ≤ −x we have Ux∗ π(a)Ux hg = α−g (1)α−g−x (a)α−g (1)hg , π(Lx (a))hg = α−g (Lx (a))hg = α−g−x (αx (Lx (a)))hg = α−g (1)α−g−x (a)α−g (1)hg where we used the inequality 0 ≤ −g − x. To finish the proof it is enough to establish the faithfulness of the representation π. But this follows from the definition of the inner product in (3.5), the definition of π (see the second line in the diagram) and the standard Gelfand-Naimark faithful representation of a C ∗ -algebra. The proof of Theorem 3.3 is complete.

4. The Banach ∗ -algebra l1 (Γ, α, A) This is a starting section for the construction of the principal object of the article, the crossed product (that will be introduced in Section 5). The algebra presented here serves as an explicitly described base of the crossed product. Hereafter we

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

413

assume that (A, Γ+ , α) is a finely representable C ∗ -dynamical system and L is the unique transfer action (this L does exist by Theorem 3.3 and is unique by Theorem 2.4). 4.1. Let l1 (Γ, α, A) be the set consisting of the elements of the form a = {ag }g∈Γ where ax ∈ Aαx (1) and a−x ∈ αx (1)A, for x ≥ 0, (4.1) P and such that g∈Γ kag k < ∞. We define the addition, multiplication by scalars and involution on l1 (Γ, α, A) in an obvious manner. Namely, let a = {ag }g∈Γ , b = {bx }x∈G ∈ l1 (Γ, α, A), and let λ ∈ C. We set (a + b)g := ag + bg ,

(4.2)

(λa)g := λag , (4.3) ∗ ∗ (a )g := a−g . (4.4) Clearly, these operations are well defined and thus we have equipped l1 (Γ, α, A) with the structure of Banach space with isometric involution, the norm taken into P account is of course the one given by kak = g∈Γ kag k. Unfortunately the multiplication of two elements from l1 (Γ, α, A) is not so nice. It generalizes the convolution multiplication in crossed products by automorphisms and by partial automorphisms. Its ’strange’ structure reflects the arising further ’antisymmetry’ between the operators Ux and Ux∗ (see Proposition 5.3) which is mainly due to the fact that in view of (3.2) one can move Ux only to the right while Ux∗ can be moved only to the left. We put X X  X ax αx−y (b−y ) + Lx (a−x by ) + ax αx (by ), if 0 ≤ g,     g=x−y g=y−x g=x+y x,y>0 x,y≥0 X X X (a·b)g := x,y>0  α (a )b + L (a b ) + αy (a−x )b−y , if g < 0. y−x x −y y −x y    g=x−y x,y>0

g=y−x x,y>0

g=−x−y x,y≥0

where x, y in sums run through Γ+ . Proposition 4.2. The above multiplication is well defined, and ka · bk ≤ kak · kbk,

for all

a, b ∈ A.

Proof. We need to show that (a · b) satisfies relations (4.1). To this end take g ∈ Γ and assume that 0 ≤ g (the case g < 0 can be considered in a similar way). Let x, y ≥ 0. If g = x − y then, since αg is a morphism, we have ax αx−y (b−y ) = ax αx−y (b−y )αx−y (1) ∈ Aαg (1). If g = y − x then, since by = by αy (1) and Lx is a transfer operator , we have Lx (a−x by ) = Lx a−x by αx (αy−x (1)) = Lx (a−x by )αy−x (1) ∈ Aαg (1). If g = x + y then, since by = by αy (1) and αx is a morphism, we have ax αx (by ) = ax αx (by αy (1)) = ax αx (by )αx+y (1) ∈ Aαg (1).

414

Kwa´sniewski and Lebedev

IEOT

Thus (a · b)g ∈ Aαx (1). Now, using the fact that kαk = kLk ≤ 1 we have X ka · bk = k(a · b)g k g∈Γ

≤ =

X X

X

kax kkb−y k +

ka−x kkby k +

g∈Γ

g=x−y

X

X kag k kbg k = kakkbk.

g∈Γ

g=y−x

X

kax kkby k

g=x+y −g=x+y

g∈Γ

Hence the multiplication is well defined and the proof is complete.

Let us provide the following notational convention: Let g0 be in Γ and let a be in Aαg0 (1) if g0 ≥ 0, or in α−g0 (1)A if g0 < 0. We shall denote by aδg0 the element given by (aδg0 )g = aδ(g0 ,g) where δ(g0 ,g) is the Kronecker symbol. Then the elements αx (1)δx , x ∈ Γ+ , and the algebra Aδ0 generate (with the help of the above defined operations) a dense subspace (in fact a ∗ -subalgebra) of l1 (Γ, α, A). Furthermore, we have the natural embedding of A into l1 (Γ, α, A) given by A 3 a 7−→ aδ0 . It is easy to check that under this embedding the unit 1 ∈ A coincides with the element 1δ0 which is neutral with respect to the multiplication we have defined. Theorem 4.3. The set l1 (Γ, α, A) with the above defined operations becomes a unital ∗ -Banach algebra. Proof. It is enough to verify the equality (a · b)∗ = b∗ · a∗ and the associativity of multiplication (the distribution laws are readily checked because α and L are linear). Let us prove the first property. Let g be in Γ. If g ≥ 0, then using the positivity of L we have X X X (a · b)∗−g = b∗−y αy−x (a∗x ) + Ly (b∗y a∗−x ) + b∗−y αy (a∗−x ), −g=x−y

−g=y−x

−g=−x−y

and simply by definition X X X (b∗ · a∗ )g = b∗−x αx−y (a∗−y ) + Ly (b∗x a∗−y ) + b∗−x αy (a∗−y ). g=x−y

g=y−x

(a · b)∗−g

∗

g=x+y ∗

Replacing x by y, one sees that = (b · a )g . Using the same argument for g ≤ 0 one obtains the desired equality: (a · b)∗ = b∗ · a∗ . Clearly, to show the associativity it suffices to consider the elements of the form aδg1 , bδg2 , cδg3 ∈ l1 (Γ, α, A) where g1 , g2 , g3 ∈ Γ are fixed. However, anyway we face a number of possibilities which have to be checked. This may be a source of

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

415

pleasure as well as a cause of a headache hence we confine ourselves to the case when g1 + g2 + g3 ≥ 0 and leave the opposite case to the enthusiasts. Suppose that g1 + g2 + g3 ≥ 0. A routine computation shows that: 1) If g1 , g2 , g3 ≥ 0, then (aδg1 · bδg2 ) · cδg3 = (aαg1 (b)δg1 +g2 ) · cδg3 = aαg1 (b)αg1 +g2 (c)δg1 +g2 +g3 = aαg1 (bαg2 (c))δg1 +g2 +g3 = aδg1 · (bαg2 (c)δg2 +g3 ) = aδg1 · (bδg2 · cδg3 ). 2) If g1 < 0 and g2 , g3 ≥ 0, then ( L−g1 (ab)αg1 +g2 (c)δg1 +g2 +g3 if g1 + g2 ≥ 0 (aδg1 · bδg2 ) · cδg3 = L−g1 −g2 (Lg2 (ab)c)δg1 +g2 +g3 if g1 + g2 < 0 ( L−g1 abα−g1 (αg1 +g2 (c)) δg1 +g2 +g3 if g1 + g2 ≥ 0 = L−g1 −g2 Lg2 (abαg2 (c)) δg1 +g2 +g3 if g1 + g2 < 0 = L−g1 (abαg2 (c))δg1 +g2 +g3 = aδg1 · (bδg2 · cδg3 ). 3) If g2 < 0 and g1 , g3 ≥ 0, then ( aαg1 +g2 (b)αg1 +g2 (c)δg1 +g2 +g3 if g1 + g2 ≥ 0 (aδg1 · bδg2 ) · cδg3 = L−g1 −g2 (α−g1 −g2 (a)bc)δg1 +g2 +g3 if g1 + g2 < 0 ( aαg1 +g2 (bc)δg1 +g2 +g3 if g1 + g2 ≥ 0 = aL−g1 −g2 (bc)δg1 +g2 +g3 if g1 + g2 < 0. and aδg1 · (bδg2

( aαg1 (L−g2 (bc))δg1 +g2 +g3 · cδg3 ) = aαg1 +g2 +g3 (Lg3 (bc))δg1 +g2 +g3

if g2 + g3 ≥ 0 if g2 + g3 < 0.

We have the four (sub)possibilities: 3a) For g1 + g2 ≥ 0 and g2 + g3 < 0 we have aδg1 · (bδg2 · cδg3 ) g +g +g = aαg1 +g2 αg3 (Lg3 (bc) 1 2 3 = aαg1 +g2 αg3 (1)bcαg3 (1) = aαg1 +g2 +g3 (1)αg1 +g2 (bc) = aαg1 +g2 (bc) = (aδg1 · bδg2 ) · cδg3 g1 +g2 +g3 . 3b) For g1 + g2 ≥ 0 and g2 + g3 ≥ 0 we have aδg1 · (bδg2 · cδg3 ) g +g +g = aαg1 +g2 α−g2 (L−g2 (bc) 1 2 3 = aαg1 +g2 α−g2 (1)bcα−g2 (1) = aαg1 +g2 (bc = (aδg1 · bδg2 ) · cδg3 g1 +g2 +g3 . 3c) For g1 + g2 < 0 and g2 + g3 ≥ 0 we have aδg1 · (bδg2 · cδg3 ) g1 +g2 +g3 = aαg1 Lg1 (L−g1 −g2 (bc) = aαg1 (1)L−g1 −g2 (bc)αg1 (1) = aL−g1 −g2 (bcα−g2 (1)) = aL−g1 −g2 (bc) = (aδg1 · bδg2 ) · cδg3 g1 +g2 +g3 .

416

Kwa´sniewski and Lebedev

IEOT

3d) For g1 + g2 < 0 and g2 + g3 < 0 we have aδg1 · (bδg2 · cδg3 ) g1 +g2 +g3 = aαg1 +g2 +g3 Lg1 +g2 +g3 (L−g1 −g2 (bc)) = aαg1 +g2 +g3 (1)L−g1 −g2 (bc)αg1 +g2 +g3 (1) = aL−g1 −g2 (bcαg3 (1)) = (aδg1 · bδg2 ) · cδg3 g +g +g . 1

2

3

4) If g3 < 0 and g1 , g2 ≥ 0, then ( aαg1 (bαg2 +g3 (c))δg1 +g2 +g3 , if g2 + g3 ≥ 0 (aδg1 · bδg2 ) · cδg3 = aαg1 +g2 +g3 (α−g2 −g3 (b)c)δg1 +g2 +g3 , if g2 + g3 < 0 = aαg1 (b)αg1 +g2 +g3 (c))δg1 +g2 +g3 = aδg1 · (bδg2 · cδg3 ). 5) If g1 , g2 < 0 and g3 ≥ 0, then as g1 + g2 + g3 ≥ 0 we have g2 + g3 > 0, and thus aδg1 · (bδg2 · cδg3 ) = L−g1 (aL−g2 (bc))δg1 +g2 +g3 = L−g1 (L−g2 (α−g2 (a)bc))δg1 +g2 +g3 = L−g1 −g2 (α−g2 (a)bc)δg1 +g2 +g3 = (aδg1 · bδg2 ) · cδg3 . 6) If g1 , g3 < 0 and g2 ≥ 0, then as g1 + g2 + g3 ≥ 0 we have g2 + g3 > 0 and g1 + g2 > 0. Thus aδg1 · (bδg2 · cδg3 ) = L−g1 (abαg2 +g3 (c))δg1 +g2 +g3 = L−g1 abα−g1 (αg1 +g2 +g3 (c)) δg1 +g2 +g3 = L−g1 (ab)αg1 +g2 +g3 (c)δg1 +g2 +g3 = (aδg1 · bδg2 ) · cδg3 . 7) If g2 , g3 < 0 and g1 ≥ 0, then as g1 + g2 + g3 ≥ 0 we have g1 + g2 > 0, and thus aδg1 · (bδg2 · cδg3 ) = aαg1 +g2 +g3 (α−g3 (b)c)δg1 +g2 +g3 = aαg1 +g2 (b)αg1 +g2 +g3 (c)δg1 +g2 +g3 = (aδg1 · bδg2 ) · cδg3 . Thus the equality aδg1 · (bδg2 · cδg3 ) = (aδg1 · bδg2 ) · cδg3 , in the case g1 + g2 + g3 ≥ 0, is proved.

5. The crossed product A oα Γ In this section we discuss the main object of the paper, the crossed product of a finely representable C ∗ -dynamical system. Definition 5.1. The crossed product of a finely representable C ∗ -dynamical system (A, Γ+ , α) (see 3.1) is the C ∗ -algebra A oα Γ obtained by taking the enveloping C ∗ -algebra of l1 (Γ, α, A) (see 4.1). We aim at the investigation of the structure of A oα Γ, but before that let us justify the above definition and show that A oα Γ is universal with respect to covariant representations.

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

417

Definition 5.2. Let (A, Γ+ , α) be finely representable. A triple (π, H, U ) consisting of a Hilbert space H, a non-degenerate representation π : A → L(H) and a semigroup homomorphism U : Γ+ → L(H), is a covariant representation of (A, Γ+ , α), if for every a ∈ A and x ∈ Γ+ we have Ux π(a)Ux∗ = π(αx (a)),

Ux∗ π(a)Ux = π(Lx (a)).

Proposition 5.3. Let (π, U, H) be a covariant representation of (A, Γ+ , α). Then the formula X X X X (π × U ) a−x δ−x + a0 δ0 + ax δx := Ux∗ a−x + a0 + ax Ux , x>0

x>0

x>0

x>0

defines a representation of l1 (Γ, α, A) and hence establishes a representation of A oα Γ. Proof. Clearly, (π × U ) is linear and preserves the involution. In order to show that (π × U ) is multiplicative let us consider the elements aδg1 , bδg2 ∈ l1 (Γ, α, A) (we recall that if aδg ∈ l1 (Γ, α, A) then a ∈ Aαg (1) when g ≤ 0 and a ∈ α−g (1)A otherwise). We have the following possibilities: I) g1 + g2 ≥ 0, in other words g1 ≥ −g2 , g2 ≥ −g1 . 1) If g2 < 0, then g1 > 0 and ∗ ∗ (π × U )(aδg1 )(π × U )(bδg2 ) = π(a)Ug1 U−g π(b) = π(a)Ug1 +g2 U−g2 U−g π(b) 2 2

= π(a)Ug1 +g2 π(α−g2 (1)b) = π(a)Ug1 +g2 π(b)Ug∗1 +g2 Ug1 +g2 = (π × U )(aαg1 +g2 (b)δg1 +g2 ) = (π × U ) (aδg1 )(bδg2 ) . 2) If g1 < 0, then g2 > 0 and ∗ ∗ (π × U )(aδg1 )(π × U )(bδg2 ) = U−g π(a)π(b)Ug2 = U−g π(ab)U−g1 Ug1 +g2 1 1

= π(L−g1 (ab))Ug1 +g2 = (π × U )(L−g1 (ab)δg1 +g2 ) = (π × U ) (aδg1 )(bδg2 ) . 3) If g1 , g2 ≥ 0, then (π × U )(aδg1 )(π × U )(bδg2 ) = π(a)Ug1 π(b)Ug2 = π(a)Ug1 π(b)Ug∗1 Ug1 Ug2 = π(a)π(αg1 (b))Ug1 +g2 = (π × U )(aαg1 (b)δg1 +g2 ) = (π × U ) (aδg1 )(bδg2 ) . II) g1 + g2 < 0, in other words g1 < −g2 , g2 < −g1 . 1) If g1 > 0, then g2 < 0 and ∗ ∗ (π × U )(aδg1 )(π × U )(bδg2 ) = π(a)Ug1 U−g π(b) = π(a)Ug1 Ug∗1 U−g π(b) 2 2 −g1 ∗ ∗ ∗ = π(aαg1 (1))U−g π(b) = U−g U−g1 −g2 π(a)U−g π(b) 1 −g2 1 −g2 1 −g2 = (π × U )(α−g1 −g2 (a)bδg1 +g2 ) = (π × U ) (aδg1 )(bδg2 ) .

2) If g1 < 0, then g2 > 0 and ∗ ∗ (π × U )(aδg1 )(π × U )(bδg2 ) = U−g π(a)π(b)Ug2 = U−g Ug∗2 π(ab)Ug2 1 1 −g2

= U−g1 −g2 π(Lg2 (ab)) = (π × U )(Lg2 (ab)δg1 +g2 ) = (π × U ) (aδg1 )(bδg2 ) .

418

Kwa´sniewski and Lebedev

IEOT

3) If g1 , g2 ≤ 0, then ∗ ∗ ∗ ∗ ∗ (π × U )(aδg1 )(π × U )(bδg2 ) = U−g π(a)U−g π(b) = U−g U−g U−g2 π(a)U−g π(b) 1 2 1 2 2 ∗ = U−g π(α−g2 (a))π(b) = (π × U )(aαg1 (b)δg1 +g2 ) 1 −g2 = (π × U ) (aδg1 )(bδg2 ) .

As we know that there exists a covariant representation (π, U, H) such that π is faithful (see Theorem 3.3) this implies the existence of a representation of l1 (Γ, α, A) which is faithful on Aδ0 and hence the algebra A is naturally embedded into the crossed product A oα Γ. The argument we have just used has in fact much stronger consequences - see item (iv) of the following Theorem 5.4. Let ux (resp. u−x ) be the element of A oα Γ corresponding to the element αx (1)δx (resp. αx (1)δ−x ) of l1 (Γ, α, A), x ∈ Γ+ . Then (i) the family {ux }x∈Γ+ forms a semigroup of partial isometries, (ii) for each x ∈ Γ+ and a ∈ A we have u∗x = u−x ,

ux au∗x = αx (a),

(iii) the elements of the form X X a= u∗x a−x + a0 + ax ux , x∈F

u∗x aux = Lx (a).

F ⊂ Γ+ \ {0}, |F | < ∞

(5.1)

x∈F ux u∗x A,

where ax ∈ Aux u∗x , a−x ∈ form a dense ∗ -subalgebra C0 of A oα Γ. (iv) if a is of the form (5.1), then we have the inequalities kag k ≤ kak,

g ∈ Γ.

In particular the ’coefficients’ ag , g ∈ Γ, of a are determined in a unique way. Proof. For x, y ∈ Γ+ we have αx (1)δx αy (1)δy = αx (1)αx (αy (1))δx+y = αx+y (1)δx+y , whence the family {ux }x∈Γ+ forms a semigroup, that is (i) holds. As (αx (1)δx )∗ = αx (1)δ−x we have u∗x = u−x . For a ∈ A we have (αx (1)δx )(aδ0 )(αx (1)δ−x ) = (αx (a)δx )(αx (1)δ−x ) = αx (a)δ0 , (αx (1)δ−x )(aδ0 )(αx (1)δx ) = (αx (1)aδ−x )(αx (1)δx ) = Lx (αx (1)aαx (1))δ0 = Lx (a)δ0 , which implies (ii),P and in particular it follows that ux is a partial isometry. P ∗ a + a + As the element u −x 0 x x∈F P x∈F ax ux corresponds to the element P a δ + a δ + a δ in l (Γ, α, A), it follows that (iii) is true. −x −x 0 0 x x 1 x∈F x∈F Now let us verify (iv) for g = 0. To this end, take any covariant representation (π, U, H) such that π is faithful. Consider the space H = l2 (Γ, H) and the representation ν : A oα Γ → L(H) given by the formulae (ν(a)ξ)g = π(a)(ξg ),

where

(ν(ux )ξ)g = Ux (ξg−x ),

a ∈ A,

l2 (Γ, H) 3 ξ = {ξg }g∈Γ ∗ (ν(ux )ξ)g = Ux∗ (ξg+x ).

;

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

419

Routine verification shows that ν(A) and ν(ux ) satisfy all the conditions of a covariant representation and thus by Proposition 5.3 ν is well defined. Now take any a ∈ A oα Γ given by (5.1) and for a given ε > 0 chose a vector η ∈ H such that kηk = 1 and kπ(a0 )ηk > kπ(a0 )k − ε.

(5.2)

Set ξ ∈ l2 (Γ, H) by ξg = δ(0,g) η, where δ(i,j) is the Kronecker symbol. We have that kξk = 1 and the explicit form of ν(a)ξ and (5.2) imply kν(a)ξk ≥ kπ(a0 )ηk > kπ(a0 )k − ε which by the arbitrariness of ε proves the desired inequality: kak ≥ kν(a)k ≥ kπ(a0 )k = ka0 k. In order to verify the corresponding inequality for an arbitrary g ∈ Γ take any x ∈ Γ+ and observe that (ux a)0 = ux u∗x a−x = a−x ,

(au∗x )0 = ax ux u∗x = ax .

Hence, we have ka−x k ≤ kux ak ≤ kak,

kax k ≤ kau∗x k ≤ kak.

Corollary 5.5. We have the one-to-one correspondence: (π, U, H) ←→ (π × U, H) between covariant representations of (A, Γ+ , α) and non-degenerate representations of A oα Γ. Proof. This follows from Proposition 5.3 and items (i), (ii) of the above theorem. 5.6. Items (iii) and (iv) of Theorem 5.4 mean that one can define the linear and continuous maps Ex : C0 → Aux u∗x and E−x : C0 → ux u∗x A, x ∈ Γ+ given by Eg (a) = ag ,

a ∈ C0 , g ∈ Γ.

By continuity these mappings can be extended to the whole of Aoα Γ thus defining the ’coefficients’ of an arbitrary element a ∈ A oα Γ. We shall show further in Theorem 6.4 that these coefficients determine a in a unique way. The next theorem shows that the norm of an element a ∈ C0 can be calculated only in terms of the elements of A (0-degree coefficients of the powers of aa∗ ). Theorem 5.7. Let a ∈ C0 ⊂ A oα Γ be of the form (5.1). Then we have q kak = lim 4k kE0 [(a · a∗ )2k ]k. k→∞

(5.3)

420

Kwa´sniewski and Lebedev

IEOT

Pm Pm 2 Proof. Applying the known equality k i=1 di k ≤ m k i=1 di d∗i k which holds for any elements d1 , ..., dm in an arbitrary C ∗ -algebra, see e.g. [1, Lemma 7.3] or [2, Lemma 22.3], we obtain

X

X

ax ux u∗x a∗x + a0 a∗0 + u∗x a−x a∗−x ux kak2 ≤ (2|F | + 1)

x∈F

x∈F

= (2|F | + 1)kE0 (aa∗ )k. On the other hand as E0 is contractive we have kak2 = kaa∗ k ≥ kE0 (aa∗ )k thus kE0 (aa∗ )k ≤ kaa∗ k = kak2 ≤ (2|F | + 1)kE0 (aa∗ )k. (5.4) ∗ k ∗ k ∗ k∗ Applying (5.4) to (aa ) and having in mind that (aa ) = (aa ) and k(aa∗ )2k k = kak4k one has kE0 (aa∗ )2k k ≤ k(aa∗ )k · (aa∗ )k∗ k = kak4k ≤ (2|F k | + 1)kE0 (aa∗ )2k k where F k is set of all elements of Γ representable as a product of k elements from F . We recall that the so called subexponential groups include the commutative groups 1 and thus limk→∞ |F k | k = 1. So q q q 4k kE0 [(aa∗ )2k ]k ≤ kak ≤ 4k 2|F k | + 1 · 4k kE0 [(aa∗ )2k ]k. Observing the equality lim

k→∞

q 2|F k | + 1 = 1

4k

we conclude that kak = lim

k→∞

q

4k

kE0 [(aa∗ )2k ]k.

The proof is complete.

6. Faithful and regular representations of the crossed product Here we give a criterion for a representation of A oα Γ to be faithful and present the regular representation of A oα Γ. In order to study faithful representations of A oα Γ we introduce the following Definition 6.1. Let (π, U, H) be a covariant representation of (A, Γ∗ , α). We shall say that (π, U, H) possesses property (∗ ) if for any element a ∈ C0 of the form (5.1) we have kE0 (a)k ≤ k(π × U )(a)k (∗ ) P P in other words ka0 k ≤ k x∈F Ux∗ π(a−x ) + π(a0 ) + x∈F π(ax )Ux k. Let us observe that if (π, U, H) possess property (∗ ) then the representation π is faithful, and the mapping N ((π × U )(a)) := E0 (a), a ∈ C0 , extends uniquely up to the positive, contractive, conditional expectation from (π × U )(A oα Γ) onto A.

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

421

Theorem 6.2. Let (π, U, H) be a covariant representation of (A, Γ+ , α). The representation (π × U ) of A oα Γ is faithful iff (π, U, H) possesses property (∗ ). Proof. Necessity follows from item (iv) of Theorem 5.4. Let us show the sufficiency. Take any a ∈ C0 . By Theorem 5.7 and the definition of property (∗ ) we have q q kak = lim 4k kE0 [(aa∗ )2k ]k = lim 4k kN [(π × U )(aa∗ )2k ]k k→∞ k→∞ q q ≤ lim 4k k(π × U )(aa∗ )2k k = lim 4k k(π × U )(aa∗ )k (π × U )(aa∗ )k k k→∞ k→∞ q 4k 4k = lim k(π × U )(a)k = k(π × U )(a)k . k→∞

Hence kak = k(π × U )(a)k on a dense subset of A oα Γ.

ˆ on A oα Γ by the automorCorollary 6.3. We have the action of the dual group Γ phisms given by λa := a,

a ∈ A,

λug := λg ug ,

ˆ λg = λ(g) g ∈ Γ, λ ∈ Γ,

(here we consider Γ as a discrete group). Proof. Suppose that A oα Γ is faithfully and nondegenerately represented on a ˆ the triple (id, λu, H) is a covariant represenHilbert space H. Then for each λ ∈ Γ tation possessing property (∗ ), whence (id×λu) is an automorphism of Aoα Γ. The next result shows that any element a ∈ A oα Γ can be recovered from its coefficients Eg (a), g ∈ Γ, see 5.6. Theorem 6.4. Let a ∈ A ×δ Γ. Then the following conditions are equivalent: (i) a = 0; (ii) Eg (a) = 0, g ∈ Γ; (iii) E0 (a∗ a) = 0. Proof. Clearly, (i) implies (ii) and (iii). We now prove (ii) ⇒ (i). Let us suppose that A oα Γ is faithfully and nondegenerately represented on a Hilbert space H. Thus to prove that a = 0 it is enough to show that for any fixed ξ, η ∈ H with kξk = kηk = 1 we have haξ, ηi = 0

(6.1)

where h , i is the inner product in H. Recall that C0 is dense in A oα Γ and hence we can choose a sequence an , n = 1, 2, . . . of elements of C0 tending to a: X X (n) (n) an = u∗x a−x + a0 + a(n) x ux , x∈F (n)

x∈F (n)

422

Kwa´sniewski and Lebedev

IEOT

(n)

(n)

where F (n) is finite subset of Γ+ \ {0} and a−x ∈ αx (1)A, ax ∈ Aαx (1). ˆ and consider the elements λan (see Corollary 6.3). We define the seLet λ ∈ Γ ˆ by quence fn , n = 1, 2, . . . , of functions on Γ X (n) X fn (λ) = hλan ξ, ηi = γ−x λ−x + γx(n) λx (6.2) x∈F (n) (n) γ−x

(n) hUx∗ a−x ξ, ηi,

(n) γx

x∈F (n)∪{0}

(n) hax Ux ξ, ηi,

where = = x ∈ F (n) ∪ {0}. It follows that fn , ˆ (since Γ is discrete Γ ˆ is compact). n = 1, 2, . . . , are continuous on Γ Let f be the function given by f (λ) = hλaξ, ηi. Then we have −→ 0 |fn (λ) − f (λ)| = |h(λan − λa)ξ, ηi| ≤ kan − ak n→∞

which means that the sequence fn of continuous functions tends uniformly to f . ˆ (µ is the Haar measure of the Thus f is continuous and therefore f ∈ L2µ (Γ) ˆ compact group Γ). Let X f (λ) = γg λg g∈Γ

ˆ it where the righthand part is the Fourier series of f . Since fn → f (in L2µ (Γ)) follows that γg(n) → γg for every g ∈ Γ (6.3) (n)

where γk

are those defined by (6.2). Now note that property (∗) implies ka(n) g k → kEg (a)k for every g ∈ Γ.

(6.4)

And also observe that |γg(n) | ≤ ka(n) g k for every g ∈ Γ which together with (6.3), (6.4) means that γg = 0 for every g ∈ Γ.

(6.5)

Now (6.5) and the continuity of f implies ˆ f (λ) = 0 for every λ ∈ Γ. In particular f (1) =< aξ, η >= 0. Thus (6.1) is true, whence a = 0. For the completion of the proof of the theorem we verify the implication (iii) ⇒ (ii). By 5.6, for the proof of (ii) it is sufficient to demonstrate that Ex (a)ux = 0,

u∗x E−x (a) = 0,

x ∈ Γ+ .

(6.6)

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

423

Selecting a sequence of elements an ∈ C0 such that an −→ a and taking account of P P (n) (n) (n) (n) the explicit form of E0 (a∗n an ) = u∗x (ax )∗ ax ux + (a−x )∗ a−x , and of the fact (n) (n) that, as n −→ ∞, for each fixed x ∈ Γ+ we have ax −→ Ex (a), a−x −→ E−x (a) and E0 (a∗n an ) −→ E0 (a∗ a), we conclude that for each finite subset F ⊂ Γ+ we have

X X

u∗x Ex (a)∗ Ex (a)ux + E−x (a)∗ ux u∗x E−x (a) ≤ E0 (a∗ a).

x∈F

x∈F

Hence equalities (6.6) follow by (iii). The proof is complete.

6.5. Regular representation of the crossed product. Now we present a faithful representation of A oα Γ that will be written out explicitly in terms of A, α, L. Keeping in mind the standard regular representations for the known various versions of crossed products we shall call it the regular representation of A oα Γ. In fact the construction of this representation has been already obtained in the proof of Theorem 3.3. Let us, however, briefly discussL it. The Hilbert space H is defined to be the direct sum f ∈F H f of Hilbert spaces H f where F is the set of all positive linear functionals on A, and every H f is in L turn the direct sum of Hilbert spaces g∈Γ Hgf . The spaces Hgf are generated by non-negative inner products h · , · ig on the initial algebra A that are given by hv, uix = f Lx (u∗ v) , hv, ui−x = f u∗ αx (1)v , x ∈ Γ+ . The semigroup homomorphism U : Γ+ → L(H) is defined L in such a manner that for an arbitrary fixed x ∈ Γ+ and for any finite sum h = g hg ∈ H f , hg ∈ Hgf L we have Ux h = g (Ux h)g where   if x ≤ g, αx (hg−x ), (Ux h)g = αx (1)αg (hg−x ), if 0 ≤ g ≤ x,   αx−g (1)hg−x , if g ≤ 0. Finally, the representation π : A → L(H) is defined in such a manner that for any a ∈ A the operator π(a) : H → H leaves invariant all the subspaces Hgf , F ∈ F , g ∈ Γ, and for hg ∈ Hgf we have ( ahg , g ≥ 0, π(a)hg = α−g (a)hg , g ≤ 0. In the process of the proof of Theorem 3.3 it was verified that the triple (H, π, U ) described above is a covariant representation of (A, Γ+ , α) (in the sense of Definition 5.2), hence by Proposition 5.3 it gives rise to a representation (π × U ) of A oα Γ. Moreover, the covariant representation (H, π, U ) possesses property (*) which can be proved by repeating the argument from the proof of item (iv) of Theorem 5.4. Thus, by Theorem 6.2, (π × U ) is a faithful representation of A oα Γ.

424

Kwa´sniewski and Lebedev

IEOT

References [1] A. Antonevich: Linear Functional Equations. Operator Approach, Operator Theory: Advances and Applications Vol. 83, Birkh¨ auser, Basel, 1996. [2] A. Antonevich, A.V. Lebedev: Functional differential equations: I. C ∗ -theory – Longman Scientific & Technical, Pitman Monographs and Surveys in Pure and Applied Mathematics 70, 1994. [3] A. Antonevich, M. Belousov, A. Lebedev: Functional differential equations: II. C*applications: Part 1: Equations with continuous coefficients. – Addison Wesley Longman, Pitman Monographs and Surveys in Pure and Applied Mathematics 94, 1998. [4] A. Antonevich, M. Belousov, A. Lebedev: Functional differential equations: II. C*applications: Part 2: Equations with discontinuous coefficients and boundary value problems. – Addison Wesley Longman, Pitman Monographs and Surveys in Pure and Applied Mathematics 95, 1998. [5] A. B. Antonevich, V. I. Bakhtin, A. V. Lebedev, Crossed product of C ∗ -algebra by an endomorphism, coefficient algebras and transfer operators, arXiv:math.OA/0502415 v1 19 Feb 2005. [6] V. I. Bakhtin, A. V. Lebedev, When a C ∗ -algebra is a coefficient algebra for a given endomorphism, arXiv:math.OA/0502414 v1 19 Feb 2005. [7] B. Blackadar: K-theory for Operator algebras, Springer-Verlag, New York, 1986. [8] O. Bratteli, D. W. Robinson: Operator algebras and Quantum Statistical Mechanics I,II, New York 1979, 1980. [9] J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185. [10] J. Cuntz, W. Krieger, A Class of C*-algebras and Topological Markov Chains, Inventiones Math. 56 (1980), 251–268. [11] R. Exel, Circle actions on C ∗ -algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequence, J. Funct. Analysis, 122 (1994), 361–401. [12] R. Exel, A new look at the crossed-product of a C ∗ -algebra by an endomorphism, Ergodic Theory Dynam. Systems, 23 (2003), 1733–1750. [13] R. Exel, A new look at the crossed product of a C ∗ -algebra by a semigroup of endomorphisms, Ergodic Theory Dynam. Systems, 28 (2008), 749–789. [14] B. K. Kwa´sniewski, Covariance algebra of a partial dynamical system, Central European Journal of Mathematics 3 (4) (2005), 718–765. [15] B. K. Kwa´sniewski, A. V. Lebedev, Crossed product of a C ∗ -algebra by a semigroup of endomorphisms generated by partial isometries, arXiv:math.OA/0507143 v1 7 Jul 2005. [16] M. Laca, I. Raeburn, A semigroup crossed product arising in number theory, J. London Math. Soc. 59 (1999), 330–344. [17] A. V. Lebedev, Topologically free partial actions and faithful representations of partial crossed products, Funct. Anal. and Its Appl., 39 (3) (2005), 207–214. [18] A. V. Lebedev, A. Odzijewicz Extensions of C ∗ -algebras by partial isometries, Matem. Sbornik, 195 (2004), No 7, 37–70. [19] J. Lindiarni, I. Raeburn, Partial-isometric crossed products by semigroups of endomorphisms, Journal of Operator Theory, 52 (1) (2004), 61–88.

Vol. 63 (2009)

Crossed Product by a Semigroup of Endomorphisms

425

[20] K. McClanachan, K-theory for partial crossed products by discrete groups, J. Funct. Analysis, 130 (1995), 77–117. [21] G. J. Murphy, Crossed products of C*-algebras by endomorphisms, Integr. Equ. Oper. Theory 24 (1996), 298–319. [22] D. P. O’Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 1–25. [23] W. L. Paschke, The crossed product of a C ∗ -algebra by an endomorphism, Proceedings of the AMS, 80 (1) (1980), 113–118. [24] A. L. T. Paterson: Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics vol. 170, Birkh¨ auser, 1999. [25] G. K. Pedersen: C ∗ -algebras and their automorphism groups, Academic Press, London, 1979. [26] P. J. Stacey, Crossed products of C ∗ -algebras by ∗ -endomorphisms, J. Austral. Math. Soc. Ser. A 54 (1993), 204–212. [27] D. P. Williams: Crossed products of C ∗ -algebras, AMS Bookstore, 2007 B.K. Kwa´sniewski Institute of Mathematics University of Bialystok ul. Akademicka 2 PL-15-267 Bialystok Poland e-mail: [email protected] A.V. Lebedev Institute of Mathematics University of Bialystok ul. Akademicka 2 PL-15-267 Bialystok Poland and Belarus State University, Minsk av. Nezavisimisti 4 220050 Minsk Belarus e-mail: [email protected] Submitted: March 26, 2008. Revised: January, 26, 2009.

Integr. equ. oper. theory 63 (2009), 427–438 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030427-12, published online February 24, 2009 DOI 10.1007/s00020-009-1666-1

Integral Equations and Operator Theory

On the Airy Reproducing Kernel, Sampling Series, and Quadrature Formula Eli Levin and Doron S. Lubinsky Abstract. We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum. Mathematics Subject Classification (2000). Primary 46E22, 47B38; Secondary 41A58. Keywords. Airy kernel, sampling series, reproducing kernel.

1. Introduction Universality limits play a central role in random matrix theory [4], [17]. A recent new approach to these [14], [15], [16], [19] involves the reproducing kernel for a suitable space of entire functions. For universality in the bulk, the reproducing kernel is the familiar sine kernel, sinπtπt . It reproduces Paley-Wiener space, the class of entire functions of exponential type at most π, that are square integrable along the real axis. More precisely, if g is an entire function of exponential type at most π, and g ∈ L2 (R), then for all complex z, Z ∞ sin π (t − z) g (z) = g (t) dt. −∞ π (t − z) For universality at the hard edge of the spectrum, one obtains instead the Bessel kernel. It is the reproducing kernel for a class of entire functions of exponential type ≤ 1, satisfying a weighted integrability condition on the real line [16]. Both the sine and Bessel kernels have also been intensively studied within the context of sampling theory, as well as Lagrange interpolation series, and quadrature formula in approximation theory [6], [9], [10], [11], [12]. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.

428

Levin and Lubinsky

IEOT

In investigating universality limits for random matrices at the soft edge of the spectrum, the authors needed to determine the class of entire functions for which the Airy kernel is a reproducing kernel. An extensive literature search did not turn up the desired results. While Paley-Wiener theorems have been obtained for some related Airy operators [21], and many identities for the Airy kernel are familiar in universality theory [2], [3], [5], [20], they do not readily yield the desired reproducing kernel formula. The associated sampling series, and quadrature formula, also could not be traced. It is the goal of this paper to present these. These will be an essential input to establishing universality at the soft edge of the spectrum for fairly general measures, in a future work. Recall that the Airy kernel Ai (·, ·) of random matrix theory, is defined by  0 0  Ai (a) Ai (b) − Ai (a) Ai (b) , a 6= b, (1.1) Ai (a, b) = a−b Ai0 (a)2 − aAi (a)2 , a = b, where Ai is the Airy function, given on the real line by [1, 10.4.32, p. 447], [18, p. 3] Z 1 ∞ 1 3 Ai (x) = t + xt dt. (1.2) cos π 0 3 The Airy function Ai is an entire function of order 32 , with only real negative zeros {aj }, where 0 > a1 > a2 > a3 > · · · (1.3) and [1, 10.4.94, p. 450], [22, pp. 15–16] 2/3 1 3πj 2/3 aj = − [3π (4j − 1) /8] 1+O = − (1 + o (1)) . (1.4) j2 2 Ai satisfies the differential equation Ai00 (z) − zAi (z) = 0.

(1.5)

We note also that standard estimates for Ai show that Ai (z, ·) ∈ L2 (R) for each fixed z. Our main result is as follows: Theorem 1.1. Let g ∈ L2 (R) be the restriction to the real axis of an entire function. The following are equivalent: (I) For all z ∈ C, Z ∞

g (z) =

Ai (z, s) g (s) ds.

(1.6)

−∞

(II) All the following are true: (a) g is an entire function of order at most 23 . (b) There exists L > 0 with the following property: whenever 0 < δ < π, there exists Cδ such that for |arg (z)| ≤ π − δ, 2 3/2 L |g (z)| ≤ Cδ (1 + |z|) e− 3 z . (1.7)

Vol. 63 (2009)

Airy Kernel

429

(c) ∞ 2 X |g (aj )| 1/2

j=1

|aj |

< ∞.

(1.8)

Remarks. (a) For |arg (z)| ≤ π − δ, with |z| → ∞, [1, 10.4.59, p. 448], [18, p. 116] 2

3/2

e− 3 z Ai (z) = (1 + o (1)) , 2π 1/2 z 1/4 so we may also formulate the bound (1.7) as L+ 14

|g (z)| ≤ Cδ (1 + |z|)

|Ai (z)| , for |arg (z)| ≤ π − δ.

(1.9)

1 2,

In fact, the representation (1.6) implies that (1.7) holds with L = as shown in the proof of Lemma 2.4. (b) At first sight, the series condition (1.8) seems to follow from the other hypotheses, especially that g ∈ L2 (R). If g was of exponential type, this would be the case, but it is not obvious for entire functions of order larger than 1. At least, the series condition (1.8) is implied by a condition on the derivative of g: Theorem 1.2. If 2

0

|g 0 (x)| dx < ∞, (1.10) −∞ 1 + |x| together with the other requirements (II) (a), (b) of Theorem 1.1, then the series in (1.8) converges. Z

We can deduce an orthogonal sampling series, of the type associated with Kramer’s lemma in signal processing and approximation theory [10], [12]. Note ∞

that Ai (aj , aj ) > 0 and

√Ai(aj ,·)

Ai(aj ,aj )

is an orthonormal sequence in L2 (R), as j=1

we shall see in Lemma 2.2 and (2.9). Corollary 1.3. Under the assumptions of Theorem 1.1 (II), we have for all complex z, ∞ X Ai (aj , z) g (z) = g (aj ) . (1.11) Ai (aj , aj ) j=1 We may also deduce a quadrature formula: Corollary 1.4. Assume that both f and g satisfy the assumptions of Theorem 1.1 (II). Then Z ∞ ∞ X (f g) (aj ) (f g) (x) dx = . (1.12) Ai (aj , aj ) −∞ j=1 It is interesting that all the abscissa {aj } lie in (−∞, 0), while the integral extends over the whole real line. We note that in the Bessel kernel case, more comprehensively formulated quadrature formulae have been developed by Dryanov, Frappier, Grozev, Olivier, and Rahman [6], [8], [9], [10].

430

Levin and Lubinsky

IEOT

2. Proofs Throughout, C, C1 , C2 , . . . denote constants indpendent of n, z, x, t. The same symbol does not necessarily denote the same constant in different occurrences. We begin with: Theorem 2.1. For a, b ∈ R, Z ∞ Ai (a, s) Ai (s, b) ds = Ai (a, b) .

(2.1)

−∞

Proof. We shall use the results of Vu Kim Tuan [21] on the Airy integral transform. To avoid confusion with the Airy function, we denote this by L. Thus, for g ∈ L2 (R), we define, for each R, S > 0, Z R L [g]RS (x) = Ai (x + t) g (t) dt. −S

As R, S → ∞, this converges in L2 (R) to a function L [g], Z ∞ L [g] (x) = Ai (x + t) g (t) dt. −∞

Since Ai itself is not in L2 (R), the integral does not converge necessarily for all x. It is is known [21, p. 525, Theorem 2] that this is an isometry on L2 (R), so Z ∞ Z ∞ 2 2 L [g] (x) dx = g (x) dx. −∞

−∞ 2

2

From this, and the relation 4gh = (g + h) − (g − h) , we immediately deduce a Parseval identity for g, h ∈ L2 (R), Z ∞ Z ∞ L [g] (x) L [h] (x) dx = g (x) h (x) dx. (2.2) −∞

−∞

L is also self-inversive. Thus if g ∈ L1 (R), and (for example) g is differentiable at x, then [21, p. 524, Theorem 1] Z ∞ g (x) = Ai (x + t) L [g] (t) dt, (2.3) −∞

where the integral is taken in a Cauchy principal value sense (integrate over [−R, R] and let R → ∞). We also use a well known identity for the Airy kernel [20, p. 165, (4.5)]. Z ∞ Ai (x, y) = Ai (x + t) Ai (y + t) dt. (2.4) 0

As shown there, this identity is an easy consequence of the differential equation (1.5). Let us now define, for a given y and all t, fy (t) = Ai (y + t) χ[0,∞) (t) ,

Vol. 63 (2009)

Airy Kernel

431

where χ[0,∞) is the characteristic function of [0, ∞). We can reformulate (2.4) as Ai (x, y) = L [fy ] (x) .

(2.5)

Here fy is certainly in L1 (R) ∩ L2 (R), as [1, (10.4.59), p. 448] 2 −1/2 |Ai (x)| ≤ C (1 + |x|) exp − x3/2 , x ∈ (0, ∞) . 3 We now apply the Parseval identity (2.2): Z Z ∞ L [fy ] (t) L [fx ] (t) dt =

∞

fy (t) fx (t) dt.

−∞

−∞

Because of (2.5), this gives Z ∞ Z Ai (t, y) Ai (t, x) dt = −∞

∞

Ai (y + t) Ai (x + t) dt,

0

and (2.4) gives the result.

Our original proof of the above result started with the reproducing kernel identity for the Hermite weight, and then involved taking appropriately scaled limits. One uses the fact that Airy universality at the edge is known for the Hermite weight. This proof is longer and more computational, but uses more well known results. We remind the reader that [1, 10.4.94, p. 450], [22, pp. 15–16] 2/3 3πj 1 2/3 aj = − [3π (4j − 1) /8] 1+O =− (1 + o (1)) , (2.6) j2 2 so 1 3π −1/2 3/2 3/2 +O ; |aj | − |aj−1 | = π |aj | (1 + o (1)) . (2.7) |aj | − |aj−1 | = 2 j Moreover, [1, (10.4.96), p. 450], [22, p. 16] j−1

π −1/2 [3π (4j − 1) /8]

j−1

π −1/2 |aj |

Ai0 (aj ) = (−1) = (−1)

1/6

1/4

(1 + o (1))

(1 + o (1))

(2.8)

and then (1.1) gives, 2

Ai (aj , aj ) = Ai0 (aj ) =

1 1/2 |aj | (1 + o (1)) . π

(2.9)

In the sequel, we let Ai (aj , z) ej (z) = p , Ai (aj , aj )

j ≥ 1,

and also let H denote the Hilbert space which is the closure of the linear span of {ej : j ≥ 1}, with the usual inner product of L2 (R), where complex valued functions are permitted: Z ∞

(f, g) =

f g¯. −∞

432

Levin and Lubinsky

IEOT

∞

Lemma 2.2. (a) Let {ak }k=1 denote the zeros of Ai (z). Then Z ∞ Ai (aj , s) Ai (s, ak ) ds = δj,k Ai (aj , aj ) .

(2.10)

−∞ ∞

(b) For any complex sequence {ck }k=1 ∈ `2 , 2 Z ∞ X ∞ X ∞ 2 = |cj | . c e j j −∞ j=1 j=1

(2.11)

Proof. (a) This follows from Theorem 2.1 and from the fact (cf. (1.1)) that Ai (aj , ak ) = 0 when j 6= k. (b) From (a), the sequence {ej } is an orthonormal sequence in L2 (R). Then (2.11) is just the Parseval identity. Lemma 2.3. Let β ∈ (0, 1). For z ∈ C, let ac denote the closest zero of Ai to Re z. (a) For z ∈ C with Re z ≥ 0 or |Im z| ≥ β |Re z|, ∞ 2 X |Ai (aj , z)| j=1

Ai (aj , aj )

≤

C

2

1/2

1 + |z|

|Ai (z)| .

(2.12)

If Re z < 0, we instead obtain ∞ 2 X |Ai (aj , z)| j=1

Ai (aj , aj )

2

≤ C |Ai (z)|

|z| +

!

1 2

|z − ac |

.

(2.13)

Moreover, the series converges uniformly for z in compact subsets of C. (b) Let g ∈ H. Then for all complex z,  1  C kgk , Re z ≥ 0 or |Imz| ≥ β |Re z|, L2 (R) |Ai (z)| 1/4 (1 + |z|) |g (z)| ≤  1/2 1 C kgk + |z−a , Re z < 0. L2 (R) |Ai (z)| |z| c| (2.14) Proof. (a) Now from (1.1), Ai (aj , z) = Ai (z) Ai0 (aj ) / (z − aj ), so (2.9) gives ∞ 2 X |Ai (aj , z)| j=1

2

= |Ai (z)|

Ai (aj , aj )

∞ X j=1

1

2.

|z − aj |

Here ∞ X

1 2

j=1

|z − aj |

≤

4

X

2

j:|aj |≤

|z| 2

−1/2

≤ C |z|

|z|

2

j:

|z| 2 ≤|aj |≤2|z|

X

+ j:

1

X

+

|z| 2 ≤|aj |≤2|z|

|z − aj | 1

2,

|z − aj |

+

X j:|aj |≥2|z|

4 a2j

Vol. 63 (2009)

Airy Kernel

433

by (2.6) and some straightforward estimation. Here, if Re z ≥ 0 or |Im z| ≥ β |Re z|, we see that 1

X j:

|z| 2 ≤|aj |≤2|z|

|z − aj |

|aj − aj−1 |

X

1/2

≤ C |z|

2

2

j: 1/2

|z| 2 ≤|aj |≤2|z|

Z

0

dt

≤ C |z|

−∞

|z − aj |

−1/2

2

|z − t|

≤ C |z|

.

Instead, if x = Re z < 0, and |Im z| < β |Re z|, choose k such that x ∈ (ak , ak−1 ). Then, as above, Z X 1 dt 1/2 2 ≤ C |x| 2 |z − aj | (−∞,0]\(ak+1 ,ak−2 ) |x − t| |z| j:

2

≤|aj |≤2|z| and |j−k|≥2

1/2

≤C

|x| ≤ C |x| ≤ C |z| . |x − ak±2 |

This directly leads to (2.13). The uniform convergence follows by easy modification of the above estimates. (b) Each g ∈ H has an orthonormal expansion g=

∞ X

cj ej

j=1

with kgkL2 (R) =

X ∞

2

1/2

|cj |

.

j=1

Then the assertion follows directly from (a) and Cauchy-Schwarz.

Lemma 2.4. Let g satisfy the hypotheses (II) (a), (b), (c) of Theorem 1.1. Then g=

∞ X j=1

g (aj ) p ej . Ai (aj , aj )

(2.15)

The series converges uniformly on compact sets, and g ∈ H. Proof. First note that H=

∞ X

g (aj ) ej Ai (aj , aj )

p j=1

(2.16)

satisfies Z

∞

2

|H (s)| ds = −∞

∞ ∞ 2 2 X X |g (aj )| |g (aj )| ≤C < ∞, 1/2 Ai (aj , aj ) j=1 j=1 |aj |

(2.17)

434

Levin and Lubinsky

IEOT

in view of (2.11) and (2.9), and by our hypothesis (1.8). Moreover, H ∈ H, and the series H (z) converges to an entire function by the last lemma. If Re z ≥ 0 or |Im z| ≥ |Re z|, |H (z)| ≤ C

|Ai (z)| 1/4

(1 + |z|)

,

(2.18)

while otherwise, 1/2 |H (z)| ≤ C |Ai (z)| |z| +

1 |z − ac |

,

(2.19)

where ac is the closest zero of Ai to Re z. Since Ai is entire of order 3/2, these inequalities show that H is also of order at most 3/2. We assumed that the same is true of g. Next, let F (z) = (g (z) − H (z)) /Ai (z) . As H (aj ) = g (aj ) for each j, F is entire. Being a ratio of entire functions of order ≤ 3/2, F is also of order ≤ 3/2 [13, p. 13, Theorem 1]. Also for |arg (z)| ≥ δ > 0, we have by (1.9), (2.18), and (2.19), L+1/2

|F (z)| ≤ C (1 + |z|)

.

By an easy application of the Phragmen-Lindel¨of principle, it follows that F is a polynomial. Indeed, we can apply the Phragmen-Lindel¨of principle on sectors of L+1 , where opening a little larger than 2π 3 , [13, Theorem 1, p. 37], to F (z) / (z − a) a is outside the sector, and we may assume that L is an integer. We deduce that throughout the plane, L+1

|F (z)| ≤ C (1 + |z|)

.

By Liouville’s theorem, F is a polynomial P , so that g (z) − H (z) = P (z) Ai (z) . Let us assume that P is of degree k ≥ 0, P (z) = bz k + · · · , Here as g ∈ L2 (R) and by (2.17), Z 0

where k ≥ 0,

b 6= 0.

2

(g − H) < ∞. −∞

We now use the asymptotic [1, (10.4.60), p. 448], [18, p. 103] 2 3/2 1 −1/2 −1/4 x − π + o x−1/4 , Ai (−x) = π x cos 3 4

x→∞

Vol. 63 (2009)

to deduce Z ∞>

Airy Kernel

0

2

Z

0

P 2 Ai2

(g − H) =

−∞ 2 1 = |b| π = ∞,

435

−∞

4k/3−2/3 Z ∞ 2 3 π 2 + o (1) (1 + o (1)) t4k/3−2/3 dt cos t − 2 4 0

so b = 0, a contradiction. Thus g = H.

Proof that (II)⇒(I) in Theorem 1.1. For m ≥ 1, let gm (s) =

m X

g (aj ) ej . Ai (aj , aj )

p j=1

Let x ∈ R. In view of Theorem 2.1, Z ∞ Ai (s, x) gm (s) ds = gm (x) .

(2.20)

−∞

Moreover, as m → ∞, we have gm (x) → g (x) pointwise by the previous lemma. That lemma Ralso shows gm → g in L2 (R), in view of the orthonormal expansion ∞ 2 (2.15). Since −∞ Ai (s, x) ds < ∞, (for example, by Theorem 2.1), we can just let m → ∞ in both sides of (2.20). This gives (1.6) for real z, and analytic continuation gives it for all z. Proof that (I)⇒(II) in Theorem 1.1. The reproducing kernel property (1.6) gives Z ∞ g (aj ) p g (s) ej (s) ds = (g, ej ) . = Ai (aj , aj ) −∞ Bessel’s inequality for orthonormal expansions in L2 (R) gives Z ∞ ∞ 2 X |g (aj )| ≤ g 2 < ∞. Ai (a , a ) j j −∞ j=1 So (1.8) follows (recall (2.9)). Next, we may define the sampling series H as in (2.16), and it has the properties (2.17)–(2.19). We use this to show g = H and hence verify the growth condition (1.9). First, observe that for fixed s not a zero of Ai, the function G (z) = Ai (s, z) satisfies all the hypotheses (II) (a), (b), (c) of Theorem 1.1. Indeed, it belongs to L2 (R) by Theorem 2.1, while G (aj ) =

Ai0 (aj ) Ai (s) , s − aj

436

Levin and Lubinsky

IEOT

so (2.9) shows that ∞ 2 X |G (aj )| 1/2

|aj |

j=1

2

≤ C |Ai (s)|

∞ X

1 2

j=1

|aj − s|

< ∞.

Finally, the growth condition and order conditions are immediate. By Lemma 2.4, applied to g = G, ∞ ∞ X X Ai (aj , x) p Ai (s, x) = ej (s) = ej (x) ej (s) , Ai (aj , aj ) j=1 j=1 and hence for real x, Z ∞

∞ 2 X Ai (aj , x)

2

Ai (s, x) ds = −∞

j=1

∞ X

=

Ai (aj , aj )

e2j (x) .

j=1

Then for n ≥ 1, and real x, our hypothesis (1.6) gives   Z ∞ n n   X X g (a ) j g (x) − p ej (s) ej (x) ds ej (x) = g (s) Ai (s, x) −   Ai (a , a ) −∞ j j j=1 j=1  1/2 Z ∞ 1/2 Z ∞ n  X 2 2 ≤ g (s) ds Ai (s, x) ds − e2j (x) → 0, n → ∞.  −∞  −∞ j=1

(We used orthonormality in the second last line.) Thus g = H on the real line, and by analytic continuation throughout the plane. Since H is of order ≤ 23 and satisfies (II)(b) of Theorem 1.1 (recall (2.18) and (2.19)), we are done. Proof of Theorem 1.2. By the fundamental theorem of calculus, and some simple estimation, Z aj 2 2 |g (aj )| ≤ inf |g| + 2 |gg 0 | . [aj+1 ,aj ]

aj+1

Using (2.7), we can continue this as 2

|g (aj )|

1/2

|aj |

≤ C (aj − aj+1 ) Z

aj

≤C

inf

|g| + C

[aj+1 ,aj ]

2

Z

aj

|g| + C aj+1

aj+1

aj

Z

2

aj+1

|gg 0 | (x) 1/2

1 + |x|

|gg 0 | (x) 1/2

1 + |x|

dx

dx.

Adding over j gives Z 0 Z 0 ∞ 2 X |g (aj )| |gg 0 | (x) 2 ≤ C |g| + C dx 1/2 1/2 −∞ −∞ 1 + |x| j=1 |aj | Z

0

≤C

2

Z

0

|g| + C −∞

2

1/2

Z

0

|g| −∞

−∞

!1/2 2 |g 0 | (x) dx , 1 + |x|

Vol. 63 (2009)

Airy Kernel

437

by Cauchy-Schwarz. So (1.8) is satisfied.

Proof of Corollary 1.3. This follows from Lemma 2.4.

Proof of Corollary 1.4. This follows easily from Corollary 1.3 and the orthonormality of {ej }. Acknowledgement The authors acknowledge very helpful comments from the referee that simplified some of the proofs.

References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. [2] J. Baik, L. Li, T. Kriecherbauer, K. McLaughlin, C. Tomei, eds., Proceedings of the Conference on Integrable Systems, Random Matrices and Applications, Contemporary Mathematics, Vol. 458, American Mathematical Society, Providence, 2008. [3] E. L. Basor and H. Widom, Determinants of Airy Operators and Applications to Random Matrices, J. Statistical. Phys., 96(1999) 1–20. [4] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Institute Lecture Notes, Vol. 3, New York University Pres, New York, 1999. [5] P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Uniform Asymptotics for Polynomials Orthogonal with respect to Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory, Communications in Pure and Applied Mathematics, 52(1999), 1335–1425. [6] D. P. Dryanov, M. A. Qazi, Q. I. Rahman, Entire Functions of Exponential Type in Approximation Theory, (in) Constructive Theory of Functions, Varna 2002, (ed. B. Bojaov), DARBA, Sofia, 2003, pp. 86–135. [7] P. J. Forrester, The Spectrum Edge of Random Matrix Ensembles, Nucl. Phys. B, 402(1993), 709–728. [8] C. Frappier and P. Olivier, A Quadrature Formula Involving Zeros of Bessel Functions, Math. Comp., 60(1993), 303–316. [9] G. R. Grozev, Q. I. Rahman, A Quadrature Formula involving Zeros of Bessel Functions as nodes, Math. Comp., 64(1995), 715–725. [10] G. R. Grozev, Q. I. Rahman, Lagrange Interpolation in the Zeros of Bessel functions by Entire Functions of Exponential Type and Mean Convergence, Methods and Applications of Analysis, 3(1996), 46–79. [11] J. R. Higgins, An Interpolation Series Associated with the Bessel-Hankel Transform, J. London Math. Soc., 5(1972), 707–714. [12] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford, 1996.

438

Levin and Lubinsky

IEOT

[13] B. Ya. Levin, in collaboration with Yu. Lyubarskii, M. Sodin, V. Tkachenko, Lectures on Entire Functions, Translations of Mathematical Monographs, Vol. 150, American Mathematical Society, Providence, 1996. [14] E. Levin and D. S. Lubinsky, Universality Limits in the Bulk for Varying Measures, Advances in Mathematics, 219(2008), 743–779. [15] D. S. Lubinsky, Universality Limits in the Bulk for Arbitrary Measures on Compact Sets, J. de’Analyse de Mathematique, 106(2008), 373–394. [16] D. S. Lubinsky, Universality Limits at the Hard Edge of the Spectrum for Measures with Compact Support, International Mathematics Research Notices, 2008 (2008):rnn099–39. [17] M. L. Mehta, Random Matrices, 2nd edn., Academic Press, Boston, 1991. [18] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, San Diego, 1974. [19] B. Simon, The Christoffel-Darboux Kernel, Perspectives in PDE, Harmonic Analysis and Applications, Proceedings of Symposia in Pure Mathematics, 79(2008), 295–335. [20] C. A. Tracy and H. Widom, Level-Spacing Distributions and the Airy Kernel, Comm. Math. Phys., 159(1994), 151–174. [21] Vu Kim Tuan, Airy Integral Transform and the Paley-Wiener Theorem, (in) Transform Methods and Special Functions, Varna, Bulgarian Academy of Sciences, 1998, pp. 523–531. [22] O. Vall´ee and M. Soares, Airy Functions and Applications to Physics, World Scientific, Singapore, 2004. Eli Levin Mathematics Department The Open University of Israel P.O. Box 808 Raanana 43107 Israel e-mail: [email protected] Doron S. Lubinsky School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160 USA e-mail: [email protected] Submitted: May 15, 2008. Revised: February 2, 2009.

Integr. equ. oper. theory 63 (2009), 439–457 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030439-19, published online February 2, 2009 DOI 10.1007/s00020-009-1655-4

Integral Equations and Operator Theory

Spectral Scattering Theory for Automorphic Forms Yoichi Uetake Abstract. We construct a scattering process for L2 -automorphic forms on the quotient of the upper half plane by a cofinite discrete subgroup Γ of SL2 (R). The construction is algebraic besides being analytic in the sense that we use some relations satisfied by real-analytic Eisenstein series with a complex parameter. Thanks to this feature, the construction of our operators and spaces is explicit. We show some properties of the Lax-Phillips generator on a scattering subspace carved out from this process. We prove that the spectrum of this operator consists only of eigenvalues, which correspond to the nontrivial zeros, counted with multiplicity, of the Dirichlet series appearing in the functional equation of the Eisenstein series. In particular, in the case of the (full) modular group SL2 (Z), the Dirichlet series reduces to the Riemann zeta function ζ, thereby we obtain a spectral interpretation of the nontrivial zeros of ζ. Mathematics Subject Classification (2000). Primary 11F72; Secondary 11F03, 47A11, 47A40. Keywords. Lax-Phillips scattering axioms, real-analytic Eisenstein series, automorphic forms, Lax-Phillips generator.

1. Introduction We construct a scattering process for L2 -class (i.e. square integrable) automorphic forms on the fundamental domain M = Γ \ H of Γ. Here H is the upper half plane with the hyperbolic metric, and Γ is a cofinite discrete subgroup of SL2 (R). By being cofinite, we mean that Γ \ H is of finite area (then Γ is also called a Fuchsian group of the first kind) and non-compact. We deal with the case of Γ \ H with only one cusp. The spectrum of the Laplacian acting on the space of L2 -forms on the fundamental domain of a cofinite discrete subgroup Γ splits into discrete and continuous parts. If Γ is an arithmetic subgroup of SL2 (R) (e.g. a congruence subgroup of

440

Uetake

IEOT

SL2 (Z)), the eigenfunctions related to the discrete spectrum can be chosen to be also eigenfunctions of the Hecke operators. The Hecke eigenvalues are used to synthesize the L-function of the corresponding simultaneous eigenfunction (cusp form). Yet it was Gelfand and his school that treated the discrete and continuous spectra on the equal footing in their automorphic representation theory from the early stages. In his 1962 ICM plenary address [6] he hinted essentially that the Dirichlet series appearing in the functional equation of Eisenstein series with a complex parameter over a linear algebraic group might be interpreted as the scattering matrix. Indeed, scattering theory concerns the continuous spectrum part. This anticipation was carried out for automorphic functions by Pavlov and Faddeev [16], using Lax-Phillips scattering theory, and was refined by Lax and Phillips themselves [12]. See also Lax [11, Chapter 37 §9]. In this paper we construct a scattering process for automorphic forms, which satisfies the axioms of Lax and Phillips, based on a simple algebraic method developed in [20]. By an algebraic method we mean the use of properties of Eisenstein series, the Eisenstein transform, and explicit solutions. To be more specific we fully use the following properties of the (0, q)-type real-analytic Eisenstein series (q = 0, 1): (1)q it satisfies the functional equation; (2)q it is a (non L2 -)eigenform of the hyperbolic Laplacian; (3)q it defines the Eisenstein transform, which is considered to be a generalized Fourier transform. This method allows one to derive an explicit construction of scattering, working directly with the L2 -subspace corresponding to the continuous spectrum of the hyperbolic spinor Laplacian. We use the Eisenstein transform for (0, 1)-forms constructed in a work of Falliero [4]. In §2.1 we set up the ambient L2 -space H for scattering, which is the L2 section of the Dolbeault complex corresponding to the continuous spectrum of the spinor Laplacian ∆S . The Dolbeault complex is seen as the spinor bundle of the Spinc structure over the fundamental domain M . In §2.2 we review the definition of the (0, 1)-Eisenstein series in two variables, derive the property (2)1 , and give hints and references for deriving the properties (1)1 and (3)1 . We also mention some isometric morphisms for identifying (0, 1)type L2 -forms with L2 -functions on a sphere tangent bundle and on SL2 (R). In §2.3 we summarize the properties (1)q –(3)q (q = 0, 1) of Eisenstein series of (0, 0)- and (0, 1)-types, which we use in our construction of scattering. In §3 we construct a scattering process for L2 -forms. First we define what we call the Fourier-Eisenstein transforms E and F that are unitary operators from L2 (R) to H. The definition of these operators as well as their inverses is constructive and explicit in the sense that the information of the zeros or poles of the Dirichlet series is not used directly (Lemmas 3.2 and 3.3). Then we see that the usual Euclidean Fourier transforms of F −1 (H) and

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

441

E −1 (H) are related by the complex function S(s), which appears in the functional equation of the Eisenstein series (Lemma 3.4 (ii)). We call this function the scattering matrix (see Remark 3.7 (i)). Using E (or F) we construct a one-parameter group of unitary operators U = {U (t)}t∈R on H (Lemma 3.5 (i)). Then we express its (infinitesimal) generator iL explicitly as a kind of pseudo-differential operator, using the Eisenstein transform (Lemma 3.5 (ii)). It turns out that L is the square root of ∆S − 41 acting on H (Lemma 3.5 (iii)). In Theorem 3.6, using these operators, we define and study the incoming, outgoing and scattering subspaces Gin , Gout and K, respectively. These subspaces satisfy the axioms of Lax-Phillips scattering. The definition of all these subspaces is constructive. In §4 we define the Lax-Phillips generator F as the generator of the LaxPhillips semigroup Z = {Z(t)}t≥0 defined by Z(t) = PK U (t)|K , t ≥ 0. Here PK is the orthogonal projection of H onto K, and U (t)|K denotes the restriction of U (t) to K. In the spectral correspondence theorem (Theorem 4.1), we study the spectral properties of A = −2F . It turns out that the spectrum of A consists only of eigenvalues. Furthermore, in the case of a congruence subgroup Γ, the spectrum coincides with the nontrivial zeros of the Dirichlet L-function associated with the Eisenstein series, counted with multiplicites. Therefore the operator A gives a spectral interpretation of the zeros of the Dirichlet L-function. In particular if Γ is the (full) modular group SL2 (Z), then the Dirichlet L-function reduces to the Riemann zeta function. We underline that the construction of the operator A and the space K, on which A acts is explicit in the sense that their construction by itself is not related to the location of the nontrivial zeros of the Dirichlet series. For the operator A, see Theorem 4.1 (ii-b); A is expressed as a composition of E, the left L2 -derivative and E −1 . For the space K, see Theorem 3.6 (iv). Scattering systems are intimately related to linear dynamical systems. This relation is described in [1]. See also [19]. Our approach can be generalized to the case that Γ \ H has many cusps. For the case of one cusp, Theorem 2.2 in [20] suffices to prove Theorem 4.1. However, for the case of many cusps, one needs a more general spectral correspondence result as developed in [1]. We use the following notation: R+ = [0, ∞), R− = (−∞, 0]. I and id stand for the identity operator on a given space. The inner product in a Hilbert space X will be denoted by hx, yiX and the norm by kxkX .

2. Preliminaries 2.1. Setup for scattering Let Γ be a cofinite (see §1) discrete subgroup of SL2 (R). We also assume that Γ has the only cusp at ∞ and is reduced there; that is the stabilizer Γ∞ of ∞ is

442

Uetake

IEOT

equal to {( 10 n1 ) ; n ∈ Z}. Our underlying Riemannian manifold is the fundamental domain for Γ denoted by M : M = Γ \ H ' Γ \ SL2 (R)/SO2 (R), H = {z = x + iy ∈ C; y > 0} with the hyperbolic metric and the area element ds2 = (dx2 + dy 2 )/y 2 ,

dµ(z) = dx ∧ dy/y 2 .

Let the C0∞ -section of (0, q)-forms on M be written as Ω0,q (M ) = Γ(M, ∧q T 00∗ M ),

T 00∗ M = (T 0,1 M )∗ ,

C0∞

q = 0, 1.

∞

Here means the compactly supported C -class. We will work with the Dolbeault complex ∂¯

∂¯

∂¯

0 −→ Ω0,0 (M ) −→ Ω0,1 (M ) −→ 0. For our purpose we need to consider the L2 -completion of each section L2 Ω0,q (M ) = L2 Γ(M, ∧q T 00∗ M ) with the Petersson inner product hfq , gq iL2 Ω0,q (M )

Z 1 fq (z) ∧ ∗(gq (z))(Im(z))2q−2 = q 2 M Z = hfq (z), gq (z)i∧q Tz00∗ M (Im(z))2q dµ(z). M

Here ∗ is the Hodge ∗-operator. To be more specific, the inner products are defined for each q = 0, 1 by Z Z hf0 , g0 iL2 Ω0,0 (M ) = hf0 (z), g0 (z)i∧0 Tz00∗ M dµ(z) = f0 (z)g0 (z)dµ(z), M

M

and for f1 = ψd¯ z , g1 = φd¯ z Z hf1 , g1 iL2 Ω0,1 (M ) = M

Z =

hψ(z)d¯ z , φ(z)d¯ z i∧1 Tz00∗ M (Im(z))2 dµ(z) ψ(z)φ(z)(Im(z))2 dµ(z).

M

The hyperbolic Laplacian ∆q : L2 Ω0,q (M ) ⊃ dom(∆q ) → L2 Ω0,q (M ) is given by ∂2 ∂2 ∂ ∂2 ∂2 ∂ 2 ∆0 = −y 2 + , ∆ = −y + + 2iy + i . 1 ∂x2 ∂y 2 ∂x2 ∂y 2 ∂x ∂y The spectral decomposition with respect to σ(∆q ) = σp (∆q ) t σc (∆q ) is given by L2 Ω0,q (M ) = L2d Ω0,q (M ) ⊕ L2c Ω0,q (M ), L2d Ω0,q (M ) ⊃ ◦L2 Ω0,q (M ), where L2d Ω0,q (M ) and L2c Ω0,q (M ) are the discrete and continuous parts, respectively, and ◦ 2

L Ω0,q (M ) = closure of linear span of {ψq,j ; j ∈ N}

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

443

is the space spaned by cusp forms ψq,j . For the spectral decomposition for q = 0 and q = 1, see Motohashi [14, Theorem 1.1, p. 13–14] (or Borel [2, Chapters 12 and 17]) and Falliero [4, §2.6, p. 274], respectively. The (ambient) Hilbert space for scattering is the direct sum of the continuous spectrum parts of the L2 -sections of the Dolbeault complex, which we write as H = L2c Ω0,0 (M ) ⊕ L2c Ω0,1 (M ). The norm of f = (f0 , f1 ) = (f0 (z), ψ(z)d¯ z ) ∈ H is defined by kf k2H = kf0 k2L2c Ω0,0 (M ) + kf1 k2L2c Ω0,1 (M ) . Here the norm k · kL2c Ω0,q (M ) is inherited from that of L2 Ω0,q (M ). 2.2. Real-analytic (0, 1)-Eisenstein series and L2 -class (0, 1)-forms We briefly recall a method to identify a Γ-invariant (1, 0)-form f dz on H with a Γ-invariant function F on SL2 (R) and on the sphere (or unit) tangent bundle S(T H). This method, along with the isometric isomorphism f dz 7→ f d¯ z from (1, 0)- to (0, 1)-forms of L2 -class, allows one to deduce three properties of the realanalytic (0, 1)-Eisenstein series, which we will use in this paper. For more details, see Falliero [4] and Kubota [9, §6.1]. If γg = n(x0 )a(y 0 )k(θ0 ) for γ = ac db ∈ Γ acting on g = n(x)a(y)k(θ) in the Iwasawa decomposition of SL2 (R) ' H × SO2 (R), then one has z 0 = γz = az+b cz+d and θ0 = θ+arg(cz+d). Hence for a function F = F (z, θ) on SL2 (R), (γ ∗ F )(z, θ) = F (γz, θ + arg(cz + d)). In particular, if F (z, θ) = f (z)η, η = Im(z)e−2iθ , then one can check that (γ ∗ F )(z, θ) = f (γz)(cz + d)−2 η since Im(γz) = Im(z)|cz + d|−2 . From this we see that F = f η is a Γ-invariant function on SL2 (R) if and only if f is an automorphic function of weight 2 on H, i.e. f (γz) = (cz + d)2 f (z). Since d(γz) = (cz + d)−2 dz for γ ∈ Γ, f (z) is an automorphic function of weight 2 on H if and only if f (z)dz is a Γ-invariant (1, 0)-form on H. Therefore a Γ-invariant (1, 0)-form f (z)dz on H is identified with a Γ-invariant function F = f (z)η on SL2 (R). Now setting f (z) = [Im(z)]s−1 , let X E(g, s)−2 = E((z, θ), s)−2 := (γ ∗ F )(z, θ) γ∈Γ∞ \Γ

=

X

s −2i(θ+arg(cz+d))

[Im(γz)] e

γ∈Γ∞ \Γ

=

X

[Im(γz)]s−1 (cz + d)−2 η, Re(s) > 1.

γ∈Γ∞ \Γ

(The notation E(g, s)−2 is used in [9, §6.1, p. 64], while it is denoted E((z, η), s)−1 with η = Im(z)e−iθ in [4].) As a function of s, E(g, s)−2 admits meromorphic continuation to the whole complex s-plane. The Laplace or Casimir operator on SL2 (R) is C = −y 2 (

∂2 ∂2 ∂2 5 ∂2 − + ) − y . ∂x2 ∂y 2 ∂x∂θ 4 ∂θ2

444

Uetake

IEOT

(In [9], it is defined as −C.) Since Cγ ∗ = γ ∗ C (∀γ ∈ Γ), and F (z, θ) = f (z)η, we see that ∆0 γ ∗ F = γ ∗ ∆0 F , where ∆0 := −y 2 (

∂2 ∗ ∂θ 2 γ F

2

∂ = γ ∗ ∂θ 2 F for

∂2 ∂2 ∂2 5 ∂2 + ) − y . = C + ∂x2 ∂y 2 ∂x∂θ 4 ∂θ2

Therefore, one obtains ∆0 E(g, s)−2 = −s(s − 1)E(g, s)−2 ,

Re(s) > 1.

This holds actually for all s ∈ C by meromorphic continuation since both sides are meromorphic. Define φ, which maps functions on H to functions on SL2 (R) by (φ(f ))((z, θ)) = f (z)η,

η = Im(z)e−2iθ .

The inverse is given by (φ−1 (F ))(z) = F (z)η −1 . Define the real-analytic (0, 1)-Eisenstein series E1 (z, s) by E1 (z, s) = (φ−1 E((z, θ), s)−2 )d¯ z = η −1 E((z, θ), s)−2 d¯ z =

z s −2i arg(cz+d) d¯

X

[Im(γz)] e

y

γ= a b ∈Γ∞ \Γ c d

! ,

Re(s) > 1 .

z iθ (In [4], it is denoted E−1 (s, z) = E(z, s)−1 d¯ y , where E(z, s)−1 = E((z, η), s)−1 e .) −1 0 It turns out that φ ∆ φ = ∆1 , where ∆1 is the weight 2 hyperbolic Laplacian acting on (1, 0)- and (0, 1)-forms on H defined in §2.1. Thus E1 (z, s) satisfies

∆1 E1 (z, s) = −s(s − 1)E1 (z, s),

s ∈ C.

The (1, 0)-forms on H identified with the functions on SL2 (R) as above are also identified with the sphere tangent bundle S(T H) to H. S(T H) can be parametrized by local coordinates (z, η), η = Im(z)e−2iθ , since the metric on each fiber is induced by the hyperbolic metric on H. The action of Γ on S(T H) is defined by γ(z, η) := (γz, (cz + d)−2 η). Thus, with the correspondence H → S(T H) → SL2 (R) z = x + iy 7→ (z, η) 7→ (z, θ), an automorphic function f of weight 2 on H carries over to a Γ-invariant function φ(f ) on S(T H) and on SL2 (R). By restricting to L2 -classes on M := Γ \ H, we have isometric maps dθ dθ ) ' L2 (Γ \ SL2 (R), dµ 2π ) L2 Ω0,1 (M ) ' L2 Ω1,0 (M ) ,→ L2 (S(T M ), dµ 2π

f (z)d¯ z 7→ f (z)dz 7→ f (z)η 7→ f (z) Im(z)e−2iθ , η = Im(z)e−2iθ . Using this correspondence, Falliero [4, §2.6] deduced the Eisenstein transform for (0, 1)- [and (1, 0)-]forms of L2 -class, which we will recall in the next section.

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

445

2.3. Properties of Eisenstein series used for the construction of scattering We summarize three properties of the real-analytic Eisenstein series. The properties of the (0, 0)-Eisenstein series are taken from Motohashi [14] (or Borel [2]). See also Hejhal [7, Chap. 6 §§9, 11, 12, Chap. 7 §§1, 2] for the functional equation and the Eisenstein transform. For a unified approach to (1)q and (2)q (q = 0, 1), see Kubota [9]. In [14], the case Γ = SL2 (Z) is treated, but the facts there carry over to the case of a cofinite discrete Γ, which is covered and complemented by [2], [7] and [9]. In all of these, proper credit is given to Selberg [17] and Roelcke (1956). The properties (especially the Eisenstein transform) of the (0, 1)-Eisenstein series are proved in Falliero [4]. The term “Eisenstein transform” is introduced in Lang [10, XIV §13, p. 346]. The real-analytic (0, 0)- and (0, 1)-Eisenstein series are defined by X E0 (z, s) = [Im(γz)]s , Re(s) > 1 γ∈Γ∞ \Γ

and E1 (z, s) =

X

[Im(γz)]s e−2i arg(cz+d)

γ= a b ∈Γ∞ \Γ c d

d¯ z , y

Re(s) > 1.

The definition of the (0, 1)-Eisenstein series is recalled in §2.2. These Eisenstein series can be meromorphically continued to the whole complex s-plane with a finite number of possible poles in ( 21 , 1]. Such continuations are expressed explicitly by using some special functions. Besides the theta function (for the Riemann zeta function part), the modified Bessel function is used for E0 (z, s) ([9, §4.4], [14, §1.1]), and the Whittaker function is used for E1 (z, s) ([9, §6.1]). At least for E0 (z, s), one can use the Kloosterman sum ([14, §1.1, pp. 5–7]). The functional equation is Eq (z, 21 + s) = Cq (s)Eq (z, 12 − s),

q = 0, 1,

(1)q

where Cq (s) is a complex function meromorphic over the whole of C. (1)q (q = 0, 1) follows from [9, Theorem 6.1.1, p. 69] by comparing Eq (z, s) with E(g, s)−2q . It also follows that 1 Γ(s + 21 )2 2 −s C1 (s) = − C0 (s). 3 1 C0 (s) = 1 Γ(s + 2 )Γ(s − 2 ) 2 +s QN σ +s C0 (s) is a product of AeBs (A, B ∈ R), j=1 σjj −s (0 < σj ≤ 12 , N < ∞) and Q ρj −s j ρj +s (Re(ρj ) > 0). See [7, Chap. 6 §12, p. 166] and [17, pp. 655–6]. C0 (s) is called a Dirichlet series (in a rather general sense) in [9, p. 16] and [17, p. 655]. (1)0 is also found in [14, Lemma 1.2, p. 6], while (1)1 in [4, Theorem 2.2]. Note that s of the functional equations in this paper is shifted to 21 + s. In the case that Γ is a congruence subgroup of SL2 (Z), a factor of C0 (s) is of the form L(2s, χ)/L(−2s, χ) ¯ −1 for a Dirichlet L-function with a Dirichlet character χ associated to Γ. Other factors of C0 (s) are of the form AeBs , Γ(αs + β)±1 or a rational function of s. See Huxley [8].

446

Uetake

Set S(s) :=

Y ρj − s ρj + s j

IEOT

(Re(ρj ) > 0).

Especially in the case of a congruence subgroup Γ, let S(s) be constituted only of ρj such that 2ρj is a nontrivial zero of L(s, χ). Let ϕq (s) := S(s)Cq (s)−1 . In this paper we call S(s) the scattering matrix (see Remark 3.7 (i) in §3) and ϕq (s) the trivial calibration factor, respectively. If Γ = SL2 (Z) then S(s) = ξ(2s)/ξ(−2s), where s ˆ ξ(s) = 21 s(s − 1)π − 2 Γ 2s ζ(s) = 12 s(s − 1)ζ(s) ˆ the completed Riemann zeta function) is the Hadamard product (e.g. Patterson (ζ: [15, §3.1]), while ϕ0 (s) = −( 21 − s)/( 21 + s) and ϕ1 (s) = −1. For a congruence subgroup Γ, one can follow the analogy of this with a completed Dirichlet Lfunction. What we call the (non L2 -)eigenform property is (∆q − 14 )Eq (z, 12 + iξ) = ξ 2 Eq (z, 12 + iξ),

q = 0, 1

(2)q

for all ξ ∈ R (actually for all ξ ∈ C). See [9] for q = 0, 1. (2)0 is found in [14, Lemma 1.2]. (2)1 is derived in §2.2 (see also [4]). Let L2 (R) be the Hilbert space R ∞of square integrable C-valued2 functions of R with inner product hY1 , Y2 iL2 (R) = −∞ Y1 (ξ)Y2 (ξ)dξ for Y1 , Y2 ∈ L (R). The inner product of L2 (R+ ) below as a subspace of L2 (R) is inherited from that of L2 (R). Then the Eisenstein transform Eisq : L2c Ω0,q (M ) → L2 (R+ ),

q = 0, 1

is defined by Z 1 1 (Eisq (fq ))(ξ) = √ fq (z) ∧ ∗(Eq (z, 21 + iξ))(Im(z))2q−2 2π 2q M Z 1 =√ hfq (z), Eq (z, 12 + iξ)i∧q Tz00∗ M (Im(z))2q dµ(z) 2π M

(3)q

for a C0∞ -class fq = fq (z) ∈ L2c Ω0,q (M ) and then extending uniquely to L2c Ω0,q (M ). This transform is unitary. For q = 0 see (1.1.46) of [14, Theorem 1.1, p. 13–14]. To compare with [14], note that E0 (z, 12 + iξ) = E0 (z, 21 − iξ). See also [2, Chapter 12, p. 122]. For q = 1 see [4, §2.6]. The inverse is given by using the formula Z ∞ 1 √ wq (ξ)Eq (z, 21 + iξ)dξ (3)−1 (Eis−1 (w ))(z) = q q q 2π 0 for wq = wq (ξ) ∈ C0∞ (R+ ) (the space of compactly supported C ∞ -functions on R+ ). For q = 0 see (1.1.45) of [14, Theorem 1.1] (or [2, Proposition 12.6]). To derive (3)−1 1 from [4, §2.6] one needs to use (1)1 and Lemma 3.1 (i)–(ii) in §3. The

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

447

difference over the constants as √12π before the integrals from those in [4], [14] is merely due to the difference over the constants in the definition of the inner product of L2 (R). Let L2 (R+ ) ⊕ L2 (R+ ) be the direct sum of L2 (R+ ), whose elements we write as (w0 , w1 ) for w0 , w1 ∈ L2 (R+ ), with the norm defined by k(w0 , w1 )k2L2 (R+ )⊕L2 (R+ ) = kw0 k2L2 (R+ ) + kw1 k2L2 (R+ ) . Using (3)q (q = 0, 1) we define the operator Eis : H → L2 (R+ ) ⊕ L2 (R+ ) by Eis(f ) = (Eis0 (f0 ), Eis1 (f1 )) for f = (f0 , f1 ) ∈ H. Then Eis is also unitary.

3. Construction of scattering Lemma 3.1. Let ϕq (s) (q = 0, 1) and S(s) be as defined in §2.3. Then (i) ϕq (iξ) = ϕq (−iξ) and ϕq (iξ)ϕq (−iξ) = 1 for all ξ ∈ R. (ii) S(iξ) = S(−iξ) and S(iξ)S(−iξ) = 1 for all ξ ∈ R. ∞ (iii) Sd (z) := S( 1−z 1+z ) is an inner function in the Hardy space H (D) on the unit disc D = {z ∈ C; |z| < 1}, which consists only of the Blaschke product with zeros accumulating at most at z = −1. Proof. Cq (s) = ϕq (s)−1 S(s) in the functional equation (1)q satisfies the similar property as in (i) and (ii). See Theorem 4.4.1 and its proof in Kubota [9, pp. 43– 44]. Note that s is shifted to 21 + s. (ii) follows since the conjugate ρj also appears Q ρ −s in the product S(s) = j ρjj +s . Then (i) follows since ϕq (s) = S(s)Cq (s)−1 . (iii) can be proved as Lemma 3.1 in [20, p. 279]. We define the Fourier transformˆ: L2 (R) → L2 (R) by Z ∞ 1 yˆ(ξ) = √ e−iξτ y(τ )dτ 2π −∞ for y in C0∞ (R) or S(R) (the Schwartz space of rapidly decreasing functions on R) and then extending uniquely to L2 (R). The Fourier transform is unitary with respect to the inner product of L2 (R) defined in §2.3, and the inverse Fourier transformˇ: L2 (R) → L2 (R) is given in a similar way by using the formula Z ∞ 1 Yˇ (τ ) = √ eiτ ξ Y (ξ)dξ 2π −∞ for Y in C0∞ (R) or S(R). Note thatˆ(resp.ˇ) is different from Fc (resp. Fc−1 ) defined in [20, p. 274]. We define Eq : L2 (R) ⊃ dom(Eq ) → L2c Ω0,q (M ) (q = 0, 1) as follows: Let dom(Eq ) = {y ∈ L2 (R); ∃ > 0 such that yˆ ∈ C0∞ (R \ [−, ])}.

448

Uetake

IEOT

This is a dense subspace of L2 (R). The reason for extracting the interval [−, ] is explained in the proof of Lemma 3.2 (i). For y ∈ dom(Eq ), let Z 0 Z ∞ 1 1 yˆ(ξ)ϕ0 (iξ)E0 (z, 12 + iξ)dξ + √ yˆ(ξ)E0 (z, 12 + iξ)dξ E0 (y) = √ 2 π −∞ 2 π 0 and 1 E1 (y) = √ 2 π

Z

0

yˆ(ξ)ϕ1 (iξ)E1 (z, −∞

1 2

1 + iξ)dξ − √ 2 π

Z 0

∞

yˆ(ξ)E1 (z, 21 + iξ)dξ.

We define E : L2 (R) ⊃ dom(E) → H by E(y) = (E0 (y), E1 (y)) for y ∈ dom(E) := {y ∈ L2 (R); ∃ > 0 such that yˆ ∈ C0∞ (R \ [−, ])}. Lemma 3.2. (i) E extends uniquely to an isometry on L2 (R). (ii) For a C0∞ -class f = (f0 , f1 ) ∈ H, y ∈ L2 (R) such that E(y) = f is given by the inverse Fourier transform of yˆ = yˆ(ξ), ξ ∈ R, where Z 1 yˆ(ξ) = √ hf0 (z), E0 (z, 12 + iξ)i∧0 Tz00∗ M dµ(z) 2 π M Z 1 − √ hf1 (z), E1 (z, 12 + iξ)i∧1 Tz00∗ M (Im(z))2 dµ(z) for ξ > 0, 2 π M Z 1 yˆ(ξ) = √ ϕ0 (−iξ) hf0 (z), E0 (z, 12 + iξ)i∧0 Tz00∗ M dµ(z) 2 π M Z 1 + √ ϕ1 (−iξ) hf1 (z), E1 (z, 12 + iξ)i∧1 Tz00∗ M (Im(z))2 dµ(z) for ξ < 0. 2 π M (iii) The extended E : L2 (R) → H is unitary. We call this operator the (outgoing) Fourier-Eisenstein transform. Proof. (i): Using the functional equations (1)q of Eisenstein series, and the inverse Eisenstein transforms (3)−1 q (q = 0, 1), we have for y ∈ dom(E) Z ∞ 1 1 √ {ˆ E0 (y) = √ y (ξ) + S(−iξ)ˆ y (−ξ)}E0 (z, 21 + iξ)dξ 2π 0 2 1 √ {ˆ y (ξ) + S(−iξ)ˆ y (−ξ)} = Eis−1 0 2 and Z ∞ 1 1 √ {−ˆ E1 (y) = √ y (ξ) + S(−iξ)ˆ y (−ξ)}E1 (z, 21 + iξ)dξ 2π 0 2 1 √ {−ˆ y (ξ) + S(−iξ)ˆ y (−ξ)} . = Eis−1 1 2

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

449

Note that yˆ(ξ) and yˆ(−ξ) are viewed as functions of ξ > 0 in C0∞ (R+ \ [0, ]) ⊂ L2 (R+ ). Define wq ∈ C0∞ (R+ \ [0, ]) (q = 0, 1) by 1 y (ξ) + S(−iξ)ˆ y (−ξ)}, w0 (ξ) := √ {ˆ 2

ξ>0

and 1 w1 (ξ) := √ {−ˆ y (ξ) + S(−iξ)ˆ y (−ξ)}, 2 Then f = (f0 , f1 ) = E(y) satisfies

ξ > 0.

kf k2H = kf0 k2L2c Ω0,0 (M ) + kf1 k2L2c Ω0,1 (M ) = kw0 k2L2 (R+ ) + kw1 k2L2 (R+ ) = k(w0 , w1 )k2L2 (R+ )⊕L2 (R+ ) . One can check easily that the map on L2 (R+ ) ⊕ L2 (R+ ) defined by 1 (w0 , w1 ) 7→ √ (w0 − w1 , S(iξ)(w0 + w1 )) =: (p1 , p2 ) 2 is invertible and preserves the norm by Lemma 3.1 (ii). It turns out that (p1 (ξ), p2 (ξ)) = (ˆ y (ξ), yˆ(−ξ)),

ξ > 0.

Therefore the map defined by (p1 , p2 ) 7→ y ∈ L2 (R) is also invertible and preserves the norm. Note that limξ→0− yˆ(ξ) 6= limξ→0+ yˆ(ξ) in general unless wq is not in C0∞ (R+ \ [0, ]) for any > 0. One can extend E uniquely to an isometry on L2 (R). (ii): From the argument in the above proof of (i), 1 yˆ(ξ) = √ {(Eis0 (f0 ))(ξ) − (Eis1 (f1 ))(ξ)} for ξ > 0, 2 1 yˆ(ξ) = √ S(−iξ){(Eis0 (f0 ))(−ξ) + (Eis1 (f1 ))(−ξ)} for ξ < 0. 2 Note that by the functional equation (1)q and Lemma 3.1 (i)–(ii), S(−iξ)hfq (z), Eq (z, 21 − iξ)i∧q Tz00∗ M = hfq (z), S(iξ)Eq (z, 21 − iξ)i∧q Tz00∗ M = hfq (z), ϕq (iξ)Eq (z, 12 + iξ)i∧q Tz00∗ M = ϕq (−iξ)hfq (z), Eq (z, 21 + iξ)i∧q Tz00∗ M , from which the expressions in (ii) follow. (iii): From the argument in the proof of (i), it is also seen that the map f 7→ y in (ii) gives an isometry from a dense subspace of H to L2 (R). By taking L2 -completion, we get the claim.

450

Uetake

IEOT

As a counterpart of Eq (q = 0, 1), we define Fq : L2 (R) ⊃ dom(Fq ) → (q = 0, 1) for u ∈ dom(Fq ) by Z 0 Z ∞ 1 1 F0 (u) = √ u ˆ(ξ)E0 (z, 12 − iξ)dξ + √ u ˆ(ξ)ϕ0 (−iξ)E0 (z, 21 − iξ)dξ 2 π −∞ 2 π 0

L2c Ω0,q (M )

and 1 F1 (u) = √ 2 π

Z

0

1 u ˆ(ξ)E1 (z, 12 − iξ)dξ − √ 2 π −∞

Z

∞

0

u ˆ(ξ)ϕ1 (−iξ)E1 (z, 12 − iξ)dξ.

Here dom(Fq ) is defined to be the same as its counterpart. We define F : L2 (R) ⊃ dom(F) → H for u ∈ dom(F) by F(u) = (F0 (u), F1 (u)), where dom(F) is also defined as its counterpart. Then the following lemma can be proved in an entirely similar manner as Lemma 3.2. Lemma 3.3. (i) F extends uniquely to an isometry on L2 (R). (ii) For a C0∞ -class f = (f0 , f1 ) ∈ H, u ∈ L2 (R) such that F(u) = f is given by the inverse Fourier transform of u ˆ=u ˆ(ξ), ξ ∈ R, where Z 1 hf0 (z), E0 (z, 12 − iξ)i∧0 Tz00∗ M dµ(z) u ˆ(ξ) = √ ϕ0 (iξ) 2 π M Z 1 − √ ϕ1 (iξ) hf1 (z), E1 (z, 12 − iξ)i∧1 Tz00∗ M (Im(z))2 dµ(z) for ξ > 0, 2 π M Z 1 u ˆ(ξ) = √ hf0 (z), E0 (z, 12 − iξ)i∧0 Tz00∗ M dµ(z) 2 π M Z 1 + √ hf1 (z), E1 (z, 12 − iξ)i∧1 Tz00∗ M (Im(z))2 dµ(z) for ξ < 0. 2 π M (iii) The extended F : L2 (R) → H is unitary. We call this operator the (incoming) Fourier-Eisenstein transform. Define S : L2 (R) → L2 (R) by (S u ˆ)(ξ) = S(−iξ)ˆ u(ξ). Define S : L2 (R) → L (R) by S =ˇSˆ, i.e. y = Su ⇔ yˆ(ξ) = S(−iξ)ˆ u(ξ). Note that Su is given by the convolution Z ∞ Z 0 1 1 1 ˇ ˇ ˇ √ √ √ (S∗u)(τ ) = S(σ)u(τ −σ)dσ = S(σ)u(τ −σ)dσ, (Su)(τ ) = 2π 2π −∞ 2π −∞ where Z ∞ 1 ˇ S(σ) =√ eiσξ S(−iξ)dξ. 2π −∞ ˇ See e.g. Taylor [18, p. 213]. Since S(s) is analytic in the right half-plane, S(σ) is supported on R− . Define the shift operator T (t) (t ∈ R) on L2 (R) by T (t)y = y(· − t) for y ∈ L2 (R). 2

Lemma 3.4.

(i) T (t)S = ST (t).

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

451

(ii) The following diagram is commutative with all unitary isomorphisms: F −1

H

−−−−→

k H

−−−−→

E −1

ˆ

L2 (R) −−−−→  Sy ˆ L2 (R) −−−−→

L2 (R)  Sy 2 L (R).

Proof. Note that (T (t)y)ˆ(ξ) = e−iξt yˆ(ξ). (i) follows from ˆT (t)Sˇ = e−iξt S(−iξ) = S(−iξ)e−iξt = ˆST (t)ˇ. (ii): Using the functional equations (1)q (q = 0, 1) and Lemma 3.1 (i), one can check that E(Su) = F(u). This is what the claim says. Let Sc (M ) = S(M ) ⊗ L1/2 be the spinor bundle over the Spinc structure of M , where L1/2 is the square root of the determinant line bundle with the trivial connection (see e.g. Friedrich [5, §3.4]). The spinor Laplacian ∆S = ∇∗ ∇ on Sc (M ) is defined by ∆S f = (∆0 f0 , ∆1 f1 ) for f = (f0 , f1 ) ∈ dom(∆S ) ∈ H. Here dom(∆S ) = {f ∈ H; ∆S f ∈ H in the sense of distribution}. Lemma 3.5. (i) Define U (t) = ET (t)E −1 for t ∈ R. Then U (t) = FT (t)F −1 . U := {U (t)}t∈R is a strongly continuous one-parameter group. (ii) Let iL be the generator of U; iL = limt→0 t−1 (U (t) − I), which is a skew selfadjoint operator on H with dense domain dom(L) by (i) and Stone’s theorem (e.g. [3, Theorem 5.6]). Let W = {f ∈ H; f = Eis−1 ((w0 , w1 )), w0 , w1 ∈ C0∞ (R+ \ [0, ]) for some > 0}. Then W ⊂ dom(L), and for f = Eis−1 ((w0 , w1 )) ∈ W , iLf is given by iLf = i Eis−1 ((ξw1 , ξw0 )). Actually, L defined on W is essentially self-adjoint, with (self-adjoint) closure L defined on dom(L). (iii) We have L2 f = (∆S − 41 )f for f ∈ W . Proof. (i): By Lemma 3.4 (i) and (ii), T (t)E −1 F = E −1 FT (t), from which the first claim follows. The second claim follows from Lemma 3.2 (iii) (or Lemma 3.3 (iii)) since {T (t)}t∈R is a strongly continuous one-parameter group. (ii): Let y be the Fourier inverse image of yˆ which is obtained from w0 and w1 in C0∞ (R+ \ [0, ]) by the map (w0 , w1 ) 7→ (p0 , p1 ) as described in the proof of Lemma 3.2 (i). Then y = E −1 (f ). Note that yˆ ∈ C0∞ (R \ [−, ]). By the functional equations of Eisenstein series, we have Z ∞ 1 1 √ {e−iξt yˆ(ξ) + S(−iξ)eiξt yˆ(−ξ)}E0 (z, 21 + iξ)dξ E0 (T (t)y) = √ 2π 0 2 1 −1 −iξt iξt √ {e = Eis0 yˆ(ξ) + S(−iξ)e yˆ(−ξ)} 2

452

Uetake

IEOT

and 1 E1 (T (t)y) = √ 2π

Z

∞

1 √ {−e−iξt yˆ(ξ) + S(−iξ)eiξt yˆ(−ξ)}E1 (z, 21 + iξ)dξ 2 0 1 −1 −iξt iξt √ {−e = Eis1 yˆ(ξ) + S(−iξ)e yˆ(−ξ)} . 2

It is easy to check that d d d U (t)f |t=0 = E0 (T (t)y)|t=0 , E1 (T (t)y)|t=0 dt dt dt is equal to the expression given in the claim. It is easy to check that W is dense in H and U (t)-invariant. Thus the other claims are direct consequences of Conway [3, Theorem 5.1 (e), p. 334] and Taylor [18, Proposition 9.6, p. 520]. (iii): Using the eigenform property (2)q (q = 0, 1), we have for f ∈ W iLf =

−1 2 2 L2 f = Eis−1 ((ξ 2 w0 , ξ 2 w1 )) = (Eis−1 0 (ξ w0 ), Eis1 (ξ w1 ))

= ((∆0 − 41 )f0 , (∆1 − 41 )f1 ) = (∆S − 41 )f.

By the Lichnerowicz formula for the Dirac operator D on Sc (M ) (see e.g. [5, §3.3]), 0 ≤ D 2 = ∆S +

(−2) 4

≤ L2 = ∆S − 14 .

Note that the scalar curvature of M is −2, which is twice the Gaussian curvature −1 of M . From Lemma 3.5 (ii), we see that f (t) = U (t)g, g ∈ dom(L) is the solution to ∂ the (Dirac type) equation ∂t f = iLf with initial condition f (0) = g. By Lemma 3.5 ∂2 S 1 (iii), it also satisfies the (Klein-Gordon type) equation ∂t 2 f = −(∆ − 4 )f provided that f (0) ∈ W . Let L2 (T) be the Hilbert space of square integrable C-valued functions on the unit circle T in C endowed with the inner product defined by Z 2π ∞ X 1 fn g¯n hf, giL2 (T) = f (eiθ )g(eiθ )dθ = 2π 0 n=−∞ P∞ P∞ for f (z) = n=−∞ fn z n , g(z) = n=−∞ gn z n , z ∈ T. Here dθ is the Lebesgue arc-length measure on T. Note that {z n }∞ n=−∞ , z ∈ T is a complete orthonormal basis of L2 (T). Define the Cayley transform C : L2 (R) → L2 (T) by √ 2 π 1 − z f (ξ) 7→ f i . z+1 1+z Note that this definition is slightly different from that in [20, p. 275]. Then C is unitary, and the inverse is given by 1 + iξ 1 C −1 f (z) 7−→ √ f . π(1 − iξ) 1 − iξ

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

453

We recall that H 2 (D) ⊂ L2 (T) is the Hardy space with the complete orthonormal basis {z n }∞ n=0 . We see that C −1

H 2 (D) 3 z n 7−→ √

(1 + iξ)n for n ≥ 0 π(1 − iξ)n+1

and

|n|−1 C −1 (1 − iξ) for n < 0. H 2 (D)⊥ := L2 (T) H 2 (D) 3 z n 7−→ √ π(1 + iξ)|n| From this we see thatˇC −1 (H 2 (D)) = L2 (R− ) andˇC −1 (H 2 (D)⊥ ) = L2 (R+ ). Note that these relations are slightly different from those in [20, p. 275].

Theorem 3.6. (i) Let Gin := F(L2 (R− )) (called an incoming subspace of H) and Gout := E(L2 (R+ )) (called an outgoing subspace of H). Then Gin and Gout are closed subspaces of H and satisfy the following properties: U (t)Gin ⊂ Gin ∀t ≤ 0 and U (t)Gout ⊂ Gout ∀t ≥ 0 \ \ U (t)Gin = {0} = U (t)Gout t<0

closure of

(S-1)in,out (S-2)in,out

t>0

[

U (t)Gin = H = closure of

t>0

[

U (t)Gout .

(S-3)in,out

t<0

(ii) Gin ⊥ Gout , that is Gin and Gout are orthogonal to each other. (iii) Define another closed subspace of H by K := H (Gin ⊕ Gout ). Then K 6= {0}. We call K a scattering subspace of H. (iv) The above scattering subspace can be expressed as K = EPL2 (R− ) E −1 F(L2 (R+ )). Here PL2 (R− ) stands for the orthogonal projection of L2 (R) onto L2 (R− ). So this is just a truncation operator. Proof. (i) follows immediately from the Lemmas 3.2 (iii), 3.3 (iii) and 3.5 (i). (ii): By Lemma 3.4 (ii), E −1 (Gin ) = E −1 F(L2 (R− )) = S(L2 (R− )). S(L2 (R− )) is isometrically isomorphic to Sd (z)H 2 (D) by Cˆ. Therefore, since a nonconstant function Sd (z) is inner by Lemma 3.1 (iii), Sd (z)H 2 (D) is a proper subspace of H 2 (D). Note that Cˆ(E −1 (Gout )) = Cˆ(L2 (R+ )) = H 2 (D)⊥ . Since E −1 is unitary by Lemma 3.2 (iii) as well as C andˆ, the claim follows. (iii): From the above argument, we also see that Cˆ(E −1 (K)) = H 2 (D) Sd (z)H 2 (D) 6= {0}. (iv): From the proof of (iii), we have E −1 (K) = L2 (R− ) S(L2 (R− )) ⊂ L2 (R− ).

454

Uetake

IEOT

Hence we have E −1 (K) ⊕ S(L2 (R− )) ⊕ L2 (R+ ) = L2 (R). Since S is unitary by Lemma 3.4 (ii), S −1 E −1 (K) ⊕ L2 (R− ) ⊕ S −1 (L2 (R+ )) = S −1 (L2 (R)) = L2 (R), from which we see that S −1 E −1 (K) ⊕ S −1 (L2 (R+ )) = L2 (R+ ). Therefore E −1 (K) ⊕ L2 (R+ ) = S(L2 (R+ )), which proves E −1 (K) = PL2 (R− ) S(L2 (R+ )) since E −1 (K) ⊂ L2 (R− ). The claim follows from this by Lemma 3.4 (ii).

Remark 3.7. (i) A collection Σ = (U, H, Gin , Gout ), where U = {U (t)}t∈R is a one-parameter group of unitary operators on a complex Hilbert space H satisfying the conditions in Theorem 3.6 (i) is called a Lax-Phillips scattering system. The unitary operators E −1 and F −1 are called outgoing and incoming translation representations of Σ, respectively. Then the commutative diagram in Lemma 3.4 (ii) gives the definitions of the scattering matrix S and the scattering operator S. For details, see [11], [12], [13]. (ii) For a compact set N of M , let HN be the subspace of H whose elements are supported on M \ N and JN = H \ HN . Since T (t1 )L2 (R+ ) ⊂ T (t2 )L2 (R+ ) for t1 ≥ t2 ≥ 0, we have JN ∩ U (t1 )Gout ⊂ JN ∩ U (t2 )Gout for t1 ≥ t2 ≥ 0. Furthermore from (S-2)out , ∩t>0 [JN ∩ U (t)Gout ] = {0}. Thus we can see that all waves in Gout fade away off N toward the cusp at ∞ as t → ∞. Similarly all waves in Gin fade away off N toward the cusp at ∞ as t → −∞.

4. The Lax-Phillips generator and the spectral correspondence theorem Since U (−t)∗ = U (t) and U (−t)Gin ⊂ Gin (t ≥ 0) by (S-1)in of Theorem 3.6 (i), we have U (t)(K ⊕ Gout ) ⊂ K ⊕ Gout (t ≥ 0). Thus U (t)K ⊂ K ⊕ Gout (t ≥ 0). Hence given k1 ∈ K, we have PK U (t2 + t1 )k1 = PK U (t2 )U (t1 )k1 = PK U (t2 )(k2 + gout ) (t1 , t2 ≥ 0) for some k2 ∈ K and gout ∈ Gout such that U (t1 )k1 = k2 + gout . Here PK is the orthogonal projection of H onto K. However, since U (t2 )Gout ⊂ Gout (t2 ≥ 0) by (S-1)out of Theorem 3.6 (i), we have PK U (t2 )(k2 + gout ) = PK U (t2 )PK (k2 + gout ) = PK U (t2 )PK U (t1 )k1 . Therefore what are called compressions defined by Z(t) := PK U (t)|K ,

t ≥ 0,

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

455

constitute a semigroup Z = {Z(t)}t≥0 of contractions on K. Here U (t)|K denotes the restriction of U (t) to K. The original definition of Z by Lax and Phillips for the case of U (t)PG⊥ [12], [13] is given by Z(t) = PK U (t)PK or Z(t) = PG⊥ out in Gin ⊥ Gout . Z is called the Lax-Phillips semigroup. See Lax [11, p. 499]. Let F be the generator of Z; F k = limt→0+ t−1 (Z(t)−I)k for k ∈ dom(F ) := {k ∈ K; F k exists in K}. We call F the Lax-Phillips (infinitesimal) generator ([19]). Theorem 4.1. Let A = −2F : K ⊃ dom(A) → K. Here dom(A) = dom(F ). The unbounded operator A has the following properties. (i) A is a closed operator with domain dom(A) dense in K. (ii-a) For k ∈ K, Z(t)k = EPL2 (R− ) T (t)E −1 k, t ≥ 0. (ii-b) For k ∈ dom(A), Ak = 2E(d` /dτ )E −1 k, where d` /dτ is the left L2 -derivative. (iii-a) The resolvent (s − A)−1 of A is meromorphic in C. (iii-b) Any s0 ∈ σ(A) is an eigenvalue of finite Riesz index, which is equal to the order of the pole s0 of the resolvent of A. By the Riesz index of s0 , we mean the smallest positive integer n such that ker (s0 −A)n = the generalized eigenspace for s0 . (iii-c) Moreover, the Riesz index of s0 = the algebraic multiplicity (:= the dimension of the generalized eigenspace) of s0 . (iv) s0 /2 is a zero of multiplicity m0 of S(s) appearing in the functional equation of the corresponding Eisenstein series if and only if s0 is an eigenvalue of algebraic multiplicity m0 of A. Thus if Γ is a congruence subgroup of SL2 (Z), then s0 is a nontrivial zero of multiplicity m0 of the Dirichlet L-function L(s, χ) associated with the functional equation of the Eisenstein series if and only if s0 is an eigenvalue of algebraic multiplicity m0 of A. In particular if Γ = SL2 (Z), the Dirichlet L-function in the claim coincides with the Riemann zeta function ζ(s). (v) The set of the generalized eigenvectors corresponding to the eigenvalues of the operator A constitutes a complete (not necessarily orthogonal) basis of K. Proof. We give the proof for the analogous claims for the Lax-Phillips generator F except for (ii-a). The claims in the theorem for A follow immediately from this since A = −2F . (i): By Lemma 3.5 (i), U (t), t ≥ 0 constitutes a strongly continuous one-parameter semigroup of contractions (actually kU (t)k = 1). Thus Z is also a strongly continuous one-parameter semigroup of contractions. The generator of a strongly continuous one-parameter semigroup is a closed operator with dense domain (e.g. [13, Appendix 1, Lemma 4]). See also the proof of Theorem 3.5 (2-i) of [20]. (ii): In the light of Lemma 3.5 (i), it suffices to consider in L2 (R), since one can then carry over to H by the unitary operator E. By Lemma 3.4 (ii), E −1 F = S.

456

Uetake

IEOT

By Theorem 3.6 (iv), if y ∈ E −1 (K) then y = PL2 (R− ) Su for some u ∈ L2 (R+ ). Now we have for t ≥ 0 PL2 (R− ) T (t)y = PL2 (R− ) T (t)PL2 (R− ) Su = PL2 (R− ) T (t)Su = PL2 (R− ) ST (t)u ∈ E −1 (K), the last equality being due to Lemma 3.4 (i). Hence, since E −1 (K) ⊂ L2 (R− ), we have PE −1 (K) T (t)y = PL2 (R− ) T (t)y. Here PE −1 (K) is the orthogonal projection from L2 (R) onto E −1 (K). Therefore for k = E(y) ∈ K Z(t)k = PK U (t)k = PK EE −1 U (t)E(y) = PK ET (t)y = EPE −1 (K) T (t)y = EPL2 (R− ) T (t)y, t ≥ 0. Thus we have (ii-a). From this F = EPL2 (R− ) (−d` /dτ )E −1 . However, since d` /dτ is the left L2 -derivative, we have PL2 (R− ) (d` /dτ )y = (d` /dτ )y for y ∈ dom(d` /dτ ) ∩ L2 (R− ). Thus we have (ii-b). (iii-a), (iii-b), (iv), (v): Recall that E −1 (Gout ) = L2 (R+ ) (Theorem 3.6 (i)), E −1 (Gin ) = E −1 F(L2 (R− )) = S(L2 (R− )) (Proof of Theorem 3.6 (ii)) and E −1 (K) = L2 (R− ) S(L2 (R− )) (Proof of Theorem 3.6 (iv)). Thus Σ = (U, H, Gin , Gout ) is unitarily equivalent (see Remark 3.6 (iii) of [20]) to the scattering model Σ1 constructed in the proof of Lemma 2.1 of [20]. To see this, set S1 = S, S2 = id in Lemma 2.1 [20]. Then by Lemma 3.4 (i) and the proof of Theorem 3.6 (iv), the properties (a)–(d) in there are all satisfied. By Lemma 3.1 (iii), one can apply Theorem 2.2 of [20]. Thus −ρj is a pole of Q ρ −s multiplicity mj of S(s) = j ρjj +s if and only if −ρj is an eigenvalue of Riesz index (and algebraic multiplicity by (iii-c) shown below) mj of F . Since A = −2F , we have (iii-a), (iii-b), (iv) and (v). (iii-c) follows since (1 − z0 z)−ν (1 ≤ ν ≤ m0 ) used in the proof of Theorem 2.2 (ii) in [20] are linearly independent (use the residue theorem). Acknowledgment The author would like to thank the referee for useful suggestions to improve the paper.

References [1] J. A. Ball, P. T. Carroll and Y. Uetake, Lax-Phillips Scattering Theory and WellPosed Linear Systems: A Coordinate-Free Approach. Mathematics of Control, Signals, and Systems 20 (2008), 37–79. [2] A. Borel, Automorphic Forms on SL2 (R). Cambridge Tracts in Math. 130, Cambridge University Press, Cambridge, 1997. [3] J. B. Conway, A Course in Functional Analysis. GTM 96, Springer-Verlag, New York, 1985.

Vol. 63 (2009)

Scattering Theory for Automorphic Forms

457

[4] T. Falliero, D´ecomposition Spectrale de 1-Formes Diff´erentielles sur une Surface de Riemann et S´eries d’Eisenstein. Math. Ann. 317 (2000), 263–284. [5] Th. Friedrich, Dirac Operators in Riemannian Geometry. Graduate Studies in Mathematics 25, AMS, Providence, 2000. [6] I. M. Gelfand, Automorphic Functions and the Theory of Representations. Proc. of the ICM, (Stockholm) (1962), 74–85. [7] D. A. Hejhal, The Selberg Trace Formula for PSL(2, R), Vol. 2. Lecture Notes in Mathematics 1001, Springer-Verlag, Berlin, 1983. [8] M. N. Huxley, Scattering Matrices for Congruence Subgroups. in: Modular Forms (Durham, 1983), R. A. Rankin (Ed.), Ellis Horwood Ltd, Chichester – Halsted Press [John Wiley & Sons], New York, 1984. pp. 141–156. [9] T. Kubota, Elementary Theory of Eisenstein Series. Kodansha Ltd., Tokyo – Halsted Press [John Wiley & Sons], New York, 1973. [10] S. Lang, SL2 (R). GTM 105, Springer-Verlag, New York, 1985. (Reprint of 1975 ed.) [11] P. D. Lax, Functional Analysis. John Wiley & Sons, New York, 2002. [12] P. D. Lax and R. S. Phillips, Scattering Theory for Automorphic Functions. Ann. of Math. Studies, No. 87, Princeton University Press, Princeton, New Jersey, 1976. [13] —, Scattering Theory. Revised Edition, Academic Press, New York, 1989. (1st ed. 1967) [14] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function. Cambridge Tracts in Math. 127, Cambridge University Press, Cambridge, 1997. [15] S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function. Cambridge studies in advanced mathematics 14, Cambridge University Press, Cambridge, 1988. [16] B. S. Pavlov and L. D. Faddeev, Scattering Theory and Automorphic Functions. Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 161–193. (Russian); English transl.: J. Soviet Math. 3 (1975), 522–548. [17] A. Selberg, Harmonic Analysis (G¨ ottingen Lecture Notes, 1954). in: A. Selberg Collected Papers. Vol. I, Springer-Verlag, Berlin, 1989. pp. 626–674. [18] M. E. Taylor, Partial Differential Equations — Basic Theory. Texts in Applied Mathematics 23, Springer-Verlag, New York, 1996. [19] Y. Uetake, The Lax-Phillips Infinitesimal Generator and the Scattering Matrix for Automorphic Functions. Ann. Polon. Math. 92 (2007), 99–122. [20] —, Lax-Phillips Scattering for Automorphic Functions Based on the Eisenstein Transform. Integral Equations Operator Theory 60 (2008), 271–288. Yoichi Uetake Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Pozna´ n, Poland e-mail: [email protected] Submitted: May 16, 2008. Revised: November 4, 2008.

Integr. equ. oper. theory 63 (2009), 459–472 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040459-14, published online February 24, 2009 DOI 10.1007/s00020-009-1667-0

Integral Equations and Operator Theory

Compact AC(σ) Operators Brenden Ashton and Ian Doust Abstract. All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts. Mathematics Subject Classification (2000). 47B40. Keywords. Functions of bounded variation, absolutely continuous functions, functional calculus, well-bounded operators, AC-operators.

1. Introduction The class of well-bounded operators was introduced to provide a theory which would allow many of the results which apply to self-adjoint operators to be extended to the Banach space setting. Since many operators which are self-adjoint on L2 have only conditionally convergent spectral expansions on the other Lp spaces, the theory needed to allow more general types of representation theorems than those available in the theory of spectral operators. The issue of the conditional convergence of spectral expansions arises most explicitly when considering compact well-bounded operators. In [6] it was shown that every compact well-bounded operator T has an expansion as ∞ X T = µj Pj (1.1) j=1

where Pj is the Riesz projection onto the eigenspace corresponding to the eigenvalue µj , and where the terms are ordered so that |µj | is decreasing. Indeed, necessary and sufficient conditions were given which ensure that any sum of the form appearing in (1.1) is a compact well-bounded operator. Even in the earliest papers (see, for example, [14]) the fact that the spectrum of a well-bounded operator is necessarily real was seen as an undesirable restriction, and various attempts at addressing this have appeared. In [5] Berkson and Gillespie

460

Ashton and Doust

IEOT

introduced the concept of an AC operator, which is one which can be written in the form T = A + iB where A and B are commuting well-bounded operators. Doust and Walden [10] showed that, as long as one takes the eigenvalues in an appropriate order, every compact AC operator has a representation in the form given in (1.1). The theory of AC operators had certain drawbacks however (see [4]) and a smaller class of operators, known as AC(σ) operators, was introduced in [1] and [3]. An advantage of the theory of AC(σ) operators is that it allows one to work with an algebra of functions defined on the spectrum of the operator, or at worst a small neighbourhood of the spectrum, rather than some rectangle in the plane. The results of [10] clearly also apply to compact AC(σ) operators, but even in the case of AC operators, there has not been a characterization of the compact operators in terms of properties of the eigenvalues and the corresponding projections. One of the main applications of the characterization result in [6] has been to enable the construction of well-bounded operators with specific properties. See, for example, [6, 8, 9]. The main aim of this paper then, is to prove Theorem 5.1 which gives sufficient conditions to ensure that an operator of the form (1.1) is a compact AC(σ) operator. To prove this theorem one has to show that (under the hypotheses of the theorem) one may sensibly define f (T ) for f ∈ AC(σ), with a norm bound kf (T )k ≤ K kf kBV (σ) . A significant challenge in working with AC(σ) operators is being able to calculate kf kBV (σ) for f ∈ AC(σ). In Section 4 we shall show that for certain sets σ, the AC(σ) norm is equivalent to a norm which is much easier to calculate. Although we will not need the full force of this result to prove Theorem 5.1, we feel that this equivalence is of independent interest. In Section 6 we show that there are compact AC(σ) operators which are not of the form constructed in Theorem 5.1. The final section includes a discussion of the properties of the splitting of an AC(σ) operator T into real and imaginary parts T = A + iB. There are many open questions regarding these splittings. In the case that T is compact however, it is possible to resolve these questions. In obtaining these results we prove a new result about rearrangements of the sum representation of a compact well-bounded operator (Corollary 7.2) which may also be of independent interest.

2. Preliminaries Throughout this paper let σ ⊂ C be compact and non-empty. We shall denote the bounded linear operators on a Banach space X by B(X). Summations over empty sets of indices should always be interpreted as having value zero. The Banach algebra of functions of bounded variation on σ, denoted BV (σ), was defined in [1]. The norm in this space is given by an expression of the form kf kBV (σ) = kf k∞ + var(f, σ) = kf k∞ + sup cvar(f, γ) ρ(γ). γ

(2.1)

Vol. 63 (2009)

Compact AC(σ) Operators

461

In (2.1) the supremum is taken over all piecewise linear curves γ : [0, 1] → C in the plane. The term cvar(f, γ) measures the variation of f as one travels along the curve γ, and ρ(γ)−1 measures how ‘sinuous’ the curve γ is. More precisely, the variation factor vf(γ) ≡ ρ(γ)−1 is defined as the maximum number of entry points of the curve γ on any line in the plane. (Heuristically one should think of this as the maximum number of times any line intersects γ.) We refer the reader to [1] for the full definitions. The affine invariance of the BV norm will be used repeatedly (with little comment) to pass between estimates for functions on R to those for functions defined on other lines. The closure of the polynomials in z and z is the subalgebra AC(σ) of absolutely continuous functions on σ. An operator T ∈ B(X) which admits an AC(σ) functional calculus is said to be an AC(σ) operator. (Note that we will often say that T is an AC(σ) operator, when we should more accurately say that T is an AC(σ) operator for some σ.) The class of AC(σ) operators includes all wellbounded operators. Indeed the well-bounded operators are precisely the AC(σ) operators with real spectrum. For this reason we prefer the more descriptive term real AC(σ) operator rather than well-bounded operator. The class of AC(σ) operators also includes all scalar-type spectral operators, and in particular, all normal operators on any Hilbert space. Theorem 2.1. Suppose that T ∈ B(X) is a compact AC(σ) operator. Then 1. T is an AC operator (in the sense of Berkson and Gillespie [5]); 2. there exist unique commuting well-bounded operators A, B ∈ B(X) such that T = A + iB; 3. A and B are compact. Proof. Statement (1) is Theorem 5.3 of [1]. Statements (2) and (3) therefore follow from [10, Theorem 6.1]. For a complex number µ = x + iy with x, y ∈ R, let |µ|∞= max{|x|, |y|}. We shall now define an order ≺ on C by setting µ1 ≺ µ2 if (i) |µ1 |∞ > |µ2 |∞ , or, (ii) if |µ1 |∞= |µ2 |∞= α and Arg µ1 > Arg µ2 . Theorem 2.1 has as an immediate corollary that compact AC(σ) operators have a spectral diagonalization analogous to that for compact normal operators, but where the sum in the representation might only converge conditionally. Corollary 2.2. [10, Theorem 4.5] Suppose that T is a compact AC(σ) operator with spectrum {0} ∪ {µj }∞ j=1 and that {µj } is ordered by ≺. Then there exists a uniformly bounded sequence of disjoint projections Pj ∈ B(X) such that T =

∞ X

µj Pj ,

j=1

where the sum converges in the norm topology of B(X).

462

Ashton and Doust

IEOT

This includes, for example, the fact that the range of the Riesz projection associated with a nonzero eigenvalue µ is exactly the corresponding eigenspace. We refer the reader to [10] for a fuller discussion of properties of compact AC operators.

3. Approximation in AC(σ) An important step in proving Corollary 2.2 is showing that the identity function λ(z) = z can be approximated in BV norm by functions whose support intersects σ(T ) at only a finite number of points. It follows from the results in [1] and [2] that this is still true if one uses the BV (σ) norm. We provide here a more direct proof, the results of which we will use later. Let σ be a nonempty compact subset of C. Given r > 0 and > 0, define gr, : σ → C by   if |z|∞ ≤ r, 0, gr, (z) = (|z|∞ − r)/, if r < |z|∞ ≤ r + ,   1, if |z|∞ > r + . Lemma 3.1. For all r > 0 and > 0, gr, ∈ AC(σ) and kgr, kBV (σ) ≤ 6. Proof. Let σR = Re(σ) = {x : x + iy ∈ σ} and let σI = Im(σ) = {y : x + iy ∈ σ}. Define u : σR → C and v : σI → C by u(x) = gr, (x) and v(y) = gr, (iy). Clearly u ∈ AC(σR ) with kukBV (σR ) ≤ 3 (and similarly for v). Let u ˆ(x + iy) = u(x), and vˆ(x + iy) = v(y). By [1, Proposition 4.4], u ˆ, vˆ ∈ AC(σ) with kˆ ukBV (σ) ≤ 3 and kˆ v kBV (σ) ≤ 3. Now it is easy to check that gr, = u ˆ ∧ vˆ, and hence, by [3, Proposition 2.10], gr, ∈ AC(σ) and kgr, kBV (σ) ≤ 6. Let x(z) = Re(z) and let y(z) = Im(z), so that λ = x + iy. Lemma 3.2. Suppose that {rn } and {n } are sequences of positive numbers which converge to 0. For n = 1, 2, . . . , let gn = grn ,n , let xn = gn x and let y n = gn y. Then xn → x and y n → y in AC(σ). Consequently gn λ → λ in AC(σ). Proof. It suffices to show that xn → x. Now xn − x = (1 − gn ) x = (1 − gn ) x ˜n where ( Re(z), if |Re(z)| ≤ rn + n , x ˜n (z) = 0 if |Re(z)| > rn + n . Now k˜ xn kBV (σ) ≤ 5(rn + n ) and so kxn − xkBV (σ) ≤ k1 − gn kBV (σ) k˜ xn kBV (σ) ≤ 30(rn + n ) → 0 as n → ∞.

Remark 3.3. It is clear that one could replace gn in the above proof with many other families of ‘cut-off’ functions. In the proof of [10, Theorem 4.5], for example, the cut-off functions are based on L-shaped regions rather than squares.

Vol. 63 (2009)

Compact AC(σ) Operators

463

4. Norm estimates in AC(σ) In order to show that an operator T admits an AC(σ) functional calculus, one often needs to find estimates for both kf (T )k and kf kBV (σ) . This can be difficult, even for quite simple functions. If σ lies inside the union of a finite number of lines through the origin, then we shall show that it is possible to decompose f ∈ AC(σ) into a sum of simpler functions in a way that allows good estimation of the norms. The main issue is the following. Suppose that suppf ⊆ σ0 ⊆ σ for some compact set σ0 . One always has that kf |σ0 kBV (σ0 ) ≤ kf kBV (σ) . The challenge is to prove an estimate of the form kf kBV (σ) ≤ C kf |σ0 kBV (σ0 ) . Even if σ ⊆ R such an estimate need not exist, so any results need to rely on geometric properties of σ0 and σ. We begin with a technical lemma. As in [1], given points z1 , z2 , . . . , zk ∈ C, let Π(z1 , z2 , . . . , zk ) denote the piecewise linear path with these points as vertices. Lemma 4.1. Suppose that k ≥ 2 and that S = z1 , z2 , . . . , zk is a list of complex numbers such that no two consecutive numbers lie in the complement of the real axis. Let J1 = {j : zj , zj+1 ∈ R}, J2 = {j : zj ∈ R, zj+1 6∈ R}, J3 = {j : zj 6∈ R, zj+1 ∈ R} have cardinalities k1 , k2 and k3 respectively. Let γS = Π(z1 , z2 , . . . , zk ). Then (k2 + k3 )ρ(γS ) ≤ 2. Proof. The conditions on S imply that |k2 − k3 | ≤ 1. The bound claimed obviously holds if k2 = k3 = 0, so we shall assume that at least one of these values is nonzero. Suppose first that k2 ≤ k3 . If j ∈ J3 , then zj+1 is an entry point of γS on the real axis. Thus ρ(γS ) ≤ 1/k3 and so (k2 + k3 )ρ(γS ) ≤ 2k3 /k3 = 2. If, on the other hand, k2 = k3 + 1, then the smallest element of J2 is less than the smallest element of J3 . Thus, in addition to the entry points associated with the elements of J3 (as in the previous paragraph), γS must have an earlier entry point on the real line. Thus ρ(γS ) ≤ 1/(k3 + 1), and so (k2 + k3 )ρ(S) ≤ (2k3 + 1)/(k3 + 1) ≤ 2. In general, if σ0 ⊆ σ, then kf |σ0 kBV (σ0 ) ≤ kf kBV (σ) , but no reverse inequality is available, even if suppf ⊆ σ0 . Such an inequality does hold however if σ0 is a line inside σ. Suppose then that σ is a compact subset of C and that σ0 = σ ∩ R 6= ∅. Lemma 4.2. Suppose that f ∈ BV (σ) and that suppf ⊆ σ0 . Then n o kf kBV (σ) ≤ 3 kf k∞ + var(f, σ0 ) = 3 kf |σ0 kBV (σ0 ) .

464

Ashton and Doust

IEOT

Proof. For an ordered finite subset S = {z1 , . . . , zk } of σ (allowing repetitions), let γS = Π(z1 , . . . , zk ). Lemma 3.5 of [1] shows that var(f, σ) = sup cvar(f, γS )ρ(γS ) S

where the supremum is taken over all such finite subsets. Indeed, by adding extra points as necessary, one sees that k−1 X var(f, σ) = sup ρ(γS ) |f (zj ) − f (zj+1 )| . S

j=1

Pk−1

Fix such a subset S, and let v(f, S) = j=1 |f (zj ) − f (zj+1 )|. Clearly v(f, S) is unchanged if we omit any consecutive elements of S which are both in σ \ σ0 . Note that omitting points never decreases the value of ρ(γS ), so we shall assume that no two consecutive elements of S are both in σ \ σ0 . We may also assume that S ∩ R 6= ∅ (or else v(f, S) = 0). Partition J = {1, 2, . . . , k − 1} into sets J1 , J2 , J3 as in Lemma 4.1. Clearly then k−1 3 X X X |f (zj ) − f (zj+1 )| = |f (zj ) − f (zj+1 )|. j=1

i=1 j∈Ji

Let S0 = {w1 , . . . , wk0 } be the sublist of S containing the elements that lie on the real axis, and let γS0 denote the corresponding piecewise linear curve. As noted above, ρ(γS ) ≤ ρ(γS0 ). Thus, using [1, Proposition 3.6], ρ(γS )

X

|f (zj ) − f (zj+1 )| ≤ ρ(γS0 )

j∈J1

kX 0 −1

|f (wj ) − f (wj+1 )| ≤ var(f, σ0 ).

j=1

On the other hand, if j ∈ J2 ∪ J3 , then |f (zj ) − f (zj+1 )| ≤ kf k∞ and so, by Lemma 4.1, X ρ(γS ) |f (zj ) − f (zj+1 )| ≤ (k2 + k3 ) kf k∞ ρ(γS ) ≤ 2 kf k∞ . j∈J2 ∪J3

Thus var(f, σ) ≤ var(f, σ0 ) + 2 kf k∞ and hence kf kBV (σ) ≤ 3 kf k∞ + var(f, σ0 ).

Note that the factor 3 in the above inequality is sharp. If σ = {i, 0, −i} and f = χ{0} then kf kBV (σ) = 3, and var(f, {0}) = 0. For 0 ≤ θ < 2π, let Rθ denote the ray {r cos θ, r sin θ) : r ≥ 0}. We shall say that σ ⊆ C is a spoke set if it is a subset of a finite union of such rays. Suppose then that σ is a nonempty compact spoke set with σ ⊆ ∪N n=1 Rθn . (We shall always assume that the angles θn are distinct.) For n = 1, . . . , N , let

Vol. 63 (2009)

Compact AC(σ) Operators

465

σn = σ ∩ Rθn . Given f ∈ BV (σ), we shall define f0 , f1 , . . . , fN ∈ BV (σ) by setting f0 (z) = f (0) and, for 1 ≤ n ≤ N , ( f (z) − f (0), if z ∈ Rθn , fn (z) = 0, otherwise. PN Then f = n=0 fn . Also, if f ∈ AC(σ), then a short limiting argument can be used to show that each fn is also in AC(σ). Define the spoke norm ||| f |||Sp = |f (0)| +

N X

kfn |σn kBV (σn ) .

n=1

Since σn lies in a line, the affine invariance of these norms and [1, Proposition 3.6] means that calculating kfn |σn kBV (σn ) is relatively easy, since this just requires an estimation of the usual variation of f along the line. Thus ||| f |||Sp is much easier to calculate than kf kBV (σ) . The following result shows that ||| · |||Sp and k·kBV (σ) are equivalent. Although we do not need the full strength of this result in the next section, it does provide a useful tool in working with sets of this sort. Proposition 4.3. Suppose that σ is a nonempty compact spoke set. Then for all f ∈ BV (σ) 1 ||| f |||Sp ≤ kf kBV (σ) ≤ 3 ||| f |||Sp . 2N + 1 Proof. Suppose that 1 ≤ n ≤ N . Then, with notation as above, kfn |σn kBV (σn ) = sup |f − f0 | + var(f − f0 , σn ) σn

≤ 2 kf k∞ + var(f, σn ) ≤ 2 kf kBV (σ) . The left hand inequality then follows from the triangle inequality. On the other hand, Lemma 4.2 (and affine invariance) shows that kf kBV (σ) ≤ |f (0)| +

n X n=1

kfn kBV (σ) ≤ |f (0)| + 3

n X

kfn |σn kBV (σn ) ≤ 3 ||| f |||Sp .

n=1

Let bv0 denote the Banach space of complex sequences of bounded variation and limit 0. The following lemma is elementary. Lemma 4.4. Suppose that {Qj }∞ j=0 is a uniformly bounded increasing family of projections on X. That is, Qi Qj = Qj Qi = Qi whenever i ≤ j and supj kQj k ≤ K. Suppose that {µj }∞ j=1 ∈ bv0 . Then Pm Pm−1 1. k j=n µj (Qj − Qj−1 ) k ≤ K |µn | + |µm | + j=n |µj − µj+1 | . P∞ 2. j=1 µj (Qj − Qj−1 ) converges in norm.

466

Ashton and Doust

IEOT

5. Constructing compact AC(σ) operators In [7], Cheng P and Doust showed that certain combinations of disjoint projections of the form λj Ej always converge and define compact well-bounded operators. In this section we shall provide some sufficient conditions for an operator of this form to be a compact AC(σ) operator. Theorem 5.1 below will allow the construction of compact AC(σ) operators which are neither scalar-type spectral, nor well-bounded operators, via a given conditional decomposition of the Banach space X. Suppose that N ≥ 1 and that θ1 , . . . , θN are distinct angles. Given • scalars {λn,m : n = 1, . . . , N, m = 1, 2, . . . } ⊂ C, and • projections {En,m : n = 1, . . . , N, m = 1, 2, . . . } ⊂ B(X) consider the following three conditions: (H1) For each n = 1, . . . , N , {λn,m }∞ m=1 ⊂ Rθn . (H2) For each n = 1, . . . , N , |λn,1 | > |λn,2 | > |λn,3 | > . . . , and λn,m → 0 as m → ∞. (H3) The operators En,m are pairwise disjoint, finite rank projections and there exists a constant K such that, for each n = 1, . . . , N , and M = 1, 2, . . . , M

X

En,m ≤ K.

m=1

The set of indices I = {(n, m) : n = 1, . . . , N, m = 1, 2, . . . } can be ordered by declaring that (n, m) (s, t) if |λn,m | < |λs,t |, or if |λn,m | = |λs,t | and θn > θs . Let σ = {0} ∪ {λn,m }(n,m)∈I , so that σ is a spoke set. Theorem 5.1. Suppose that the scalars {λn,m : n = 1, . . . , N, m = 1, 2, . . . } ⊂ C and projections {En,m : n = 1, . . . , N, m = 1, 2, . . . } ⊂ B(X) satisfy (H1), (H2) and (H3). Then X T = λn,m En,m n,m

converges in operator norm (in the order ) to a compact AC(σ) operator. Proof. Define Ψ : AC(σ) → B(X) by X Ψ(f ) = f (0)I + (f (λn,m ) − f (0))En,m .

(5.1)

n,m

The first thing to verify is that Ψ is well-defined, that is, that the sum on the right-hand side of (5.1) converges for all f ∈ AC(σ). Suppose then that f ∈ AC(σ). Let µn,m = f (λn,m ) − f (0). Fix > 0. As f is continuous at 0, if (n0 , m0 ) is large enough, then |µn,m | < /4N,

for all (n, m) (n0 , m0 ).

(5.2)

As f is absolutely continuous, for every n = 1, 2, . . . , N , the variation along Rθn , var(f |σn ), is finite. Hence, if m0 is large enough, t X m=s

|µn,m − µn,m+1 | < /2N,

whenever m0 ≤ s ≤ t.

(5.3)

Vol. 63 (2009)

Compact AC(σ) Operators

467

As N is finite we can choose n0 ∈ {1, . . . , N } and m0 ≥ 1 such that (5.2) holds and such that (5.3) holds for all n at once. Suppose then that (n1 , m1 ) (n0 , m0 ). For each n, let In be the (possibly empty) set In = {m : (n0 , m0 ) (n, m) ≺ (n1 , m1 )}. If In 6= ∅, let sn = min In and tn = max In . The difference in the partial sums from index (n0 , m0 ) to index (n1 , m1 ) is therefore given by ∆=

tn X X n In 6=∅

µn,m En,m .

m=sn

Thus, by the Lemma 4.4, and using (5.2) and (5.3), tn

X

X µn,m En,m k∆k ≤

In 6=∅

≤

X

m=sn tX n −1 K |µn,sn | + |µn,tn | + |µn,m − µn,m+1 | < K m=sn

In 6=∅

It follows that the partial sums are Cauchy and hence the series converges. Note that in particular, this implies that the sum defining T = Ψ(λ) converges. Since each En,m is finite rank, T is compact. It is clear that Ψ is linear. For 1 ≤ n ≤ N , let σn = σ∩Rθn = {0}∪{λn,m }∞ m=1 as in Section 4, and define f0 , f1 , . . . , fN as before Proposition 4.3. Note that, using the affine invariance of AC(σ) and Lemma 3.2 of [7], kΨ(fn )k ≤ K kfn |σn kBV (σn ) ,

for 1 ≤ n ≤ N .

Then, by Proposition 4.3, kΨ(f )k ≤

N X

kΨ(fn )k

n=0

≤ |f (0)| + K

N X

kfn |σn kBV (σn )

n=1

≤ K||| f |||Sp ≤ (2N + 1)K kf kBV (σ) . It is easy to verify that Ψ(f g) = Ψ(f )Ψ(g) if f and g are constant on a disk around 0. The continuity of Ψ then implies that Ψ is multiplicative on AC(σ). Finally, since T = Ψ(λ), it follows that T has an AC(σ) functional calculus, and hence that T is a compact AC(σ) operator.

6. Examples As one might expect, Theorem 5.1 is far from giving a characterization of compact AC(σ) operators. Here we shall give some examples which show that there are many ways of producing compact AC(σ) operators whose spectra do not lie in a finite number of lines through the origin.

468

Ashton and Doust

IEOT

Proposition 6.1. Let σ = {0, λ1 , λ2 , . . . } be a countable set of complex numbers whose only limit point is 0. Define T on AC(σ) by T f (z) = z f (z). Then T is a compact AC(σ) operator. Proof. That T has the required functional calculus is an immediate consequence of that fact that AC(σ) is a Banach algebra. Let rn = n = n1 and define λn = gn λ as in Lemma 3.2. It follows that T = limn→∞ λn (T ). But λn (T ) is a finite rank operator, and hence T is compact. An important class of examples is given in [3, Example 3.9]. Proposition 6.2. Let A be a closed operator on a Banach space X, and suppose that for some x ∈ ρ(A), the resolvent (xI − A)−1 is compact and well-bounded. Then (µI − A)−1 is a compact AC(σµ ) operator for all µ ∈ ρ(A). Proof. Let R(µ, A) = (µI − A)−1 . The resolvent identity clearly implies that if one resolvent is compact then every resolvent is compact. If we fix µ 6∈ σ(T ), then R(µ, A) = f (R(x, A)) where f (t) = t/(1 + (µ − x)t) is a M¨obius transformation. If J is any compact interval containing σ(R(x, A)) then ρ(f (J)) = 21 . It follows from [3, Theorem 3.5] that R(µ, A) is an AC(f (J)) operator. It was shown in [8] that there exist compact AC operators (in the sense of Berkson and Gillespie) for which the sum (1.1) fails to converge if the eigenvalues are listed in order of decreasing modulus. It is not clear however whether that construction always produces an AC(σ) operator. In Example 6.3 we adapt the construction from [8] to produce an AC(σ) operator with this property. Certain aspects require more care here however, due to the nature of the BV (σ) norm. Example 6.3. Let θ = tan−1 (1/6). For k = 1, 2, . . . , let ei jθ/k , k λk,j + λk,j−1 = , 2

λk,j =

j = 0, 1, . . . , k,

µk,j

j = 1, 2, . . . , k.

Thus dk = |µk,j | is independent of j, and dk < |λk,j | = k1 for all k, j. Let σ = {λk,j }k,j ∪ {µk,j }k,j ∪ {0}. Then σ is compact, and hence by Proposition 6.1, the operator P∞ T ∈ B(AC(σ)), T f (z) = zf (z) is a compact AC(σ) operator. Thus T = j=1 λj P (λj ) where {λj } is a listing of the nonzero elements of σ according to the order ≺ defined in Section 2, and P (λj ) is the projection P (λj )f = χ{λj } f . P For r > 0, let Sr = |λj |≥r λj P (λj ), so that Sr is a partial sum of the above series for T when the terms are ordered according to modulus. We shall show that the series does not converge in this order by showing that this sequence of partial sums is not Cauchy. Fix k. Then

Vol. 63 (2009)

Compact AC(σ) Operators

Sdk − S1/k =

k X

469

λk,j P (λk,j )

j=0

= λk,0

k X

P (λk,j ) +

j=0

k X

(λk,j − λk,0 )P (λk,j ).

j=1

Now λk,0 = 1/k. Elementary trigonometry ensures that for all j, 1 1 . |λk,j − λk,0 | ≤ |λk,k − λk,0 | ≤ tan θ = k 6k

As AC(σ) is a Banach algebra, kP (λk,j )k = χ{λk,j } BV (σ) ≤ 3 for all j. Thus k

X

1

(λk,j − λk,0 )P (λk,j ) ≤ . 2 j=1

(6.1)

Pk Now j=0 P (λk,j ) is the projection of multiplication by the characteristic function of the set Λk = {λk,0 , . . . , λk,k } and so k

X

P (λk,j ) = kχΛk kBV (σ) .

j=0

Let γk denote the piecewise linear curve in C joining the elements of Λk in order. Note that γk passes through each of the points µk,j . Clearly any line in the plane has at most two entry points on γk and so ρ(γk ) = 1/2. Thus 1 cvar(χΛk , γk )ρ(γk ) = 2(k − 1) = k − 1 2 and so kχΛk kBV (σ) = kχΛk k∞ + sup cvar(χΛk , γ)ρ(γ) ≥ 1 + (k − 1) = k. γ

Thus, using (6.1), k k

X

X

1 1

Sd − S1/k ≥ 1 P (λ ) − (λ − λ )P (λ ) = .

k,j k,j k,0 k,j ≥ 1 − k k j=0 2 2 j=1

It follows that the partial sum sequence is not Cauchy and hence the infinite sum does not converge.

7. Other properties As was noted in Section 2, every AC(σ) operator T admits a splitting into real and imaginary parts T = A + iB, where A and B are commuting well-bounded operators. On nonreflexive spaces this splitting might not be unique [3, Example 4.5]. Even on nonreflexive spaces however, one does get a unique splitting when T is compact. As is shown in the proof of [4, Theorem 6.1], this is because in this

470

Ashton and Doust

IEOT

case the real and imaginary parts are determined by the familyP of Riesz projections associated with the nonzero eigenvalues of T . That is, if T = µj Pj , then X X A= xi Pj i

Re(µj )=xi

where {xi } is the set of nonzero real parts of eigenvalues of T , ordered so that |x1 | ≥ |x2 | ≥ . . . . Given an AC(σ) operator T and ω = α + iβ ∈ C, the operator ωT is an AC(ωσ) operator. A longstanding open question is whether a splitting of αT must be given by ωT = (αA − βB) + i(αB + βA).

(7.1)

Of course, if every real linear combination of A and B is again well-bounded, then this must be a splitting. It is well-known however that the sum of two commuting well-bounded operators A and B need not be well-bounded (see [12]). If T is compact, then ωT is also obviously compact, and hence has a unique splitting as αT = U + iV . The main issue in showing that (7.1) holds is in rearranging the conditionally convergent sums that arise. The following lemma shows that while rearrangements of the sum in (1.1) may fail to converge, they cannot converge to a different limit. Lemma 7.1. Suppose that {cj }∞ j=1 is a sequence of real numbers whose only limit point is 0. Suppose that {Pj } is a sequence of disjoint finite rank projections and that there is a constant K such that

P

• for all t > 0, cj ≥t Pj ≤ K,

P

• for all t < 0, cj ≤t Pj ≤ K. P∞ Then j=1 cj Pj is well-bounded if the sum converges. Proof. Without loss we may assume P∞ that all the scalars cj and projections Pj are nonzero. Suppose that U = j=1 cj Pj converges. Let σ = {0} ∪ {cj }. Then U is clearly a compact operator with σ ⊆ σ(U ). If β ∈ σ(U ) \ σ then it is an isolated P eigenvalue with corresponding Riesz projection Pβ . But then U Pβ = βPβ = cj Pj Pβ = 0 which is impossible. Thus σ(U ) = σ. It is easy to confirm that the Riesz projection corresponding to cj is Pj . Let π be a permutation of the positive integers so that |cπ(j) | is nonincreasing. P∞ It follows from [6, Theorem 3.3] that V = j=1 cπ(j) Pπ(j) converges to a wellbounded operator (with σ(V ) = σ). Let AC c (σ) = {f ∈ AC(σ) : f is constant on a neighbourhood of 0}. Then AC c (σ) is dense in AC(σ). Let A denote the algebra of functions f which are analytic on a neighbourhood of σ, and for which the restriction of f to σ lies in AC c (σ). Note that every f ∈ AC c (σ) has an extension to a locally constant element of A. Suppose then that f ∈ A. Write σ = σ1 ∪ σ2 , where σ2 is the

Vol. 63 (2009)

Compact AC(σ) Operators

471

component of the spectrum containing 0 on which f is constant, and where its complement σ1 is finite. The Riesz functional calculus for U and V gives that X X f (U ) = f (cj )Pj + f (0) I − Pj = f (V ). cj ∈σ1

cj ∈σ1

But V is well-bounded and so kf (U )k = kf (V )k ≤ K kf kBV (σ) for some K. The density of AC c (σ) now implies that U is well-bounded. Note in particular that in the above proof, if fn ∈ AC c (σ), and fn → λ in AC(σ), then U = limn fn (U ) = limn fn (V ) = V . This proves the following result. Corollary P 7.2. Suppose that T is a compact well-bounded operator with sum representation j µj Pj with |µ1 | ≥ |µ2 | ≥ . . . . Let π be a permutation of the positive integers. Then X T = µπ(j) Pπ(j) j

if the sum on the right-hand side converges. It might be noted that we have been unable to prove the corresponding result for compact AC(σ) operators. We return now to the question raised at the beginning of this section. Theorem 7.3. Let T be a compact AC(σ) operator with splitting T = A + iB, and let ω = α + iβ ∈ C. The unique splitting of αT is ωT = (αA − βB) + i(αB + βA). P

Proof. Write T = µj Pj via Corollary 2.2. Let x = Re(λ) and y = PIm(λ). The proof of Corollary 2.2 (see Section 3) shows that the sums x(T ) = Re(µj )Pj P and y(T ) = Im(µj )Pj both converge. Thus X X T = A + iB = Re(µj )Pj + i Im(µj )Pj

and so ωT = α

X

Re(µj )Pj + iβ

X

=

X

Re(ωµj )Pj + i

Re(µj )Pj + iα

X

Im(ωµj )Pj .

X

Im(µj )Pj − β

X

Im(µj )Pj

(7.2)

The AC(σ) functional calculus for T now provides the bounds on the norms of sums of the P Riesz projections needed to so that we may apply Lemma 7.1 and P deduce that Re(ωµj )Pj and Im(ωµj )Pj are well-bounded. Since these operators clearly commute, Equation P P (7.2) gives the unique splitting of ωT . But Re(ωµ )P = αA − βB and j j Im(ωµj )Pj = αB + βA so the proof is complete.

472

Ashton and Doust

IEOT

The known examples of AC operators which are not AC(σ) operators share the property that they can be written as T = A+iB where A and B are commuting well-bounded operators whose sum is not well-bounded. The previous theorem shows that, at least for compact operators, the well-boundedness of A + B is necessary for T to be an AC(σ) operator. It would of course be interesting to know whether it is sufficient. Corollary 7.4. Let T = A + iB be a compact AC(σ) operator. Then A + B is well-bounded.

References [1] B. Ashton and I. Doust, Functions of bounded variation on compact subsets of the plane, Studia Math. 169 (2005), 163–188. [2] B. Ashton and I. Doust, A comparison of algebras of functions of bounded variation, Proc. Edinb. Math. Soc. (2) 49 (2006), 575–591. [3] B. Ashton and I. Doust, AC(σ) operators, preprint 2008. (arXiv:0807.1052) [4] E. Berkson, I. Doust and T. A. Gillespie, Properties of AC operators, Acta Sci. Math. (Szeged) 6 (1997), 249–271. [5] E. Berkson and T. A. Gillespie, Absolutely continous functions of two variables and well-bounded operators, J. London Math. Soc. (2) 30 (1984), 305–321. [6] Q. Cheng and I. Doust, Compact well-bounded operators, Glasg. Math. J. 43 (2001), 467–475. [7] Q. Cheng and I. Doust, Well-bounded operators on nonreflexive Banach spaces II, Quaest. Math. 24 (2001), 183–191. [8] I. Doust and T. A. Gillespie, An example in the theory of AC operators, Proc. Amer. Math. Soc. 129 (2001), 1453–1457. [9] I. Doust and G. Lancien, The spectral type of sums of operators on non-Hilbertian Banach lattices, J. Austral. Math. Soc. 84 (2008), 193–198. [10] I. Doust and B. Walden, Compact AC operators, Studia Math. 117 (1996), 275–287. [11] H. R. Dowson, Spectral theory of linear operators, Academic Press, 1978. [12] T. A. Gillespie, Commuting well-bounded operators on Hilbert spaces, Proc. Edinburgh Math. Soc. (2) 20 (1976), 167–172. [13] K. B. Laursen and M. M. Neumann, An introduction to local spectral theory, The Clarendon Press, Oxford University Press, New York, 2000. [14] J. R. Ringrose, On well-bounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613–638. Brenden Ashton Silverbrook Research, 3 Montague St, Balmain 2041, Sydney, Australia e-mail: [email protected] Ian Doust School of Mathematics and Statistics, University of New South Wales UNSW Sydney 2052, Australia e-mail: [email protected] Submitted: July 21, 2008. Revised: October 30, 2008.

Integr. equ. oper. theory 63 (2009), 473–499 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040473-27, published online February 12, 2009 DOI 10.1007/s00020-009-1659-0

Integral Equations and Operator Theory

Riesz Bases of Root Vectors of Indefinite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions. II ´ Paul Binding and Branko Curgus Abstract. We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions affinely dependent on the eigenparameter. We give sufficient conditions under which the root vectors of this SturmLiouville problem can be selected to form a Riesz basis of a corresponding weighted Hilbert space. Mathematics Subject Classification (2000). Primary 34B10; Secondary 34B09, 47B25, 47B50. Keywords. Indefinite Sturm-Liouville problem, Riesz basis, eigenvalue dependent boundary conditions, Krein space, definitizable operator.

1. Introduction Consider the following eigenvalue problem −f 00 (x) = λ (sgn x)f (x),

x ∈ [−1, 1],

0

f (1) = λ f (−1), 0

−f (−1) = λ f (1). Lengthy but straightforward calculations show that all eigenvalues are real, simple, and they accumulate only at −∞ and +∞. Details can be found at the second author’s web-site. To our knowledge the following related question, which presents interesting mathematical challenges, has not been addressed: Is it possible to select eigenvectors of the given eigenvalue problem to form a Riesz basis of the Hilbert space L2 (−1, 1) ⊕ C2 ? In this article we answer such questions for a wide class

474

´ Binding and Curgus

IEOT

of indefinite Sturm-Liouville problems with λ-dependent boundary conditions. In particular, our Theorem 5.2 applies to the above example. We consider a regular indefinite Sturm-Liouville eigenvalue problem of the form −(p f 0 )0 + q f = λ r f on [−1, 1]. (1.1) We assume throughout that the coefficients 1/p, q, r in (1.1) are real and integrable over [−1, 1], p(x) > 0, and x r(x) > 0 for almost all x ∈ [−1, 1]. We impose the following eigenparameter dependent boundary conditions on equation (1.1): Mb(f ) = λ Nb(f ),

(1.2)

where M and N are 2 × 4 matrices and the boundary mapping b is defined for all f in the domain of (1.1) by T b(f ) = f (−1) f (1) (pf 0 )(−1) (pf 0 )(1) . For our opening example " 0 0 0 M= 0 0 −1

# 1 0

" ,

N=

1

0

0

# 0

0

1

0

0

.

We remark that more general boundary conditions have been studied by many authors, recently for example in [4] and [5], but expansion theorems were not considered. Expansion theorems for polynomial boundary conditions and more general operators, but with weight r = 1, were given in [12] and [22]. In this article we study the problem (1.1), (1.2) in an operator theoretic setting established in [6]. Under Condition 2.1 below, a definitizable self-adjoint operator A in the Krein space L2,r (−1, 1) ⊕ C2∆ (actually A is quasi-uniformly positive as defined in [11]) is associated with the eigenvalue problem (1.1), (1.2). Here ∆ is a 2 × 2 nonsingular Hermitean matrix which is determined by M and N; see Section 2 for details. We remark that the topology of this Krein space is that of the corresponding Hilbert space L2,|r| ⊕ C2|∆| . Here, and in the rest of the paper, we abbreviate L2,r (−1, 1) to L2,r and L2,|r| (−1, 1) to L2,|r| . For more details about Krein spaces and their operators see the standard reference [16] and [2] for recent developments. Our main goal in this paper is to provide sufficient conditions on the coefficients in (1.1), (1.2) under which there is a Riesz basis of the above Hilbert space consisting of the union of bases for all the root subspaces of the above operator A. This will be referred to for the remainder of this section as the Riesz basis property of A. We remark that the Riesz basis property of A is equivalent, modulo a finite dimensional subspace, to similarity of A to a self-adjoint operator in a Hilbert space. The latter similarity has been the subject of several recent papers (see for example [17] and [18]) involving Sturm-Liouville expressions on R without boundary conditions. Existence of Riesz bases and expansion theorems with a stronger topology, but in a smaller space corresponding to the form domain of the operator A (which in our case is a Pontryagin space), have been considered by many authors; see

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

475

[6, 23] and the references there. The results in [6] turned out to be independent of the number and the nature of the boundary conditions and the coefficients p and r. In contrast, the Riesz basis property depends nontrivially on the problem data even for the case when the boundary conditions are λ-independent (corresponding to N = 0 in our notation). Sufficient conditions on r (near the turning point 0) for the Riesz basis property when N = 0 can be found in [3, 10, 13, 15, 20, 21], for example. That some condition is necessary, even in the case p = 1, was shown by Volkmer [24] who proved the existence of an odd r for which the Dirichlet problem (1.1) does not have this property. Explicit examples of such functions r were constructed in [1] and [14]. Recently Parfenov [19] gave a necessary and sufficient condition on an odd weight function r, near its turning point 0, for the Dirichlet problem (1.1) to have the Riesz basis property. In [7] we constructed an odd r for which the Dirichlet problem (1.1) has the Riesz basis property but the anti-periodic problem does not. This example shows that an additional condition on r near the boundary of [−1, 1] (which in some cases behaves as a second turning point, in addition to 0, for (1.1)) is needed for the general case of (1.2). Such conditions are given in [10] for λ-independent boundary conditions and in [8] for exactly one λ-dependent boundary condition (i.e., when N has rank 1). In this paper we consider the more difficult case of two λ-dependent boundary conditions. The method we use has its origins in the work of Beals [3]. Subsequently it was developed in [9] into a criterion (given below as Theorem 2.2) equivalent to the Riesz basis property of A. This criterion involves a positive homeomorphism W of the Krein space L2,r ⊕ C2∆ with the form domain of A as an invariant subspace. The explicit description of the form domain of A (given in Section 2) depends entirely on the number k ∈ {0, 1, 2} of boundary conditions which do not include derivatives in the λ-terms. We call such boundary conditions essential. Note that this differs from the usual terminology for λ-independent conditions. For example, in our terminology y 0 (1) = λy(1) is an essential boundary condition. The direct sum structure of the Krein space L2,r ⊕ C2∆ naturally leads us to consider the homeomorphism W as a block operator matrix, the top left entry W11 being an operator on L2,r . Since it is clear from Section 2 that the functional components of the vectors in the form domain of A are (absolutely) continuous, we see that W11 induces a boundary matrix B satisfying f (−1) (W11 f )(−1) B = . f (1) (W11 f )(1) An important hurdle, with analogues in several of the above references, is to solve the inverse problem of finding a suitable W11 for a given matrix B. For example, in [8] (see also Section 3 below) such operators W11 were constructed with special diagonal B under one-sided Beals type conditions at −1 or 1. In Section 4 we use conditions at −1, at 1, and a condition connecting −1 and 1 to produce W11 with an arbitrary prescribed boundary matrix B.

´ Binding and Curgus

476

IEOT

In Sections 5 and 6 we complete the construction of W , thus establishing our sufficient conditions for the Riesz basis property. When there are no essential boundary conditions (k = 0), it turns out that the one-sided Beals type condition at 0 suffices; see Theorem 5.1. In other cases, however, we need conditions near the boundary of [−1, 1]. Conditions at 0, and at −1 or 1, are sufficient if k = 2 and ∆ is definite. If ∆ is indefinite, then we also need the condition linking −1 and 1. In these cases it suffices to construct W as a block diagonal matrix. This is carried out in Theorem 5.2. The most difficult case is k = 1 which we tackle in Section 6. In this case we need not only off-diagonal blocks for W , but also a perturbation K of W11 , where K is an integral operator whose construction is rather delicate. Our final result, Theorem 6.1, is as follows. If only one boundary point −1 or 1 appears with λ in the essential boundary condition, then a Beals type condition at that point and at 0 are sufficient. Otherwise we need conditions at both boundary points and at 0, as well as the condition linking −1 and 1. To conclude this introduction we remark that our conditions simplify drastically if p is even and r is odd, a case which has been studied by several authors [7, 19, 24]. In fact all the conditions that we impose on the boundary are then equivalent; see Example 4.3 and Corollary 6.5.

2. Operators associated with the eigenvalue problem The maximal operator Smax in L2,r associated with (1.1) is defined by 1 Smax : f 7→ `(f ) := −(pf 0 )0 + qf , f ∈ D(Smax ), r where D(Smax ) = Dmax = f ∈ L2,r : f, pf 0 ∈ AC[0, 1], `(f ) ∈ L2,r . We define the boundary mapping b by b(f ) = f (−1) f (1) (pf 0 )(−1)

T (pf 0 )(1) ,

and the concomitant matrix Q corresponding to  0 0 −1 0 0 0  Q = i 1 0 0 0 −1 0

f ∈ D(Smax ).

b by  0 1  . 0 0

The significance of Q is captured by the following identity: Z 1 Smax f g − f Smax g r = i b(g)∗ Qb(f ), f, g ∈ Dmax . −1

We note that Q = Q−1 . Throughout, we shall impose the following nondegeneracy and self-adjointness condition on the boundary data.

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

477

Condition 2.1. The boundary matrices M and N in (1.2) satisfy the following: M (1) the 4 × 4 matrix is nonsingular; N (2) MQM∗ = NQN∗ = 0; (3) the 2 × 2 matrix iMQ−1 N∗ is self-adjoint and invertible and we define −1 . ∆ := −i MQ−1 N∗ Clearly the boundary value problem (1.1),(1.2) will not change if row reduction is applied to the coefficient matrix M N . (2.1) In what follows we will assume that the matrix in (2.1) is row reduced to row echelon form (starting the reduction at the bottom right corner). In particular the matrix N has the form " # Ne 0 . N= N1 Nn The matrix 0 in the formula for N is k × 2 with k ∈ {0, 1, 2}. The k × 2 matrix Ne and the (2 − k) × 2 matrix Nn are of maximal ranks. There are three possible cases for N in (2.1): (a) Nn is a 2 × 2 identity matrix (so k = 0); (b) Ne and Nn are nonsingular 1 × 2 (row) matrices (so k = 1); (c) Ne is a 2 × 2 identity matrix (so k = 2). In case (a), both boundary conditions in (1.2) are non-essential, that is both rows on the right hand side of (1.2) contain derivatives. In case (b), the boundary condition corresponding to the first row in (1.2) is essential, that is no derivatives appear in this row on the right hand side; the second boundary condition in (1.2) is non-essential. In case (c), both boundary conditions in (1.2) are essential. Evidently k is the number of essential boundary conditions. Next we define a Krein space operator associated with the problem (1.1),(1.2). We consider the linear space L2,r ⊕ C2∆ , equipped with the inner product Z 1 f g := f gr + v∗ ∆u, f, g ∈ L2,r , u, v ∈ C2 . , u v −1 Then L2,r ⊕ C2∆ , [ · , · ] is a Krein space. A fundamental symmetry on this Krein space is given by " # J0 0 J := , 0 sgn(∆) where 2 × 2 matrix sgn(∆) and J0 : L2,r → L2,r are defined by sgn(∆) = |∆|−1 ∆

and (J0 f )(t) := f (t) sgn(r(t)), t ∈ [−1, 1].

Then h · , · i := [J · , · ] is a positive definite L2,r ⊕ C2∆ inner product which turns 2 2 into a Hilbert space L2,|r| ⊕ C|∆| , h · , · i . The topology of L2,r ⊕ C∆ is defined to

´ Binding and Curgus

478

IEOT

be that of L2,|r| ⊕C2|∆| , and a Riesz basis of L2,r ⊕C2∆ is defined as a homeomorphic image of an orthonormal basis of L2,|r| ⊕ C2|∆| . We define the operator A in the Krein space L2,r ⊕ C2∆ on the domain (" # ) f D(A) = ∈ K : f ∈ D Smax Nb(f ) by "

# " # f Smax f A := , Nb(f ) Mb(f )

f ∈ D(A).

Using [6, Theorems 3.3 and 4.1] we see that this operator is definitizable with discrete spectrum in the Krein space L2,r ⊕ C2∆ . As in [8, Theorem 2.2], we then obtain the following, which is our basic tool. Theorem 2.2. Let F(A) denote the form domain of A. Then there exists a Riesz basis of L2,r ⊕ C2∆ which consists of root vectors of A if and only if there exists a bounded, boundedly invertible, positive operator W in L2,r ⊕ C2∆ such that W F(A) ⊂ F(A). In order to apply this result, we need to characterize the form domain F(A). To this end, let Fmax be the set of all functions f in L2,r which are absolutely R1 continuous on [−1, 1] and such that −1 p |f 0 |2 < +∞. By [6, Theorem 4.2], there are three possible cases for the form domain F(A) of A, corresponding to cases (a), (b) and (c) above. 1 0 (a) If Nn = , then 0 1 " #  L2,r  f  F(A) = ∈ ⊕ : f ∈ Fmax , v ∈ C2 . (2.2)  v  C2∆ (b) If Ne = [u v] with u, v ∈ C and |u|2 + |v|2 6= 0, then      f L2,r     F(A) = uf (−1) + vf (1) ∈ ⊕ : f ∈ Fmax , z ∈ C .     C2∆ z 1 (c) If Ne = 0

0 , then 1      f L2,r     F(A) = f (−1) ∈ ⊕ : f ∈ Fmax .    f (1)  C2∆

(2.3)

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

479

To construct an operator W as in Theorem 2.2 we need to impose conditions (to be given in the next two sections) on the coefficients p and r in (1.1). In all cases we need Condition 3.5 in a neighborhood of 0, and in some cases we need one of two Conditions, 3.7 or 3.8, on r in neighborhoods of −1 or 1. These will be discussed in Section 3. In some cases we also need Condition 4.1 connecting the boundary points −1 and 1. This is developed in Section 4.

3. Conditions at 0, −1 and 1 In this section we recall the remaining concepts and results from [8, Sections 3, 4 and 5] which we need in this paper. A closed interval of non-zero length is said to be a left half-neighborhood of its right endpoint and a right half-neighborhood of its left endpoint. Let ı be a closed subinterval of [−1, 1]. By Fmax (ı) we denote the set of R all functions f in L2,r (ı) which are absolutely continuous on ı and such that ı p |f 0 |2 < +∞. With this notation we have Fmax = Fmax [−1, 1]. Definition 3.1. Let p and r be the coefficients in (1.1). Let a, b ∈ [−1, 1] and let ha and hb , respectively, be half-neighborhoods of a and b which are contained in [−1, 1]. We say that the ordered pair (ha , hb ) is smoothly connected if there exist (a) positive real numbers and τ , (b) non-constant affine functions α : [0, ] → ha and β : [0, ] → hb , (c) non-negative real functions ρ and $ defined on [0, ] such that (i) (ii) (iii) (iv)

α(0) = a and β(0) = b, p ◦ α and p ◦ β are locally integrable on the interval (0, ], ρ ◦ α−1 ∈ Fmax α([0, ]) , 1/τ < $ < τ a.e. on [0, ], r β(t) p β(t) and $(t) = for t ∈ (0, ]. (v) ρ(t) = r α(t) p α(t)

The numbers α0 , β 0 (the slopes of α, β, respectively) and ρ(0) are called the parameters of the smooth connection. A broad class of examples satisfying this definition can be given via the following one. Definition 3.2. Let ν and a be real numbers and let ha be a half-neighborhood of a. Let g be a function defined on ha . Then g is of order ν on ha if there exists g1 ∈ C 1 (ha ) such that g(x) = |x − a|ν g1 (x)

and

g1 (x) 6= 0,

x ∈ ha .

(The absolute value is missing in the corresponding definition in [8]).

´ Binding and Curgus

480

IEOT

Example 3.3. Let a, b ∈ {−1, 0, 1}. Let ha and hb be half-neighborhoods of a and b, respectively, and contained in [−1, 1]. For simplicity assume that p = 1. If r in (1.1) has order ν (> −1 to ensure integrability) on both half-neighborhoods ha and hb then as noted in [8] the half-neighborhoods ha and hb are smoothly connected. Moreover the parameters of the smooth connection are nonzero numbers. We remark that that p can be much more general—see [8, Example 3.4]. Theorem 3.4. Let ı and  be closed intervals, ı,  ∈ [−1, 0], [0, 1] . Let a be an endpoint of ı and let b be an endpoint of . Denote by a1 and b1 , respectively, the remaining endpoints. Assume that the half-neighborhoods ı of a and  of b are smoothly connected with parameters α0 , β 0 and ρ(0). Then there exists an operator S : L2,|r| (ı) → L2,|r| () such that the following hold: (S-1) S ∈ L L2,|r| (ı), L2,|r| () , S ∗ ∈ L L2,|r| (), L2,|r| (ı) ; (S-2) (Sf )(x) = 0, |x − b1 | ≤ 21 for all f ∈ L2,|r| (ı) and (S ∗ g)(x) = 0, |x − a1 | ≤ 12 for all g ∈ L2,|r| (); (S-3) SFmax (ı) ⊂ Fmax (), S ∗ Fmax () ⊂ Fmax (ı); (S-4) for all f ∈ Fmax (ı) and all g ∈ Fmax () we have lim (Sf )(y) = |α0 | x→a lim f (x),

y→b y∈

x∈ı

lim (S ∗ g)(x) = |β 0 |ρ(0) lim g(y).

x→a x∈ı

y→b y∈

This is [8, Theorem 3.6]. Condition 3.5 (Condition at 0). Let p and r be coefficients in (1.1). Denote by h0− a generic left and by h0+ a generic right half-neighborhood of 0. We assume that at least one of the four ordered pairs of half-neighborhoods (h0− , h0− ),

(h0− , h0+ ),

(h0+ , h0− ),

(h0+ , h0+ ),

is smoothly connected with the connection parameters α00 , β00 and ρ0 (0) such that |α00 | = 6 |β00 |ρ0 (0). We note from Example 3.3 that this condition is automatically satisfied if p = 1 and r is of order ν on some half-neighborhood of 0. Theorem 3.6. Assume that the coefficients p and r satisfy Condition 3.5. Then there exists an operator W0 : L2,r → L2,r such that the following hold: (a) W0 is bounded on L2,|r| ; (b) J0 W0 > I, in particular W0−1 is bounded and W0 is positive on the Krein space L2,r ; (c) (W0 f )(x) = (J0 f )(x), 21 ≤ |x| ≤ 1, f ∈ L2,r ; (d) W0 Fmax ⊂ Fmax . This is [8, Theorem 4.2].

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

481

Condition 3.7 (Condition at −1). Let p and r be coefficients in (1.1). We assume that a right half neighborhood of −1 is smoothly connected to a right half neigh0 0 borhood of −1 with the connection parameters α−1 , β−1 and ρ−1 (0) such that 0 0 |α−1 | = 6 |β−1 |ρ−1 (0). Condition 3.8 (Condition at 1). Let p and r be coefficients in (1.1). We assume that a left half-neighborhood of 1 is smoothly connected to a left half0 0 neighborhood of 1 with the connection parameters α+1 , β+1 and ρ+1 (0) such that 0 0 |α+1 | = 6 |β+1 |ρ+1 (0). Again, we note from Example 3.3 that these conditions are automatically satisfied if p = 1 and r is of order ν−1 and ν+1 on some half-neighborhood (in [−1, 1]) of −1 and 1, respectively. The following two propositions appear in [8] as Propositions 5.3 and 5.4, respectively. Proposition 3.9. Assume that the coefficients p and r satisfy Condition 3.7. Let b be an arbitrary complex number. Then there exists an operator W−1 : L2,r → L2,r such that the following hold: (a) W−1 is bounded on L2,|r| ; (b) J0 W−1 > I, in particular (W−1 )−1 is bounded and W−1 is positive on the Krein space L2,r ; (c) (W−1 f )(x) = (J0 f )(x), − 12 ≤ x ≤ 1, f ∈ L2,r ; (d) W−1 Fmax ⊂ Fmax [−1, 0] ⊕ Fmax [0, 1]; (e) (W−1 f )(−1) = bf (−1) for all f ∈ Fmax . Proposition 3.10. Assume that the coefficients p and r satisfy Condition 3.8. Let b be an arbitrary complex number. Then there exists an operator W+1 : L2,r → L2,r such that the following hold: (a) W+1 is bounded on L2,|r| ; (b) J0 W+1 > I, in particular (W+1 )−1 is bounded and W+1 is positive on the Krein space L2,r ; (c) (W+1 f )(x) = (J0 f )(x), −1 ≤ x ≤ 12 , f ∈ L2,r ; (d) W+1 Fmax ⊂ Fmax [−1, 0] ⊕ Fmax [0, 1]; (e) (W+1 f )(1) = bf (1) for all f ∈ Fmax .

4. Mixed condition at ±1 and associated operator In this section we establish analogues of the above results for a new condition involving both endpoints of the interval [−1, 1].

´ Binding and Curgus

482

IEOT

Condition 4.1 (Condition at −1,1). Let p and r be the coefficients in (1.1). We assume that at least one of the following three conditions is satisfied. (A) There are two smooth connections each connecting a right half-neighborhood 0 of −1 to a left half-neighborhood of 1 with the connection parameters αmj , 0 βmj and ρmj (0), j = 1, 2, such that 0 0 |αm1 | |αm2 | (4.1) 6= 0. |β 0 |ρm1 (0) |β 0 |ρm2 (0) m1

m2

(B) There are two smooth connections each connecting a left half-neighborhood of 0 0 , βmj 1 to a right half-neighborhood of −1 with the connection parameters αmj and ρmj (0), j = 1, 2, such that (4.1) holds. (C) A right half-neighborhood of −1 is smoothly connected to a left half-neigh0 0 borhood of 1 with the connection parameters αm1 , βm1 and ρm1 (0), and a left half-neighborhood of 1 is smoothly connected to a right half-neighborhood 0 0 of −1 with the connection parameters αm2 , βm2 and ρm2 (0), such that 0 0 |αm1 | |βm2 |ρm2 (0) 6= 0. |β 0 |ρm1 (0) |α0 | m1

m2

Example 4.2. From Example 3.3 it follows that this condition is satisfied if p = 1 and r has the same order ν on a right half-neighborhood of −1 and a left halfneighborhood of 1. Example 4.3. If p is an even function and r is odd, then it turns out that Conditions 3.7, 3.8 and 4.1 are equivalent. The first equivalence is clear. For the second, assume that Condition 3.8 is satisfied. Let α+1 and β+1 be the corresponding affine functions from Definition 3.1 defined on [0, ]. Now define αm1 (t) = α+1 (t), βm1 (t) = −β+1 (t), t ∈ [0, ), so ρm1 = ρ+1 . Note that p is locally integrable on [α +1 (), 1) by Definition 3.1 (ii). Then define αm2 (t) = 1 − t, βm2 (t) = −1 + t, t ∈ 0, 1 − α+1 () and so ρm2 = 1. Then Condition 4.1(B) is satisfied since (4.1) takes the form 0 0 0 |αm1 | |αm2 | |α+1 | 1 = |β 0 |ρm1 (0) |β 0 |ρm2 (0) |β 0 |ρ+1 (0) 1 m1

m2

+1

which is nonzero by Condition 3.8. The proof of the converse is similar. Example 4.4. We call a function g : [−1, 1] → C nearly odd (nearly even) if there exists a positive constant c 6= 1 such that g(−x) = −c g(x) (g(−x) = c g(x)) for almost all x ∈ (0, 1]. We note that if p is a nearly even function and r is nearly odd, both Conditions 3.5 and 4.1 are satisfied. Also, Conditions 3.7 and 3.8 are equivalent. The verification is straightforward. Example 4.5. Let p = 1 and r(x) = −1 for x ∈ [−1, 0) and r(x) = 1 − x for x ∈ [0, 1]. It is not difficult to verify directly that these functions satisfy Conditions 3.5,

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

483

3.7 and 3.8, but not Condition 4.1. In addition notice that r is of order 0 in a right half-neighborhood of −1 and of order 1 in a left half-neighborhood of 1. The proof of the following theorem occupies the remainder of this section. Theorem 4.6. Assume that the coefficients p and r satisfy Conditions 3.7, 3.8 and 4.1. Let bjk , j, k = 1, 2, be arbitrary complex numbers. Then there exists an operator Ws1 : L2,r → L2,r such that the following hold: (a) Ws1 is bounded on the Hilbert space L2,|r| ; −1 (b) J0 Ws1 > I, in particular Ws1 is bounded and Ws1 is positive on the Krein space L2,r ; (c) (Ws1 f )(x) = (J0 f )(x), − 12 ≤ x ≤ 21 , f ∈ L2,r ; (d) Ws1 Fmax ⊂ Fmax [−1, 0] ⊕ Fmax [0, 1]; " # " #" # (e) (Ws1 f )(−1) b11 b12 f (−1) = . (Ws1 f )(1) b21 b22 f (1) Proof. We construct Ws1 in the form ∗ Ws1 = J0 Xs1 Xs1 + I , where

"

Xs1

X11 = X21

X12 X22

#

is a block operator matrix corresponding to the decomposition L2,|r| = L2,|r| (−1, 0) ⊕ L2,|r| (0, 1). We split the proof into three parts. The off-diagonal and diagonal entries of Xs1 are constructed in the first and second parts, respectively. In the third part we establish the stated properties of Ws1 . 1. To construct the off-diagonal operators we treat each case (A), (B), (C) of Condition 4.1 separately. Case (A). By Theorem 3.4 there exist operators Smj : L2,|r| (−1, 0) → L2,|r| (0, 1), j = 1, 2, which satisfy (S-1)-(S-4) in Theorem 3.4 with ı = [−1, 0],  = [0, 1], a = −1 and b = 1. In particular, for f ∈ Fmax [−1, 0] and j = 1, 2, 0 (Smj f )(1) = |αmj | f (−1),

∗ 0 (Smj f )(−1) = |βmj | ρmj (0) f (1).

To simplify the formulas we use the following notation 0 0 | |αm2 | |αm1 Υ := . |β 0 | ρm1 (0) |β 0 | ρm2 (0) m1 m2

´ Binding and Curgus

484

IEOT

Define X21 : L2,|r| (−1, 0) → L2,|r| (0, 1) by Sm1 Sm2 . |β 0 | ρm1 (0) |β 0 | ρm2 (0) m1 m2

−1

X21 := b21 Υ

Here and below we write such determinants as abbreviations for corresponding linear combinations of operators. For all f ∈ Fmax [−1, 0] we have 0 0 |α | f (−1) |αm2 | f (−1) −1 m1 (X21 f )(1) = b21 Υ = b21 f (−1). |β 0 | ρm1 (0) |β 0 | ρm2 (0) m1

m2

Also for all g ∈ Fmax [0, 1] we have 0 0 |βm1 | ρm1 (0) g(1) |βm2 | ρm2 (0) g(1) ∗ (X21 g)(−1) = b21 Υ−1 = 0. 0 |β 0 | ρm1 (0) |βm2 | ρm2 (0) m1 Now define the opposite off-diagonal corner X12 : L2,|r| (0, 1) → L2,|r| (−1, 0) by 0 ∗ 0 ∗ X12 := −b12 Υ−1 −|αm2 | Sm1 + |αm1 | Sm2 = −b12 Υ−1

0 |αm1 | S∗ m1

0 |αm2 | . ∗ Sm2

Then for all f ∈ Fmax [0, 1] we have 0 0 |αm1 | |αm2 | −1 (X12 f )(−1) = −b12 Υ = −b12 f (1). |β 0 | ρm1 (0) f (1) |β 0 | ρm2 (0) f (1) m1 m2 Also ∗ (X12 f )(1) = −b12 Υ−1

0 0 |αm1 | |αm2 | = 0. |α0 | f (−1) |α0 | f (−1) m1 m2

Case (B). By Theorem 3.4 there exist operators Smj : L2,|r| (0, 1) → L2,|r| (−1, 0), j = 1, 2, which satisfy (S-1)-(S-4) in Theorem 3.4 with ı = [0, 1],  = [−1, 0], a = 1 and b = −1. In particular, for all f ∈ Fmax [0, 1] and j = 1, 2, 0 (Smj f )(−1) = |αmj | f (1),

∗ 0 (Smj f )(1) = |βmj | ρmj (0) f (−1).

To simplify the formulas we continue to use the notation 0 0 |αm1 | |αm2 | Υ := . |β 0 | ρm1 (0) |β 0 | ρm2 (0) m1

m2

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

485

Define X12 : L2,|r| (0, 1) → L2,|r| (−1, 0) by Sm1 Sm2 . |β 0 | ρm1 (0) |β 0 | ρm2 (0) m1 m2

−1

X12 = −b12 Υ

Then for all f ∈ Fmax [0, 1] we have 0 0 |αm1 | f (1) |αm2 | f (1) −1 (X12 f )(−1) = −b12 Υ = −b12 f (1) |β 0 | ρm1 (0) |β 0 | ρm2 (0) m1 m2 and for all g ∈ Fmax [−1, 0] we have 0 0 |β | ρm1 (0) g(−1) |βm2 | ρm2 (0) g(−1) ∗ −1 m1 (X12 g)(1) = −b12 Υ = 0. 0 |β 0 | ρm1 (0) |βm2 | ρm2 (0) m1 Now define the opposite off-diagonal corner X21 : L2,|r| (−1, 0) → L2,|r| (0, 1) by −1

X21 = b21 Υ

0 |αm1 | S∗ m1

0 |αm2 | . ∗ Sm2

Then for all f ∈ Fmax [−1, 0] we have 0 0 |αm1 | |αm2 | −1 (X21 f )(1) = b21 Υ = b21 f (−1) |β 0 | ρm1 (0) f (−1) |β 0 | ρm2 (0) f (−1) m1 m2 and for all g ∈ Fmax [0, 1] we have ∗ (X21 g)(−1)

−1

= b21 Υ

0 0 |αm1 | |αm2 | = 0. |α0 | g(1) |α0 | g(1) m1 m2

Case (C). By Theorem 3.4 there exists an operator Sm1 : L2,|r| (−1, 0) → L2,|r| (0, 1) with the properties listed in Case (A) of this proof and there exists an operator Sm2 : L2,|r| (0, 1) → L2,|r| (−1, 0) with the properties listed in Case (B). To simplify the formulas in this part of the proof we use the notation 0 0 |αm1 | |βm2 | ρm2 (0) Υ := . 0 |β 0 | ρm1 (0) |αm2 | m1 Define X12 : L2,|r| (0, 1) → L2,|r| (−1, 0)

´ Binding and Curgus

486

IEOT

by 0 |αm1 | S∗

−1

X12 = −b12 Υ

m1

0 |βm2 | ρm2 (0) . Sm2

Then for all f ∈ Fmax [0, 1] we have 0 0 |αm1 | |βm2 | ρm2 (0) −1 (X12 f )(−1) = −b12 Υ = −b12 f (1), 0 |β 0 | ρm1 (0) f (1) |αm2 | f (1) m1 and for all g ∈ Fmax [−1, 0] we have ∗ (X12 g)(1) = −b12 Υ−1

sm1 θm2 (0) = 0. sm1 g(−1) θm2 (0)g(−1)

The other off-diagonal operator X21 : L2,|r| (−1, 0) → L2,|r| (0, 1) is defined as X21 = bm21 Υ−1

∗ Sm1 Sm2 . |β 0 | ρm1 (0) |α0 | m1 m2

Then for all f ∈ Fmax [−1, 0] we have 0 0 |αm1 | f (−1) |βm2 | ρm2 (0) f (−1) (X21 f )(1) = b21 Υ−1 = b21 f (−1), |β 0 | ρm1 (0) |α0 | m1

m2

and for all g ∈ Fmax [0, 1] we have ∗ (X21 g)(−1)

−1

= b21 Υ

0 0 |βm1 | ρm1 (0) g(1) |αm2 | g(1) = 0. 0 |β 0 | ρm1 (0) |αm2 | m1

We conclude this part of the proof by summarizing that in each of the three cases above we have defined operators X12 : L2,|r| (0, 1) → L2,|r| (−1, 0)

and X21 : L2,|r| (−1, 0) → L2,|r| (0, 1)

such that X12 Fmax [0, 1] ⊂ Fmax [−1, 0],

∗ X12 Fmax [−1, 0] ⊂ Fmax [1, 0],

∗ X21 Fmax [0, 1] ⊂ Fmax [−1, 0],

X21 Fmax [−1, 0] ⊂ Fmax [1, 0],

and for all f ∈ Fmax [0, 1] and g ∈ Fmax [−1, 0] we have (X12 f )(−1) = −b12 f (1),

∗ (X12 g)(1) = 0,

∗ (X21 f )(−1)

(X21 g)(1) = b21 f (−1).

= 0,

This completes the construction of the off-diagonal entries of Xs1 .

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

487

2. To construct the diagonal entries we need two self-adjoint operators P1,− and P1,+ defined as follows. Let φ1 : [−1, 1] → [0, 1] be an even function with φ1 ∈ C 1 [−1, 1] and such that φ1 (−1) = 1,

φ1 (x) = 0

for 0 ≤ |x| ≤ 1/2,

φ1 (1) = 1.

We now define P1,− : L2,|r| (−1, 0) → L2,|r| (−1, 0)

and P1,+ : L2,|r| (0, 1) → L2,|r| (0, 1)

by (P1,− f )(x) = f (x) φ1 (x),

f ∈ L2,|r| (−1, 0),

x ∈ [−1, 0],

(4.2)

(P1,+ f )(x) = f (x) φ1 (x),

f ∈ L2,|r| (0, 1),

x ∈ [0, 1].

(4.3)

and

These operators enjoy the following properties: (P1,− f )(x) = 0,

f ∈ L2,|r| (−1, 0),

(P1,+ f )(x) = 0,

f ∈ L2,|r| (0, 1),

P1,− Fmax [−1, 0] ⊂ Fmax [−1, 0],

− 12 ≤ x ≤ 0, 0 ≤ x ≤ 21 ,

P1,+ Fmax [0, 1] ⊂ Fmax [0, 1],

and (P1,− f )(−1) = f (−1),

f ∈ Fmax [−1, 0], f ∈ Fmax [0, 1].

(P1,+ f )(1) = f (1),

Now we use Condition 3.7 to construct the operator X11 . As in Proposition 3.9, Theorem 3.4 implies that there exists an operator S−1 : L2,|r| (−1, 0) → L2,|r| (−1, 0) with the properties listed there. In particular for all f ∈ Fmax [−1, 0] we have 0 (S−1 f )(−1) = |α−1 | f (−1),

∗ 0 (S−1 f )(−1) = |β−1 | ρ−1 (0) f (−1).

0 0 Since |α−1 |= 6 |β−1 | ρ−1 (0) we can choose complex numbers γ1 and γ2 such that 0 γ 1 |β−1 | ρ−1 (0) + γ 2 = 1.

0 γ1 |α−1 | + γ2 = −b11 − 1,

Let P1,− be the operator defined in (4.2). Put X11 = γ1 S−1 + γ2 P1,− . Then for all f ∈ Fmax [−1, 0] we have (X11 f )(−1) = (−b11 − 1) f (−1),

∗ (X11 f )(−1) = f (−1).

Note also that X11 Fmax [−1, 0] ⊂ Fmax [−1, 0]

∗ and X11 Fmax [−1, 0] ⊂ Fmax [−1, 0].

To construct X22 we use Condition 3.8. By Theorem 3.4 there exists a bounded operator S+1 : L2,|r| (0, 1) → L2,|r| (0, 1)

´ Binding and Curgus

488

IEOT

such that ∗ S+1 Fmax [0, 1] ⊂ Fmax [0, 1],

S+1 Fmax [0, 1] ⊂ Fmax [0, 1], and for all f ∈ Fmax [0, 1], 0 (S+1 f )(1) = |α+1 | f (1),

∗ 0 (S+1 f )(1) = |β+1 | ρ+1 (0)f (−1).

0 0 Since |α+1 |= 6 |β+1 | ρ+1 (0) we can choose complex numbers δ1 and δ2 such that 0 δ1 |α+1 | + δ2 = −b11 − 1,

0 δ 1 |β+1 | ρ+1 (0) + δ 2 = 1.

Let P1,+ be the operator defined in (4.3). Put X22 = δ1 S+1 + δ2 P1,+ . Then for all f ∈ Fmax [0, 1] we have (X22 f )(1) = (b22 − 1) f (1)

and

∗ (X22 f )(1) = f (1).

Note also that X22 Fmax [0, 1] ⊂ Fmax [0, 1]

∗ and X22 Fmax [0, 1] ⊂ Fmax [0, 1].

∗ 3. Now we formally define Ws1 := J0 (Xs1 Xs1 + I) where " # X11 X12 Xs1 = . X21 X22

To complete the proof, we verify the properties of Ws1 stated in the theorem. Indeed, (a) and (b) are immediate, and since (Xij f )(x) = 0 whenever − 21 ≤ x ≤ 21 , (c) follows. Moreover, each of the operators Xij maps Fmax [−1, 0] or Fmax [0, 1] to Fmax [−1, 0] or Fmax [0, 1] according to its position in the matrix, so (d) holds. Finally, we check the effect of the individual components at the boundary points −1 and 1. Evidently Xs1 Fmax ⊂ Fmax ,

∗ Xs1 Fmax ⊂ Fmax .

Moreover for f, g ∈ Fmax we have " # " # " # (Xs1 f )(−1) (X11 f )(−1) + (X12 f )(−1) (−b11 − 1)f (−1) − b12 f (1) = = (Xs1 f )(1) (X21 f )(1) + (X22 f )(1) b21 f (−1) + (b22 − 1)f (1) and "

# ∗ g)(−1) (Xs1 ∗ (Xs1 g)(1)

" =

# ∗ ∗ g)(−1) (X11 g)(−1) + (X21 ∗ ∗ (X12 g)(1) + (X22 g)(1)

" =

g(−1) + 0 0 + g(1)

Substituting g = Xs1 f ∈ Fmax , we get # " # " ∗ (−b11 − 1)f (−1) − b12 f (1) (Xs1 Xs1 f )(−1) = . ∗ (Xs1 Xs1 f )(1) b21 f (−1) + (b22 − 1)f (1)

# .

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

489

∗ With Ys1 = Xs1 Xs1 + I we have # " " # " −b11 f (−1) − b12 f (1) −b11 (Ys1 f )(−1) = = b21 f (−1) + b22 f (1) b21 (Ys1 f )(1)

−b12

#"

b22

# f (−1) f (1)

which proves (e) since Ws1 = J0 Ys1 .

,

Remark 4.7. Notice that the operators W−1 and W+1 from Propositions 3.9 and 3.10 satisfy " # " #" # " # " #" # (W−1 f )(−1) b 0 f (−1) (W+1 f )(−1) −1 0 f (−1) = and = , (W−1 f )(1) 0 1 f (1) (W+1 f )(1) 0 b f (1) respectively, with arbitrary b ∈ C. A stronger conclusion is contained in Theorem 4.6 (e) under stronger assumptions.

5. Two essential or two non-essential boundary conditions The first theorem of this section deals with the case of two non-essential boundary conditions. Theorem 5.1. Assume that the following two conditions are satisfied. 1 0 (a) Nn = . 0 1 (b) The coefficients p and r satisfy Condition 3.5. Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . Proof. By (2.2), the form domain of A is given as " #  L2,r  f  F(A) = ∈ ⊕ : f ∈ Fmax , v ∈ C2 .  v  C2∆ Recalling W0 from Theorem 3.6, we easily see that the operator " # L2,r L2,r W0 0 ⊕ ⊕ W = : → −1 0 ∆ C2 C2 ∆

∆

is bounded, boundedly invertible and positive in the Krein space L2,r ⊕ C2∆ . A simple verification shows that W F(A) ⊂ F(A) so the theorem follows from Theorem 2.2. We now consider the case of two essential conditions. Theorem 5.2. Assume that the following three conditions are satisfied. 1 0 (a) Ne = . 0 1 (b) The coefficients p and r satisfy Condition 3.5.

490

(c) One (i) (ii) (iii)

´ Binding and Curgus

IEOT

of the following holds: ∆ > 0 and the coefficients p and r satisfy Condition 3.7; ∆ < 0 and the coefficients p and r satisfy Condition 3.8; the coefficients p and r satisfy Conditions 3.7, 3.8 and 4.1.

Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . Proof. Define the following two Krein spaces: ˙ 2,r ( 12 , 1). K1 := L2,r (−1, − 21 )[+]L K0 := L2,r − 12 , 12 , Extending functions in K0 and K1 by zero, we consider the spaces K0 and K1 as subspaces of L2,r . Then ˙ 1. L2,r = K0 [+]K As in the previous proof our goal is to construct W : L2,r ⊕ C2∆ → L2,r ⊕ C2∆ . The first step is to define W01 : L2,r → L2,r . We proceed by considering each case in (c) separately. (i) Let W0 be the operator constructed in Theorem 3.6 and let W−1 be the operator constructed in Proposition 3.9 with b = 1. Property (c) in Theorem 3.6 and Proposition 3.9 imply that K0 and K1 are invariant under W0 and W−1 . Since we chose b = 1, we have (W−1 f )(−1) = f (−1) and (W−1 f )(1) = f (1). Define ˙ −1 |K . W01 := W0 |K0 [+]W 1

(5.1)

Since W0 and W−1 are bounded, boundedly invertible and positive in the Krein space L2,r , so is the the operator W01 . Also, W01 Fmax ⊂ Fmax and " # " # (W01 f )(−1) f (−1) = . (5.2) (W01 f )(1) f (1) (ii) Instead of W−1 in (i), we use the operator W+1 constructed in Proposition 3.10 with b = −1. Redefining the operator W01 as ˙ +1 |K , W01 := W0 |K0 [+]W 1

(5.3)

we see that it is again bounded, boundedly invertible, and positive in the Krein space L2,r , W01 Fmax ⊂ Fmax and (since we use b = −1) " # " # (W01 f )(−1) f (−1) =− . (5.4) (W01 f )(1) f (1) (iii) This time we replace W−1 from (i) by Ws1 from Theorem 4.6, so we define the operator ˙ s1 |K , W01 := W0 |K0 [+]W (5.5) 1

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

491

which is again bounded, boundedly invertible and positive in the Krein space L2,r . Also, W01 Fmax ⊂ Fmax and " # " # (Ws1 f )(−1) f (−1) = ∆−1 . (5.6) (Ws1 f )(1) f (1) Finally we define W : L2,r ⊕ C2∆ → L2,r ⊕ C2∆ by " # W01 0 W = 0 I

(5.7)

in case (c)(i), " W =

W01

0

0

−I

# (5.8)

in case (c)(ii), and " W =

W01

0

0

∆−1

# (5.9)

in case (c)(iii). By (2.3), the form domain of A is    f   L2,r     F(A) = f (−1) ∈ ⊕ : f ∈ Fmax .     f (1) C2∆ A straightforward verification shows that in each of the cases (5.7), (5.8), and (5.9), W is a bounded, boundedly invertible, positive operator in the Krein space L2,r ⊕ C2∆ . Moreover W F(A) ⊂ F(A) via (5.2), (5.4) or (5.6). Now the theorem follows from Theorem 2.2. Example 5.3. Consider the eigenvalue problem −f 00 = λ r f f 0 (1) = λf (−1) −f 0 (−1) = λf (1), where r(x) =sgn x, x ∈ [−1, 1], as in our example in the introduction. Then 1 0 clearly Ne = , giving (a) in Theorem 5.2 and (b) follows from the note 0 1 0 1 after Condition 3.5. Moreover, an easy computation gives ∆ = , which is 1 0 indefinite. Condition (c) now follows from Examples 4.2 and 4.3, so Theorem 5.2 applies. On the other hand, if instead we take r as in Example 4.5, then as we have seen, Condition 4.1 fails and hence so does (c)(iii) in Theorem 5.2. Therefore Theorem 5.2 gives no conclusion about a Riesz basis for this amended case.

492

´ Binding and Curgus

IEOT

6. One essential and one non-essential boundary condition The main result of this section is the following theorem. Its proof will occupy the most of the section, and then we will proceed to some examples. Theorem 6.1. Assume that the following three conditions are satisfied. " # " # u v 0 0 u v 0 0 (a) N = or N = , where |u|2 + |v|2 > 0 and the ∗ ∗ ∗ 1 ∗ ∗ 1 0 asterisks stand for arbitrary complex numbers. (b) The coefficients p and r satisfy Condition 3.5. (c) One of the following holds. (i) u = 1, v = 0 and the coefficients p and r satisfy Condition 3.7. (ii) u = 0, v = 1 and the coefficients p and r satisfy Condition 3.8. (iii) uv 6= 0 and the coefficients p and r satisfy Conditions 3.7, 3.8 and 4.1. Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . Proof. It follows from (a) that the form domain of A is      f L2,r     F(A) = uf (−1) + vf (1) ∈ ⊕ : f ∈ Fmax , z ∈ C .     C2∆ z

(6.1)

It is no restriction if we scale the first boundary condition so that |u|2 + |v|2 = 1.

(6.2)

As in the previous proofs we shall construct W : L2,r ⊕ blocks. We divide the proof into three parts and two lemmas.

C2∆

→ L2,r ⊕ C2∆ in

1. First we define a bounded operator W01 : L2,r → L2,r such that J0 W01 > I,

(6.3)

W01 Fmax ⊂ Fmax ,

(6.4)

u(W01 f )(−1) + v(W01 f )(1) = 0,

f ∈ Fmax .

(6.5)

We distinguish the three cases in (c) above. (i) As in the proof of Theorem 5.2(i), we define W01 by (5.1), but now using b = 0 instead of b = 1. Then W01 is a bounded operator in the Krein space L2,r , and it satisfies (6.4) and (W01 f )(−1) = 0, (W01 f )(1) = f (1) and hence (6.5). Inequality (6.3) follows from (5.1), Theorem 3.6(b) and Proposition 3.9(b). (ii) This time we define W01 by (5.3), but now using b = 0 instead of b = −1. Then W01 is a bounded operator in the Krein space L2,r , it satisfies (6.4) and (W01 f )(−1) = −f (−1), (W01 f )(1) = 0 and hence (6.5). In this case inequality (6.3) follows from (5.3), Theorem 3.6(b) and Proposition 3.10(b). (iii) We now define W01 as in the proof of Theorem 5.2(iii), but instead of using ∆−1 in (5.6) we use the zero 2×2 matrix 0. Then W01 is a bounded operator in the

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

493

Krein space L2,r , it satisfies (6.4) and (W01 f )(−1) = 0, (W01 f )(1) = 0 and hence (6.5). Inequality (6.3) follows from (5.5), Theorem 3.6(b) and Theorem 4.6(b). 2. Next we define an integral operator K which will be a perturbation of W01 . 2.1. We start by writing the inverse of the matrix ∆ in the form " # η11 η12 −1 ∆ = η 12 η22 and setting η := max{|η11 |, |η12 |} > 0, with δ2 ≥ δ1 > 0 as the eigenvalues of |∆|. We also define three positive constants δ2 , 1 + 2krk1 δ2 η 2 r α δ1 c := , 2δ2 2 2δ2 η 2 krk1 = 2 αη 2 krk1 . κ := 1 + 2krk1 δ2 η 2

α :=

Notice that 1−κ=

α 1 = . 2 1 + 2krk1 δ2 η δ2

2.2. Since r is integrable over [−1, 1], there exists γ ∈ [0, 1) such that 2 Z −γ Z 1 c − r+ . r≤ αη −1 γ

(6.6) (6.7)

(6.8)

(6.9)

Noting that p−1/2 ∈ L2 (0, 1) ⊂ L1 (0, 1) we can define Z x φ(x) = p−1/2 χ[γ,1] , x ∈ [0, 1]. 0

Extending φ as an even function over [−1, 1] we see that φ ∈ Fmax . Since φ(1) is a positive real number, we define ψ = φ/φ(1). Clearly ψ : [−1, 1] → [0, 1] is an even function in Fmax such that ψ(−1) = 1,

ψ(0) = 0,

ψ(1) = 1,

(6.10)

and, by (6.9), kψk2,|r| ≤ 2.3. Define ψj (x) =

c . αη

(6.11)

 α η1j u ψ(x),

x ∈ [−1, 0),

α η v ψ(x), 1j

x ∈ [0, 1].

(6.12)

Since ψ ∈ Fmax and ψ(0) = 0, the functions ψ1 and ψ2 belong to Fmax . Set ω(x) := η11 ψ1 (x) + η12 ψ2 (x),

x ∈ [−1, 1],

´ Binding and Curgus

494

and define k : [−1, 1] × [−1, 1] → C by  u ω(x)     v ω(t) k(x, t) =  v ω(x)    u ω(t)

t ≤ −|x|, x > |t|, t ≥ |x|, x < −|t|.

if if if if

IEOT

(6.13)

By the definitions of ψ1 , ψ2 and ω, since ψ is a nonnegative even function, for all x ∈ [0, 1] we have u ω(−x), v ω(x) ∈ R,

and v ω(−x) = u ω(x).

(6.14)

Since ω is continuous, it follows from (6.14) and (6.13) that k is a continuous function. Moreover, by (6.2) and (6.12), |ω(t)| < η ηα + η ηα = 2η 2 α. Therefore (6.7) shows that |k(x, t)| ≤ 2η 2 α =

κ . krk1

(6.15)

The first of our two lemmas is as follows. Lemma 6.2. Let K : L2,r → L2,r be the integral operator defined by Z 1 (Kf )(x) := k(x, t) f (t) r(t) dt, f ∈ L2,r . −1

Then (I) the operator K is bounded and self-adjoint on L2,r and kKk2,|r| ≤ κ; (II) the range of K is contained in Fmax . Proof. (I) We first note that for f in L2,r the function f r is integrable on (−1, 1). In fact Z 1 Z 1 |f r| = |r|1/2 |f ||r|1/2 −1

−1

Z

1

≤

1/2 Z |r|

−1

1

−1

1/2 1/2 |f | |r| = krk1 kf k2,r .

For f ∈ L2,|r| we calculate Z 1Z 1 Z 2 kKf k2,|r| ≤ |k(x, t)| |f (t)| |r(t)|dt ≤

−1

−1

κ2 krk21

Z

(6.16)

2

1

|k(x, s)| |f (s)| |r(s)| ds |r(x)|dx

−1

1

−1

Z

1

−1

2 |f | |r| |r(x)|dx ≤ κkf k22,|r| ,

by virtue of (6.15) and (6.16). Thus kKk2,|r| ≤ κ, so K is bounded, and selfadjointness follows from (6.13) since k(x, t) = k(t, x), x, t ∈ [−1, 1].

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

495

(II) Let f ∈ L2,r . By definition, for −1 ≤ x < 0, Z x Z −x Z (Kf )(x) = u ω(x) (f r)(t)dt + u ωf r (t)dt + v ω(x) −1

1

(f r)(t)dt

−x

x

and, for 0 < x ≤ 1, Z

−x

(Kf )(x) = u ω(x)

Z

x

ωf r (t)dt + v ω(x)

(f r)(t)dt + v −1

−x

Z

1

(f r)(t)dt. x

The function f r is integrable on (−1, 1) by (6.16). Since ω ∈ Fmax , the function ωf r is also integrable on (−1, 1). Moreover, lim(Kf )(x) = lim(Kf )(x) = (Kf )(0) = 0. x↑0

x↓0

Therefore for each f ∈ L2,|r| the function Kf is absolutely continuous on [−1, 1]. For almost all x ∈ [−1, 0), we have Z x Z 1 0 0 0 (f r)(t)dt + v ω (x) (f r)(t)dt (Kf ) (x) = u ω (x) −1

−x

+ u ω(x) (f r)(x) − u ω(x) f r (x) − u ω(−x) f r (−x) + v ω(x) (f r)(−x), and, for almost all x ∈ (0, 1], Z −x Z 1 (Kf )0 (x) = uω 0 (x) (f r)(t)dt + vω 0 (x) (f r)(t)dt −1

x

− u ω(x) (f r)(−x) + v ω(−x) f r (−x) + v ω(x) f r (x) − v ω(x) (f r)(x). By (6.14) the terms not involving integrals in the above two equations cancel in pairs. Thus Kf ∈ Fmax for all f ∈ L2,|r| since ω ∈ Fmax . This completes the proof of the lemma. 3. We create off-diagonal blocks for W by means of the operator Z : C2∆ → L2,r which we define by " # a1 Za := a1 ψ1 + a2 ψ2 , a = ∈ C2 . a2 The adjoint Z [∗] : L2,r → C2∆ of Z is given by " # [∗] −1 [f, ψ1 ] Z f =∆ , [f, ψ2 ]

f ∈ L2,r .

Equalities (6.2), (6.11) and (6.12) yield kψ1 k2,|r| ≤ c and kψ2 k2,|r| ≤ c. Therefore Z 1 2 Za |r| ≤ 2 |a1 |2 kψ1 k2 + |a2 |2 kψ2 k2 2,|r| 2,|r| −1

≤ 2c2 a∗ a ≤ 2

c2 ∗ a |∆|a. δ1

´ Binding and Curgus

IEOT

r α 2 kZk = kZ [∗] k ≤ c = . δ1 2δ2

(6.17)

496

Consequently, by (6.6),

The second lemma we need is as follows. Lemma 6.3. Let the operator W : L2,r ⊕ C2∆ → L2,r ⊕ C2∆ be defined by " # W01 + K Z W := . Z [∗] α ∆−1 Then (I) W is bounded and uniformly positive on L2,r ⊕ C2∆ ; (II) W F(A) ⊂ F(A). Proof. (I) The operator W is bounded since each of its components is bounded. To prove that W is uniformly positive, we shall show that the operator J W is uniformly positive in the Hilbert space L2,|r| ⊕ C2|∆| . From Lemma 6.2, kKk = kJKk ≤ κ and

0

α Z Z

= J 0

Z [∗] 0 Z [∗] 0 ≤ 2δ2 follows from (6.17). Thus f f J0 W01 0 f f , JW , = a a 0 α|∆|−1 a a 0 Z f J0 K 0 f f f , + , + J 0 0 a a a Z [∗] 0 a 0 Z f f = hJ0 W01 f, f i + α a∗ a + hJ0 Kf, f i + J , a Z h∗i 0 a α α ∗ ∗ ≥ hf, f i + hf, f i + a |∆|a a |∆|a − κhf, f i − δ2 2δ2 α α α ≥ 1−κ− hf, f i + − a∗ |∆|a 2δ2 δ2 2δ2 α α α ∗ = − hf, f i + a |∆|a (by (6.8)) δ2 2δ2 2δ2 α = hf, f i + a∗ |∆|a 2δ2 α f f = , , a a 2δ2 as required. (II) We start with the identity u(Kf )(−1) + v(Kf )(1) = η11 [f, ψ1 ] + η12 [f, ψ2 ],

f ∈ L2,|r| ,

(6.18)

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

497

which follows from the calculation u(Kf )(−1) + v(Kf )(1) Z Z 1 k(−1, t) f (t) r(t) dt + v =u −1 Z 1

=u

1

k(1, t) f (t) r(t) dt

−1

u η11 ψ 1 (t) + η12 ψ 2 (t) f (t) r(t) dt

−1

Z

1

+v

v η11 ψ 1 (t) + η12 ψ 2 (t) f (t) r(t) dt

−1

= |u|2 η11 [f, ψ1 ] + |u|2 η12 [f, ψ2 ] + |v|2 η11 [f, ψ1 ] + |v|2 η12 [f, ψ2 ] = η11 [f, ψ1 ] + η12 [f, ψ2 ]. By (6.1), the general element of F(A) takes the form   f   uf (−1) + vf (1) z where f ∈ Fmax and z ∈ C. Applying W to this vector, we obtain   g   w := η11 [f, ψ1 ] + η12 [f, ψ2 ] + α η11 uf (−1) + vf (1) + α η12 z  , ∗ where g := W01 f + Kf + uf (−1) + vf (1) ψ1 + z ψ2 ∈ Fmax by (6.4) and Lemma 6.2. Thus to prove that w ∈ F(A), it is enough to show that u g(−1)+v g(1) = η11 [f, ψ1 ]+η12 [f, ψ2 ]+α η11 uf (−1)+vf (1) +α η12 z. (6.19) To this end we calculate u g(−1) = u (W01 f )(−1) + (Kf )(−1) + uf (−1) + vf (1) ψ1 (−1) + z ψ2 (−1) = u (W01 f )(−1) + u (Kf )(−1) + α |u|2 η11 uf (−1) + vf (1) + α |u|2 η12 z from (6.10) and (6.12). Similarly v g(1) = v (W01 f )(1) + (Kf )(1) + uf (−1) + vf (1) ψ1 (1) + z ψ2 (1) = v (W01 f )(1) + v (Kf )(1) + α |v|2 η11 uf (−1) + vf (1) + α |v|2 η12 z. Adding and using (6.5), (6.18) and (6.2), we obtain (6.19). This completes the proof of the lemma. The theorem now follows from Theorem 2.2 and Lemma 6.3.

We now specialize Theorems 5.1, 5.2 and 6.1 to some of our earlier examples. First we consider Example 3.3 (cf. Example 4.2).

´ Binding and Curgus

498

IEOT

Corollary 6.4. Assume that p = 1 and r is of order ν0 > −1 on a half-neighborhood of 0, and of order ν1 > −1 on both a right half-neighborhood of −1 and a left halfneighborhood of 1. Then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . Now we consider Examples 4.3 and 4.4. Corollary 6.5. Assume that p is even, r is odd and that Condition 3.5 holds. If k = 0 or Condition 3.7 holds, then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| . As a simple illustration of this corollary we could consider the eigenvalue problem stated in Example 5.3 but with r odd and of order ν0 at 0 and ν1 at 1 (and hence of order ν1 at −1, since r is odd). Corollary 6.6. Assume that p is nearly even and r is nearly odd. If k = 0 or Condition 3.7 holds, then there is a basis for each root subspace of A, so that the union of all these bases is a Riesz basis of L2,|r| ⊕ C|∆| .

References [1] N. Abasheeva, S. Pyatkov, Counterexamples in indefinite Sturm-Liouville problems. Siberian Advances in Mathematics. Siberian Adv. Math. 7 (1997), 1–8. [2] T. Azizov, J. Behrndt, C. Trunk, On finite rank perturbations of definitizable operators. J. Math. Anal. Appl. 339 (2008), no. 2, 1161–1168. [3] R. Beals, Indefinite Sturm-Liouville problems and half-range completeness. J. Differential Equations 56 (1985), 391–407. [4] J. Behrndt, P. Jonas, Boundary value problems with local generalized Nevanlinna functions in the boundary condition, Integral Equations Operator Theory 56 (2006), 453–475. [5] J. Behrndt, C. Trunk, Sturm-Liouville operators with indefinite weight functions and eigenvalue depending boundary conditions, J. Differential Equations 222 (2006), 297–324. ´ [6] P. Binding, B. Curgus, Form domains and eigenfunction expansions for differential equations with eigenparameter dependent boundary conditions. Canad. J. Math. 54 (2002), 1142–1164. ´ [7] P. Binding, B. Curgus, A counterexample in Sturm-Liouville completeness theory. Proc. Roy. Soc. Edinburgh Sect. 134A (2004), 244–248. ´ Riesz basis of root vectors of indefinite Sturm-Liouville prob[8] P. Binding, B. Curgus, lems with eigenparameter dependent boundary conditions. I. Oper. Theory Adv. Appl. 163 (2006), 75–96. ´ [9] B. Curgus, On the regularity of the critical point infinity of definitizable operators. Integral Equations Operator Theory 8 (1985), 462–488. ´ [10] B. Curgus, H. Langer, A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differential Equations 79 (1989), 31–61.

Vol. 63 (2009)

Riesz Bases of Root Vectors. II

499

´ [11] B. Curgus, B. Najman, Quasi-uniformly positive operators in Krein space. Oper. Theory Adv. Appl. 80 (1995), 90–99. [12] A. Dijksma, Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), no. 1–2, 1–27. [13] A. Fleige, The “turning point condition” of Beals for indefinite Sturm-Liouville problems. Math. Nachr. 172 (1995), 109–112. [14] A. Fleige, A counterexample to completeness properties for indefinite SturmLiouville problems. Math. Nachr. 190 (1998), 123–128. [15] G. Freiling, M. Vietri, V. Yurko, Half-range expansions for an astrophysical problem, Lett. Math. Phys. 64 (2003), 65–73. [16] H. Langer, Spectral function of definitizable operators in Krein spaces. Functional Analysis, Proceedings, Dubrovnik 1981. Lecture Notes in Mathematics 948, SpringerVerlag, 1982, 1-46. [17] I. Karabash, A. Kostenko, M. Malamud, The similarity problem for J-nonnegative Sturm-Liouville operators, J. Differential Equations (2008), to appear. d2 [18] I. Karabash, M. Malamud, Indefinite Sturm-Liouville operators (sgn x) − dx 2 +q(x) with finite-zone potentials. Oper. Matrices 1 (2007), no. 3, 301–368. [19] A. Parfenov, On an embedding criterion for interpolation spaces and application to indefinite spectral problems, Siberian Math. J. 44 (2003), 638–644. [20] S. Pyatkov, Interpolation of some function spaces and indefinite Sturm-Liouville problems. Oper. Theory Adv. Appl. 102 (1998), 179–200. [21] S. Pyatkov, Some properties of eigenfunctions and associated functions of indefinite Sturm-Liouville problems, in: Nonclassical Problems of Mathematical Physics, Sobolev Institute of Mathematics, Novosibirsk, 2005, 240–251. [22] E. Russakovskii, The matrix Sturm-Liouville problem with spectral parameter in the boundary condition: algebraic operator aspects, Trans. Moscow Math. Soc. 1996 (1997), 159–184. [23] C. Tretter, Nonselfadjoint spectral problems for linear pencils N −λP of ordinary differential operators with λ-linear boundary conditions: completeness results. Integral Equations Operator Theory 26 (1996), no. 2, 222–248. [24] H. Volkmer, Sturm-Liouville problems with indefinite weights and Everitt’s inequality. Proc. Roy. Soc. Edinburgh Sect. 126A (1996), 1097–1112. Paul Binding Department of Mathematics and Statistics, University of Calgary Calgary, Alberta, T2N 1N4, Canada e-mail: [email protected] ´ Branko Curgus Department of Mathematics, Western Washington University Bellingham, WA 98225, USA e-mail: [email protected] Submitted: July 31, 2008. Revised: December 17, 2008.

Integr. equ. oper. theory 63 (2009), 501–520 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040501-20, published online April 2, 2009 DOI 10.1007/s00020-009-1674-1

Integral Equations and Operator Theory

Carleson Measures via BMO Boo Rim Choe, Hyungwoon Koo and Michael Stessin Abstract. We obtain new characterizations of Carleson measures via uniform boundedness of BMO norms of certain mass functions associated with the given measure in a natural way. Mathematics Subject Classification (2000). Primary 30D55; Secondary 30D50. Keywords. Carleson measure, BMO, Carleson approach region.

1. Introduction Let D be the unit disk in the complex plane and T be the unit circle, the boundary of D. Given an arc I ⊂ T, we denote by SI the Carleson window consisting of all points rζ ∈ D such that 1 − |I| ≤ r < 1 and ζ ∈ I where |I| denotes the normalized arclength of I. Let µ be a finite positive Borel measure on D (hereafter we simply write µ ≥ 0). Given α > 0, we say that µ ≥ 0 is an α-Carleson measure if kµkα := sup I

µ(SI) < ∞. |I|α

The name “Carleson measure” comes from the celebrated work [C] of L. Carleson who proved that µ ≥ 0 is a 1-Carleson measure if and only if H 1 (D), the Hardy space on D, is contained in L1 (µ) and applied it in the solution of the Corona Theorem. P. Duren [D] then noticed, as a special case of his generalization, that the containment H 1 (D) ⊂ L1 (µ) in Carleson’s characterization can be replaced by H p (D) ⊂ Lp (µ) with arbitrary 0 < p < ∞. It is now known that to each α > 0 This research was performed during M. Stessin’s visit to Korea University. He thanks the Mathematics Department of Korea University and the “Brain Pool” program for their hospitality and support. The first two authors were supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-314-C00012).

502

Choe, Koo and Stessin

IEOT

correspond Banach spaces or F -spaces Xαp of analytic functions on D such that αCarleson measures are precisely those measures µ ≥ 0 for which Xαp ⊂ Lp (µ) (with or without restriction on p), or said differently via the Closed Graph Theorem, the embedding Xαp ⊂ Lp (µ) is continuous. As mentioned above, we have X1p = H p (D), 0 < p < ∞. Later W. Hastings [H] showed X22 = A2 (D), the Bergman space on D. This characterization was then extended to all α > 1 by D. Stegenga [S] who proved Xα2 = A2α−2 (D), the Bergman space with respect to the weighted measure (1−|z|2 )α−2 dA(z) where dA is the area measure on D. Proofs of these results, as well as analogues for arbitrary weighted Bergman spaces Apα−2 (D) with 0 < p < ∞, now can be found in standard references [CM, Theorem 2.38] or [Z, Theorem 7.4]. For 0 < α < 1, R. Zhao and K. Zhu [ZZ, Theorem 46] have recently shown that a class of weighted Bergman-Sobolev spaces plays the role of Xαp . More precisely, they showed Xαp = R−s Apps+α−2 (D) with 0 < p ≤ 1 and ps + α > 1, where R−s denotes the radial differentiation of order −s < 0. Given s > 0, recall that a function f ∈ L1loc ((0, s]) is said to belong to BMO((0, s]) if Z 1 |f (x) − fJ | dx < ∞ kf k∗,(0,s] := sup J |J| J where the supremum is taken over R all intervals J ⊂ (0, s]. Here, fJ denotes the 1 mean of f over J, i.e., fJ = |J| f (x)dx. Note that we are abusing the notation J |J| for the length of J. This should cause no confusion from the context. Note that kf k∗,(0,s] decreases, as s does. See [G] or [T] for more information on BMO functions. We let kf k∗ = kf k∗,(0,1] for short. In this paper we introduce a couple of families of functions with parameter ζ ∈ T, naturally associated with given µ, and characterize α-Carleson measures by means of uniform boundedness of BMO norms of functions in each of those families. These families are based on two “mass” functions. The first one, denoted by Fµα , measures certain weighted total masses on compact sets and the second one, denoted by Gα µ , measures scaled total masses outside compact sets. Precise definitions of these two functions are given below. Given α > 0 and µ ≥ 0, we define Z dµ(z) Fµα (t) = α D1−t (1 − |z|) and µ (D \ D1−t ) tα for 0 < t ≤ 1. Here, Dr = {z ∈ D : |z| < r} for 0 < r ≤ 1 and D0 = ∅. Note that Fµα is lower semicontinuous and Gα µ is upper semicontinuous. In particular, they are both measurable functions on (0, 1]. Also, we define Gα µ (t) =

α α Φα µ = Fµ + Gµ .

Vol. 63 (2009)

Carleson Measures

503

Our first result is the following estimates for each α > 0: α α ||Gα µ ||∞ ≈ µ(D) + ||Fµ ||∗ ≈ µ(D) + ||Φµ ||∗ ;

(1.1)

see Theorem 2.3. Here, ||Gα µ ||∞ denotes the supremum (not the essential supremum) of Gα over (0, 1]. In the course of proofs we prove some properties of deµ creasing BMO functions. In order to introduce our families of functions associated with each α > 0 and µ ≥ 0, we first recall nontangential approach regions. Given ζ ∈ T and β > 1, let Γ(β; ζ) be the nontangential approach region with vertex at ζ and aperture β consisting of all points z ∈ D such that |1 − zζ| < β(1 − |z|). In the rest of the paper we fix β = 2, unless otherwise specified; our results and proofs remain valid for arbitrary β > 1 with obvious modifications. So, we simply let Γ(ζ) = Γ(2; ζ). Given ζ ∈ T and µ ≥ 0, we denote by µζ := µ|Γ(ζ) the restriction of µ to Γ(ζ). Now, {Fµαζ }ζ∈T and {Φα µζ }ζ∈T are the families of functions to be used in our characterizations of α-Carleson measures. Our next result, based on the estimates (1.1), is the quantitative characterization of α-Carleson measures by means of BMO-norms of Fµαζ or Φα µζ . Namely, we show α kµkα ≈ µ(D) + sup ||Φα µζ ||∗ ≈ µ(D) + sup ||Fµζ ||∗ ; ζ∈T

(1.2)

ζ∈T

see Theorem 3.2. This result is reminiscent in form of, but seems to have no connection with, the BMO characterization of radially bounded measures; see [K, p. 245]. We also obtain the compact version in Theorem 3.3. These results are restricted to the case α > 1 and an example is constructed to illustrate the failure for 0 < α ≤ 1. However, if we consider modified mass functions Feµαζ measuring eα weighted total masses away from ζ and G µ measuring scaled total masses near ζ, ζ

then the analogue of (1.2), as well as its compact analogue, turns out to hold for all α > 0. See the remarks at the end of Section 3. In Section 4 we show that the choice of Γ(ζ) as the basic region for our characterization of Carleson measures is natural. Theorem 4.1 shows that every convex approach region which admits this characterization must lie inside some of Γ(β, ζ). We do not know whether convexity hypothesis is essential and are led to a problem posed in Section 5. Constants. Throughout the paper we use the same letter C to denote various constants which may change at each occurrence. For nonnegative quantities X and Y , we often write X . Y or Y & X if X is dominated by Y times some inessential positive constant that may depend on allowed parameters. Also, we say that X and Y are comparable and write X ≈ Y if X . Y . X.

504

Choe, Koo and Stessin

IEOT

2. Estimates of ||Gαµ ||∞ and ||Fµα ||∗ In this section we establish the estimates in (1.1). We begin with a couple of facts, Lemma 2.1 and Proposition 2.2 below, which play key roles in our proofs. These facts may have been known, but we are not aware of any reference and so include proofs for completeness. Lemma 2.1. Let f, g ∈ L1 (J) for an interval J ⊂ R. If f, g and f − g are real valued decreasing functions on J, then Z Z 1 1 |g(x) − gJ | dx ≤ |f (x) − fJ | dx. |J| J |J| J Proof. Being a decreasing function on J, the function f has one-sided limits at each interior point of J. We denote by f (x+ ) and f (x− ), respectively, the righthand limit and the left-hand limit of f at an interior point x of J. Also, g(x+ ) and g(x− ) are defined similarly. Since g is decreasing, we have gJ = θg(a− ) + (1 − θ)g(a+ ) for some a in the interior of J and θ ∈ [0, 1]. Consider a function h on J defined by h(x) = g(x) − gJ + c where c = θf (a− ) + (1 − θ)f (a+ ). Note that f (x) − h(x) = f (x) − g(x) + gJ − c = θ[f (x) − g(x) + g(a− ) − f (a− )] + (1 − θ)[f (x) − g(x) + g(a+ ) − f (a+ )] for all x ∈ J. Since f − g is decreasing, it follows that f − h ≥ 0 on J ∩ (−∞, a)

(2.1)

h − f ≥ 0 on J ∩ (a, ∞).

(2.2)

and

Note that we also have f (b− ) ≥ fJ ≥ f (b+ ) for some b in the interior of J. We now consider two cases hJ ≥ fJ and fJ ≥ hJ separately. First, consider the case hJ ≥ fJ . Since f (a− ) ≥ c = hJ ≥ fJ ≥ f (b+ ), we may assume a ≤ b so that J1 ⊃ J2 where J1 = J ∩ (−∞, b) and J2 = J ∩ (−∞, a). Since f ≥ f (b− ) ≥ fJ on J1 , we obtain Z Z Z |f (x) − fJ | dx = 2 (f (x) − fJ ) dx ≥ 2 (f (x) − fJ ) dx. J

J1

J2

Vol. 63 (2009)

Carleson Measures

505

Meanwhile, since f − fJ ≥ h − hJ = g − gJ on J2 by (2.1) and assumption that hJ ≥ fJ , we have Z Z Z 1 (f (x) − fJ ) dx ≥ (g(x) − gJ ) dx = |g(x) − gJ | dx. 2 J J2 J2 So our assertion holds if hJ ≥ fJ . Next, consider the case hJ ≤ fJ . This time we have f (b− ) ≥ fJ ≥ hJ = c ≥ f (a+ ). Thus we may assume b ≤ a so that J3 ⊃ J4 where J3 = J ∩ (b, ∞) and J4 = J ∩ (a, ∞). Now, as above, we obtain Z Z Z |f (x) − fJ | dx = 2 (fJ − f (x)) dx ≥ 2 (f (x) − fJ ) dx. J

J3

J4

Also, since fJ − f ≥ hJ − h = gJ − g on J4 by (2.2) and assumption that hJ ≤ fJ , we have Z Z Z 1 |g(x) − gJ | dx. (gJ − g(x)) dx = (fJ − f (x)) dx ≥ 2 J J4 J4 So , our assertion also holds if hJ ≤ fJ . This completes the proof.

Proposition 2.2. The inequalities 1 ||f ||∗,(0,s] ≤ sup [f (x) − f (2x)] 6 x∈(0,s/2]

(2.3)

[f (x) − f (2x)] ≤ 6||f ||∗,(0,s]

(2.4)

and sup x∈(0,s/3]

hold whenever s > 0 and f : (0, s] → R is a decreasing function. We remark that the range (0, s/3] over which the supremum taken in (2.4) cannot be extended to the whole (0, s/2]. To see an example (for s = 1), consider a sequence f1 , f2 , . . . of decreasing functions defined by 1 1 fj (x) = 1 + log2 1 − x + j , 0 < x ≤ 1. j 2 Note fj (1/2) − fj (1) = fj (1/2) → 1 but kfj k∗ → 0 as j → ∞. In order to see the last convergence we recall (see [T, p. 201]) that log x belongs to BMO over (0, ∞) (with BMO norm at most 2/ log 2 < 3). Proof. Fix s > 0 and a decreasing function f : (0, s] → R . We first prove (2.3). Let a, b ∈ (0, s] with a < b be given. Choose an integer k ≥ 0 such that 2−k−1 b ≤ a < 2−k b. Put M1 =

sup x∈(0,s/2]

[f (x) − f (2x)] .

506

Choe, Koo and Stessin

IEOT

Since f (a) ≤ f (2−k−1 b) and k < log2 (b/a), we have f (a) − f (b) ≤ f (2−k−1 b) − f (b) =

k X

[f (2−j−1 b) − f (2−j b)]

j=0

≤ (k + 1)M1 ≤ 2M1 log2 (b/a). This means that the function 2M1 log(1/x) − f (x) is a decreasing function on (0, s]. So, we obtain kf k∗,(0,s] ≤ 2M1 k log2 (1/x)k∗,(0,s] ≤ 6M1 by Lemma 2.1 and conclude (2.3). We now show (2.4). Put M2 =

sup

[f (x) − f (2x)] .

x∈(0,s/3]

Fix a ∈ (0, s/3] and let J = [a, 2a]. Since f (2a) ≤ fJ ≤ f (a), we have f (a) − f (2a) ≤ 2 max{f (a) − fJ , fJ − f (2a)}. Thus it is sufficient to show max{f (a) − fJ , fJ − f (2a)} ≤ 3kf k∗,(0,s] .

(2.5)

Let J 0 = (0, 2a]. Since f is decreasing, we have fJ 0 ≥ fJ . Thus we have Z 2a Z 2a 1 1 |f (x) − fJ 0 | dx ≥ 0 (fJ 0 − fJ ) − |f (x) − fJ | dx |J 0 | a |J | a 1 ≥ ((fJ 0 − fJ ) − kf k∗ ). 2 Also, since f is decreasing, we have Z a 1 f (a) − fJ 0 (f (x) − fJ 0 ) dx ≥ . |J 0 | 0 2 Combining these observations, we obtain Z 2 |f (x) − fJ 0 | dx ≥ f (a) − fJ − kf k∗,(0,s] . 2kf k∗,(0,s] ≥ 0 |J | J 0 Thus we conclude f (a) − fJ ≤ 3kf k∗,(0,s] , which is one half of the estimate (2.5). For the other half of the estimate (2.5), we consider the interval J 00 = [a, 3a] ⊂ (0, s]. Since f is decreasing, we have fJ ≥ fJ 00 . Thus Z 2a Z 2a 1 1 00 |f (x) − fJ | dx ≥ 00 (fJ − fJ 00 ) − |f (x) − fJ | dx |J 00 | a |J | a 1 ≥ ((fJ − fJ 00 ) − kf k∗,(0,s] ). 2

Vol. 63 (2009)

Carleson Measures

507

Also, since f is decreasing, we have Z 3a 1 1 (fJ 00 − f (x)) dx ≥ (fJ 00 − f (2a)). 00 |J | 2a 2 Combining these observations, we obtain Z 2 |f (x) − fJ 00 | dx ≥ fJ − f (2a) − kf k∗,(0,s] 2||f ||∗,(0,s] ≥ 00 |J | J 00 and therefore conclude fJ − f (2a) ≤ 3kf k∗,(0,s] , which completes the proof.

We are now ready to prove the main result of this section. Theorem 2.3. Given α > 0, the estimate α ||Gα µ ||∞ ≈ µ(D) + ||Fµ ||∗ α holds for all µ ≥ 0. The same estimate holds with Φα µ in place of Fµ . α Proof. Fix α > 0 and µ ≥ 0. Put F = Fµα , G = Gα µ and Φ = Φµ for simplicity. Note µ(D1−r \ D1−s ) µ(D1−r \ D1−s ) ≤ F (r) − F (s) ≤ (2.6) sα rα for 0 < r < s < 1. Given 0 < t ≤ 1, we have ∞ X µ (D1−2−j−1 t \ D1−2−j t ) G(t) = 2jα · (2−j t)α j=0 (2.7) ∞ X F (2−j−1 t) − F (2−j t) ≤ 2jα j=0

where the inequality comes from (2.6). Thus, taking supremum over t ∈ (0, 2/3], we obtain ||F ||∗ sup G(t) . 1 − 2−α 0
508

Choe, Koo and Stessin

IEOT

Since kGk∗ ≤ 2kGk∞ , the second inequality with Φ in place of F follows from the second inequality for F . For 0 < r < s ≤ 1, we have by (2.6) Φ(r) − Φ(s) = F (r) − F (s) + G(r) − G(s) r α ≥ G(s) − G(r) + G(r) − G(s) h s r α i . = G(r) 1 − s This shows that Φ is a decreasing function. Also, substituting r = t and s = 2t, we have Φ(t) − Φ(2t) ≥ 1 − 2−α G(t), 0 < t ≤ 1/2. Thus, taking the supremum over t ∈ (0, 1/3], we obtain by Proposition 2.2 sup G(t) . kΦk∗ . 0
This yields the first inequality with Φ in place of F , as above. The proof is complete. As a by-product of the proof we have the following corollary. Corollary 2.4. Given α > 0, the estimates α lim sup Gα µ () ≈ lim ||Fµ ||∗,(0,]

(2.9)

→0

→0

and ! lim sup →0

sup Gα µζ () ζ∈T

! ≈ lim

→0

sup ||Fµαζ ||∗,(0,] ζ∈T

(2.10)

α α α hold for all µ ≥ 0. The same estimates hold with Φα µ (Φµζ ) in place of Fµ (Fµζ ).

Proof. Fix α > 0. Let ∈ (0, 1] and µ be given. Using (2.7) and (2.8), we have ||Fµα ||∗,(0,/3] .

sup

α Gα µ (t) . ||Fµ ||∗,(0,]

0
by Proposition 2.2. This yields (2.9). Also, using the above inequalities with µζ , ζ ∈ T, we have (2.10). Similarly, the second part holds.

3. Carleson measures and BMO In this section we establish the estimates in (1.2), as well as their compact analogues. We also construct an example showing that the parameter range α > 1 is sharp. Given µ ≥ 0 and ζ ∈ T, recall µζ = µ|Γ(ζ) . Also, recall that the functions Fµαζ and Gα µζ are given by Z dµ(z) Fµαζ (t) = α Γ(ζ)∩D1−t (1 − |z|)

Vol. 63 (2009)

Carleson Measures

509

and µ (Γ(ζ) \ D1−t ) tα for 0 < t ≤ 1. In what follows we let Gα µζ (t) =

Γt (ζ) = Γ(ζ) \ D1−t for simplicity. We first prove the following pointwise estimates. Proposition 3.1. Given α > 0, the estimate ||µζ ||α ≈ µ(Γ(ζ)) + ||Fµαζ ||∗ α holds for all µ ≥ 0 and ζ ∈ T. The same estimate holds with Φα µζ in place of Fµζ .

Proof. By Theorem 2.3 we only need to show the first part of the proposition. Fix α α > 0, ζ ∈ T and µ ≥ 0. Note Gα µζ (t) = µ (Γt (ζ)) /t . Thus, by Theorem 2.3, it is sufficient to show µ (Γt (ζ)) . (3.1) ||µζ ||α ≈ sup tα 0
µ(Γct (ζ)) µ(Γt (ζ)) = cα sup . tα tα 0
(3.2)

Also, we have µ(Γ(ζ)) ||µζ ||α µ(Γt (ζ)) ≤ ≤ . α α t c cα c
So, we obtain the estimate for & in (3.1). This completes the proof.

Closely examining the proof of Proposition 3.1, we also see that the estimate (α > 0 fixed) ! ! µ(SI ∩ Γ(ζ)) α lim sup sup Gµζ () ≈ lim sup sup (3.3) |I|α →0 →0 ζ∈T ζ∈T,|I|≤ holds for all µ ≥ 0. This will be used later in the proof of Example 3.4. We now prove the following quantitative characterizations of α-Carleson measures for α > 1.

510

Choe, Koo and Stessin

IEOT

Theorem 3.2. Let α > 1. Given µ ≥ 0, the following statements are equivalent: (a) µ is an α-Carleson measure; (b) supζ∈T ||Φα µζ ||∗ < ∞; (c) supζ∈T ||Fµαζ ||∗ < ∞. Moreover, the quantities α kµkα , µ(D) + sup ||Φα µζ ||∗ , µ(D) + sup ||Fµζ ||∗ ζ∈T

ζ∈T

are comparable to one another. Proof. Fix µ ≥ 0. Since µζ (SI) ≤ µ(SI) for all ζ and I, we clearly have supζ∈T kµζ kα ≤ kµkα . We will prove the estimate for the converse direction: sup kµζ kα ≥ Ckµkα

(3.4)

ζ∈T

for some constant C > 0 independent of µ and ζ. The second part of the theorem easily follows from this estimate and Proposition 3.1, because supζ∈T µ(Γ(ζ)) ≤ µ(D) ≤ kµkα . We now turn to the proof of (3.4). Let I ⊂ T with |I| = t be given. Decompose SI = ∪∞ k=0 Qk where Qk = {z ∈ SI : 2−k−1 t < 1 − |z| ≤ 2−k t}. It is not hard to find an absolute positive integer N such that Q0 is covered by N regions Γt (ζ0j ) with ζ0j ∈ I for j = 1, . . . , N and the next region Q1 is then covered by 2N regions Γt/2 (ζkj ) with ζ1j ∈ I for j = 1, . . . , 2N , and so on. In general each Qk is covered by 2k N regions Γ2−k t (ζkj ) with ζkj ∈ I for j = 1, . . . , 2k N . Put k N sk = ∪2j=1 Γ2−k t (ζkj ). Setting Mt =

µ(Γs (ζ)) , sα ζ∈T,0<s≤t sup

we have µ (sk ) ≤

k 2X N

µ (Γ2−k t (ζkj )) ≤ Mt · 2k N · (2−k t)α =

j=1

Mt N t α 2k(α−1)

for each k. Now, since SI ⊂ ∪∞ k=0 sk , we obtain µ(SI) ≤

∞ X

µ (sk ) ≤

k=0

Mt N tα . 1 − 21−α

(3.5)

Since I is arbitrary, this yields the estimate kµkα ≤ CM for some C = C(α) where M = sup Mt = sup kGα µζ k∞ . 0
ζ∈T

Now the estimate (3.4) follows from Proposition 3.1 and Theorem 2.3.

Vol. 63 (2009)

Carleson Measures

511

We also obtain the compact version of Theorem 3.2. We say that µ ≥ 0 is a vanishing α-Carleson measure if µ(SI) = 0. |I|→0 |I|α lim

For α > 1, it is well known that compact α-Carleson measures are precisely those measures for which the inclusion A2α−2 (D) ⊂ L2 (µ) is compact; see, for example, [Z, Theorem 7.7] where hyperbolic disks are used in place of Carleson windows. Theorem 3.3. Let α > 1 and µ ≥ 0. Then the following statements are equivalent. (a) µ is a vanishing α-Carleson measure; ! (b) lim

→0

sup kFµαζ k∗,(0,]

= 0;

ζ∈T

! (c) lim

→0

sup kΦα µζ k∗,(0,]

= 0.

ζ∈T

Proof. Fix α > 1. Let µ ≥ 0. By Corollary 2.4 we only need to show that (a) and (b) are equivalent. By Corollary 2.4 we have ! ! µ (Γ (ζ)) α = 0. (3.6) lim sup kFµζ k∗,(0,] = 0 ⇐⇒ lim sup →0 ζ∈T →0 ζ∈T α Let c be the constant introduced in the proof of Proposition 3.1. Note that c depends only on the aperture of nontangential approach regions. Then, proceeding as in the proof of (3.2), we have ! µζ (SI) µ (Γt (ζ)) µ (SI) ζ sup ≤ sup ≤ c−α sup (3.7) α α tα 0
|I|→0

µ (SI) = 0, |I|α

(3.9)

i.e., µ is a vanishing α-Carleson measure. Since µζ (SI) ≤ µ(SI) for all ζ and I, that (3.9) implies (3.8) is trivial. For the converse, note that the estimate (3.5) is valid for all I and ζ. It follows from (3.5) and (3.7) that µ(Γs (ζ)) µζ (SI) µ (SI) . sup . sup α α α |I| s ζ∈T,0<s≤c ζ∈T,0<|I|≤ |I| |I|≤c sup

for all small > 0. Thus (3.8) implies (3.9). This completes the proof.

512

Choe, Koo and Stessin

IEOT

Note that the proof of Theorem 3.2 works only for α > 1. This naturally raises a question whether the uniform estimates as in Theorem 3.2 remains true for α ≤ 1. The answer is negative as the following example shows. Example 3.4. Given 0 < α ≤ 1, there exists µ ≥ 0 such that µ is not an α-Carleson measure, but ! sup ||Fµαζ ||∗ < ∞

and

ζ∈T

lim

→0

sup ||Fµαζ ||∗,(0,]

= 0.

ζ∈T

Proof. Fix 0 < α ≤ 1. We actually construct a measure µ ≥ 0 such that µ is not an α-Carleson measure, but −1 1 α (3.10) sup µ(SI ∩ Γ(ζ)) ≤ C|I| log |I| ζ∈T for some constant C > 0 independent of µ and I ⊂ T. This implies our assertion by Proposition 3.1, Corollary 2.4 and (3.3). We introduce some notation first. Given a positive integer k and an odd √ −k integer j with 1 ≤ j < 2k , we let ζk,j = ej2 i ∈ T where i = −1. To each ζk,j we assign the arc Λk,j = {z ∈ Γ(ζk,j ) : |z| = 1 − 2−k−2 , 0 < arg z < 1}. Note Λk,j ∪ Λk,` = ∅ for j 6= `. In fact, if z ∈ Λk,j ∩ Λk,` with j 6= `, then we have 7 1 |ζ − ζ k,j | < |z||ζ k,` − ζ k,j | < 4(1 − |z|) < k 8 k,` 2 and thus

−k+1 2 1 1 4 i · k−1 < |1 − e2 | ≤ |ζk,` − ζk,j | < · k−1 , π 2 7 2 which is impossible. Given k, pick the largest integer Nk such that Nk ≤ (2k /k)α . Set

Λ=

∞ [

Λk

k=1

where

Λk =

Nk [

Λk,j

j=1

and consider a measure µ supported on Λ such that µ is the arclength measure on each Λk,j , 1 ≤ j ≤ Nk , scaled to have total length 1/(k2kα ) (independent of j). It is easy to see that µ is a finite measure. Now, fix a positive integer m and let Im = {eiθ : |θ| ≤ 2−m+1 }. Assume z ∈ Λm+k,j for some k and j < 2k . Write z = reiθ where r = 1 − 2−(m+k+2) . Since 2(1 − r) > |1 − reiθ ζ k+m,j | ≥ |1 − eiθ | − (1 − r) − r|1 − ζ k+m,j | and |1 − ζ k+m,j | < j/2m+k , we have |θ| 3 + 4j < |1 − eiθ | < 3(1 − r) + |1 − ζ k+m,j | < m+k+2 2 2

Vol. 63 (2009)

Carleson Measures

513

and thus

3 + 4j 2k+2 − 1 1 ≤ < m−1 . 2m+k+1 2m+k+1 2 This shows that Λm+k,j ⊂ SIm whenever j < 2k . Moreover, if k ≤ 2m − m, then 2k ≤ 2m+k /(m + k) and thus j ≤ Nm+k for j < 2kα . Note 2kα ≤ 2k , because α ≤ 1. It follows that |θ| <

µ(SIm ) ≥

2m −m X

X

µ(Λm+k,j ) ≈

k=1 j<2kα

so that

2m −m X k=1

2kα−1 (m + k)2(m+k)α

m

mα

2

m 2 X 1 2 µ(SIm ) & ≈ log →∞ ` m+1 `=m+1

as m → ∞. This shows that µ is not an α-Carleson measure. We now show (3.10). Let I ⊂ T with |I| small and ζ ∈ T be given. Note that if SI ∩ Γ(ζ) 6= ∅, then the length of the smallest arc with center at ζ containing I is comparable to |I|. Thus we may assume I is centered at ζ. Choose m such that 2−(m+3) < |I| ≤ 2−(m+2) . First, consider the case where ζ = ζk,j for some k and j. Our construction shows Γ(ζk,j )∩Λ` = ∅ for ` > k. Note SI ∩Λ` = ∅ for ` < m. Thus SI ∩Γ(ζk,j ) = ∅ if k < m. So, assume k ≥ m. Our construction also shows that the set Γ(ζk,j ) can intersect with at most one arc from each Λ` . Accordingly, we have −1 k X 1 1 α (3.11) . |I| log , µ(SI ∩ Γ(ζk,j )) ≤ `2`α |I| `=m

which yields (3.10). Next, consider the general case. We may assume 0 < arg ζ < 1. Choose a large k > m such that   −1 [ 1 µ Λ \ Γ(ζk,j ) < |I|α log (3.12) |I| j and an odd integer j1 < 2k − 2 such that j1 2−k < arg ζ ≤ (j1 + 2)2−k . Note Λ ∩ Γ(ζ) ∩ [∪j Γ(ζk,j )] is contained in Γ(ζk,j1 ) ∪ Γ(ζk,j1 +2 ). Thus we have (3.10) by (3.11) and (3.12). The proof is complete. As we’ve seen from Example 3.4, the failure of Theorem 3.2 for α ≤ 1 is caused by the fact that supζ∈T kµζ kα can be much smaller than kµkα . In conjunction with this observation, we note kµkα =

µ(SIt (ζ)) tα ζ∈T,0
(3.13)

where It (ζ) denotes the arc centered at ζ whose normalized arclength is t. It is easy to see that the sets SIt (ζ) can be replaced by equivalent choices Dt (ζ) = {z ∈ D : |ζ − z| < t}.

514

Choe, Koo and Stessin

IEOT

Now, we can define modified mass functions by means of the sets Dt (ζ) as follows. Given α > 0, µ ≥ 0 and ζ ∈ T, we define Z dµ(z) α Fµ,ζ (t) = α D\Dt (ζ) |1 − zζ| and Gα µ,ζ (t) =

µ (Dt (ζ)) tα

for 0 < t ≤ 1. Also, we define α α Φα µ,ζ = Fµ,ζ + Gµ,ζ .

It is easily seen that µ(Ds (ζ) \ Dr (ζ)) µ(Ds (ζ) \ Dr (ζ)) α α ≤ Fµ,ζ (r) − Fµ,ζ (s) ≤ sα rα and h r α i α α Φα . µ,ζ (r) − Φµ,ζ (s) ≥ Gµ,ζ (r) 1 − s for 0 < r < s ≤ 1. Thus, easily modifying the proof of Theorem 3.2, we see that the quantities sup ||Gα µ,ζ ||∞ , ζ∈T

α µ(D) + sup ||Fµ,ζ ||∗ , ζ∈T

µ(D) + sup ||Φα µ,ζ ||∗ ζ∈T

are comparable to one another. Note that the first quantity above is comparable to the one on the right side of (3.13). Thus we conclude that the analogue of Theorem 3.2 with modified mass functions holds for all α > 0. Similarly, the analogue of Theorem 3.3 with modified mass functions holds for all α > 0.

4. Approach regions In this section we show that the choice of nontangential approach regions for our BMO characterization of Carleson measures is natural. Note kµkα ≤ kGα µ k∞ for all α > 0 and µ ≥ 0. Thus we see from Theorem 2.3 that if Fµα ∈ BMO, then µ is an α-Carleson measure. The converse is also true by Proposition 3.1 for µ supported in a nontangential approach region. Motivated by these results, we are led to the following definition. Definition. Let ζ ∈ T and Ω be a domain in D. We say that a domain Ω ⊂ D is a Carleson approach region at ζ, if Ω has the following properties: (A1) ∂Ω ∩ T = {ζ}; (A2) There exists some α > 0 such that if µ ≥ 0 is an α-Carleson measure supported in Ω, then Fµα ∈ BMO.

Vol. 63 (2009)

Carleson Measures

515

For example, any domain satisfying (A1) and contained in a nontangential approach region are all Carleson approach regions. As far as convex regions are concerned, it turns out that Carleson approach regions are essentially the same as nontangential approach regions. In what follows we say that Ω is convex near ζ, if Ω satisfies (A1) and Ω ∩ U is convex for some neighborhood U of ζ. Theorem 4.1. Let ζ ∈ T and Ω ⊂ D be a domain convex near ζ. Then Ω is a Carleson approach region at ζ if and only if Ω ⊂ Γ(β; ζ) for some β. We do not know whether the convexity hypothesis, which plays an essential role in the proof below, is redundant. Proof. For simplicity we may assume ζ = 1. The sufficiency follows from Proposition 3.1, because Γ(β, 1) ∩ D ⊂ Γ(β 0 ; 1) for β 0 > β > 1. For the necessity, which is the harder part, we assume that Ω is not contained in any Γ(β; 1) and derive a contradiction. We let θ(z) = arg z. In order to derive a contradiction we need to construct α-Carleson measures / BMO. By (A1) points in Ω µ = µ(α), α > 0, supported in Ω such that Fµα ∈ and away from 1 stay in a compact set, which we may ignore in the construction of α-carleson measures. Also, since Ω is not contained in any Γ(β, 1), the same is true for either Ω ∩ {θ(z) > 0} or Ω ∩ {θ(z) < 0}. So, passing to the intersection with a neighborhood of 1 if necessary, we may assume that Ω itself is convex and is contained in {0 < θ(z) < 1}. Before defining our measures, we need some preparation. Let ω ∈ (0, 1) be the maximal argument of points in ∂Ω. For θ ∈ [0, ω], denote by z θ the point in ∂Ω closest to T and let E be the set of all such z θ . For θ ∈ [0, ω], define δ(θ) by δ(θ) = 1 − |z θ |. Note that δ is a strictly increasing convex function, and δ(0) = 0. Moreover, since Ω is not contained in any angular sector with vertex 1, we have δ(θ) − δ(θ0 ) = o(|θ − θ0 |)

(4.1)

0

as θ, θ → 0. Fix a positive integer k0 such that 2−k0 < δ(ω). Then, given an integer k ≥ k0 there is a unique point zk in E such that |zk | = 1 − 2−k . Put θk = θ(zk ) so that δ(θk ) = 2−k . Note that θk is decreasing to 0 as k goes to infinity. Further, denoting by nk the largest integer which is not bigger than (θk − θk+1 )2k+1 , we see from (4.1) that nk ≈ (θk − θk+1 )2k+1 → ∞,

as k → ∞.

(4.2)

For α > 1, put sk = nk . For 0 < α ≤ 1, let sk be the largest integer not bigger than α/2 nk . We may assume sk > 0. Break the interval [θk+1 , θk ] into sk subintervals of equal length (θk − θk+1 )/sk , and denote by θk,j , j = 0, . . . , sk , the partition points, so that θk,0 = θk+1 and θk,sk = θk . Also, let zk,j be the unique point in E such that θ(zk,j ) = θk,j for j = 0, . . . , sk . Denote by Ed,k the set of all points zk,j with 0 ≤ j < sk and let Ed = ∪k Ed,k .

516

Choe, Koo and Stessin

IEOT

We now define a discrete measure µ supported in Ed , and thus supported in Ω, by α µ(zk,j ) = [δ(θk,j )] for each allowed pair of k and j. Note that our construction above shows that contribution from each atom to Fµα is precisely 1. Thus we have by (4.2) 1 1 α − F = sk → ∞ Fµα µ k+1 2 2k / BMO by Proposition 2.2. Now, as k → ∞. Since Fµα is decreasing, this yields Fµα ∈ we show that µ is an α-Carleson measure. For that purpose we split the rest of the proof into two cases; α > 1 and 0 < α ≤ 1. (Case 1) α > 1: First, we check that µ is a finite measure. Indeed, since δ(θk,j ) ≤ δ(θk ) = 2−k , we have for each k ≥ k0 µ(Ed,k ) =

sX k −1

α

[δ(θk,j )] ≤

j=0

sk 2k+1 (θk − θk+1 ) 2 ≤ ≤ (α−1)k , αk αk 2 2 2

(4.3)

which yields µ(Ed ) < ∞, because α > 1. We now show that µ is an α-Carleson measure. By Theorem 3.2 it suffices to show that BMO norms of functions Fµαη are uniformly bounded as η runs over T. Note that each Fµαη is bounded by ][Γ(η) ∩ Ed ], the number of points in Γ(η) ∩ Ed . Thus it is sufficient to show sup ][Γ(η) ∩ Ed ] < ∞.

(4.4)

η

The estimate is clear if θ(η) = 0 or θ(η) stays away from 0. So, assume θ(η) > 0 is sufficiently small. Assume θ(η) ∈ [θk+1 , θk ] for some k. To complete the proof of (4.4), we show below that Γ(η) ∩ Ed can contain at most an absolute number of points for all large k. Let w1 = w1 (η) and w2 = w2 (η) be the points of the intersection of ∂Γ(η) and E. Assume θ(w1 ) < θ(w2 ). Then all points in Γ(η) ∩ Ed have their arguments inside of the interval Iη := [θ(w1 ), θ(w2 )]. Note |θ(wj ) − θ(η)| ≤ |η − wj | + 1 − |wj (η)| = 3δ(θ(wj ))

(4.5)

for j = 1, 2. Thus we have by (4.1) 1−

δ(θ(η)) δ(θ(w2 ) − δ(θ(η)) ≤3· →0 δ(θ(w2 )) θ(w2 ) − θ(η)

as k → ∞. This yields δ(θ(w2 )) < 2δ(θ(η)) ≤ 2δ(θk ) = δ(θk−1 ) and thus θ(w2 ) < θk−1 for all large k. Similarly, θ(w1 ) > θk+2 for all large k. Fix such large k. Then we have by (4.5) 3 θ(w2 ) − θ(η) < 3δ(θk−1 ) = k−1 . 2 This implies the interval [θ(η), θ(w2 )] may contain at most 8 points in θ(Ed ), at most 4 points from each of the sets θ(Ed,k ) and θ(Ed,k−1 ). Similarly, the interval

Vol. 63 (2009)

Carleson Measures

517

[θ(w1 ), θ(η)] may contain at most 8 points in θ(Ed ). Thus we conclude that the set Γ(η) ∩ Ed , with θ(η) sufficiently small, may contain at most 16 points. This completes the proof for α > 1. (Case 2) 0 < α ≤ 1: In this case, instead of (4.3), we have α/2 [2k+1 (θk − θk+1 )]α/2 θk − θk+1 α/2 (4.6) µ(Ed,k ) ≤ =2 2αk 2k for each k ≥ k0 . Thus, for n > m ≥ k0 , we have " n−1 #(2−α)/2 n−1 X θk − θk+1 α/2 X 1 α/2 ≤ (θm − θn ) 2k 2kα/(2−α) k=m k=m 1 1 α/2 ≈ (θm − θn ) − nα/2 . 2mα/2 2 where the first inequality holds by H¨older’s inequality. On the other hand, since α ≤ 1, we have α/2 1 1 1 1 − ≤ − 2m 2n 2mα/2 2nα/2 α/2 α/2 δ(θm ) − δ(θn ) = (θm − θn ) . θm − θn Note that the expression inside of the bracket above is uniformly bounded by (4.1). Combining these observations with (4.6), we conclude n−1 X

µ(Ed,k ) . (θm − θn )α

(4.7)

k=m

for n > m ≥ k0 . In particular, taking m = k0 and n → ∞, we see that µ is a finite measure. We now show that µ is an α-Carleson measure. We need to establish the estimate µ(SI) ≤ C|I|α

(4.8)

for some constant C > 0 independent of µ and I ⊂ T. Denote by λ(I) be the total mass from atoms associated with partition points contained in I. Clearly, µ(SI) ≤ λ(I). We only need to consider I of the form I = {η ∈ T : 0 ≤ a ≤ θ(η) ≤ b ≤ 1}. First, consider the case where a, b ∈ [θm+1 , θm ]. The worst case is when b is a partition point and |I| ≥ δ(b) so that SI contains the atom associated with that partition point. Assume so and let ` be the number of partition points, other than b, contained in I. In case ` = 0, we have µ(SI) = [δ(b)]α ≤ |I|α . So, assume ` > 0. We have 2/α−1 `(θm − θm+1 ) `sm |I| ≥ ≈ sm 2m

518

Choe, Koo and Stessin

IEOT

and λ(I) ≤ (` + 1)[δ(θm )]α = Thus we have

`+1 ` ≈ αm . 2αm 2

λ(I) `1−α 1 . 2−α ≤ ≤ 1, α |I| sm sm

which yields (4.8). Next, assume b ∈ [θm+1 , θm ] and a ∈ [θn+1 , θn ] with n > m. Let I1 = I ∩ [θn+1 , θn ), I2 = I ∩ [θm+1 , θm ], and I3 = [θn , θm+1 ). If I2 does not contain any partition point other than θm+1 , we may shift I until b meets θm+1 . Such shift only increases λ(I) without changing |I|. So, we may assume that I2 contains at least one partition point in (θm+1 , θm ]. Now, we have λ(Ij ) . |Ij |α for j = 1, 2 by the proof above and also λ(I3 ) . |I3 |α by (4.7). Thus we obtain λ(I) . |I1 |α + |I2 |α + |I3 |α ≤ 31−α |I|α so that (4.8) holds. The proof for a = 0 is similar. This completes the proof.

Let Ω be a Carleson approach region at 1. Then Ω(ζ) := ζΩ is a Carleson approach region at ζ for every ζ ∈ T. Given a measure µ ≥ 0, in the way similar to what we did in the previous sections, we can consider its restriction, denoted by µΩ ζ , to Ω(ζ). As a consequence of Theorem 4.1, we have the following analogue of Theorem 3.2: Let α > 1. If Ω is a convex Carleson approach region at 1, then µ ≥ 0 is an α-Carleson measure if and only if the BMO norms of FµαΩ are uniformly ζ

bounded. The necessity follows from Theorem 4.1 and Theorem 3.2 (with arbitrary but fixed aperture), because Ω must lie inside of some Γ(β; 1). For the sufficiency we note that convexity of Ω implies that it contains a triangle with one vertex at 1 and the other two inside of D. Now, the proof for the sufficiency is similar to that of Theorem 3.2. However, the above may fail to hold for Carleson approach regions that are not necessarily convex. For example, given α > 1, take 2 − α < β < 1 and consider the weighted measure dµ(z) = (1 − |z|)−β dA(z) where A is the area measure on D. Then µ is a finite measure, but not an α-Carleson measure. Also, if Ω is a Carleson approach region at 1 of sufficiently sharp cusp shape, then one may check that functions FµαΩ are uniformly bounded. ζ

5. Open Problem We would like to pose an open problem which was implicitly mentioned in Section 4. Theorem 4.1 was proved under the assumption of local convexity of the Carleson approach region. Though this assumption was heavily used in the proof presented above, in many non-convex cases the construction of Theorem 4.1 may

Vol. 63 (2009)

Carleson Measures

519

be adjusted, so the result is still valid. Moreover, we failed to construct a nonconvex Carleson approach region which would not lie inside of any nontangential approach region. Thus we are tempted to pose the following problem. Problem. Does every Carleson approach region at ζ lie inside of some non-tangential approach region Γ(β; ζ)? We believe that most of our results may have extensions to a wide class of domains in Cn . So, we think that the solution of this problem is potentially important for the function theory in general domains in Cn such as hyperconvex domains.

References [C]

L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. Math. 76 (1962), 547–559.

[CM] C. Cowen and B. MaCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, 1995. [D]

P. Duren, Extension of a theorem of Carleson, Bull. Amer. Math. Soc. 75 (1969), 143–146.

[G]

J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981.

[H]

W.W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), 237–241.

[K]

P. Koosis, Introduction to Hp spaces, Cambridge University Press, Cambridge, 1998.

[S]

D. A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), 113–139.

[T]

A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, New York, 1986.

[ZZ]

R. Zhao and K. Zhu, Theory of Bergman spaces in the unit ball of Cn , Mem. Soc. Math. France, to appear.

[Z]

K. Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs Volume 138, Amer. Math. Soc., Providence, 2007.

Boo Rim Choe and Hyungwoon Koo Department of Mathematics Korea University Seoul 136-713 Korea e-mail: [email protected] [email protected]

520

Choe, Koo and Stessin

Michael Stessin Department of Mathematics SUNY 1400 Washington Avenue Albany, NY 12222 USA e-mail: [email protected] Submitted: October 28, 2008. Revised: February 26, 2009.

IEOT

Integr. equ. oper. theory 63 (2009), 521–531 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040521-11, published online February 12, 2009 DOI 10.1007/s00020-009-1660-7

Integral Equations and Operator Theory

Maximal Lp–Lq Regularity for Parabolic Partial Differential Equations on Manifolds with Cylindrical Ends Thomas Krainer

Abstract. We give a short, simple proof of maximal Lp –Lq regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent. Mathematics Subject Classification (2000). Primary 35K40; Secondary 58J05. Keywords. Maximal regularity, R-boundedness, pseudodifferential operators.

1. Introduction and main results Let (M, g) be a Riemannian manifold with cylindrical ends, i.e., there exists a relatively compact, open subset K ⊂ M such that M \ K is isometric to [0, ∞)r × Y , where (Y, gY ) is a closed compact Riemannian manifold (not necessarily connected), and the cylinder [0, ∞)r ×Y is equipped with the product metric dr2 +gY . By employing the change of variables x = 1/r on [0, ∞)r × Y for large values of r and attaching a copy of Y at x = 0, we obtain a compactification of M to a smooth compact manifold M with boundary ∂M = Y . The interior of M is diffeomorphic to the original manifold M , and in a collar neighborhood [0, ε)x × Y of 2 the boundary the Riemannian metric now takes the form cu g = dx x4 + gY , which is the form of a cusp metric (see [11]). Differential operators on a manifold with cylindrical ends with a ‘reasonably nice’ coefficient behavior at infinity correspond in this way to cusp differential operators on a compact manifold with boundary. Similarly, function spaces on the original manifold (M, g) with cylindrical ends correspond to function spaces on (M , cu g).

522

Krainer

IEOT

It is of course possible to consider compactifying functions for the cylindrical ends other than x = 1/r. For example, the choice x = e−r leads to totally √ characteristic operators on M (see [10]), and the choice x = 1/ n r, n ∈ N, leads to generalized cusp operators (see [7]). The results of this paper and their proofs have natural counterparts in those cases as well. We will henceforth stick to our initial choice x = 1/r, mainly because it allows for a simple exposition. Let E → M be a smooth vector bundle. We will prove the following result: Theorem 1.1. Let At ∈ C([0, T ], cu Diff m (M , E)), m > 0, 0 < T < ∞, and assume that At is cusp-elliptic with parameter in Λ = {λ ∈ C; <(λ) ≥ 0} for every 0 ≤ t ≤ T . Then, for every 1 < p, q < ∞, the parabolic evolution equation  ∂u − At u = f  ∂t (1.2) u| =u  t=0

0

has a unique solution u ∈ W 1,p ([0, T ], cu Lq (M , E)) ∩ Lp ([0, T ], cu H m,q (M , E)) for every f ∈ Lp ([0, T ], cu Lq (M , E)) m(1−1/p) u0 ∈ cu Bq,p (M , E) =

and cu

Lq (M , E), cu H m,q (M , E) 1−1/p,p .

The mapping u 7→ (f, u0 ) given by (1.2) is a topological isomorphism of these function spaces. In particular, the solution u satisfies optimal Lp –Lq a priori estimates with respect to the data f and u0 . By general results from [13] (see also [1] for recent improvements), we need to prove Theorem 1.1 only in the autonomous case At ≡ A and for u0 = 0, i.e., we need to prove that if A ∈ cu Diff m (M , E), m > 0, is cusp-elliptic with parameter in Λ = {λ ∈ C; <(λ) ≥ 0}, then A has maximal Lp –Lq regularity on [0, T ]. By [14] this is the case if the resolvent (A − λ)−1 : cu Lq (M , E) → cu H m,q (M , E) exists for λ ∈ Λ with |λ| ≥ R sufficiently large and if, in addition, the family {λ(A − λ)−1 ; λ ∈ Λ, |λ| ≥ R} ⊂ L cu Lq (M , E) (1.3) is R-bounded. Thus Theorem 1.1 is a consequence of the following result, which is of interest in its own right. Theorem 1.4. Let Λ ⊂ C be a closed sector, and let A ∈ cu Diff m (M , E), m > 0, be cusp-elliptic with parameter in Λ. Let 1 < q < ∞. For λ ∈ Λ with |λ| ≥ R sufficiently large, the operator A − λ : cu H m,q (M , E) → cu Lq (M , E) is invertible, and the set (1.3) is R-bounded.

Vol. 63 (2009)

Maximal Lp –Lq Regularity

523

To prove Theorem 1.4 we will employ a parameter-dependent parametrix of A − λ in the calculus of cusp pseudodifferential operators on M to approximate the resolvent. The parametrix is then further analyzed making use of the results from [4] on R-boundedness of families of pseudodifferential operators. We note that Mazzucato and Nistor [9] proved maximal Lp –Lq regularity for uniformly elliptic semibounded second order operators on complete manifolds with bounded geometry by analyzing the corresponding heat kernel. They make use of functional calculus and finite propagation speed arguments, extending earlier results for the Laplacian as obtained, e.g., in [2]. The R-boundedness approach pursued in this paper has several advantages compared with such a heat kernel analysis. In particular, it is applicable to not necessarily selfadjoint uniformly elliptic operators of arbitrary order where finite propagation speed and (selfadjoint) functional calculus are unavailable. Moreover, since Weis [14] has obtained a characterization of maximal Lp regularity in terms of the R-boundedness of the resolvent, it is natural to strive for proving the latter. In recent years, the literature on Fourier analysis of operators in vectorvalued function spaces and applications to parabolic partial differential equations has been developing rapidly, to a large extent with R-boundedness arguments at its core (see, e.g., [3, 6]). In [4] several results about pseudodifferential operators were obtained that make it possible to analyze the resolvents of differential operators effectively. It is one of the main objectives of this work to demonstrate this effectiveness as well as the simplicity of this approach for proving maximal Lp –Lq regularity. The structure of this paper is as follows: In Section 2 we review the definition and necessary results about R-boundedness of operator families that we need. For a comprehensive account on general aspects of R-boundedness and its applications to parabolic equations we refer to the monograph [3] or the survey paper [6], for R-boundedness of families of pseudodifferential operators see [4] (see also [5] for related work). Section 3 is devoted to cusp differential and pseudodifferential operators on manifolds with boundary (see [8, 12]). Finally, Section 4 contains the proof of Theorem 1.4.

2. R-boundedness and families of pseudodifferential operators Definition 2.1. Let E and F be Banach spaces. A subset T ⊂ L (E, F ) is called R-bounded, if for some constant C ≥ 0 the inequality ! N N

X

X

X X

εj Tj ej ≤ C εj ej (2.2)

ε1 ,...,εN ∈{−1,1} j=1

ε1 ,...,εN ∈{−1,1} j=1

holds for all choices of T1 , . . . , TN ∈ T and e1 , . . . , eN ∈ E, N ∈ N.

524

Krainer

The best constant ( C = sup

X

IEOT

N

X

εj Tj ej ; N ∈ N, T1 , . . . , TN ∈ T ,

ε1 ,...,εN ∈{−1,1} j=1

X

) N

X

εj ej = 1

ε1 ,...,εN ∈{−1,1} j=1

in (2.2) is called the R-bound of T and will be denoted by R(T ). The general properties of R-bounded sets yield to the following result about functions with R-bounded range (see [4], Propositions 2.11 and 2.13): Proposition 2.3. Let Γ be a nonempty set. Define `∞ R (Γ, L (E, F )) as the space of all functions f : Γ → L (E, F ) with R-bounded range and norm kf k`∞ := R f (Γ) . (2.4) R ∞ Then `∞ R (Γ, L (E, F )), k · k`R is a Banach space, and the embeddings ∞ ˆ π L (E, F ) ,→ `∞ `∞ (Γ)⊗ R (Γ, L (E, F )) ,→ ` (Γ, L (E, F ))

are well defined and continuous. The norm in `∞ R is submultiplicative, i.e. kf · gk`∞ ≤ kf k`∞ · kgk`∞ R R R whenever the composition f ·g makes sense, and we have k1k`∞ = 1 for the constant R map 1 ≡ IdE . Corollary 2.5 (Corollary 2.14 in [4]). i) Let M be a smooth manifold, and let K ⊂ M a compact subset. Let f ∈ C ∞ (M, L (E, F )). Then the range f (K) is an R-bounded subset of L (E, F ). ii) Let f ∈ S (Rn , L (E, F )). Then the range f (Rn ) ⊂ L (E, F ) is R-bounded. Proof. The assertion follows from Proposition 2.3 in view of ∼ C ∞ (M )⊗ ˆ π L (E, F ), C ∞ (M, L (E, F )) = ˆ π L (E, F ). S (Rn , L (E, F )) ∼ = S (Rn )⊗

In what follows all Banach spaces are assumed to be of class (HT ) and to satisfy Pisier’s property (α) (this is of relevance for the validity of Theorem 2.9 further below). We do not supply these definitions here, but merely note that all Banach spaces that are isomorphic to a scalar Lq -space for some 1 < q < ∞ have both properties, and so do vector-valued Lq -spaces provided that the target space satisfies both properties. This will be sufficient for our purposes. Definition 2.6. Let ` ∈ N be fixed. For every µ ∈ R we define the anisotropic µ;` R-bounded symbol class SR (Rn × Rq ; E, F ) to consist of all operator functions ∞ n q a ∈ C (R × R , L (E, F )) such that for all α ∈ Nn0 and β ∈ Nq0 n q (1 + |ζ| + |λ|1/` )−µ+|α|+`|β| ∂ζα ∂λβ a(ζ, λ) ∈ `∞ (2.7) R (R × R , L (E, F )).

Maximal Lp –Lq Regularity

Vol. 63 (2009)

525

We equip this symbol class with the Fr´echet topology whose seminorms are the `∞ R -norms of the functions in (2.7). These spaces have all the usual properties of symbol spaces (see [4], Section 3). Moreover, we have µ;` µ;` Scl (Rn × Rq ; E, F ) ,→ SR (Rn × Rq ; E, F ) ,→ S µ;` (Rn × Rq ; E, F )

with continuous embeddings, where S µ;` (Rn ×Rq ; E, F ) denotes the standard space ∞ of anisotropic operator-valued symbols of order µ (replace `∞ R in (2.7) by ` ), and µ;` Scl (Rn ×Rq ; E, F ) is the subspace of anisotropic classical symbols, i.e., those that P∞ admit an asymptotic expansion a ∼ j=0 aj with aj (%ζ, %` λ) = %µ−j aj (ζ, λ) for |(ζ, λ)| ≥ 1 and % ≥ 1. Furthermore, \ µ;` SR (Rn × Rq ; E, F ) = S (Rn × Rq , L (E, F )). µ∈R

For what we have in mind, the parameter space Rq needs to be replaced by a closed sector Λ ⊂ R2 . As is customary, the symbol spaces in this case consist by definition of the restrictions of symbols defined in the full space, and we equip those spaces with the quotient topology. Now split Rn = Rd × Rn−d in the (co-)variables ζ = (η, ξ), where 1 ≤ d ≤ n (in the case d = n the Rn−d -factor just drops out), and consider symbols µ;` 0 a(y, η, ξ, λ) ∈ Scl Rdy , SR (Rdη × Rn−d × Λ; E, F ) . (2.8) ξ With a(y, η, ξ, λ) we associate the family of pseudodifferential operators A(ξ, λ) = opy (a)(ξ, λ) : S (Rd , E) → S (Rd , F ), where −d

[opy (a)(ξ, λ)u](y) = (2π)

ZZ

0

ei(y−y )η a(y, η, ξ, λ)u(y 0 ) dy 0 dη.

The following is a consequence of Theorem 3.18 in [4]. Theorem 2.9. For ν ≥ µ the family of pseudodifferential operators opy (a)(ξ, λ) extends by continuity to opy (a)(ξ, λ) : H s,q (Rd , E) → H s−ν,q (Rd , F ) for every s ∈ R and 1 < q < ∞, and the operator function Rn−d × Λ 3 (ξ, λ) 7→ opy (a)(ξ, λ) ∈ L H s,q (Rd , E), H s−ν,q (Rd , F ) 0

µ ;` belongs to the R-bounded symbol space SR (Rn−d ×Λ; H s,q (Rd , E), H s−ν,q (Rd , F )) 0 0 with µ = µ if ν ≥ 0, or µ = µ − ν if ν < 0. The mapping opy : a(y, η, ξ, λ) 7→ opy (a)(ξ, λ) is continuous in µ;` µ0 ;` 0 Scl Rd , S R (Rd ×Rn−d ×Λ; E, F ) → SR Rn−d ×Λ; H s,q (Rd , E), H s−ν,q (Rd , F ) .

526

Krainer

IEOT

Let Y be a closed compact manifold, and let E, F → Y be smooth (finite dimensional) vector bundles. Let Lµ;` (Y, Rn−d × Λ; E, F ) be the class of families of pseudodifferential operators A(ξ, λ) : C ∞ (Y, E) → C ∞ (Y, F ) that are locally modelled on symbols (2.8), and global remainders on Y that are integral operators with C ∞ -kernels that depend rapidly decreasing (together with all µ;` derivatives) on the parameters (ξ, λ) ∈ Rn−d ×Λ. We write Lcl (Y, Rn−d ×Λ; E, F ) if the symbols in (2.8) are in addition required to be classical. The following is an immediate consequence of Corollary 2.5 (applied to global remainders) and Theorem 2.9 (applied to families supported in a chart). Corollary 2.10. Let A(ξ, λ) ∈ Lµ;` (Y, Rn−d × Λ; E, F ). For ν ≥ µ and every s ∈ R, 1 < q < ∞, the operator family A(ξ, λ) : H s,q (Y, E) → H s−ν,q (Y, F ) is continuous, and the operator function Rn−d × Λ 3 (ξ, λ) 7→ A(ξ, λ) ∈ L H s,q (Y, E), H s−ν,q (Y, F ) 0

µ ;` belongs to the R-bounded symbol space SR (Rn−d × Λ; H s,q (Y, E), H s−ν,q (Y, F )) 0 0 with µ = µ if ν ≥ 0, or µ = µ − ν if ν < 0. The embedding µ0 ;` Lµ;` (Y, Rn−d × Λ; E, F ) ,→ SR Rn−d × Λ; H s,q (Y, E), H s−ν,q (Y, F )

is continuous.

3. Analysis of cusp operators on manifolds with boundary Let M be a smooth n-dimensional compact manifold with boundary. Let U ∼ = [0, ε) × Y be a collar neighborhood of the boundary Y = ∂M , and fix a smooth defining function x for Y (i.e. x ≥ 0 on M , x = 0 precisely on Y , and dx 6= 0 on Y ) that coincides in U with the projection to the first coordinate. With these choices we let cu V = {V ∈ C ∞ (M , T M ); V x ∈ x2 C ∞ (M )} be the Lie algebra of cusp vector fields on M . Let cu Diff ∗ (M ) be the envoloping algebra of cusp differential operators generated by cu V and C ∞ (M ). In coordinates near the boundary, an operator A ∈ cu Diff m (M ) takes the form X A= ak,α (x, y)(x2 Dx )k Dyα (3.1) k+|α|≤m ∞

with C -coefficients ak,α that are smooth up to x = 0. The vector fields cu V are a projective finitely generated module over C ∞ (M ), hence by Swan’s theorem there is a smooth vector bundle cu T M on M , the cusptangent bundle, whose space of C ∞ -sections is cu V. Locally near the boundary, the vector fields x2 ∂x and ∂yj , j = 1, . . . , n − 1, form a frame for this bundle. There

Maximal Lp –Lq Regularity

Vol. 63 (2009)

527

is a canonical homomorphism φ : cu T M → T M that restricts to an isomorphism over the interior of M . Let cu T ∗ M be the cusp-cotangent bundle, the dual bundle to cu T M , and let t φ : T ∗ M → cu T ∗ M be the dual map to φ. Let σ(A) be the principal symbol of A ∈ cu Diff m (M ), defined on T ∗ M \ 0. Over the interior of M we set −1 cu σ(A) = σ(A) ◦ t φ . This function extends by continuity to a smooth function on all of cu T ∗ M \ 0, and it is homogeneous of degree m in the fibres. cuσ(A) is called the cusp-principal symbol of A. In coordinates near the boundary, the cusp-principal symbol of the operator A in (3.1) is given by X cu ak,α (x, y)ξ k η α . σ(A) = k+|α|=m

More generally, if E is a smooth vector bundle on M , we let cu Diff m (M ; E) denote the space of cusp differential operators A of order (at most) m. Initially, we consider A an operator A : C˙ ∞ (M , E) → C˙ ∞ (M , E), (3.2) where C˙ ∞ (M , E) denotes the space of smooth sections of E that vanish to infinite order on the boundary. The cusp-principal symbol of A is a section cuσ(A) ∈ C ∞ cu T ∗ M \ 0, End( cuπ ∗ E) , where cuπ : cu T ∗ M \ 0 → M is the canonical projection. Definition 3.3. Let Λ ⊂ C be a closed sector. We call A ∈ elliptic with parameter in Λ if

cu

Diff m (M ; E) cusp-

spec( cuσ(A)) ∩ Λ = ∅ everywhere on

cu

T ∗ M \ 0.

The operator (3.2) extends by continuity to an operator A : cu H s,q (M , E) → cu H s−m,q (M , E) for every s ∈ R and 1 < q < ∞. Here cu H 0,q (M , E) = cu Lq (M , E), the Lq -space of sections of E with respect to a Hermitian metric on E and the Riemannian density induced by a cusp metric, i.e., a Riemannian metric cu g in the interior of M that 2 in a collar neighborhood of the boundary Y takes the form cu g = dx x4 + gY . For m ∈ N0 we have cu

H m,q (M , E) = {u ∈ cu Lq (M , E); Au ∈ cu Lq (M , E) for all A ∈ cu Diff m (M ; E)},

and for general s ∈ R the cusp Sobolev spaces cu H s,q (M , E) are defined by duality and interpolation.

528

Krainer

IEOT

Parameter-dependent cusp pseudodifferential operators By cu Ψµ;` (M , Λ) we denote the space of parameter-dependent families of cusp pseudodifferential operators A(λ) : C˙ ∞ (M , E) → C˙ ∞ (M , E) of order µ ∈ R and anisotropy ` ∈ N, where Λ is as usual a closed sector in C. This operator class can be described as follows: Let ω, ω ˜ ∈ Cc∞ ([0, ε)) be cut-off functions, i.e., ω, ω ˜ ≡ 1 near x = 0. We consider ω and ω ˜ functions on M that are supported in the collar neighborhood U ∼ = [0, ε) × Y of the boundary. • Whenever ω and (1 − ω ˜ ) have disjoint supports, then the operator families ωA(λ)(1 − ω ˜ ) and (1 − ω ˜ )A(λ)ω are integral operators with kernels 0 k(z, z , λ) ∈ S Λ, C˙ ∞ (M z × M z0 , E E ∗ ) . • In the collar neighborhood U of the boundary, the bundle E|U is isomorphic to π ∗ E|Y , where π : [0, ε) × Y → Y is the projection on the second factor. Hence sections of E on U can be interpreted as functions of x ∈ [0, ε) taking values in sections of E|Y on Y , e.g., C˙ ∞ (U, E) ∼ = C˙ ∞ [0, ε), C ∞ (Y, E|Y ) . The operator family ωA(λ)˜ ω : C˙ c∞ (U, E) → C˙ c∞ (U, E) now is of the form Z Z ε dy 1 u 7→ ei(1/y−1/x)ξ a(x, ξ, λ)u(y) 2 dξ 2π R 0 y ∞ ∞ ˙ for u ∈ Cc [0, ε), C (Y, E|Y ) , where a(x, ξ, λ) ∈ C ∞ [0, ε)x , Lµ;` cl (Y, Rξ × Λ; E|Y ) . • The operator family (1 − ω)A(λ)(1 − ω ˜ ) belongs to the class Lµ;` cl (2M , Λ) of classical anisotropic parameter-dependent pseudodifferential operators of order µ on the double 2M of M (acting in sections of an extension of the bundle E to the double). Every A(λ) ∈ cu Ψµ;` (M , Λ) extends by continuity to a family of continuous operators A(λ) : cu H s,q (M , E) → cu H s−µ,q (M , E) for every s ∈ R and 1 < q < ∞. The class of parameter-dependent families of cusp pseudodifferential operators forms an algebra filtered by order. Theorem 3.4. Let A ∈ cu Diff m (M ; E), m > 0, be cusp-elliptic with parameter in Λ. We have A − λ ∈ cu Ψm;m (M , Λ), and there exists a parameter-dependent parametrix P (λ) ∈ cu Ψ−m;m (M , Λ), i.e., (A − λ)P (λ) − 1, P (λ)(A − λ) − 1 ∈ cu Ψ−∞ (M , Λ). Theorem 3.4 is an instance of a standard result in the calculus of pseudodifferential operators. The structural result that it entails about parameter-dependent parametrices of A − λ is the key to the proof of Theorem 1.4 which is given in the next section.

Vol. 63 (2009)

Maximal Lp –Lq Regularity

529

4. Proof of Theorem 1.4 Every R(λ) ∈ cu Ψ−∞ (M , Λ) gives rise to an operator function R(λ) ∈ S Λ, L (cu H s,q (M , E)) for every s ∈ R and 1 < q < ∞. Consequently, by Theorem 3.4, the operator family A − λ : cu H m,q (M , E) → cu Lq (M , E) is invertible for λ ∈ Λ with |λ| ≥ R sufficiently large and, moreover, (A − λ)−1 − P (λ) : cu Lq (M , E) → cu Lq (M , E) belongs to S ΛR , L (cu Lq (M , E)) , where ΛR = {λ ∈ Λ; |λ| ≥ R}. Here we are making use of the fact that the parameter-dependent parametrix P (λ) is tempered as a function of λ ∈ Λ taking values in the bounded operators on cu Lq (M , E). By Corollary 2.5 we get that {λ(A − λ)−1 − λP (λ); λ ∈ ΛR } ⊂ L cu Lq (M , E) is R-bounded. To complete the proof it thus remains to show that {λP (λ); λ ∈ ΛR } ⊂ L cu Lq (M , E) is R-bounded. To see this, let ω ˆ , ω, ω ˜ ∈ Cc∞ ([0, ε)) be cut-off functions, i.e., ω ˆ , ω, ω ˜ ≡ 1 near x = 0, and suppose that ω ≡ 1 in a neighborhood of the support of ω ˆ , and that ω ˜ ≡ 1 in a neighborhood of the support of ω. We consider ω ˆ , ω, ω ˜ as functions on M that are supported in the collar neighborhood U ∼ = [0, ε) × Y of the boundary. Write P (λ) = ωP (λ)˜ ω + (1 − ω)P (λ)(1 − ω ˆ ) + R(λ). The operator family R(λ) ∈ cu Ψ−∞ (M , Λ) and so the set {λR(λ); λ ∈ Λ} ⊂ L cu Lq (M , E) is R-bounded in view of Corollary 2.5. The family (1 − ω)P (λ)(1 − ω ˆ ) can be −m;m regarded as an element of Lcl (2M , Λ) (supported in the interior of the original −m;m copy of M ) and thus furnishes an element in SR Λ; cu Lq (M , E), cu Lq (M , E) in view of Corollary 2.10. Consequently, {λ(1 − ω)P (λ)(1 − ω ˆ ); λ ∈ Λ} ⊂ L cu Lq (M , E) is R-bounded. So it remains to show that {λωP (λ)˜ ω ; λ ∈ Λ} ⊂ L

cu

Lq (M , E)

(4.1)

is R-bounded. Consider ωP (λ)˜ ω : cu Lq (U, E) → cu Lq (U, E), where U ∼ = [0, ε)x × Y . Under the change of variables r = −1/x, this operator family can by definition of the

530

Krainer

IEOT

cusp calculus be regarded as the pull-back of an operator family Q(λ) : Lq Rr , Lq (Y, E|Y ) → Lq Rr , Lq (Y, E|Y ) ZZ 0 1 [Q(λ)u](r) = ei(r−r )% a(r, %, λ)u(r0 ) dr0 d% for u ∈ S Rr , C ∞ (Y, E|Y ) , 2π 0 where a(r, %, λ) ∈ Scl Rr , L−m;m (Y, R% × Λ; E|Y ) . In view of Corollary 2.10 we cl have −m;m L−m;m (Y, R × Λ; E|Y ) ,→ SR R × Λ; Lq (Y, E|Y ), Lq (Y, E|Y ) , cl and so −m;m 0 Rr , S R R% × Λ; Lq (Y, E|Y ), Lq (Y, E|Y ) . a(r, %, λ) ∈ Scl Thus, by Theorem 2.9, −m;m Q(λ) = opr (a)(λ) ∈ SR Λ; Lq R, Lq (Y, E|Y ) , Lq R, Lq (Y, E|Y ) , which shows that {λQ(λ); λ ∈ Λ} ⊂ L Lq R, Lq (Y, E|Y ) is R-bounded. Consequently, the set (4.1) is R-bounded, and the proof of Theorem 1.4 is complete.

References [1] W. Arendt, R. Chill, S. Fornaro, and C. Poupaud, Lp -maximal regularity for nonautonomous evolution equations, J. Differential Equations 237 (2007), 1–26. [2] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15–53. [3] R. Denk, M. Hieber, and J. Pr¨ uss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs of the American Mathematical Society, vol. 788, 2003. [4] R. Denk and T. Krainer, R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators, Manuscripta Math. 124 (2007), 319–342. [5] T. Hyt¨ onen and P. Portal, Vector-valued multiparameter singular integrals and pseudodifferential operators, Adv. Math. 217 (2008), 519–536. [6] P.C. Kunstmann and L. Weis, Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus, Functional analytic methods for evolution equations, pp. 65–311, Lecture Notes in Math., vol. 1855, Springer, 2004. [7] R. Lauter, B. Monthubert, and V. Nistor, Invariance spectrale des alg`ebres d’op´erateurs pseudodiff´erentiels, C. R. Math. Acad. Sci. Paris 334 (2002), 1095–1099. [8] R. Lauter and S. Moroianu, The index of cusp operators on manifolds with corners, Ann. Global Anal. Geom. 21 (2002), 31–49.

Vol. 63 (2009)

Maximal Lp –Lq Regularity

531

[9] A. Mazzucato and V. Nistor, Mapping properties of heat kernels, maximal regularity, and semi-linear parabolic equations on noncompact manifolds, J. Hyperbolic Differ. Equ. 3 (2006), 599–629. [10] R. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A. K. Peters, Ltd., Wellesley, MA, 1993. [11] , Geometric Scattering Theory, Stanford Lecture Notes in Mathematics, Cambridge University Press, 1995. [12] R. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, Preprint 1996 (funct-an/9606005 on arXiv.org). [13] J. Pr¨ uss and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl. 256 (2001), 405– 430. [14] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity, Math. Ann. 319 (2001), 735–758. Thomas Krainer Mathematics and Statistics Penn State Altoona 3000 Ivyside Park Altoona, PA 16601 U.S.A. e-mail: [email protected] Submitted: August 29, 2008. Revised: December 4, 2008.

Integr. equ. oper. theory 63 (2009), 533–545 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040533-13, published online March 25, 2009 DOI 10.1007/s00020-009-1669-y

Integral Equations and Operator Theory

Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization Heinz Langer, Alexander Markus and Vladimir Matsaev Abstract. In this note we continue the study of spectral properties of a selfadjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval ∆0 and is boundedly invertible in the endpoints of ∆0 , then the ‘embedding’ of the original Hilbert space H into the Hilbert space F , where the linearization of A(z) acts, is in fact an isomorphism between a subspace H(∆0 ) of H and F . As a consequence, properties of the local spectral function of A(z) on ∆0 and a so-called inner linearization of the operator function A(z) in the subspace H(∆0 ) are established. Mathematics Subject Classification (2000). Primary 47A56, 47B50; Secondary 47A11, 47A48, 47A68. Keywords. Self-adjoint analytic operator function, linearization, Krein space, spectrum of positive type, local spectral function.

1. Introduction Let H be a Hilbert space with inner product (·, ·), and denote by L(H) the set of all bounded linear operators in H. We consider a bounded simply connected domain D ⊂ C, that is symmetric with respect to the real axis R, and an L(H)-valued function A(z) on D which is analytic and self-adjoint, i.e. A(z) = A(z)∗ , z ∈ D; in particular, A(λ) = A(λ)∗ , λ ∈ D ∩ R. The spectrum σ(A), the point spectrum σp (A), and the resolvent set ρ(A) of the operator function A(z) are defined in the usual way (see [5], [7]). A real point λ0 ∈ σ(A) is said to be a spectral point of positive type of the operator function A(z), if for each sequence (xn ), satisfying kxn k = 1 and kA(λ0 )xn k → 0 if n → ∞, we have lim inf n→∞ (A0 (λ0 )xn , xn ) > 0;

534

Langer, Markus and Matsaev

IEOT

(see [5]). The set of all spectral points of positive type of A(z) is denoted by σ+ (A). For the rest of the paper we fix some real interval ∆0 = [α0 , β0 ] ⊂ D and suppose that ∆0 ∩ σ(A) ⊂ σ+ (A), (1.1) and that α0 , β0 ∈ ρ(A). Because of (1.1) we can choose a complex neighborhood U (⊂ D) of ∆0 such that U \ ∆0 ⊂ ρ(A) (see [4]). According to [4], [5] (see also [1]), A(z) admits a linearization Λ in a Krein space F. Here F = L2+ (γ0 , H) A(z)L2+ (γ0 , H) (1.2) where γ0 (⊂ U) is a sufficiently smooth simple positive oriented curve which surrounds ∆0 and passes through the points α0 , β0 . The inner product in F is defined by the relation I 1 A(t)−1 f (t), g(t) dt, f, g ∈ L2+ (γ0 , H), (1.3) hf, gi := 2πi γ0 followed by the factorization (1.2). The condition (1.1) implies that the Krein space F with the inner product induced by (1.3) is even a Hilbert space. In the space L2+ (γ0 , H) we consider the operator Λ0 of the multiplication by the independent variable. The corresponding operator Λ in the factor space F is called linearization of the operator function A(z); this operator Λ is self–adjoint in the Hilbert space F. Let P denote the mapping of the space H into the space F which associates with the element g ∈ H the equivalence class in F which contains the vector function u(t) ≡ g ∈ L2+ (γ0 , H), and let P ∗ be the adjoint operator from F into H. The basic relation which connects A(z) and Λ is A(z)−1 − B(z) = −P ∗ (Λ − z)−1 P,

z ∈ U \ σ(A),

(1.4)

where B(z) is an analytic in U operator function which is uniquely defined by the condition that the expression on the left hand side is the principal part of the operator function A(z)−1 with respect to γ0 . The linearization is minimal in the sense that (see [4, Theorem 3.2]) F = span (Λ − z)−1 P H : z ∈ U ∩ ρ(Λ) , (1.5) where U ∩ ρ(Λ) can be replaced by any of its nonempty open subsets, and the spectrum of the operator Λ coincides with σ(A) ∩ ∆0 (see [4]). The relation I I 1 ∗ (Λ − t)−1 dt 1 A(t)−1 dt −1 ∗ −1 P =− , P (Λ − z) (Λ − ζ) P = − P 2πi γ0 (t − z)(t − ζ) 2πi γ0 (t − z)(t − ζ) z, ζ ∈ U and outside of γ0 , implies for these z, ζ the formula I

A(t)−1 x, y dt 1 −1 −1 (Λ − z) P x, (Λ − ζ) P y = − 2πi γ0 (t − z)(t − ζ) for the inner product in F. Because of (1.5), in the Krein space situation (without assumption (1.1)) this implies that the linearization Λ is uniquely determined up to a weak isomorphism (see [4, Remark 2]), in our situation (with assumption

Vol. 63 (2009)

Self-adjoint Analytic Operator Functions

535

(1.1)) this implies the uniqueness of the linearization Λ in the Hilbert space F up to unitary equivalence. It is worth to mention that the quintuple {Λ, P, P ∗ ; F, H} is a spectral node in the sense of [2]. Since the spectrum of the self-adjoint operator Λ is contained in ∆0 , the operator Λ has a spectral function E which is supported on ∆0 and is defined for all Borel subsets Γ of R. In [5] the L(H)-valued function Q(Γ) := P ∗ E(Γ)P,

Γ Borel set in ∆0 ,

was called the local spectral function of the operator function A(z) on ∆0 (in fact, in [5] this notion was used for the function Qt := Q([α0 , t]), t ∈ ∆0 ); in the following we call the range ran Q(Γ) the spectral subspace of A(z) corresponding to Γ. Clearly, the values Q(Γ) of the local spectral function are nonnegative operators in H but in general not projections. Under the general assumption (1.1) the local spectral function does not have some of the properties that one usually associates with the term ‘spectral function’, e.g. its ranges on disjoint intervals can have nonempty intersection. This is excluded if instead of (1.1) on ∆0 the Virozub–Matsaev condition (VM) is imposed: (VM)

∃ε, δ > 0 : λ ∈ ∆0 , f ∈ H, kf k = 1, |(A(λ)f, f )| < ε =⇒ (A0 (λ)f, f ) > δ.

In [5] it was shown that (VM) is a natural condition for a comprehensive spectral theory of the self-adjoint operator function A(z). This condition is always supposed to be fulfilled in the present paper; besides, as was already mentioned, we also assume that A(α0 ), A(β0 ) are boundedly invertible. It was shown in [5] that under the condition (VM) the operator Q(∆0 ) has closed range and hence it has the remarkable property that it is uniformly positive on its range. In particular, the spectral subspace ran Q(∆0 ) =: H(∆0 ) of A(z) corresponding to the interval ∆0 is a closed subspace of H, which admits the decomposition (see [5, Theorem 7.3]) H = ran A(β0 )− u H(∆0 ) u ran A(α0 )+ . We mention that the condition (VM) in the case of a finite dimensional space H reduces to the simpler condition (vm) (see [6]): (vm) λ0 ∈ ∆0 , (A(λ0 )f, f ) = 0 for some f ∈ H, f 6= 0 =⇒ (A0 (λ0 )f, f ) > 0. The basic result of the present note is that if the condition (VM) is satisfied on ∆0 and A(z) is boundedly invertible in the endpoints of ∆0 , then always ran P = F, and P ∗ is a bijection from F onto H(∆0 ).

(1.6)

This implies that the spectral subspaces ran Q(Γ) of A(z) for any Borel set Γ ⊂ ∆0 are closed. It also allows us to prove Theorem 7.111 and the following results of [5] in a more compact way and sometimes in a more general form. Moreover, we construct a so-called inner linearization of the analytic operator function A(z), which acts in the subspace H(∆0 ) of the originally given Hilbert space H. 1 The

proof of [5, Theorem 7.11 (1)] is not correct, but now this claim follows trivially from (1.6).

536

Langer, Markus and Matsaev

IEOT

We mentioned already that that the space F and the linearization Λ were intoduced in [2] without the restriction (1.1), i.e. for an arbitrary self–adoint analytic operator function A(z) such that its spectrum is a compact subset of the domain D. In general the space F is a Krein space and it is ‘much larger’ than the given space H (with respect to the mapping P ). E.g. for a self-adjoint monic self-adjoint operator polynomial A(z) of degree n and D = C, the space F can be chosen to be Hn (and Λ can be choosen to be the companion operator of A(z)). By enlarging the space H we gain that the linear operator Λ in F represents in D the spectral properties of the analytic operator function A(z). However, under the assumption of the present paper that the operator function A(z) satisfies the condition (VM) on the interval [α0 , β0 ] and that the operators A(α0 ) and A(β0 ) are boundedly invertible, the subspace H(∆0 ) of H, which is mapped by P bijectively onto F, can be even smaller than H, that is, also F can be ‘smaller’ than H; in this situation P will have a nontrivial kernel. In the particular case that A(α0 ) 0, A(β0 ) 0, the space F is of the ‘same size’ as H, that is, the mapping P is a bijection between H and F. Clearly, if mathcalH is finite–dimensional, F ‘smaller’ than H means dim F ≤ dim H. The claims (1.6) are proved in Section 2. In Section 3 we establish some properties of the local spectral function and of the spectral subspaces of A(z). Finally, in Section 4 we introduce in H(∆0 ) the inner linearization S of the operator function A(z) corresponding to the interval ∆0 , which is just an isomorphic copy of the linearization Λ in F. As an application, in the case H(∆0 ) = H, which is equivalent to A(α0 ) 0 and A(β0 ) 0, we give a simple proof of a factorization result of Virozub-Matsaev from [8] and explicit expressions for the factors in terms of the local spectral function of A(z).

2. The operators P and P ∗ Theorem 2.4 below is the crucial result of this paper. In its proof we need some lemmata which we prove first. Recall that the condition (VM) is supposed to hold on the interval ∆0 and that the endpoints of ∆0 are regular points for the operator function A(z). For an interval ∆ ⊂ ∆0 we introduce the following subspaces of H: H∆ := P ∗ E(∆)F,

H(∆) := P ∗ E(∆)P H.

(2.1)

If ∆ = ∆0 the closure in the last relation is superfluous (see [5, Theorem 7.3]); later we will see that all closures are superfluous. From the definition in (2.1) it follows immediately that H(∆) ⊂ H∆ ; (2.2) below (Corollary 2.5) it will be shown that in (2.2) always equality holds. Lemma 2.1. If ∆ ⊂ ∆0 , then H∆ = span {P ∗ E(∆0 )P H : ∆0 ⊂ ∆}.

(2.3)

Vol. 63 (2009)

Self-adjoint Analytic Operator Functions

537

Proof. Clearly, P ∗ E(∆0 )P H ⊂ P ∗ E(∆0 )F ⊂ P ∗ E(∆)F, and hence for the two sets in (2.3) the inclusion ⊃ follows. Conversely, from the minimality (1.5) of the linearization Λ we have P ∗ E(∆)F = span {P ∗ E(∆)(Λ − z)−1 P H : z ∈ U ∩ ρ(Λ)}. Observing that ∗

−1

P E(∆)(Λ − z)

Z

(λ − z)−1 P ∗ dE(λ)P

P = ∆

and approximating the integral by finite Riemann-Stieltjes sums, we see that each element x ∈ P ∗ E(∆)F belongs to the set on the right hand side of (2.3), and the inclusion ⊂ for the two sets in (2.3) follows. The following lemma is a partial extension of [5, Corollary 5.2], where it was proved for x ∈ H(∆). Lemma 2.2. Let ∆ = [α, β] ⊂ ∆0 . If x ∈ H∆ , then (A(α)x, x) ≤ 0,

(A(β)x, x) ≥ 0.

(2.4)

Proof. Consider two intervals ∆j = [αj , βj ], j = 1, 2, such that α ≤ α1 ≤ β1 ≤ α2 ≤ β2 ≤ β, and an element x = P ∗ E(∆1 )P x01 + P ∗ E(∆2 )P x02 =: x1 + x2 ,

x01 , x02 ∈ H.

By [5, Corollary 5.2] we have (A(β1 )x1 , x1 ) ≥ 0, and hence, by [5, Lemma 4.1 (d)]2 (A(α2 )x1 , x1 ) ≥ 0. According to [5, Lemma 5.1] this implies (A(β2 )(x1 + x2 ), x1 + x2 ) ≥ 0, and, again by [5, Lemma 4.1 (d)], the relation (A(β)(x1 + x2 ), x1 + x2 ) ≥ 0 follows. By induction, we obtain the second inequality in (2.4) for all x ∈ H of the form n X x= P ∗ E(∆j )P x0j , x0j ∈ H, j = 1, 2, . . . , n, (2.5) j=1

where ∆j = [αj , βj ], α ≤ α1 , βj ≤ αj+1 , j = 1, . . . , n − 1, βn ≤ β. If an element x of the form (2.5) with arbitrary closed intervals ∆j is given, we can choose a new decomposition of the set ∪nj=1 ∆j into closed intervals such that not any two of them have common inner points, and obtain a representation of x as required in the line after (2.5). By Lemma 2.1 the set of elements x of the form (2.5) with arbitrary closed intervals ∆j is dense in H∆ , and the second inequality in (2.4) follows. The proof of the first inequality in (2.4) is analogous. 2 We

indicate here two misprints in [5, Lemma 4.1]: in items (b) and (d) the absolute value signs should be removed.

538

Langer, Markus and Matsaev

IEOT

Lemma 2.3. Let ∆ =: [α, β] ⊂ ∆0 , denote by P∆ the orthogonal projection onto H∆ and set A∆ (λ) := P∆ A(λ) H . (2.6) ∆

If H is separable, then σp (A∆ ) is at most countable. Proof. We can suppose that ∆ lies strictly inside ∆0 , i.e. α0 < α,

β0 > β.

(2.7)

Indeeed, if e.g. α0 = α, we replace the point α0 by a point α00 such that α00 < α0 , [α00 , α0 ] ⊂ ρ(A) and (VM) holds on [α00 , α0 ]. By Lemma 2.2, A∆ (α) ≤ 0, A∆ (β) ≥ 0, and then from (2.7) and [5, Lemma 4.1 (e)] it follows that A∆ (α0 ) 0, A∆ (β0 ) 0. Since (VM) holds for A∆ (λ) on ∆0 and α0 , β0 ∈ ρ(A∆ ), the operator function A∆ (λ) on ∆0 has a linearization Λ∆ which is a selfadjoint operator in a Krein space F∆ (see [4, Section 3]), and since the whole spectrum of Λ∆ is of positive type (see [4, Theorem 5.3]), the space F∆ is uniformly positive, i.e. it is in fact a Hilbert space (see [3, Theorem 3.1]). The separability of H∆ implies the separability of the space F∆ , and hence the point spectrum of the self-adjoint operator Λ∆ is at most countable. By [4, Theorem 3.1], σp (A∆ ) = σp (Λ∆ ). Theorem 2.4. If the condition (VM) holds on the interval ∆0 and the endpoints of ∆0 are regular points for the operator function A(z), then the operators P : H 7→ F and P ∗ : F 7→ H have the properties ran P = F,

ran P ∗ = H∆0 ,

and P ∗ is a bijection between F and H∆0 . Proof. Since ran P ∗ is closed (see [5, Corollary 7.10]) and since E(∆0 ) = I, from the definition of H∆0 we have ran P ∗ = H∆0 . If we show that ran P = F, then it follows that P ∗ is injective and the theorem is proved. According to the definition of F and P , ran P = F means that for any vector function f (t) ∈ L2+ (γ0 , H) there exists an element g ∈ H such that f (t) − g ∈ A(t)L2+ (γ0 , H).

(2.8)

In the first step of the proof we show that without loss of generality we can suppose that the space H is separable. To this end we choose a dense countable subset T = {tj : j = 1, 2, . . . } of D and, with the given function f (t) in (2.8), we b of the linear span of all elements consider the closure H A(tj1 )A(tj2 ) · · · A(tjn )f (tj ),

tj , tj1 , tj2 , . . . , tjn ∈ T , j, n ∈ N.

b for all z ∈ D and that H b is an invariant subspace It is easy to check that f (z) ∈ H b of the operators A(z), z ∈ D. So the restriction A(z) of A(z) to the separable b is an operator function A(z) b b with the same properties as Hilbert space H in H

Vol. 63 (2009)

Self-adjoint Analytic Operator Functions

539

b can be considered in a natural way as A(z) in H, and the Hilbert space L2+ (γ0 , H) 2 b such that a subspace of L+ (γ0 , H). Therefore, if we find an element gb in H b f (t) − gb ∈ A(t)L2+ (γ0 , H), then for the element g in (2.8) we can choose g = gb. So we suppose in the rest of this proof that the space H is separable. The relation ran P = F will be proved if we show that ker P ∗ = {0}. Assume x e0 ∈ F, P ∗ x e0 = 0. Choose a point λ0 ∈ / σp (A∆0 ) that is close to the point α0 + β0 , and set ∆1,1 := [α0 , λ0 ] and ∆1,2 = [λ0 , β0 ]. Denote x1 := P ∗ E(∆1,1 )e x0 . 2 ∗ The relation P x e0 = 0 implies P ∗ E(∆1,2 )e x0 = −P ∗ E(∆1,1 )e x0 , hence x1∈ H∆1,1∩H∆1,2 . By Lemma 2.2, (A(λ0 )x1 , x1) ≥ 0 and (A(λ0 )x1 , x1) ≤ 0, i.e. (A(λ0 )x1 , x1 ) = 0.

(2.9)

Since, according to Lemma 2.2, A∆1,1 (λ0 ) ≥ 0, relation (2.9) implies that A∆1,1 (λ0 )x1 = 0. Using x1 ∈ H∆1,2 , we obtain by precisely the same argument that A∆1,2 (λ0 )x1 = 0. So, the vector A(λ0 )x1 is orthogonal to both H∆1,j , j = 1, 2, and therefore it is orthogonal to H∆0 . Hence A∆0 (λ0 )x1 = 0, and the condition λ0 ∈ / σp (A∆0 ) implies that x1 = 0. This means that P ∗ E(∆1,1 )˜ x0 = P ∗ E(∆1,2 )˜ x0 = 0. Now choose points λ01 , λ02 which are close to the middle points of ∆1,1 and ∆1,2 respectively, and such that λ0j ∈ / σp (A∆1,j ), j = 1, 2. We denote the corresponding subintervals of ∆0 by ∆2,j , j = 1, 2, 3, 4. By the same arguments as above we find P ∗ E(∆2,j )e x0 = 0, j = 1, 2, 3, 4. Continuing this procedure we obtain a sequence of partitions {∆n,j , j = 1, 2, . . . , 2n } of ∆0 , n = 1, 2, . . . , such that P ∗ E(∆n,j )e x0 = 0,

j = 1, 2, . . . , 2n , n = 1, 2, . . . .

Hence with tn,j ∈ ∆n,j and z ∈ C \ ∆0 we find X 2n x e0 , (tn,j − z)−1 E(∆n,j )P H = {0}. j=1

Passing to the limit n → ∞ we obtain

x e0 , (Λ − z)−1 P H = {0}, and the minimality of the linearization (see (1.5)) implies that x e0 = 0. Thus, ker P ∗ = {0}, and hence ran P = F.

540

Langer, Markus and Matsaev

IEOT

Corollary 2.5. For all intervals ∆ ⊂ ∆0 , P ∗ E(∆)F = P ∗ E(∆)P H,

(2.10)

and these sets are closed; in particular H∆ = H(∆). Proof. The equality (2.10) follows from the relation ran P = F. Since P ∗ is an isomorphism and the spectral subspace E(∆)F is closed, the subspace on the left hand side in (2.10) is closed as well.

3. The local spectral function and the spectral subspaces of A Recall that Q(Γ) = P ∗ E(Γ) P, Γ Borel set of ∆0 , (3.1) is the local spectral function of the analytic operator function A(z), the range ran Q(Γ) is the spectral subspace of the analytic operator function A(z) corresponding to Γ. Clearly, Q(Γ) is a nonnegative operator in H which, because of Corollary 2.5, is uniformly positive on its range. If ∆ ⊂ ∆0 is an interval such that the endpoints α, β of ∆ are not eigenvalues of the operator function A(z), then the operator Q(∆) can be expressed directly through A(z) as follows: Z 0 1 A(z)−1 dz, (3.2) Q(∆) = 2πi γ(∆) where γ(∆) is a smooth contour in U which surrounds ∆ and crosses the real axis in α and β orthogonally, the prime at the integral denotes the Cauchy principal value at α and β. The relation (3.2) follows from the representation of the spectral function of the linearization Λ by means of the resolvent of Λ and the relation (1.4) (see also [5, (3.3)]). The properties of the operator valued set function Q(Γ) are summarized in the following theorem. Theorem 3.1. Let Γ, Γ1 , Γ2 , . . . ⊂ ∆0 be Borel sets. Then: (1) Q(Γ) = P ∗ E(Γ)F, and ran Q(Γ) is closed. (2) If Γ1 ⊂ Γ2 , then ran Q(Γ1 ) ⊂ ran Q(Γ2 ). (3) If Q(Γ1 ∩ Γ2 ) = 0, then ran Q(Γ1 ) ∩ ran Q(Γ2 ) = {0},

ran Q(Γ1 ) u ran Q(Γ2 ) = ran Q(Γ1 ∪ Γ2 ).

(Γj )∞ 1

(4) If is an infinite sequence such that Q(Γj ∩ Γk ) = 0 for all j 6= k, j, k = 1, 2, . . . , then ∞ [∞ X Q Γj = Q(Γj ), j=1

j=1

where the sum on the right hand side converges strongly and unconditionally. (5) The point λ0 ∈ ∆0 is a regular point of the operator function A(z) if and only if there exists a neighbourhood Γ of λ0 such that Q(Γ) = 0.

Vol. 63 (2009)

Self-adjoint Analytic Operator Functions

541

(6) The point λ0 ∈ ∆0 is an eigenvalue of the operator function A(z) if and only if Q({λ0 }) 6= 0; in this case kerA(λ0 ) = ran Q({λ0 }).

(3.3)

(7) For an open interval ∆ the following two statements are equivalent: (a) dim ran Q(∆) = n; (b) ∆ contains only a finite number of points of σ(A), all of them are eigenvalues and the sum of their multiplicities is n. In this case ran Q(∆) = span {ker A(λj ) : λj ∈ ∆ ∩ σ(A)} . (8) The eigenvectors of the operator function A(z), corresponding to different eigenvalues in ∆0 , are linearly independent. If there is an infinite number of such eigenvalues, then the corresponding eigenvectors form a Riesz basis in their closed linear span. (9) If ∆ is a subinterval of ∆0 and A∆ is the operator function in H(∆) (= H∆ ) defined by (2.6), then σ(A∆ ) ∩ ∆0 ⊂ ∆. Proof. The first equality in (1) is a consequence of (2.10). All the other statements in (1)–(8) follow from the fact that ran P = F and that P ∗ is an isomorphic embedding of F into H, see Theorem 2.4, and the corresponding properties of the spectral function E of the self–adjoint operator Λ in F. For the proof of (9) we observe that with ∆ =: [α, β] by Lemma 2.2 we have A∆ (α) ≤ 0, A∆ (β) ≥ 0. Then, by [5, Lemma 4.1 (e)], A∆ (α0 ) 0 for all α0 ∈ [α0 , α) and A∆ (β 0 ) 0 for all β 0 ∈ (β, β0 ], therefore the intervals [α0 , α) and (β, β0 ] belong to ρ(A∆ ). Remark 3.2. The spectral function E of the operator Λ exists also for intervals ∆ with ∆ ∩ σ(A) ⊂ σ+ (A), see [5], and hence also the local spectral function Q of the operator function A can be defined by (3.1) on ∆. In this situation e.g. item (3) of Theorem 3.1 does not hold. However, the relation (3.3) remains true in this case, that is we have kerA(λ0 ) = ran P ∗ E({λ0 })P. Indeed, multiply (1.4) by z0 − z and let z → z0 (parallel to imaginary axis). Then the left hand side converges strongly to Q({z0 }), and we can also apply it to elements y(z) which converge strongly for z → z0 . Now for any x we find P ∗ E({z0 })P x = lim (z − z0 )A(z)−1 x, z→z0

and the limit on the right hand side exists; it is easy to check that this element is in kerA(z0 ). The converse inclusion follows if in the above reasoning we choose A(z)x0 y(z) = with x0 such that A(z0 )x0 = 0. z − z0 Remark 3.3. If H is finite–dimensional and the condition (vm) holds on ∆0 , then dim F ≤ dim H. This follows from the relation dim F = dim H(∆0 ) and Theorem 3.1 (8).

542

Langer, Markus and Matsaev

IEOT

4. The inner linearization S in H(∆0 ) In this section it is convenient to consider P ∗ as an operator which acts from F to H(∆0 ). We will use for this operator the special notation P0∗ . By Theorem 2.4, the operator P0∗ is boundedly invertible, and it is easy to see that (P0∗ )∗ x = P x,

x ∈ H(∆0 ).

(4.1)

Furthermore, in this section also the operators Q(Γ) will be considered as operators in H(∆0 ). Then Q(∆0 ) is a positive and boundedly invertible operator. Under the bijection P0∗ from F onto H(∆0 ) the linearization Λ in F of the operator function A(z) becomes an operator S which is self-adjoint in H(∆0 ) with respect to a suitable Hilbert inner product (·, ·)0 . More exactly, we equip H(∆0 ) with the positive definite inner product (x, y)0 := (Q(∆0 )−1 x, y),

x, y ∈ H(∆0 ),

(4.2)

and define in H(∆0 ) the operator S := P0∗ Λ(P0∗ )−1 .

(4.3)

Since S acts in a subspace of the originally given space H (equipped with the inner product (4.2)) we call S the inner linearization of A(z). Theorem 4.1. The operator S in (4.3) is self-adjoint in H(∆0 ) with respect to the inner product (·, ·)0 from (4.2), and, if ES denotes the spectral function of S, for each Borel subset Γ of ∆0 we have ES (Γ) = Q(Γ)Q(∆0 )−1 ,

ran ES (Γ) = ran Q(Γ).

(4.4)

Proof. Under the mapping P ∗ the (positive definite) inner product h·, ·i on F becomes a positive definite inner product (·, ·)0 on H(∆0 ). In fact, if x ˜, y˜ ∈ F, P ∗x ˜ = x, P ∗ y˜ = y with x, y ∈ H(∆0 ), then, observing (4.1), we find

(x, y)0 = h˜ x, y˜i = (P0∗ )−1 x, (P0∗ )−1 y = (P0∗ )−∗ (P0∗ )−1 x, y = (P0∗ P )−1 x, y = Q(∆0 )−1 x, y . Further, Q(∆0 ) = P ∗ P implies (P0∗ )−1 = P Q(∆0 )−1 , and we obtain Z ∗ ∗ −1 S = P Λ(P0 ) = λ dQ(λ) Q(∆0 )−1 .

(4.5)

∆0

Clearly, S is self-adjoint in the inner product (·, ·)0 and for any Borel subset Γ ⊂ ∆0 the spectral projection ES (Γ) of S in H(∆0 ) is given by ES (Γ) = P ∗ E(Γ)P Q(∆0 )−1 = Q(Γ)Q(∆0 )−1 , and also the second equality in (4.4) follows.

Remark 4.2. Note that the connection (4.4) between the spectral subspaces of A(z) and of the self-adjoint operator S implies again the statements (1)-(8) of Theorem 3.1. The inner linearization S in H(∆0 ) of the analytic operator function A(z) has on ∆0 the same spectral properties as A(z). The relation (4.4) yields also the following corollary.

Vol. 63 (2009)

Self-adjoint Analytic Operator Functions

543

Corollary 4.3. If Γ1 , Γ2 are Borel subsets of ∆0 and Q(Γ1 ∩ Γ2 ) = 0, then the spectral subspaces ran Q(Γ1 ) and ran Q(Γ2 ) are orthogonal in the inner product (., .)0 . Finally we show that a factorization result from [8] can be obtained in a simple way from the above considerations, and we obtain an explicit form of the factors in terms of the spectral function of A(z). Theorem 4.4. Suppose that, additionally to the above assumptions, A(α0 ) 0 and A(β0 ) 0. Then, in a neighbourhood of ∆0 , the operator function A(z) admits the unique factorization A(z) = A1 (z)(S − z). (4.6) Here the operator S ∈ L(H) is self–adjoint with respect to the inner product (·, ·)0 on H and such that σ(S) = σ(A) ∩ ∆0 , the operator function A1 (z) is boundedly invertible in a neighbourhood of ∆0 , and such that A1 (z) and its inverse A1 (z)−1 are analytic there. The operator S and the operator function A1 (z) admit the representations Z Z A(λ) − A(z) dQ(λ)Q(∆0 )−1 . S= λ dQ(λ) Q(∆0 )−1 , A1 (z) = − λ − z ∆0 ∆0 Proof. According to [5, Corollary 7.4], H(∆0 ) = H. Hence P is an invertible operator from H to F, P ∗ is an invertible operator from F to H, and Q(∆0 ) is an invertible operator in H. Consider the operator S = P ∗ Λ P −∗ from (4.5) (now P0∗ = P ∗ ). The relation (1.4) and the invertibility of P ∗ imply A(z)−1 = −(S − z)−1 P ∗ P + B(z).

(4.7)

for all z ∈ U \ ∆0 , where U is a neighbourhood of ∆0 . Multiplying (4.7) from the left by A(z) gives −A(z)(S − z)−1 P ∗ P = I − A(z)B(z).

(4.8)

On the other hand, multiplying (4.7) from the left by S − z implies (S − z)A(z)−1 = −P ∗ P + (S − z)B(z).

(4.9)

Denote A1 (z) := A(z)(S − z)−1 . Taking into account that P ∗ P is an invertible operator we see from (4.8) and (4.9) that A1 (z) is invertible in a neighbourhood of ∆0 and that A1 (z)−1 is analytic there. Moreover, Z −1 A1 (z) = A(z)(S − z) = A(z) (λ − z)−1 dQ(λ)Q(∆0 )−1 ∆0

Z = ∆0

A(z) − A(λ) dQ(λ)Q(∆0 )−1 . λ−z

544

Langer, Markus and Matsaev

IEOT

To prove the uniqueness of the factorization we consider another factorization b1 (z)(Sb − z) with the same properties as in (4.6). It follows that A(z) = A b1 (z)−1 A1 (z). Sb − z (S − z)−1 = A The operator function on the left hand side is analytic outside ∆0 , the function on the right hand side is analytic in a neighbourhood of ∆0 . Thus both operator functions can be extended by analytic continuation to the whole complex plane. b If z → ∞ the left hand side tends to the identity operator, and the claim S = S, b1 (z) follows from Liouville’s theorem. A1 (z) = A Remark 4.5. Taking adjoints in (4.6) we obtain a factorization with a linear left hand factor: A(z) = (S ∗ − z)A1 (z)∗ , S ∗ = P −1 ΛP. Acknowledgement. The authors thank the referee for valuable suggestions.

References [1] I. Gohberg, M.A. Kaashoek, D.C. Lay: Equivalence, linearization and decomposition of holomorphic operator functions. J. Funct. Anal. 28 (1978), 102–144. [2] M.A. Kaashoek, C.V.M. van der Mee, L. Rodman: Analytic operator functions with compact spectrum, I. Spectral nodes, linearization and equivalence. Integral Equations Operator Theory 4 (1981), 504–547. [3] H. Langer, A. Markus, V. Matsaev: Locally definite operators in indefinite inner product spaces. Math. Ann. 308 (1997), 405 –424. [4] H. Langer, A. Markus, V. Matsaev: Linearization and compact perturbation of self– adjoint analytic operator functions. Oper. Theory: Adv. Appl. 118 (2000), 255–285. [5] H. Langer, A. Markus, V. Matsaev: Self-adjoint analytic operator functions and their local spectral function. J. Funct. Anal. 235 (2006), 193–225. [6] H. Langer, M. Langer, A. Markus, C. Tretter: The Virozub–Matsaev condition and spectrum of definite type for self-adjoint operator functions. Compl. Anal. Oper. Theory 2 (2008), 99–134. [7] A.S. Markus: Introduction to the Spectral Theory of Polynomial Operator Pencils. AMS Translations of Mathematical Monographs, vol. 71, 1988. [8] A.I. Virozub, V.I. Matsaev: The spectral properties of a certain class of selfadjoint operator–valued functions. Funct. Anal. Appl. 8 (1974), 1–9. Heinz Langer Institut f¨ ur Analysis und Scientific Computing Technische Universit¨ at Wien 1040 Wien Austria e-mail: [email protected]

Vol. 63 (2009)

Self-adjoint Analytic Operator Functions

Alexander Markus Department of Mathematics Ben-Gurion-University of the Negev Beer Sheva, 84105 Israel e-mail: [email protected] Vladimir Matsaev School of Mathematical Sciences Tel Aviv University Ramat Aviv, 69978 Israel e-mail: [email protected] Submitted: November 7, 2008. Revised: February 25, 2009.

545

Integr. equ. oper. theory 63 (2009), 547–555 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040547-9, published online February 12, 2009 DOI 10.1007/s00020-009-1661-6

Integral Equations and Operator Theory

Finite-Rank Products of Toeplitz Operators in Several Complex Variables Trieu Le Abstract. For any α > −1, let A2α be the weighted Bergman space on the unit ball corresponding to the weight (1 − |z|2 )α . We show that if all except possibly one of the Toeplitz operators Tf1 , . . . , Tfr are diagonal with respect to the standard orthonormal basis of A2α and Tf1 · · · Tfr has finite rank, then one of the functions f1 , . . . , fr must be the zero function. Mathematics Subject Classification (2000). 47B35. Keywords. Toeplitz operator, weighted Bergman space, finite-rank product.

1. Introduction As usual, let Bn denote the open unit ball in Cn . Let ν denote the Lebesgue measure on Bn normalized so that ν(Bn ) = 1. Fix a real number α > −1. The weighted Lebesgue measure να on Bn is defined by dνα (z) = cα (1 − |z|2 )α dν(z), where cα is a normalizing constant so that να (Bn ) = 1. A direct computation shows that Γ(n + α + 1) cα = . Γ(n + 1)Γ(α + 1) Let L2α denote L2 (Bn , dνα ) and L∞ denote L∞ (Bn , dνα ), which is the same as L∞ (Bn , dν). We denote the inner product in L2α by h·, ·iα and the corresponding norm by k · k2,α . The weighted Bergman space A2α consists of all functions in L2α which are holomorphic on Bn . It is well known that A2α is a closed subspace of L2α . For any multi-index m = (m1 , . . . , mn ) ∈ Nn (here N denotes the set of all non-negative integers), we write |m| = m1 + · · · + mn and m! = m1 ! · · · mn !. For ¯m = z¯1m1 · · · z¯nmn . The any z = (z1 , . . . , zn ) ∈ Cn , we write zm = z1m1 · · · znmn and z 2 n standard orthonormal basis for Aα is {em : m ∈ N }, where h Γ(n + |m| + α + 1) i1/2 em (z) = zm , m ∈ Nn , z ∈ Bn . m! Γ(n + α + 1)

548

Le

IEOT

For a more detailed discussion of A2α , see Chapter 2 in [8]. Since A2α is a closed subspace of the Hilbert space L2α , there is an orthogonal projection Pα from L2α onto A2α . For any function f ∈ L2α the Toeplitz operator with symbol f , denoted by Tf , is densely defined on A2α by Tf ϕ = Pα (f ϕ) for bounded holomorphic functions ϕ on Bn . If f is a bounded function, then Tf is a bounded operator on A2α with kTf k ≤ kf k∞ and (Tf )∗ = Tf¯. However, there are unbounded functions f that give rise to bounded operators Tf . Let P be the space of holomorphic polynomials in the variable z = (z1 , . . . , zn ) n in C . For any f ∈ L2α and holomorphic polynomials p, q ∈ P we have hTf p, qiα = R p¯ q f dνα . This shows that Tf can be viewed as an operator from P into the space Bn L∗ (P, C) of conjugate-linear functionals on P. More generally, for any compactly supported regular Borel measure µ on Cn , we define Lµ : P −→ L∗ (P, C) by the R formula (Lµ p)(q) = Cn p¯ q dµ, for p, q ∈ P. For f ∈ L2α , if we let dµ = f dνα , then Tf = Lµ on P. It follows from Stone-Weierstrass’s Theorem that if Lµ = 0, then µ = 0. It is also immediate that if µ is a linear combination of point masses, then Lµ has finite rank. That the converse is also true is the content of the following theorem, which had been an open conjecture for about twenty years. See [1, 6, 7]. Theorem 1.1. Lµ has finite rank if and only if µ is a (finite) linear combination of point masses. Theorem 1.1 for the case n = 1 was proved by D. Luecking in [6]. Using a refined version of Theorem 1.1 in this case, the current author was able to show that if f1 , . . . , fr are bounded measurable functions on the disk, all except possibly one of them are radial functions and Tf1 · · · Tfr has finite rank, then one of these functions is the zero function. See [4] for more detail. To the best of the author’s knowledge, Theorem 1.1 in high dimensions has been proved in at least two papers. In [7], G. Rozenblum and N. Shirokov give a proof by induction on the dimension n. In the base case (n = 1), they use the above Luecking’s result. In [1], B. Choe follows Luecking’s scheme with modifications (to the setting of several variables) to prove Theorem 1.1 for all n ≥ 1. In this note, we modify Choe’s proof to obtain a refined version of Theorem 1.1. We then apply the refined theorem to solve the problem about finite-rank products of Toeplitz operators in all dimensions, when all but possibly one of the operators are (weighted) shifts. This result is the content of Theorem 3.2, which is a generalization of the main result in [4].

2. A Refined Luecking’s Theorem in High Dimensions For any 1 ≤ j ≤ n, let σj : N × Nn−1 −→ Nn be the map defined by the formula σj (s, (r1 , . . . , rn−1 )) = (r1 , . . . , rj−1 , s, rj , . . . , rn−1 ) for all s ∈ N and

Vol. 63 (2009)

Finite-Rank Products of Toeplitz Operators

549

(r1 , . . . , rn−1 ) ∈ Nn−1 . If S is a subset of Nn and 1 ≤ j ≤ n, we define X 1 n−1 e Sj = ˜r = (r1 , . . . , rn−1 ) ∈ N : =∞ . s+1 s∈N σj (s,˜ r)∈S

The following definition is given in [5]. For completeness, we recall it here. Definition 2.1. We say that S has property (P) if one of the following statements holds: 1. S = ∅, or P 1 < ∞, or 2. S 6= ∅, n = 1 and s∈S s+1 3. S = 6 ∅, n ≥ 2 and for any 1 ≤ j ≤ n, the set Sej has property (P) as a subset of Nn−1 . 1.

2. 3. 4. 5.

6.

With the above definition, the following statements are immediate. P 1 = ∞. If S ⊂ Nn If S ⊂ N and S does not have property (P), then s∈S s+1 with n ≥ 2 and S does not have property (P), then Sej does not have property (P) as a subset of Nn−1 for some 1 ≤ j ≤ n. If S1 and S2 are subsets of Nn that both have property (P), then S1 ∪ S2 also has property (P). If S ⊂ Nn has property (P) and l ∈ Zn , then (S + l) ∩ Nn also has property (P). Here, S + l = {m + l : m ∈ S}. If S ⊂ Nn has property (P), then N × S also has property (P) as a subset of Nn+1 . This follows by induction on n. The set Nn does not have property (P) for all n ≥ 1. This together with (2) shows that if S ⊂ Nn has property (P), then Nn \S does not have property (P). For any m = (m1 , . . . , mn ) and k = (k1 , . . . , kn ) in N, we write m k if mj ≥ kj for all 1 ≤ j ≤ n and write m k if otherwise. Then for any fixed k ∈ Nn , the set S = {m ∈ Nn : m k} has property (P). This follows from (2), (4) and the fact that n [ S⊂ N × · · · × N × {0, . . . , kj − 1} × N × · · · × N. j=1

The following proposition shows that if the zero set of a holomorphic function (under certain additional assumptions) does not have property (P), then the function is identically zero. The proof is presented in Section 3 in [5]. Proposition 2.2 (Proposition 3.2 in [5]). Let K denote the right half of the complex plane. Let F : Kn → C be a holomorphic function. Suppose there exists a polynomial p such that |F (z)| ≤ p(|z|) for all z ∈ Kn . Put Z(F ) = {r ∈ Nn : F (r) = 0}. If Z(F ) does not have property (P), then F is identically zero in Kn . We are now ready for the statement and proof of a refined version of Theorem 1.1.

550

Le

IEOT

Theorem 2.3. Suppose S ⊂ Nn is a set that has property (P). Let N be the linear space spanned by the monomials {zm : m ∈ Nn \S}. Let L∗ (N , C) denote the space of all conjugate-linear functionals on N . Suppose µ is a complex regular Borel measure on Cn with compact support. Let Lµ : N −→ L∗ (N , C) be the operator R defined by (Lµ f )(g) = Cn f g¯dµ for f, g ∈ N . If Lµ has finite rank, then µ ˜ is a linear combination of point masses, where d˜ µ(z) = |z1 | · · · |zn |dµ(z) for z ∈ Cn . As a consequence, if µ is absolutely continuous with respect to the Lebesgue measure on Cn , then µ is the zero measure. Proof. Suppose Lµ has rank strictly less than N , where N ≥ 1. Arguing as in pages 2 and 3 in [1], for any polynomials f1 , . . . , fN and g1 , . . . , gN in N , we have Y N

Z Cn×N

fj (zj ) det(¯ gi (zj ))dµN (z1 , . . . , zN ) = 0,

(2.1)

j=1

where µN is the product of N copies of µ on Cn×N . Let m1 , . . . , mN and k1 , . . . , kN be multi-indices in Nn . Let L = {l ∈ Nn : l + mj ∈ / S and l + kj ∈ / S for all 1 ≤ j ≤ N } [ N N [ n (S − kj ) . (S − mj ) ∪ =N \ j=1

j=1

Since S has property (P), Nn \L has property (P). This implies that L does not have property (P). For any l ∈ L, the monomials fj (z) = zmj +l and gj (z) = zkj +l are in N for j = 1, . . . , N . Equation (2.1) then implies that Z 0= Cn×N

Z = Cn×N

Z = Cn×N

Z = Cn×N

Y N

m +l zj j

det((¯ zkj i +l ))dµN (z1 , . . . , zN )

j=1

Y N

m zj j

j=1

dµN (z1 , . . . , zN )

mj

Y n N Y ki 2ls |zj,s | dµN (z1 , . . . , zN ) det((¯ zj ))

Y N n Y 2ls ki det((¯ zj )) |zj,s | dµN (z1 , . . . , zN ),

zj

j=1 s=1

j=1

Y N

¯lj zlj z

j=1

j=1

Y N

Y N

det((¯ zkj i ))

mj

zj

s=1 j=1

where l = (l1 , . . . , ln ) and zj = (zj,1 , . . . , zj,n ) for 1 ≤ j ≤ N . Suppose that µ is supported in the ball B(0, R) of radius R centered at 0 in Cn . Then µN is supported in the product of N copies of B(0, R) in Cn×N . For any

Vol. 63 (2009)

Finite-Rank Products of Toeplitz Operators

551

ζ = (ζ1 , . . . , ζn ) ∈ Cn with <(ζ1 ), . . . , <(ζn ) > 0, define Y N

Z F (ζ) = Cn×N

m zj j

Y n Y N

det((¯ zkj i ))

Y N

= (B(0,R))N

2ζs

dµN (z1 , . . . , zN )

s=1 j=1

j=1

Z

|zj,s |R

−1

mj

zj

Y n Y N 2ζs dµN (z1 , . . . , zN ). det((¯ zkj i )) |zj,s |R−1 s=1 j=1

j=1

Then F is holomorphic and bounded on its defining domain and F (l) = 0 for all l in L. Since L does not have property (P), Proposition 2.2 implies that F (ζ) = 0 for all ζ = (ζ1 , . . . , ζn ) ∈ Cn with <(ζ1 ), . . . , <(ζn ) > 0. In particular, we have F ( 12 , . . . , 21 ) = 0. This shows that Y N

Z 0= Cn×N

=

Y n Y N

det((¯ zkj i ))

Cn×N

|zj,s | dµN (z1 , . . . , zN )

s=1 j=1

j=1

Y N

Z

m zj j

mj

zj

det((¯ zkj i ))d˜ µN (z1 , . . . , zN ),

j=1

where µ ˜N is the product of N copies of µ ˜ (recall that d˜ µ(z) = |z1 | · · · |zn |dµ(z) for z = (z1 , . . . , zn ) in Cn ). Since m1 , . . . , mN and k1 , . . . , kN were arbitrary, by taking finite sums, we conclude that Z Cn×N

Y N

fj (zj ) det(¯ gi (zj ))d˜ µN (z1 , . . . , zN ) = 0,

(2.2)

j=1

for all f1 , . . . , fN and g1 , . . . , gN in P. Now following Choe’s proof on pages 3–6 in [1], we see that µ ˜ is supported in a set of less than N points. Remark 2.4. Suppose n = 1. That µ ˜ is a linear combination of point masses implies that µ is also a linear combination of point masses. Thus, we recover Theorem 3.1 in [4]. Remark 2.5. Suppose n ≥ 2. Let S = {m = (m1 , . . . , mn ) ∈ Nn : m1 · · · mn = 0}. Then S has property (P). Let W = {z = (z1 , . . . , zn ) ∈ Cn : z1 · · · zn = 0}. If µ is any complex regular Borel measure supported on W , then for any f, g in N (recall that N = Span{zm : m ∈ Nn \S}), we have Z Z f g¯dµ = f g¯dµ = 0, Cn

W

because f and g vanish on W . This shows that Lµ is the zero operator from N into L∗ (N , C). However, since W is an infinite set, µ may not be a linear combination of point masses.

552

Le

IEOT

3. Finite-Rank Toeplitz Products In the first part of this section, we use Theorem 2.3 to show that under certain conditions on the bounded operators S1 and S2 on A2α , if f ∈ L2α so that S2 Tf S1 is a finite-rank operator, then f must be zero almost everywhere on Bn . Theorem 3.1. Let S1 , S2 be two bounded operators on A2α . Suppose there is a set S ⊂ Nn which has property (P) such that ker(S2 ) ⊂ M and N ⊂ ran(S1 ). Here M (respectively N ) is the linear subspace of A2α spanned by {zm : m ∈ S} (respectively {zm : m ∈ Nn \S}). Suppose f ∈ L2α is such that the operator S2 Tf S1 has finite rank, then f is the zero function. Proof. Since S2 Tf S1 has finite rank and N ⊂ S1 (A2α ), we see that S2 Tf (N ) is a finite-dimensional linear subspace of A2α . Let {u1 , . . . , uN } be a basis for this subspace. Let vj ∈ A2α such that S2 vj = uj for 1 ≤ j ≤ N . It then follows that Tf (N ) is contained in Span(ker(S2 ) ∪ {v1 , . . . , vN }), which is a subspace of Span(M ∪ {v1 , . . . , vN }). Let PM denote the orthogonal projection from A2α onto M. Replacing vj by vj − PM vj if necessary, we may assume that vj ⊥ M for 1 ≤ j ≤ N . Using the Gram-Schmidt process if necessary, we may assume that {v1 , . . . , vN } is an orthonormal set in A2α (we may have fewer vectors after using the Gram-Schmidt process but let us still denote by N the total number of these vectors). For any p in N we have Tf p = PM Tf p +

N X

hTf p, vj iα vj = PM Tf p +

j=1

N X

hf p, vj iα vj .

j=1

Now for any q in N , since q ⊥ M, we have Z f p¯ q dνα = hTf p, qiα Bn

= hPM Tf p, qiα +

N X

hf p, vj iα hvj , qiα

j=1

=

N X

hf p, vj iα hvj , qiα .

j=1

Let dµ = f dνα . Then the above identities show that the map Lµ from N into the space R R of all conjugate-linear functionals on N defined by (Lµ p)(q) = p¯ q dµ = p¯ q f dνα has rank at most N . Theorem 2.3 then implies that µ is Bn Bn the zero measure. Thus f is zero almost everywhere on Bn . Suppose f˜ ∈ L∞ such that f˜(z1 , . . . , zn ) = f˜(|z1 |, . . . , |zn |)

for almost all z ∈ Bn .

(3.1)

Vol. 63 (2009)

Finite-Rank Products of Toeplitz Operators

553

Then for any m, k in Nn , we have Z hTf˜em , ek iα = f˜(z)em (z)¯ ek (z)dνα (z) Bn Z = f˜(|z1 |, . . . , |zn |)em (z)¯ ek (z)dνα (z) Bn Z ¯k dνα (z) = C(m, k, n, α) f˜(|z1 |, . . . , |zn |)zm z Bn

=

 0 C(m, n, α)

if m 6= k, Z

¯ dνα (z) f˜(|z1 |, . . . , |zn |)z z m m

if m = k.

Bn

The last equality follows from the invariance of the measure να under the action of the n-torus (see also Corollary 3.5 in [5] for more detail). This implies that Tf˜ is a diagonal operator with respect to the standard orthonormal basis of A2α . The eigenvalues of Tf˜ are given by ωα (f˜, m) = hTf˜em , em iα ,

m ∈ Nn .

(3.2)

If the set Z(f˜) = {m ∈ Nn : ωα (f˜, m) = 0} does not have property (P), then, by Lemma 3.3 in [5], it is all of Nn . This implies that Tf˜ is the zero operator, hence f˜ is the zero function on Bn . Therefore, if f˜ is not the zero function on Bn , then Z(f˜) has property (P). ¯t f˜(z) for z ∈ Bn . For m, k in Nn , Now suppose s, t are in Nn . Let f (z) = zs z we have hTf em , ek iα = C(m, s, k, t, n, α)hf˜em+s , ek+t iα ( 0 if m + s 6= k + t, = ˜ C(m, s, k, t, n, α)ωα (f , m + s) if m + s = k + t. This shows that ( 0 Tf em = C(m, s, m + s − t, t, n, α)ωα (f˜, m + s)em+s−t

if m t − s, if m t − s.

(3.3)

Note that the constant C(m, s, m + s − t, t, n, α) is positive. Now suppose f˜1 , . . . , f˜r are non-zero functions in L∞ satisfying (3.1). Let s1 , . . . , sr and t1 , . . . , tr be multi-indices in Nn . For each 1 ≤ j ≤ r, let fj (z) = P ¯tj f˜j (z) for z ∈ Bn . Let S = Tfr · · · Tf1 . For any multi-index m rj=1 (sj +tj ), zs j z using (3.3), we see that there is a positive constant C depending on m, s1 , . . . , sr , t1 , . . . , tr , n and α so that Y j−1 r X Sem = C · ωα f˜j , m + (si − ti ) + sj em+Prj=1 (sj −tj ) . (3.4) j=1

i=1

554

Le

IEOT

Define r n o X J = m:m (sj + tj ) j=1 j−1 r n o Y X ∪ m: ωα f˜j , m + (si − ti ) + sj = 0 j=1

i=1

j−1 r r n o [ X X = m:m (sj + tj ) ∪ Z(f˜j ) − (si − ti ) + sj . j=1

j=1

i=1

Since none of the functions f1 , . . . , fr is the zero function, the set J has property (P). For m ∈ Nn \J , we see that Sem 6= 0 and em+Prj=1 (sj −tj ) is a multiple of Sem . Suppose ϕ ∈ A2α such that Sϕ = 0. Then we have X X hϕ, em iα Sem . 0 = Sϕ = S hϕ, em iα em = m∈Nn

m∈Nn n

So (3.4) implies that for any m ∈ N \J , hϕ, em iα = 0. Therefore ker(S) is contained in the closure in A2α of the linear span of {em : m ∈ J }. Now put r r n o \ X X I = k ∈ Nn : k (sj − tj ) ∪ Nn J + (sj − tj ) . j=1

j=1

Pr Then I has property (P) and for any k ∈ Nn \I, m = k − j=1 (sj − tj ) belongs to Nn \J . It then follows that ek = em+Prj=1 (sj −tj ) is a multiple of Sem . So the linear span of {ek : k ∈ Nn \I} is contained in the range of S. We now prove a result about products of Toeplitz operators on A2α . Theorem 3.2. Let r1 and r2 be two positive integers. Let f˜1 , . . . , f˜r1 +r2 be bounded measurable non-zero functions satisfying (3.1). Let s1 , . . . , sr1 +r2 and t1 , . . . , tr1 +r2 ¯tj f˜j (z) for be multi-indices in Nn . For each 1 ≤ j ≤ r1 + r2 , define fj (z) = zsj z 2 z ∈ Bn . If f ∈ Lα such that the operator Tfr1 +r2 · · · Tfr1 +1 Tf Tfr1 · · · Tf1 (which is densely defined on A2α ) has finite rank, then f is the zero function. Proof. Let S1 = Tfr1 · · · Tf1 and S2 = Tfr1 +r2 · · · Tfr1 +1 . From the discussion preceding the theorem, there are subsets J and I of Nn that have property (P) such that ker(S2 ) is contained in the closure in A2α of Span({em : m ∈ J }) and Span({ek : k ∈ Nn \I}) is a subspace of S1 (A2α ). Let S = J ∪ I. Then S has property (P), ker(S2 ) ⊂ M and N ⊂ S1 (A2α ), where M (respectively N ) is the linear subspace of A2α spanned by {zm : m ∈ S} (respectively {zm : m ∈ Nn \S}). If f ∈ L2α such that S2 Tf S1 has finite rank, then Theorem 3.1 implies that f is the zero function. Remark 3.3. Suppose n = 1. The functions fj in the hypothesis of Theorem 3.2 are called quasihomogeneous functions. Each Tfj is a weighted forward or backward shift. It was shown in [2] (Theorem 2) that if the product of finitely many

Vol. 63 (2009)

Finite-Rank Products of Toeplitz Operators

555

Toeplitz operators whose symbols are bounded quasihomogeneous functions is of finite rank, then one of the functions must be zero. This result is a special case of our Theorem 3.2 above. In [3], K. Guo, S. Sun and D. Zheng showed that if f and g are bounded harmonic functions on the unit disk and Tf Tg has finite rank, then either f = 0 or g = 0. However, for arbitrary bounded measurable functions f and g, it is still not known whether Tf Tg = 0 implies that one of these functions must be the zero function. Acknowledgement The paper was completed when the author was a postdoctoral fellow at the Department of Mathematics and the Fields Institute for Research in Mathematical Sciences, University of Toronto. The author is grateful for their support.

References [1] B. R. Choe, On higher dimensional Luecking’s theorem, J. Math. Soc. Japan 61 (2009), no. 1, 213–224. ˇ Cuˇ ˇ ckovi´c and I. Louhichi, Finite rank commutators and semicommutators of quasi[2] Z. homogeneous Toeplitz operators, Complex Anal. Oper. Theory 2 (2008), no. 3, 429– 439. [3] 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. [4] T. Le, A refined Luecking’s theorem and finite rank products of Toeplitz operators, preprint, available at http://www.math.uwaterloo.ca/~t29le/Papers.htm. , The commutants of certain Toeplitz operators on weighted Bergman spaces, [5] J. Math. Anal. Appl. 348 (2008), no. 1, 1–11. [6] D. Luecking, Finite rank Toeplitz operators on the Bergman space, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1717–1723. [7] G. Rozenblum and N. Shirokov, Finite rank Bergman-Toeplitz and Bargmann-Toeplitz operators in many dimensions, preprint, available at http://front.math.ucdavis. edu/0802.0192. [8] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005. Trieu Le Department of Pure Mathematics University of Waterloo Waterloo, Ontario, N2L 3G1 Canada e-mail: [email protected] Submitted: April 1, 2008

Integr. equ. oper. theory 63 (2009), 557–569 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040557-13, published online April 1, 2009 DOI 10.1007/s00020-009-1672-3

Integral Equations and Operator Theory

Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions Pascal Lef`evre Abstract. We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H ∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators. Mathematics Subject Classification (2000). Primary 47B33; Secondary 46E15, 47B10. Keywords. Composition operators, essential norm, analytic functions, weakly compact operators, Dunford-Pettis.

Introduction The aim of this paper is to investigate the potential default of complete continuity or weak compactness of composition operators on uniform algebras of analytic functions. This includes the case of the space H ∞ of bounded analytic functions on the open unit disk D of the complex plane, the case of the disk algebra A(D), and more generally the polydisk and polyball algebras. The composition operators have been investigated in many ways. The context of H p spaces (1 < p < ∞), in particular, has been intensively investigated. But, on such spaces, their weak compactness and complete continuity are trivial problems (because of reflexivity). The monographs [6] and [17] are good surveys on this topic. Another interesting survey is the paper [13]. The characterization of weak compactness for composition operators on the classical spaces H ∞ and A(D) is not new: this is due to Aron, Galindo and Lind¨ str¨ om [1] (see also Ulger [18]). The author recovered this result in [16] as a consequence of a characterization of (general) weakly compact operators on H ∞ . The

558

Lef`evre

IEOT

characterization of complete continuity of composition operators was settled in [12] (see also [11]). On the other hand, weak compactness and complete continuity of the multiplication operator were considered in [14], but for C(K), L1 or H 1 spaces. Here we use very elementary techniques to estimate the essential norm relatively to Dunford-Pettis operators and weakly compact operators. This generalizes the result of Zheng [19] on the (classical) essential norm of a composition operator on H ∞ . Moreover, the results cited above ([1],[12]) are recovered in a simple way. We also generalize all these results, mixing classical composition operators with multiplication operators, i.e. studying weighted composition operators. The compact case was already considered on the disk algebra by Kamowitz in [15]. In that direction too, the results are generalized. Weighted composition operators were also studied by Contreras and Hern´andez-D´ıaz in [5], but on H p spaces, with 1 ≤ p < ∞. In the first part, we treat the classical case of analytic functions of one variable. In the second part, we adapt the method to treat the case of general analytic functions on the open unit ball of a complex Banach space. This is, to the best of our knowledge, new even for the Calkin algebra. In the sequel, we use different classical classes of operators ideals. We refer to [10] for details on the matter. Recall that an operator T : X → Y is a weakly compact operator if T maps the unit ball of X into a relatively weakly compact set. Recall also the following Banach spaces properties. Definition 0.1. Let X be a Banach space. X has the Dunford-Pettis property if for every weakly null sequence (xn ) in X and every weakly null sequence (x∗n ) in X ∗ , x∗n (xn ) tends to zero. Equivalently, for every Banach space Y and every operator T : X → Y which is weakly compact, T maps a weak Cauchy sequence in X into a norm Cauchy sequence. A good survey on the subject (up to the early eighties) is the paper of Diestel [8]. Definition 0.2. Let X be a Banach space. X has the property (V ) of Pelczy´ nski if, ∗ for every non-relatively weakly compact bounded set K ⊂ X , there exists a weakly P unconditionaly series xn in X such that inf n sup{|k(xn )|; k ∈ K} > 0. Equivalently, for every Banach space Y and every operator T : X → Y which is not weakly compact, there exists a subspace Xo of X isomorphic to co such that T|Xo is an isomorphic embedding. Here we extend results, known for the Calkin algebra, to some other ideals of operators. Let us precise the terminology. Definition 0.3. Let X, Y be Banach spaces and I be a closed subspace of the space B(X, Y ) of bounded operators from X to Y . The essential norm (relatively to I) of T ∈ B(X, Y ) is the distance from T to I: kT ke,I = inf{kT + Sk; S ∈ I}.

Vol. 63 (2009) Generalized Essential Norm of Weighted Composition Operators

559

This is the canonical norm on the quotient space B(X, Y )/I. If moreover I is an ideal of the space B(X) then B(X)/I is a Banach algebra. The classical case corresponds to the case of compact operators I = K(X, Y ) (in this case, the above quotient space is the Calkin algebra). In the sequel, we study the case of weakly compact operators, I = W(X, Y ), and the case of completely continuous operators (sometimes also refered to as Dunford-Pettis operators), I = DP (X, Y ). Recall that a completely continuous operator maps a weak Cauchy sequence into a norm Cauchy sequence. Compact operators are both weakly compact and completely continuous. Note that X = H ∞ (resp. the disk algebra, the polydisk algebra and the polyball algebra) has the Dunford-Pettis property (see [4], [2] and [3]), and this can be reformulated as W(X, Y ) ⊂ DP (X, Y ), for every Banach space Y . On the other hand, X = H ∞ (resp. the disk algebra, the polydisk algebra and the polyball algebra) has the Pelczy´ nski property. This implies that DP (X, Y ) ⊂ W(X, Y ) (see [7], [2] and [3]). So actually, in this framework, the two notions of weak compactness and complete continuity coincide. Nevertheless, we give self-contained proofs, hence we do not require the (non-trivial) results cited above. That is why we are able to treat the case of more general spaces, where such properties may not hold. Given an analytic bounded function u on the unit ball B of the Banach space E and an analytic function ϕ from B to B, we introduce the following weighted composition operator: Tu,ϕ (f ) = u.f ◦ ϕ

where f is analytic on B.

We shall estimate in the paper the (generalized) essential norm of Tu,ϕ . Of course, when u = 1, this operator is a classical composition operator and is simply denoted by Cϕ . When ϕ = IdB , this operator is the multiplication operator by u, denoted by Mu . Observe that Tu,ϕ is always bounded from the space of bounded analytic functions on B to the space of analytic bounded functions on B. Moreover, we have kTu,ϕ k = kuk∞ , where kuk∞ = sup{|u(z)|; z ∈ B}. The following quantity plays a crucial role to estimate the essential norm. Let us define lim − |u(z)| = lim− sup |u(z)|; z ∈ B and kϕ(z)k ≥ r nϕ (u) = kϕ(z)k→1 z∈B

r→1

which is a finite number since u is bounded. When kϕk∞ < 1, i.e. ϕ(B) ∩ ∂B = ∅, then nϕ (u) = 0 (i.e. the supremum over the empty set is taken as 0). In this area, it is common to consider analytic functions, so we shall assume in the paper that u is analytic. Nevertheless all the results can be easily adapted assuming only continuity. Under such an assumption, the operators are not defined anymore into H ∞ , but into the space of continuous bounded functions on B. It is actually easier to study these operators, removing the analyticity of u.

560

Lef`evre

IEOT

1. The one variable case In this section, we compute the essential norm relatively both to the completely continuous operators and to the weakly compact operators, on H ∞ and A(D). We first establish the following characterization, which is a generalization of [15], Th.1.2. With the previous notations: Theorem 1.1. The following assertions are equivalent: 1. 2. 3. 4.

Tu,ϕ : A(D) → H ∞ is completely continuous. Tu,ϕ : A(D) → H ∞ is weakly compact. nϕ (u) = 0. Tu,ϕ : H ∞ → H ∞ is compact.

In the previous statement, the third assertion clearly implies that we have u∗|ϕ∗−1 (T) = 0 (a.e.), where, as usual, we denote by ϕ∗ , resp. u∗ , the boundary values of ϕ, resp. u (defined almost everywhere on T by radial limit). An important point of the proof is that we avoid the requirement to the Dunford-Pettis property, as this will be done in the general case (see the second section below). Proof. Obviously 4 implies 1 and 2. 1 ⇒ 3. Assume that kϕk∞ = 1 and nϕ (u) > ε0 > 0. Choose any sequence (zj )j in the unit disk D such that |ϕ(zj )| converges to 1 and |u(zj )| ≥ ε0 . Extracting a subsequence if necessary, we may suppose that ϕ(zj ) converges to some a, belonging to the torus. Now, we consider the sequence of functions fn (z) = 2−n (¯ az + 1)n , which lies in the unit ball of the disk algebra. This is clearly a weak Cauchy sequence: for every z ∈ D \ {a}, fn (z) → 0; and fn (a) = 1. Since the operator Tu,ϕ is completely continuous, the sequence u.fn ◦ ϕ n∈N is norm-Cauchy, hence converges to some σ ∈ H ∞ . But for every fixed z ∈ D, u(z).fn ◦ ϕ(z) converges both to 0 and σ(z), so that σ = 0. Fixing ε > 0, this gives n0 such that supz∈D u(z)fn0 ◦ ϕ(z) ≤ ε. Choosing z = zj0 with j0 large enough to have fn0 ◦ ϕ(zj0 ) ≥ 1 − ε, we have: ε ≥ u zj0 (1 − ε) ≥ (1 − ε)ε0 . As ε is arbitrary, this gives a contradiction. 2 ⇒ 3. Assume that kϕk∞ = 1 and nϕ (u) > ε0 > 0. The idea is very close to the previous argument. Choose any sequence zj ∈ D such that |ϕ(zj )| converges to 1 and |u(zj )| ≥ ε0 . We may assume that ϕ(zj ) converges to some a ∈ T. Now, we consider the same sequence of functions fn (z) = 2−n (¯ az + 1)n . Since the operator Tu,ϕ is a weakly compact operator, there exists a sequence of integers (nk ) such that u.fnk ◦ ϕ k∈N is weakly convergent to some σ ∈ H ∞ . Testing the weak convergence on the Dirac δz ∈ (H ∞ )∗ , for every fixed z ∈ D, we obtain that σ = 0.

Vol. 63 (2009) Generalized Essential Norm of Weighted Composition Operators

561

By the Mazur Theorem, there exists a convex combination of these functions which is norm convergent to 0, X ck u.fnk ◦ ϕ → 0 k∈Im

P

where ck ≥ 0 and k∈Im ck = 1. Now, fixing ε ∈ 0, ε0 /2 , we have for a suitable m0 and every j X X ck u(zj ).fnk ϕ(zj ) ck .fnk ϕ(zj ) ≤ ε0 k∈Im0

k∈Im0

X ck u(z).fnk ϕ(z) ≤ sup z∈D k∈Im0

≤ ε. Letting j tend to infinity, we have fnk ϕ(zj ) → fnk (a) = 1 for each k so that X ε 0 = ε0 ck ≤ ε. k∈Im0

This gives a contradiction. 3 ⇒ 4. We point out that Tu,ϕ = Mu ◦ Cϕ . If kϕk∞ < 1 then Cϕ is compact (see the remark below). If kϕk∞ = 1 and lim

|ϕ(z)|→1− z∈D

u(z) = 0

then Tu,ϕ is compact. Indeed, given a sequence in the unit ball of H ∞ , we can extract a subsequence (fn )n converging on every compact subsets of D. Given ε > 0, we choose a compact disk K ⊂ D such that |u(z)| ≤ ε when ϕ(z) ∈ / K. Then we have ku.(fn − fm ) ◦ ϕk∞ ≤ max kuk∞ . sup |(fn − fm )(z)|; 2ε ϕ(z)∈K

which is less than 2ε when n, m are large enough, due to the uniform convergence on the compact set ϕ(K). Remark 1.2. Note that the preceding result implies that a composition operator Cϕ on H ∞ is completely continuous if and only if it is weakly compact if and only if kϕk∞ < 1. Indeed, if kϕk∞ < 1, it is actually compact (and even nuclear) and if Cϕ is completely continuous (resp. weakly compact) on H ∞ then its restriction to the disk algebra shares the same property. The result follows from the preceding theorem applied to u = 1. We have the same results when the operators act on A(D) (under the extra assumption that ϕ ∈ A(D)).

562

Lef`evre

IEOT

Corollary 1.3. Let u : D → C be a bounded analytic map. 1. Assume that ϕ∗ −1 (T) has positive measure. Then Tu,ϕ is weakly compact or completely continuous if and only if u = 0. 2. In particular, if we assume that Mu : A(D) → H ∞ is weakly compact or completely continuous, then u = 0. Proof. If Tu,ϕ is weakly compact or completely continuous, it follows immediatly from the preceding theorem that u∗ = 0 on a set of positive measure. As u ∈ H ∞ , we obtain that u = 0. In this section, X denotes either A(D) or H ∞ . In the sequel, we shall adapt our argument to compute essential norms. We generalize the proposition in the following way. This is also a generalization of the result of Zheng [19] in several directions. We first have a majorization Lemma 1.4. Let ϕ : D → D be an analytic map and u : D → C be a bounded analytic function. Then kTu,ϕ ke ≤ inf{2nϕ (u), kuk∞ }. Proof. Obviously, kTu,ϕ ke ≤ kTu,ϕ k = kuk∞ . Fix ε > 0. There exists r ∈ (0, 1) such that sup |u(z)| ≤ nϕ (u) + ε. |ϕ(z)|≥r z∈D

Denoting by FN the F´ej`er kernel, we consider the operator defined by N X n fˆ(n)ϕn (z), S(f )(z) = u(z).(FN ∗ f ) ϕ(z) = u(z) 1− N + 1 n=0 PN where N is chosen large enough to satisfy rN ≤ ε(1 − r) and N1+1 n=1 nrn ≤ ε. S is a finite rank operator and the lemma is proved as soon as the following inequality holds: kTu,ϕ − Sk ≤ max 2nϕ (u) + 2ε, 2εkuk∞ . Clearly, for every f in the unit ball of H ∞ , k(Tu,ϕ − S)(f )k is less than max sup |u(z)|. (f − FN ∗ f ) ϕ(z) ; sup |u(z)|. (f − FN ∗ f ) ϕ(z) |ϕ(z)|≥r z∈D

|ϕ(z)|≤r z∈D

We have sup |u(z)|. (f − FN ∗ f ) ϕ(z) ≤ (nϕ (u) + ε) sup |(f − FN ∗ f )(w)|, |ϕ(z)|≥r z∈D

w∈D

which is less than 2(nϕ (u) + ε) by the properties of the F´ej`er kernel and the maximum modulus principle.

Vol. 63 (2009) Generalized Essential Norm of Weighted Composition Operators

563

On the other hand, for every z ∈ D such that |ϕ(z)| ≤ r, we have X N ∞ X n ˆ n n ˆ f (n)ϕ (z) f (n)ϕ (z) + |u(z)|. (f − FN ∗ f ) ϕ(z) ≤ kuk∞ N +1 n=0 n=N +1

so that X N sup |u(z)|. (f − FN ∗ f ) ϕ(z) ≤ kuk∞

n rN rn + N +1 1−r n=0

|ϕ(z)|≤r z∈D

≤ 2εkuk∞ .

This gives the result.

Another kind of proof could be given. This will be done in the next section (see Lemma 2.4). On the other hand, we have the lower estimate: Lemma 1.5. Let u ∈ H ∞ and ϕ : D → D be an analytic map. We assume that I ⊂ W(X, H ∞ ) = DP (X, H ∞ ). Then nϕ (u) ≤ kTu,ϕ ke,I . Proof. The idea of the proof is a mix of the one of Theorem 1.1 and the one of Zheng [19]. We already know that kTu,ϕ ke,I = 0 if and only if Tu,ϕ is completely continuous if and only if nϕ (u) = 0 if and only if Tu,ϕ is compact. We assume now that Tu,ϕ is not completely continuous and this implies that kϕk∞ = 1. We choose a sequence zj ∈ D such that ϕ(zj ) converges to some a ∈ T and |u(zj )| converges to nϕ (u). We introduce the sequence of functions (where n ≥ 2) fn (z) =

n¯ az − (n − 1) , n − (n − 1)¯ az

which lies in the unit ball of the disk algebra. Obviously, fn (z) → −1 for every z ∈ D \ {a} and fn (a) = 1. Now, let S ∈ I. As the sequence (fn )n is a weak Cauchy sequence, the sequence S(fn ) n is a norm Cauchy sequence, hence converging to some σ ∈ H ∞ . Observe that for every n, k(Tu,ϕ − S)(fn )k∞ ≥ kTu,ϕ (fn ) − σk∞ − kS(fn ) − σk∞ and we already know that kS(fn ) − σk∞ → 0. For every z ∈ D \ {a}, we have fn (z) → −1 so that for every z ∈ D \ {a}, |u(z).fn ◦ ϕ(z) − σ(z)| → |u(z) + σ(z)|. The proof splits into two cases: Case 1. If |u(w0 ) + σ(w0 )| > nϕ (u) for some w0 ∈ D, then kTu,ϕ − Sk ≥ limk(Tu,ϕ − S)(fn )k∞ ≥ lim|u(w0 ).fn ◦ ϕ(w0 ) − σ(w0 )| ≥ |u(w0 ) + σ(w0 )| ≥ nϕ (u).

564

Lef`evre

IEOT

Case 2. If not, then ku + σk∞ ≤ nϕ (u) and it follows that for every z ∈ D, |u(z) − σ(z)| ≥ 2|u(z)| − nϕ (u). We have for every n ≥ 2 and every integer j: kTu,ϕ − Sk ≥ |u(zj ).fn ◦ ϕ(zj ) − σ(zj )| − kS(fn ) − σk∞ ≥ 2|u(zj )| − nϕ (u) − |u(zj )|.|fn ◦ ϕ(zj ) − 1| − kS(fn ) − σk∞ . Letting first j tend to infinity (and not forgetting that u is bounded on D), we obtain for every integer n ≥ 2 kTu,ϕ − Sk ≥ nϕ (u) − kS(fn ) − σk∞ . Then, letting n tend to infinity, we have kTu,ϕ − Sk ≥ nϕ (u).

Remark 1.6. We could have given another proof for weak compactness (avoiding the argument of the coincidence of W(X, H ∞ ) and DP (X, H ∞ )): this will be done below when we shall treat the general case (see Lemma 2.3). The following theorem gives a generalization of previously known results on the subject. Theorem 1.7. Let u ∈ H ∞ and ϕ : D → D be an analytic map. We assume that K(X, H ∞ ) ⊂ I ⊂ W(X, H ∞ ) = DP (X, H ∞ ). Then kTu,ϕ ke,I ≈ nϕ (u). More precisely nϕ (u) ≤ kTu,ϕ ke,I ≤ inf{2nϕ (u), kuk∞ }. As a particular case, when nϕ (u) = kuk∞ , the following equality holds: kTu,ϕ ke,I = kTu,ϕ ke = kuk∞ . Proof. We obviously have kTu,ϕ ke,I ≤ kTu,ϕ ke . The result follows from the two preceding lemmas. As an immediate consequence we have: Corollary 1.8. Let u : D → C be a bounded analytic map. Then kMu ke,I = kMu ke = kuk∞ . We are able to obtain the exact value of the essential norms of the operators Tu,ϕ when ϕ has some contacts with T on thin sets, for instance on a finite set. More generally, we define the property T for ϕ : D → D (analytic) by • There exists a compact set K ⊂ T, with Haar measure 0, such that for every sequence (zn ) in D converging to k ∈ D, with |ϕ(zn )| → 1, we have k ∈ K. Clearly, if ϕ ∈ A(D), this property means that ϕ−1 (T) has Haar measure 0. Proposition 1.9. Let u ∈ A(D) and ϕ : D → D be an analytic map with property T . We assume that K(X, H ∞ ) ⊂ I ⊂ W(X, H ∞ ) = DP (X, H ∞ ). Then kTu,ϕ ke,I = kTu,ϕ ke = nϕ (u).

Vol. 63 (2009) Generalized Essential Norm of Weighted Composition Operators

565

Actually, the proof shows that we could assume only that u is bounded and u|K is continuous. Proof. By the Rudin-Carleson Theorem, there exists some v ∈ A(D), with v = u on K and kvk∞ ≤ ku|K k∞ . We claim that Tu−v,ϕ is compact since nϕ (u − v) = 0. Indeed, we have nϕ (u − v) ≤ sup{|(u − v)(z)|; z ∈ K} by property T . Obviously, kTu,ϕ ke ≤ kTu,ϕ − Tu−v,ϕ k = kTv,ϕ k ≤ kvk∞ ≤ ku|K k∞ = nϕ (u).

2. The several variable case We fix a general frame: B denotes in this section the open unit ball of a complex Banach space (E, k.k). A function f : B → C is analytic if it is Fr´echet differentiable. The space H ∞ (B) is then the space of bounded analytic functions on B (see [9] to know more on the subject). The space A(B) is the space of uniformly continuous analytic functions on B. These two spaces are equipped with the uniform norm kf k∞ = sup |f (z)|. z∈B

The case E = C and B = D corresponds clearly to the classical case. When d ≥ 2, we have the following two special cases, which we are particularly interested in: • When Cd is equipped with the sup-norm k(z1 , . . . , zd )k∞ = max |zj |, 1≤j≤d

the framework corresponds to the polydisk algebra B = Dd . • When Cd is equipped with the hermitian norm k(z1 , . . . , zd )k22 =

d X

|zj |2 ,

j=1

the framework corresponds to the polyball algebra B = Bd . Let ϕ be an analytic function from B into itself and Cϕ the associated composition operator. In the sequel, kϕk∞ = sup kϕ(z)k. z∈B

We also consider u ∈ H ∞ (B) and we study the operator Tu,ϕ . We shall denote by X either A(B) or H ∞ (B). The results are essentially the same than the one obtained in the one variable case. The proofs mainly use the same ideas and tools. Nevertheless, we first need the following specific lemma. Lemma 2.1. Let ξ be in the unit ball of the dual of E. The sequence of functions z ∈ B 7−→ fn (z) =

nξ(z) − (n − 1) n − (n − 1)ξ(z)

is a weak Cauchy sequence in the space A(B).

566

Lef`evre

IEOT

Proof. First, (fn ) clearly belongs to the unit ball of A(B). Actually, the key point is that fn = Fn ◦ ξ, where Fn (t) = nt−(n−1) n−(n−1)t is a weak Cauchy sequence in the space of continuous function on D. Obviously, Fn (t) → −1 for every t ∈ D with t 6= 1. We have fn (1) = 1. Let ν ∈ A(B)∗ . By the Hahn-Banach theorem, we obtain ν˜ belonging to the dual of C(B), the space of continuous functions on B (the norm closure of B). We can define a linear continuous functional χ on C(D) in the following way: h ∈ C(D) 7−→ ν˜(h ◦ ξ). The classical Riesz representation R theorem gives us a Borel measure µ on D such that for every h ∈ C(D), χ(h) = D h dµ. But we have by the Lebesgue domination theorem, ν(fn ) = χ(Fn ) → µ {1} − µ D \ {1} . This implies that the sequence fn is a weak Cauchy sequence in the space A(B). The following result is a consequence of Montel’s Theorem, similar to Th.1.1 (3 ⇒ 4). Lemma 2.2 ([1]). Let ϕ : B → B be an analytic map. We assume that kϕk∞ < 1 and that ϕ(B) is relatively compact. Then Cϕ is compact. Proof. See [1], Prop. 3.

We have the following lemma, similar to Lemma 1.4. Lemma 2.3. Let u ∈ H ∞ (B) and ϕ : B → B be analytic with kϕk∞ = 1 and ϕ(B) relatively weakly compact. We assume that I ⊂ W(X, H ∞ (B)) + DP (X, H ∞ (B)). Then nϕ (u) ≤ kTu,ϕ ke,I . When E is reflexive (e.g. finite dimensional), the relative weak compactness assumption on ϕ(B) is automatically fullfiled. Proof. First, we begin with some preliminary remarks. There exists zj ∈ B such that kϕ(zj )k converges to 1. Up to an extraction, we may suppose that ϕ(zj ) is weakly converging to some a ∈ B. Actually, kak = 1. We choose ξ in the unit sphere of the dual of E such that ξ(a) = kak = 1. nξ(z)−(n−1) We introduce the sequence of functions z ∈ B 7−→ fn (z) = n−(n−1)ξ(z) , which clearly lies in the unit ball of A(B). Let S ∈ I. The assumption gives S = D + W , where D ∈ DP (X, H ∞ (B)) and W ∈ W(X, H ∞ (B)). By Lemma 2.1, the sequence (fn )n is a weak-Cauchy sequence, the sequence D(fn ) n is a norm Cauchy sequence, hence converges to some ∆ ∈ H ∞ (B). On the other hand, a subsequence of W (fn ) is weakly convergent to some w ∈ H ∞ (B), so by the Mazur Theorem, there exist some

Vol. 63 (2009) Generalized Essential Norm of Weighted Composition Operators

567

P P ck ≥ 0 with Im ck = 1, where Im ⊂ N; and Im ck W (fk ) → w. Moreover, we can assume that sup Im < inf Im+1 . P Writing f˜m = c f , we have: for every z ∈ B, f˜m (z) → −1 and for Im k k ˜ every m, fm ϕ(zj ) → 1. It is clear that D(f˜m ) m is norm convergent to ∆, so S(f˜m ) m is norm convergent to σ = ∆ + w. The argument now follows the lines of the proof of Lemma 1.4 and we obtain kTu,ϕ − Sk ≥ nϕ (u).

For the upper estimate, we have a result similar to Lemma 1.3. We present here an alternative argument. Lemma 2.4. Let ϕ : B → B be analytic with ϕ(B) relatively compact and u : D → C be a bounded analytic function. Then kTu,ϕ ke ≤ inf{2nϕ (u), kuk∞ }. Proof. Fix ε > 0. There exists r ∈ (0, 1) such that |u(z)| ≤ nϕ (u) + ε.

sup kϕ(z)k≥r z∈B

We consider the operator defined by S(f )(z) = u(z).f ρϕ(z) , where ρ is chosen P in (0, 1), close enough to 1 to satisfy n≥0 1 − ρn rn ≤ ε. By Lemma 2.2, S is a compact operator since kρϕk∞ ≤ ρ < 1 and ϕ(B) is relatively compact. For every f in the unit ball of X and every z ∈ B, we have the Taylor expansion X 1 f (z) = dn f (0).(z)(n) n! n≥0

n

where d f (0) denotes the nth differential in the point 0, and (z)(n) = (z, . . . , z). Hence, X 1 (n) 1 − ρn dn f (0). ϕ(z) . Tu,ϕ − S (f )(z) = u(z) n! n≥0

Since kdn f (0)k/n! ≤ kf k∞ ≤ 1, we obtain, when kϕ(z)k ≤ r, X

Tu,ϕ − S (f )(z) ≤ kuk∞ 1 − ρn kϕ(z)kn ≤ εkuk∞ . n≥0

On the other hand, when kϕ(z)k ≥ r, we have

Tu,ϕ − S (f )(z) ≤ (nϕ (u) + ε) f ϕ(z) + f ρϕ(z) ≤ 2(nϕ (u) + ε). Finally,

Tu,ϕ − S ≤ max εkuk∞ , 2(nϕ (u) + ε) . As ε > 0 is arbitrary, we conclude kTu,ϕ ke ≤ 2nϕ (u). This gives the result. The following theorem is the main result of this section.

568

Lef`evre

IEOT

Theorem 2.5. Let u ∈ H ∞ (B) and ϕ : B → B be analytic. We assume that ϕ(B) is relatively compact and K(X, H ∞ (B)) ⊂ I ⊂ W(X, H ∞ (B)) + DP (X, H ∞ (B)). Then kTu,ϕ ke,I ≈ nϕ (u). More precisely nϕ (u) ≤ kTu,ϕ ke,I ≤ inf{2nϕ (u), kuk∞ }. As a particular case, when nϕ (u) = kuk∞ , the following equality holds: kTu,ϕ ke,I = kTu,ϕ ke = kuk∞ . Of course, when E is finite dimensional, the compactness assumption can be forgotten. Proof. This is an immediate consequence of the preceding lemmas.

We specify two particular cases. Corollary 2.6. Let u ∈ H ∞ (B) and ϕ : B → B be analytic with ϕ(B) relatively compact. We assume that K(X, H ∞ (B)) ⊂ I ⊂ W(X, H ∞ (B)) + DP (X, H ∞ (B)). 1. kMu ke,I = kMu ke = kuk∞ . 2. kCϕ ke,I = 1 if kϕk∞ = 1, and kCϕ ke,I = 0 if kϕk∞ < 1.

References [1] R. Aron, P. Galindo, M. Lindstr¨ om, Compact homomorphisms between algebras of analytic functions, Studia Math. 123 (1997) No. 3, 235–246. [2] J. Bourgain, The Dunford-Pettis property for the ball algebra, the polydisk algebras and Sobolev spaces, Studia Math. 77 (1984), 245–253. [3] J. Bourgain, H ∞ is a Grothendieck space, Studia Math. 75 (1983), 193–216. [4] J. Chaumat, Une g´en´eralisation d’un th´eor`eme de Dunford-Pettis, Analyse harmonique. Orsay Paris XI (preprint 1974). [5] M. Contreras, A. Hern´ andez-D´ıaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001) No. 1, 224–233. [6] C. Cowen, B. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. [7] F. Delbaen, Weakly compact operators on the disk algebra, J. Algebra 45 (1977), 284–294. [8] J. Diestel, A survey of results related to the Dunford-Pettis property, Contemp. Math., vol. 2 (1980), 15–60. [9] S. Dineen, Complex analysis on infinite dimensional spaces, Springer Monographs in Mathematics. Springer-Verlag London, 1999. [10] J. Diestel, H. Jarchow, A. Tonge, Absolutely summing operators, Cambridge University Press, Cambridge, 1995. [11] P. Galindo, T.W. Gamelin, M. Lindstr¨ om, Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets, Trans. Amer. Math. Soc. 359 (2007) No. 5, 2109–2121.

Vol. 63 (2009) Generalized Essential Norm of Weighted Composition Operators

569

[12] P. Galindo, M. Lindstr¨ om, R. Ryan, Weakly compact composition operators between algebras of bounded analytic functions, Proc. Amer. Math. Soc. 128 (2000) No. 1, 149–155. [13] H. Jarchow, Special operators on classical spaces of analytic functions, Extr. Math. 19 (2004) No. 1, 21–53. [14] H. Jarchow, E. Labuschagne, Pointwise multipliers on L1 and related spaces, Note Mat. 25(2005-2006) No. 1, 221–229. [15] H. Kamowitz, Compact operators of the form uCϕ , Pacific J. Math. 80 (1979), 205–211. [16] P. Lef`evre, Some characterizations of weakly compact operators on H ∞ and on the disk algebra. Application to composition operators, J. Oper. Theory 54 (2005) No. 2, 229–238. [17] J. Shapiro, Composition Operators, Springer-Verlag, New York, 1993. ¨ [18] A. Ulger, Some results about the spectrum of commutative Banach algebras under the weak topology and applications, Monat. Math. 121 (1996), 353–376. [19] L. Zheng, The essential norms and spectra of composition operators on H ∞ , Pacific J. Math. 203 (2002) No. 2, 503–510. Pascal Lef`evre Laboratoire de Math´ematiques de Lens. Facult´e Jean Perrin Universit´e d’Artois. rue Jean Souvraz S.P. 18. 62307 Lens cedex France e-mail: [email protected] Submitted: November 26, 2007. Revised: February 16, 2009.

Integr. equ. oper. theory 63 (2009), 571–590 c 2009 Birkhäuser Verlag Basel/Switzerland

0378-620X/040571-20, published online April 1, 2009 DOI 10.1007/s00020-009-1671-4

Integral Equations and Operator Theory

Classes of Operators Similar to Partial Isometries Mostafa Mbekhta and Laurian Suciu Abstract. The present paper deals with operators similar to partial isometries. We get some (necessary and) sufficient conditions for the similarity to (adjoints of) quasinormal partial isometries, or more general, to power partial isometries. We illustrate our results on the class of n-quasi-isometries, obtaining that a n-quasi-isometry is similar to a power partial isometry if and only if the ranges R(T j ) (1 ≤ j ≤ n) are closed. In particular if n = 2, these conditions ensure the similarity to quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasi-isometries. The case when a n-quasi-isometry is a partial isometry is also studied, and a structure theorem for n-quasiisometries which are power partial isometries is given. Mathematics Subject Classification (2000). Primary 47A05; Secondary 47A10, 47A15, 47A63. Keywords. Partial isometry, quasi-isometry, nilpotent operator, similarity, Duggal transform, Aluthge transform.

Introduction and preliminaries Throughout this paper H stands for a complex Hilbert space and B(H) is the Banach algebra of all bounded linear operators on H where I = IH is the identity operator. For T ∈ B(H) we write T ∗ for its adjoint, and R(T ), N (T ) denote the range and the kernel of T , respectively. For a class C of operators on a Hilbert space, we say that an operator T ∈ B(H) is similar to an element of C if there exists C ∈ C ∩ B(H) and an invertible operator A ∈ B(H) such that AT = CA. In this case we say that T is similar to C by A, or that A is a similarity between T and C. In this paper we give certain results concerning the similarity to (adjoints of) quasinormal or power partial isometries for some classes of operators. Recall that T ∈ B(H) is quasinormal if T ∗ T 2 = T T ∗ T . The second author was partially supported by Romanian 2-CEX Research grant. no. 06-1134/2006.

572

Mbekhta and Suciu

IEOT

Let A, T ∈ B(H), A 6= 0 being a positive operator. We say that T is an A-contraction if T ∗ AT ≤ A. When the equality occurs in this inequality we say that T is an A-isometry. Such operators were studied by many authors in different contexts (see [2], [3], [4], [16, 17]). Here we refer to a class of A-contractions which generalizes the quasi-isometries. According to [3] we call T a n-quasicontraction (for n ≥ 1) if T is a T ∗n T n contraction. So T is a n-quasicontraction if and only if T |R(T n ) is a contraction. Particularly, T is called a n-quasi-isometry if it is a T ∗n T n -isometry, or equivalently, if T is an isometry on R(T n ). Briefly, 1-quasicontractions are called quasicontractions and 1-quasi-isometries are called quasi-isometries. The quasiisometries were studied by S. M. Patel in [13, 14] (see also [16, 17]). In particular Patel remarked that any left invertible quasi-isometry is similar to an isometry. Since any n-quasicontraction is similar to a contraction (by Theorem 4.1 [3]), a natural question arises: which n-quasicontractions are similar to partial isometries? We find such n-quasicontractions and, in particular, we infer that a quasiisometry having the range closed is similar to a quasinormal partial isometry. Recall that in [1] the attention is focused on operators similar to partial isometries whose generalized spectrum (see [11]) does not contain the scalar 0. By contrast, we refer here to some n-quasi-isometries which have 0 in their generalized spectrum. In fact, we remark that any n-nilpotent operator T (that is satisfying T n = 0) is a n-quasi-isometry and, on the other hand, in the structure of each n-quasi-isometry appears a n-nilpotent operator on N (T ∗n ) (see Section 2 below). In general, nilpotent operators are not similar to partial isometries on infinite dimensional Hilbert spaces. The similarity of a nilpotent operator to power partial isometries is studied by L. R. Williams [20]. More generally, L. A. Fialkow [6] obtained some partial results concerning the similarity to partial isometries of operators having the spectrum in the open unit disc. We refer now to the content of the paper. In Section 1 we impose some conditions on the range of an operator to obtain its similarity to (an adjoint of) a quasinormal partial isometry. We give characterizations for similarity to such partial isometries, which are related to some corresponding results obtained in [15]. In Section 2 we find a class of A-isometries which are similar to quasinormal partial isometries, whence we infer that a left invertible n-quasi-isometry is similar to an isometry. We prove that, in general, a n-quasi-isometry is similar to a direct sum between an isometry and a nilpotent operator of order n if and only if R(T n ) is closed, and in the case n = 2 we get a “model” up to a similarity for such operators. We establish that, under some conditions on the range, Duggal and Aluthge transforms of a 2-quasi-isometry are similar to a quasinormal partial isometry. In Section 3 we characterize the n-quasi-isometries which are partial isometries. We completely describe the structure of such an operator if it is a power partial isometry, using the decompositions of Neumann-Wold ([19], [10]) and HalmosWallen [9].

Vol. 63 (2009)

Operators Similar to Partial Isometries

573

1. Operators similar to quasinormal partial isometries Recall that any operator T ∈ B(H) has a matrix representation on H = R(T ) ⊕ N (T ∗ ) of the form C S T = (1.1) 0 0 with C = T |R(T ) ∈ B(R(T )) and S ∈ B(N (T ∗ ), R(T )). In [3] it was proved that T is a n-quasicontraction (n-quasi-isometry) on H if and only if C is a (n − 1)-quasicontraction ((n − 1)-quasi-isometry) on R(T ), for n ≥ 2. It is known (see for instance [5], [3]) that T is similar to a contraction if and only if C in (1.1) is similar to a contraction. So, in order to see the similarity of T to a partial isometry, it is natural to investigate conditions for T on its range. Firstly we have the following Theorem 1.1. Let T ∈ B(H) have the matrix representation (1.1). If C = T |R(T ) is similar to a partial isometry and R(S) ⊂ R(C), then T is similar to a partial isometry W on H such that R(T ) is a reducing subspace for W and N (T ∗ ) ⊂ N (W ). Proof. Consider C and S to be as in the matrix (1.1) of T such that R(S) ⊂ R(C). Suppose first that C is a partial isometry on R(T ). In this case we will show that T is similar to the partial isometry W on H = R(T ) ⊕ N (T ∗ ) defined by the matrix C 0 W = . 0 0 Thus, we desire to find an invertible operator L in B(H) with the property LT = W L, L having the form L0 L1 L= L2 L3 with respect to the above decomposition of H. Then the relation LT = W L leads to the following equations: L0 C = CL0 ,

L0 S = CL1 ,

L2 C = 0,

L2 S = 0.

Since R(T ) ⊂ R(C) = R(S) and L2 ∈ B(R(T ), N (T ∗ )), we infer from the previous equations that L2 = 0. If we put L0 = IR(T ) and L3 = IN (T ∗ ) , then we can choose L1 = C ∗ S in the second equation from above, because R(S) ⊂ R(C) = R(CC ∗ ) and C is a partial isometry. Thus we get the invertible operator I C ∗S L= , 0 I such that W = LT L−1 is a partial isometry. This yields that R(T ) is closed, and the matrix form of W shows that R(T ) reduces W and N (T ∗ ) ⊂ N (W ).

574

Mbekhta and Suciu

IEOT

Suppose now that C is similar to a partial isometry, preserving the inclusion R(S) ⊂ R(C). Hence there exist a partial isometry W0 on R(T ) and an invertible f operator B0 ∈ B(R(T )) such that B0 C = W0 B0 . Then we define the operators W ∗ and B on H = R(T ) ⊕ N (T ) by W0 B0 S B0 0 f W = , B= , 0 0 0 I f B. Since B is invertible in B(H) it follows that and it is easy to see that BT = W f by B. T is similar to W Next using the fact that R(S) ⊂ R(C) we obtain R(B0 S) = B0 R(S) ⊂ B0 R(C) = W0 R(B0 ) ⊂ R(W0 ) = R(W0 W0∗ ), whence one infers that B0 S = W0 W0∗ B0 S because W0 W0∗ is an orthogonal projecf is similar to tion. Using this remark, one can proceed as above to obtain that W ∗ e on H = R(T ) ⊕ N (T ) U = W0 ⊕ 0 by the invertible operator B I W0∗ B0 S e B= , 0 I eW f = U B. e Finally, we have BBT e e that is, one has B = U BB, which means that e = BB e on T is similar to the partial isometry U by the invertible operator L H = R(T ) ⊕ N (T ∗ ) B0 W0∗ B0 S e L= . 0 I Clearly, R(T ) is a closed reducing subspace for U and N (T ∗ ) ⊂ N (U ). This ends the proof. f| In the previous proof, we have that W = W0 is a partial isometry, but R(T ) f| W

f) R(W

f ) = R(W0 ), in general. So, = W0 |R(W0 ) is not a partial isometry on R(W

f is similar to a we cannot directly infer from the first part of the proof that W partial isometry. Proposition 1.2. An operator T ∈ B(H) is similar to the adjoint of a quasinormal partial isometry if and only if T |R(T ) is similar to a coisometry. If this is the case, then T is similar to the adjoint of a quasinormal partial isometry W on H such that R(T ) = R(W ) and this subspace reduces W . Proof. Suppose that T |R(T ) is similar to a coisometry V ∗ on R(T ). Clearly one has R(T ) = R(V ∗ ) = R(C) where C = T |R(T ) , so R(S) ⊂ R(C), S being as in (1.1). Then from the preceding proof we infer that T is similar to the partial isometry W = V ∗ ⊕ 0, such that R(W ) = R(V ∗ ) = R(T ), this subspace being closed and reducing for W . Also, W ∗ is a quasinormal partial isometry. Conversely, we assume that T is similar to a partial isometry W ∗ of the form ∗ W = V ∗ ⊕ 0 on H = R(W ∗ ) ⊕ N (W ), where V ∗ is a coisometry on R(W ∗ ). Thus

Vol. 63 (2009)

Operators Similar to Partial Isometries

575

T ∗ is similar to the quasinormal partial isometry W , and by Proposition 3.14 [15] it follows that C ∗ = PR(T ) T ∗ |R(T ) is similar to an isometry. Finally, C = T |R(T ) is similar to a coisometry. Corollary 1.3. A quasicontraction T on H with T |R(T ) a coisometry is similar to the adjoint of a quasinormal partial isometry W such that R(T ) = R(W ), and this subspace reduces W . A characterization of similarity to quasinormal partial isometries is mentioned in the following theorem which completes some results from [15]. Theorem 1.4. Let T ∈ B(H). Then T is similar to a quasinormal partial isometry if and only if R(T ) ∩ N (T ) = {0} and T ∗ |R(T ∗ ) is similar to a partial isometry. In this case, T is similar to a quasinormal partial isometry W on H such that N (W ) = N (T ) and this is a reducing subspace for W . Proof. Suppose that R(T ) ∩ N (T ) = {0} and that T ∗ |R(T ∗ ) is similar to a partial isometry. Let T = U |T | be the polar decomposition of T , U being the partial isometry with N (U ) = N (T ) and |T | = (T ∗ T )1/2 . If Te = |T |U , then Te|T | = |T |T , or equivalently T ∗ |T | = |T |Te∗ . Clearly, N (T ) ⊂ N (Te) and the converse inclusion also holds because R(T )∩N (T ) = {0}. So N (T ) = N (Te), that is R(T ∗ ) = R(Te∗ ), hence R(T ∗ ) reduces Te (having in view the definition of Te). and Te1 = Te| We put T1 = T ∗ | ∗ ∗ . Then the above intertwining reR(T )

R(T )

lation implies T1 A1 = A1 Te1∗ where A1 = |T ||R(T ∗ ) , and also A1 T1∗ = Te1 A1 . Since H = N (Te) ⊕ R(T ∗ ) one has that Te1 is an injective operator, and the previous equality implies that T1∗ is also an injective operator. Then the hypothesis gives that T1∗ is similar to an injective partial isometry, that is to an isometry, or equivalently, T1 is similar to a coisometry. In this case T ∗ has a matrix representation of the form T1 T2 T∗ = 0 0 on H = R(T ∗ ) ⊕ N (T ), where T2 ∈ B(N (T ), R(T ∗ )). By Proposition 1.2 it follows that T ∗ is similar to a partial isometry W ∗ on H such that R(W ∗ ) = R(T ∗ ) reduces W to an isometry. So N (W ) = N (T ) reduces W , and T is similar to W which is a quasinormal partial isometry. Conversely, we assume that T is similar to a quasinormal partial isometry W on H. If L ∈ B(H) is an invertible operator such that LT = W L, then LR(T ) = R(W ), so R(T ) is closed and LN (T ) = N (W ), whence L(R(T ) ∩ N (T )) ⊂ R(W ) ∩ N (W ). Since W is quasinormal one has N (W ) ⊂ N (W ∗ ), hence R(W ) ∩ N (W ) = {0}, and so R(T ) ∩ N (T ) = {0}. On the other hand, since T ∗ is similar to W ∗ it follows that T ∗ |R(T ∗ ) is similar to W ∗ |R(W ∗ ) , R(W ∗ ) being a reducing subspace

576

Mbekhta and Suciu

IEOT

for W . Hence W ∗ |R(W ∗ ) is a partial isometry and so T ∗ |R(T ∗ ) is similar to a partial isometry. The argument from this proof can be used to obtain the following result which completes Proposition 3.14 [15]. Proposition 1.5. Let T ∈ B(H) such that R(T ) is closed. Then T is similar to a quasinormal partial isometry if and only if T |R(T ) is similar to an isometry. Proof. We preserve the notations from the previous proof. Since R(T ∗ ) is closed, the operator A1 = |T ||R(T ∗ ) is positive and invertible. Furthermore, A1 T1∗ = Te1 A1 where T1 = T ∗ |R(T ∗ ) , Te = |T |U and Te1 = Te|R(T ∗ ) . Also, the partial isometry U from the polar decomposition of T satisfies U R(T ∗ ) = R(T ) and N (U ) = N (T ), hence U0 = U |R(T ∗ ) is unitary from R(T ∗ ) onto R(T ). Since U Te = T U , one has U0 Te1 = T0 U0 where T0 = T |R(T ) , and hence U0 A1 T1∗ = T0 U0 A1 , which means that T1∗ is similar to T0 . From this relation we infer that T0 is similar to an isometry on R(T ) if and only if T1 is similar to a coisometry on R(T ). In this case we have also R(T ) ∩ N (T ) = N (T0 ) = {0}, and by Theorem 1.4, T is similar to a quasinormal partial isometry on H. Conversely, if T is similar to a quasinormal partial isometry W , then T0 = T |R(T ) will be similar to W |R(W ) . As R(W ) ⊂ R(W ∗ ) because W is quasinormal and W |R(W ∗ ) is an isometry, it follows that W |R(W ) is an isometry, consequently T0 is similar to an isometry. Recall that the operator Te = |T |U from the above proofs is called the Duggal transform of T (see [7]). So, we infer from the previous proof the following Corollary 1.6. An operator T ∈ B(H) with R(T ) closed is similar to a quasinormal partial isometry if and only if Te|R(T ∗ ) is similar to an isometry. In this case Te is similar to a quasinormal partial isometry. Proof. We know that T |R(T ) and Te|R(T ∗ ) are unitarily equivalent, and also Te = 0 ⊕ Te|R(T ∗ ) , R(T ∗ ) being a reducing subspace for Te. So we can apply Proposition 1.5. Corollary 1.7. A quasi-isometry T on H with closed range is similar to a quasinormal partial isometry on H which preserves the kernel of T as a reducing subspace. Proof. Since T |R(T ) is an isometry, we can apply Proposition 1.5 and Theorem 1.4. We remark that if T is similar to a quasi-isometry, then T |R(T ) is similar to an isometry, but the converse statement is not true, in general. In fact, if R(T ) is closed and T |R(T ) is similar to an isometry, then T is just similar to a quasiisometry (by Proposition 1.5). Other consequences of the above results can be derived as follows:

Vol. 63 (2009)

Operators Similar to Partial Isometries

577

Corollary 1.8. Let T ∈ B(H) such that R(T ) ∩ N (T ) = {0} and T ∗ |R(T ∗ ) is a partial isometry. Then T ∗ |R(T ∗ ) is a coisometry. Proof. Theorem 1.4 implies that T is similar to a quasinormal partial isometry, hence R(T ) and R(T ∗ ) are closed. Also, by Proposition 1.2, T1∗ = (T ∗ |R(T ∗ ) )∗ is similar to an isometry, and so T1∗ is injective. As T1∗ is a partial isometry it follows that T1∗ is an isometry, that is T1 = T ∗ |R(T ∗ ) is a coisometry. Corollary 1.9. Let T ∈ B(H) such that R(T ) ∩ N (T ) = {0} and T ∗ is a quasiisometry on H. Then T ∗ |R(T ∗ ) is unitary, T |R(T ) is similar to a unitary operator, and (T ∗ T )2 ≥ T ∗ T. Proof. By hypothesis T1 = T ∗ |R(T ∗ ) is an isometry and by Corollary 1.8, T1 is a coisometry. So, T1 is a unitary operator on R(T ∗ ) and by the proof of Proposition 1.5, T |R(T ) is similar to a unitary operator. Next, if we consider the matrix representation of T ∗ from the proof of Theorem 1.4, we have I + T2∗ T2 0 ∗ T T = . 0 0 Since (I + T2∗ T2 )2 ≥ I + T2∗ T2 we infer that (T ∗ T )2 ≥ T ∗ T .

As a consequence, we can obtain Theorem 3.6 [14]. Corollary 1.10. Suppose that T and T ∗ are quasi-isometries on H. Then T |R(T ) and T ∗ |R(T ∗ ) are unitary operators, and (T ∗ T )2 ≥ T ∗ T , (T T ∗ )2 ≥ T T ∗ . Moreover, T is similar to a normal partial isometry. Proof. This follows from Corollary 1.9.

Remark 1.11. The last two corollaries give quasi-isometries whose restrictions to their ranges are unitary operators. Also, we remark from Theorem 1.1 that a quasi-isometry T of the form (1.1) such that R(S) ⊂ R(C) is similar to a partial isometry W with R(W ) = R(T ) a reducing subspace for W . In this case, R(T ) is closed and C = T |R(T ) is unitary, because C is an isometry, and as R(S) ⊂ R(C) we have R(T ) = R(C).

2. The case of n-quasi-isometries In general a n-quasi-isometry, for n ≥ 2, is not similar to a partial isometry (see Theorem 2.8 below). But some partial results in this sense can be derived from the following more general result. Theorem 2.1. Let T be an A-isometry on H. Suppose that N (A) ⊂ N (T ) and that either R(A) is closed, or AT = A1/2 T A1/2 . Then T is similar to a quasinormal partial isometry W on H such that N (A) = N (T ) = N (W ), and this subspace reduces W .

578

Mbekhta and Suciu

IEOT

Proof. Let T be an A-isometry with N (A) ⊂ N (T ). Since T ∗ AT = A we have also N (T ) ⊂ N (A), hence N (A) = N (T ), or equivalently R(A) = R(T ∗ ). Then T has a matrix representation on H = R(T ∗ ) ⊕ N (T ) of the form T0 0 T = . T1 0 Assume first that R(A) is closed. Since kA1/2 T hk = kA1/2 hk for h ∈ H, it follows that there exists an isometry V on R(A) = R(T ∗ ) such that V A1/2 h = A1/2 T h, h ∈ H. If we define the operator A0 = A|R(A) , then A0 is invertible in B(R(A)) because R(A) = R(A1/2 ) and A is injective on R(A). Also, it is easy to 1/2 1/2 see that V A0 = A0 T0 , which means that T0 is similar to the isometry V . Then ∗ ∗ T0 = T |R(T ∗ ) is similar to a coisometry, and by Proposition 1.2, T ∗ is similar to the adjoint of a quasinormal partial isometry W on H such that R(T ∗ ) = R(W ∗ ) and this subspace reduces W . Hence T is similar to W and N (T ) = N (W ), this being a reducing subspace for W . Suppose now that the condition AT = A1/2 T A1/2 holds. Using this relation and the above matrix of T one infers that 1/2

1/2

1/2

1/2

A0 V A0 h = A0 T0 A0 h 1/2

(h ∈ R(A)).

1/2

But this implies V A0 = T0 A0 because A0 is injective, and so V = T0 on R(A) = R(T ∗ ). Thus T ∗ is a quasicontraction on H with T ∗ |R(T ∗ ) = V ∗ a coisometry. Then as above, we can apply Corollary 1.3 to conclude that T is similar to a quasinormal partial isometry W such that N (A) = N (T ) = N (W ), this subspace being reducing for W . The proof is finished. Some consequences of this theorem can be derived. Corollary 2.2. A n-quasi-isometry T on H with the range R(T n ) closed and satisfying N (T ) = N (T n ) is similar to a quasinormal partial isometry W on H such that N (T ) = N (W ), and this subspace reduces W . Proof. We apply Theorem 2.1 to the A-isometry T with A = T ∗n T n .

Without the condition N (T ) = N (T n ), when n ≥ 2, we have only a weaker result than the previous corollary: Corollary 2.3. If T is a n-quasi-isometry on H and the range R(T n ) is closed, then T n is similar to a quasinormal partial isometry W on H such that N (T n ) = N (W ), and this subspace reduces W . Proof. If T is a n-quasi-isometry with R(T n ) closed, then T n is also a n-quasiisometry, and one can apply the previous corollary to T n . A n-quasi-isometry T with R(T ) closed and n ≥ 2 is not similar to a quasinormal partial isometry, in general (by Proposition 1.5, or Theorem 1.4). But, under supplementary conditions, T can be similar even to an isometry as below, where we obtain a result which generalizes Theorem 3.7 [14].

Vol. 63 (2009)

Operators Similar to Partial Isometries

579

Corollary 2.4. A left invertible n-quasi-isometry is similar to an isometry. Proof. Let T be a left invertible quasi-isometry on H. Then R(T ) is closed and N (T ) = N (T n ) = {0}. By Corollary 2.2, T is similar to a partial isometry W on H with N (W ) = N (T ) = {0}. Hence W is an isometry on H, and this proves the result. Corollary 2.5. If T is a n-quasi-isometry with R(T ) closed such that T m |R(T ) is left invertible for some m ≥ n ≥ 2, then T is similar to a quasinormal partial isometry. Proof. Since T0 = T |R(T ) is a (n − 1)-quasi-isometry (by Theorem 3.3 [3]), for m ≥ n as above we have that T0m−1 = T m |R(T ) is a left invertible quasi-isometry. So, by Corollary 2.5, T0m−1 is similar to an isometry. Then Corollary 3.13 [2] ensures that T0 is similar to an isometry, and by Proposition 1.5, T is similar to a quasinormal partial isometry. Concerning the Duggal transform we have the following Corollary 2.6. If T is a n-quasi-isometry for n ≥ 2, then its Duggal transform Te is a (n − 1)-quasi-isometry. Proof. We know (Theorem 3.3 [3]) that if T is a n-quasi-isometry then T0 = T |R(T ) is a (n−1)-quasi-isometry. Since T0 is unitary equivalent to Te1 = Te| ∗ , it follows R(T )

that Te1 , and hence Te = Te1 ⊕ 0 are (n − 1)-quasi-isometries.

To see the model of similarity for the n-quasi-isometries, we first make the following Remark 2.7. Let T be a n-quasi-isometry on H, that is V = T |R(T n ) is an isometry. Then T has a matrix representation on H = R(T n ) ⊕ N (T ∗n ) of the form V R T = , 0 Q

(2.1)

where R ∈ B(N (T ∗n ), R(T n )) and Q ∈ B(N (T ∗n )). Moreover, Qn = 0 because T n is a quasi-isometry and so T n needs to have the matrix of the form (1.1) with respect to the above decomposition of H. Conversely, any operator T having a matrix representation of the form (2.1) with V an isometry and Q a n-nilpotent operator, is a n-quasi-isometry. We discuss in the following theorem about the similarity of T of the form (2.1). This result also follows from Corollary 2.3 before and Lemma 3.9 [15], but we give here the details for completeness. Theorem 2.8. Let T be a n-quasi-isometry on H (n ≥ 2). Then T is similar to the direct sum of an isometry and a nilpotent operator of order n if and only if R(T n ) is closed. If furthermore Q in (2.1) is similar to a partial isometry on N (T ∗n ), then T is similar to a partial isometry W on H such that R(T n ) reduces W .

580

Mbekhta and Suciu

IEOT

Proof. First we suppose that R(T n ) is closed, and we want to prove that T having the form (2.1) is similar to the operator W1 = V ⊕ Q on H = R(T n ) ⊕ N (T ∗n ). This means to find an invertible operator L ∈ B(H) of the form L0 L1 L= L2 L3 on H with the above decomposition, satisfying LT = W1 L. This equality leads to the equations L0 V = V L0 ,

L0 R + L1 Q = V L1 ,

L2 V = QL2 ,

L2 R + L3 Q = QL3 .

Choosing L0 = IR(T n ) , L3 = IN (T ∗n ) and L2 = 0, it remains to get L1 such that R + L1 Q = V L1 , which implies L1 = V ∗ R + V ∗ L1 Q. This last relation gives (by recurrence) n−1 X L1 = V ∗(j+1) RQj . j=0

Now, with the above choices for L0 , L1 , L2 and L3 we have I −L1 V 0 I L1 V V L1 − L1 Q V = = 0 I 0 Q 0 I 0 Q 0

R , Q

hence LT = W1 L, that is T and W1 are similar by L. Now, if Q is similar to a partial isometry Q0 on N (T ∗n ), it follows that T is similar to the partial isometry W = V ⊕ Q0 on H, and then R(T n ) is a reducing subspace for W . The converse statement of theorem is obvious, that is if T is similar to an operator of the form V 0 ⊕ Q0 on H with V 0 an isometry and Q0n = 0, then T n is similar to V 0n ⊕ 0, hence R(T n ) is closed. The proof is finished. We remark that W in the above proof is not quasinormal, in general. The argument which ends this proof shows also that if R(T n ) is not closed, then T cannot be similar to the operator W1 = V ⊕ Q. Therefore the condition on R(T n ) to be closed is effective for the similarity of T to W1 . Some conditions on the operators with spectral radius strictly less than one (in particular, nilpotent operators) which are similar to partial isometries are given in [6] (respectively [20]), whence it follows that a nilpotent partial isometry is not necessary similar to a power partial isometry, for n ≥ 2. In our context the case concerning the similarity with power partial isometries is contained in the following Corollary 2.9. If T is a n-quasi-isometry on H then T is similar to a power partial isometry if and only if the ranges R(T j ) for 1 ≤ j ≤ n are closed. When these ranges are closed, T is similar to a power partial isometry W on H such that R(T n ) reduces W . Proof. Suppose that R(T j ) are closed for j = 1, ..., n. By Theorem 2.8, T is similar to V ⊕ Q where V is an isometry on R(T n ) and Q is a n-nilpotent operator on N (T ∗n ). Since R(T j ) is closed, R(Qj ) will be also closed for j = 1, ..., n − 1, hence

Vol. 63 (2009)

Operators Similar to Partial Isometries

581

R(Qm ) is closed for any m ≥ 1. Then by Theorem 2 [20], Q is similar to a Jordan operator, and so to a power partial isometry Q0 on N (T ∗n ). Hence T is similar to the power partial isometry W = V ⊕ Q0 on H = R(T n ) ⊕ N (T ∗n ), and R(T n ) is a reducing subspace for W . The converse statement is obvious. Concerning the last statement of Theorem 2.8, we can see now an example of a 3-nilpotent partial isometry which is not a power partial isometry. Example. Let V1 and V2 be two partial isometries on H such that V1 V2 is not a partial isometry, and let T be the operator on H ⊕ H ⊕ H given by the matrix   0 V1 0 T = 0 0 V2  . 0 0 0 Then T is a partial isometry with T 3 = 0, and one has   0 0 V1 V2 0  T 2 = 0 0 0 0 0 which is not a partial isometry because T ∗2 T 2 = 0 ⊕ (V1 V2 )∗ V1 V2 is not an orthogonal projection on (H ⊕ H) ⊕ H ((V1 V2 )∗ V1 V2 is not idempotent on H). Next we can complete Theorem 2.8 in the case n = 2 as follows: Proposition 2.10. For a 2-quasi-isometry T on H the following are equivalent: (i) the range R(T 2 ) is closed; (ii) T is similar to a direct sum of an isometry and a nilpotent operator of order 2; (iii) the Duggal transform Te of T is similar to a quasinormal partial isometry. Proof. The equivalence between (i) and (ii) is given by Theorem 2.8. Suppose now that R(T 2 ) is closed. If T = U |T | is the polar decomposition of T , then f1 ⊕ 0 where T f1 = Te| , R(T ∗ ) being a reducing subspace for Te = |T |U = T R(T ∗ ) Te. Also U0 = U | is a unitary operator from R(T ∗ ) onto R(T ) and we have ∗ R(T )

f1 = T0 U0 where T0 = T | U0 T . Since R(T0 ) = T R(T ) = R(T 2 ) is closed, it R(T ) f1 ) is closed, hence R(Te) is closed. As Te is a quasi-isometry by follows that R(T Corollary 2.6, from Corollary 1.7 we infer that Te is similar to a quasinormal partial isometry. Thus (i) implies (iii). The converse implication also holds, because by f1 ) closed, and from U0 T f1 = T0 U0 (iii) the range R(Te) is closed which yields R(T one infers that R(T 2 ) = R(T0 ) is closed, that is the statement (i). The case n = 2 in Corollary 2.9 can be also related to the Aluthge transform ∆(T ) of T which is defined by ∆(T ) = |T |1/2 U |T |1/2 , U being as in the previous proof. We have the following

582

Mbekhta and Suciu

IEOT

Proposition 2.11. For an operator T on H the following statements hold: (i) If T is a 2-quasi-isometry with R(T ) and R(T 2 ) closed, then ∆(T ) is similar to a quasinormal partial isometry. (ii) If R(T ) is closed and ∆(T ) is similar to a partial isometry, then R(T 2 ) is closed. Proof. It is clear that R(T ∗ ) reduces ∆(T ), hence one has ∆(T ) = T∗ ⊕ 0 with T∗ = ∆(T )|R(T ∗ ) . If U1 = U |T |1/2 |R(T ∗ ) ∈ B(R(T ∗ ), R(T )), then we have U1 T∗ = T0 U1 with T0 = T |R(T ) . When R(T ) is closed, U1 is invertible from R(T ∗ ) onto R(T ) and so T∗ and T0 are similar by U1 in this case. Now the hypotheses from (i) ensure that T0 is a quasi-isometry with R(T0 ) = R(T 2 ) closed, hence T0 and so T∗ are similar to quasinormal partial isometries. Thus ∆(T ) = T∗ ⊕ 0 is similar to a quasinormal partial isometry, which proves (i). Next the hypotheses from (ii) yield that R(T∗ ) is closed and that T∗ is similar (by U1 ) with T0 , hence R(T 2 ) = R(T0 ) is closed. This proves (ii). Remaining in the context of 2-quasi-isometries, we obtain now another version of Theorem 2.8 of similarity to a particular 2-quasi-isometry. Theorem 2.12. Let T be a 2-quasi-isometry on H with the range R(T 2 ) closed. Then T is similar to a 2-quasi-isometry W on H = R(T ) ⊕ N (T ∗ ) of the form W 0 S0 W = , (2.2) 0 0 where W0 is a quasinormal partial isometry and S0 ∈ B(N (T ∗ ), R(T )) such that W0∗ S0 = 0. Furthermore, if R(S) is closed, S being as in the matrix (1.1) of T , then R(T )is closed. Proof. Suppose that T has the representation (1.1). Then C is a quasi-isometry on R(T ) and R(C) = R(T 2 ) is closed. By Corollary 1.7, C is similar to a quasinormal partial isometry W0 on R(T ) by an invertible operator B0 ∈ B(R(T )) and, as in the proof of Theorem 1.1, T is similar to the operator V on H = R(T ) ⊕ N (T ∗ ) W0 B0 S V = . 0 0 Next, if P |R(W0 ) ∈ B(R(T )) is the orthogonal projection onto R(W0 ), then the operator W0 P |R(W0 ) B0 S V0 = 0 0 is similar to the partial isometry W0 ⊕ 0 by the invertible operator B on H = R(T ) ⊕ N (T ∗ ) I W0∗ P |R(W0 ) B0 S B= . 0 I

Vol. 63 (2009)

Operators Similar to Partial Isometries

583

So BV0 = (W0 ⊕0)B, and it is easy to see that one has also B(V −V0 ) = (V −V0 )B. Hence we have BV = BV0 + B(V − V0 ) = (W0 ⊕ 0 − V + V0 )B = W B, where the operator W on H = R(T ) ⊕ N (T ∗ ) is given by W 0 S0 W = 0 0 with S0 = P |N (W0∗ ) B0 S. Also W0∗ S0 = 0, and T is similar to W by the invertible operator B(B0 ⊕ I) ∈ B(H). Next, using the fact that W0∗ S0 = 0 and that W0 is a quasinormal partial isometry, one can obtain by the above matrix representation of W that ∗ 0 W0 W0 W ∗3 W 3 = = W ∗2 W 2 , 0 S0∗ W0∗ W0 S0 that is W is a 2-quasi-isometry on H. Let us assume now that R(S) is closed. Then obviously R(S0 ) is closed, and R(W ) = R(W0 ) ⊕ R(S0 ) is closed. Since T is similar to W , R(T ) is also closed. This ends the proof. Remark 2.13. In the context of n-quasi-isometries two natural questions arise: When is a n-quasi-isometry T a partial isometry, and which from these have the form T = V ⊕ 0 with V an isometry on R(T n )? We answer this problem in the sequel.

3. n-quasi-isometries and partial isometries We begin with the following Theorem 3.1. Let T be a n-quasi-isometry on H having the matrix representation (2.1). The following statements hold: (i) T is a partial isometry if and only if V ∗ R = 0 and R∗ R+Q∗ Q is a orthogonal projection in B(N (T ∗n )). In this case, one has R(T ∗ ) = R(T n ) ⊕ R(R∗ R + Q∗ Q),

N (T ) = N (R) ∩ N (Q).

(3.1)

(ii) T is quasinormal if and only if T is a partial isometry and N (T ) reduces Q. In this case T is a quasi-isometry. Proof. (i) Suppose that T is a partial isometry. Therefore T = T T ∗ T , and using this relation and the matrix (2.1) of T we infer that RR∗ V = 0, that is R∗ V = 0, and also R(R∗ R + Q∗ Q) = R, Q(R∗ R + Q∗ Q) = Q. Clearly, these relations give (R∗ R + Q∗ Q)2 = R∗ R + Q∗ Q, which means that R∗ R + Q∗ Q is an orthogonal projection in B(N (T ∗n )). But in this case one has I 0 ∗ T T = 0 R∗ R + Q∗ Q

584

Mbekhta and Suciu

IEOT

on H = R(T n ) ⊕ N (T ∗n ), and so the first relation in (3.1) follows. Since the above relations between R and Q yield that R(R∗ ), R(Q∗ ) ⊂ R(R∗ R + Q∗ Q), and using the first relation in (3.1) one infers that N (T ) = N (R∗ R + Q∗ Q) ⊂ N (R) ∩ N (Q). As the converse inclusion is obvious, we get the second relation in (3.1). Suppose now that V ∗ R = 0 and that R∗ R + Q∗ Q is an orthogonal projection in B(N (T ∗n )). Then T ∗ T has the above quoted matrix representation, hence T ∗ T is an orthogonal projection in B(H), that is T is a partial isometry. (ii) Assume that T is quasinormal, that is T and T ∗ T commute. Since T ∗(n+1) T n+1 = T ∗n T n , we have equivalently |T |n T ∗ T |T |n = |T |2n , which means that T is an isometry on R(|T |n ) = R(|T |) = R(T ∗ ). Hence T is a partial isometry. Also, since T is quasinormal one has R(T ) ⊂ R(T ∗ ) (the ranges being closed), and so T is an isometry on R(T ), that is T is a quasi-isometry. Furthermore, since T ∗ T 2 = T in this case, we deduce that (R∗ R + Q∗ Q)Q = Q, and hence (R∗ R + Q∗ Q)Q = Q(R∗ R + Q∗ Q) having in view the above relation between R and Q (in the proof of (i)). This means that N (R∗ R + Q∗ Q) = N (T ) reduces Q. Conversely, we suppose now that T is a partial isometry and that N (T ) reduces Q. Then as above N (T ) = N (R∗ R + Q∗ Q), and so (R∗ R + Q∗ Q)Q = Q(R∗ R+Q∗ Q) = Q, and since V ∗ R = 0 (by (i)) it follows that T ∗ T 2 = T = T T ∗ T . Hence T is quasinormal, which ends the proof. Remark 3.2. It is possible to have T n = 0 for n ≥ 2 in the preceding theorem, but in this case R = 0 and N (T ) become N (Q) in (3.1). For instance, the operator 0 0 T = 1 0 is a partial isometry and 2-quasi-isometry on C2 , but T is not a quasi-isometry. In the case n = 1 we obtain from the above result a more complete version of Corollary 2.3 [13]. Corollary 3.3. The following statements are equivalent for a quasi-isometry T on H: (i) T is a partial isometry; (ii) T is quasinormal; (iii) V ∗ R = 0 and R is a partial isometry, V and R being as in the corresponding matrix (2.1) of T . Proof. Clearly, Q = 0 in the matrix (2.1) of T (the case n = 1). Thus, the equivalences (i) ⇔ (iii) and (i) ⇔ (ii) follow from the assertions (i) and (ii) respectively, in the preceding theorem. Corollary 3.4. A quasi-isometry T is normal if and only if V is unitary and R = 0 in the corresponding matrix (2.1) of T .

Vol. 63 (2009)

Operators Similar to Partial Isometries

585

Proof. If T is a normal quasi-isometry, by the above corollary T is a partial isometry, and so T ∗ is an isometry on R(T ) = R(T ∗ ). This means that T ∗ is also a quasi-isometry, hence V needs to be unitary and R = 0. This corollary shows that normal quasi-isometries have the form U ⊕ 0, with U a unitary operator. But for a quasinormal quasi-isometry one has R 6= 0, in general, in the matrix of T on H = R(T ) ⊕ N (T ∗ ). Obviously we have T = W ⊕ 0 on H = R(T ∗ ) ⊕ N (T ) with W an isometry, because T is a partial isometry in this case. Remark 3.5. For a n-quasi-isometry T with n ≥ 2 we have Q = 0 if and only if R(T n ) = R(T ) and T is a quasi-isometry (as in the case when T is quasinormal). On the other hand, R = 0 if and only if R(T n ) reduces T , and then T n = V n ⊕ 0 is a power partial isometry. If R = 0, T is a (power) partial isometry if and only if Q is a (power) partial isometry. Clearly, a quasi-isometry which is a partial isometry, is a power partial isometry. We also remark that, if a n-quasi-isometry is a power partial isometry, it is not regular in the sense of [1], because 0 belongs to its generalized spectrum. This fact also follows from Theorem 2.3 [1] and the result below which is based on the decomposition of Neumann-Wold ([19], [10]) and Halmos-Wallen [9]. Recall Lmthat a truncated shift of index m ≥ 1 on H means an operator Jm on H = j=1 H0 (the sum of m copies of a closed subspace H0 of H) defined by JL m (h1 , ..., hm ) = (0, Lhn1 , ..., hm−1 ), where J1 = 0. An operator of the form n J = m=1 Jm on H = m=1 Hm (Hm subspaces of H) is called a Jordan operator of order n, n ≥ 1. Theorem 3.6. Let T be a n-quasi-isometry on H which is a power partial isometry. The following statements hold: (i) The maximum subspace which reduces T to a unitary operator is Hu =

∞ \

T j R(T n )

(3.2)

j=0

and R(T n ) = N (I − T n T ∗n ). (ii) The maximum subspace which reduces T to an isometry is R(T ∗n ) = N (I − T ∗n T n ). Moreover, T is a shift on R(T

∗n

) Hu =

∞ M

T j R(Pn ),

(3.3)

j=0

where Pn is the orthogonal projection in B(N (T ∗n )) given by Pn =

n−1 X

Q∗j R∗ RQj ,

j=0

with R and Q as in the matrix (2.1) of T .

(3.4)

586

Mbekhta and Suciu

IEOT

(iii) N (T n ) reduces T to a Jordan operator of order n. Proof. (i) Since T n is a partial isometry, T ∗n is an isometry on R(T n ), and so R(T n ) ⊂ N (I − T n T ∗n ). In fact, R(T n ) = N (I − T n T ∗n ), the converse inclusion being trivial. Now, the subspace Hu of above is invariant for T , and T is an isometry on Hu because T is an isometry on R(T n ). This also implies T ∗ T j R(T n ) = T j−1 R(T n ) for j ≥ 1, hence T ∗ Hu ⊂ Hu . Since T is a contraction one has R(T n ) = N (I − T n T ∗n ) ⊂ N (I − T T ∗ ), and so it follows that Hu reduces T to a unitary operator. Now, if G ⊂ H is another reducing subspace for T such that T is unitary on G, then G also reduces T ∗n to an isometry, therefore G ⊂ N (I − T n T ∗n ). This leads to G = T j G ⊂ T j R(T n ) for j ≥ 1, and finally to G ⊂ Hu . Thus, Hu has the desired property in (i). (ii) As above we have (T n being a partial isometry) R(T ∗n ) = N (I − T ∗n T n ) ⊂ N (I − T ∗ T ), and clearly, R(T ∗n ) is invariant for T ∗ . Since T ∗(n+1) T n+1 = T ∗n T n , for h ∈ N (I − T ∗n T n ) one has kT n T hk = kT n hk = khk = kT hk, that is T h ∈ N (I − T ∗n T n ). Thus R(T ∗n ) reduces T to an isometry. Now if G ⊂ H is another subspace which reduces T , and so T n , to an isometry, then it needs G ⊂ N (I − T ∗n T n ). Hence R(T ∗n ) has the desired property relative to T . Clearly, Hu ⊂ R(T ∗n ) and R(T ∗n ) Hu reduces T to a shift with its ambulant subspace L = R(I − Tn Tn∗ ), where Tn = T |R(T ∗n ) Hu . We will see that L = R(Pn ), Pn being as in (3.4). To prove this, we use some arguments from the proof of Theorem 3.1. Thus, as V ∗ R = 0 and Qn = 0, we get by recurrence that I 0 T ∗n T n = 0 Pn on H = R(T n ) ⊕ N (T ∗n ), where Pn is the orthogonal projection in B(N (T ∗n )) from (3.4). This leads to R(T ∗n ) = R(T n ) ⊕ R(Pn ), and next one has R(T ∗n ) Hu =

∞ M

T n L = (R(T n ) Hu ) ⊕ R(Pn ),

n=0

whence we find L = R(Pn ). (iii) Obviously, N (T n ) = H R(T ∗n ) reduces T , and by the L Halmos-Wallen decomposition [9] T is a direct sum between a coshift C and S = m≥1 Jm with Jm a truncated shift of index m. We suppose that M ⊂ N (T n ) is a subspace which reduces T to a coisometry. Then T ∗n is an isometry on M, and so M ⊂ N (I − T n T ∗n ) = R(T n ) ⊂ R(T ∗n ),

Vol. 63 (2009)

Operators Similar to Partial Isometries

587

hence M = {0}. Thus, C = 0, therefore T |N (T n ) = S on N (T n ). But N (T n ) ⊂ N (T ∗n ), hence S n = 0 on N (T n ), therefore it follows that Jm = 0 for m > n Ln (see also Theorem 3.3 [8]). Thus S = m=1 Jm that is T |N (T n ) = S is a Jordan operator of order n on N (T n ). Remark 3.7. For T as in the preceding theorem we have ∞ \ Hu ⊆ T j N (I − T T ∗ ) =: H1 j=0 ∗

and N (I − T T ) = R(T ). Also H1 is invariant for T but T is not an isometry on H1 , in general when n ≥ 2. Thus, it is possible to have Hu $ H1 . When n = 1 one has Hu = H1 , this being also justified by the fact that T is quasinormal in this case (see [16, 17]). Notice also that a n-quasi-isometry which is a power partial isometry is an element of the class C1 defined in [15], being a direct sum of an isometry and of a Jordan operator. Some consequences can be inferred from the preceding theorem. Corollary 3.8. Let T be a n-quasi-isometry on H. Then T is a power partial isometry if and only if T = Tu ⊕ Ts ⊕ Tn with Tu unitary, Ts a shift and Tn a Jordan operator of order n. Moreover, T is a power partial isometry with no shift part in the above decomposition if and only if R = 0 and Q, Q2 , . . . , Qn−1 are partial isometries, R and Q being as in the matrix (2.1) of T . Proof. The first statement is clear. Now, T = Tu ⊕ Tn if and only if Pn = 0 in (3.4), or equivalently R = 0 and Tn = Q. As Qn = 0, we have in view only the powers Qj for 1 ≤ j ≤ n − 1. Corollary 3.9. A nilpotent operator T ∈ B(H) is a power partial isometry if and only if T is a Jordan operator. Proof. Since a n-nilpotent operator is a n-quasi-isometry, the conclusion follows from the previous corollary. Corollary 3.10. Let T be a quasi-isometry which is a partial isometry. Then we have ∞ ∞ \ M ∗ ∗ j ∗ R(T ) = N (I − T T ) = T N (I − T T ) ⊕ T j L, (3.5) j=0 ∗

j=0

∗

where L = R(R R) = N (T ) N (T ). Moreover, N (T ) = N (R). Proof. In this case (n = 1) one has Q = 0 and P1 = R∗ R in (3.4). Thus, the decomposition (3.5) follows from Theorem 3.6. Also, by (3.1) we have R(T ∗ ) = R(T ) ⊕ R(R∗ R), whence we infer that L = R(R∗ R) = N (T ∗ ) N (T ). Also, from (3.1) one has N (T ) = N (R) because Q = 0.

588

Mbekhta and Suciu

IEOT

Clearly, (3.5) is just the Wold decomposition in R(T ∗ ) of the isometry T |R(T ∗ ) . We remark that the adjoint of a quasi-isometry is not a n-quasi-isometry, in general. The following result related to this fact generalizes the above Corollaries 1.9 and 1.10. Theorem 3.11. Let T be a quasi-isometry on H such that T ∗ is a n-quasi-isometry. Then T ∗ is a quasi-isometry, T |R(T ) and T ∗ |R(T ∗ ) are unitary operators, and (T ∗ T )2 ≥ T ∗ T , (T T ∗ )2 ≥ T T ∗ . Proof. Suppose that T has the form (1.1) on H = R(T )⊕N (T ∗ ), with C = T |R(T ) an isometry. Since T ∗ is a n-quasi-isometry one has T n+1 T ∗(n+1) = T n T ∗n , or equivalently C n+1 C ∗(n+1) + C n SS ∗ C ∗n = C n C ∗n + C n−1 SS ∗ C ∗(n−1) . As C is an isometry, it follows that C 2 C ∗2 + CSS ∗ C ∗ = CC ∗ + SS ∗ , which means T 2 T ∗2 = T T ∗ , that is T ∗ is a quasi-isometry. Also, the previous equality implies N (C ∗ ) ⊂ N (S ∗ ) and CC ∗ + SS ∗ = I + C ∗ SS ∗ C, which gives N (C ∗ ) ⊂ N (I + C ∗ SS ∗ C) = {0}. So, C is unitary on R(T ). Then we have (T T ∗ )2 = (CC ∗ + SS ∗ )2 ≥ I + SS ∗ = T T ∗ , and by symmetry (or by Corollary 1.10) one obtains that T ∗ |R(T ∗ ) is unitary and (T ∗ T )2 ≥ T ∗ T . Corollary 3.12. Suppose that T and T ∗ are quasi-isometries. Then T is a partial isometry if and only if T is normal.

Acknowledgements We are grateful to Catalin Badea for useful discussions concerning this paper.

References [1] C. Badea, M. Mbekhta, Operators similar to partial isometries. Acta Sci. Math. (Szeged), 71 (2005), 663–680.

Vol. 63 (2009)

Operators Similar to Partial Isometries

589

[2] G. Cassier, Autour de quelques interactions récentes entre l’analyse complexe et la théorie des opérateurs. Operator Theory and Banach Algebras, Proceedings of the International Conference in Analysis Rabat (Morocco), April 12–14, 1999, Theta, Bucharest 2003. [3] G. Cassier, L. Suciu, Mapping theorems and similarity to contractions for classes of A-contractions. Hot Topics in Operator Theory, Theta Series in Advanced Mathematics, 2008, 39–58. [4] R.G. Douglas, On the operator equation S ∗ XT = X and related topics. Acta. Sci. Math. (Szeged) 30 (1969), 19–32. [5] G. Eckstein, A. Racz, On operator similar to contractions. Preprint SLOHA, West University of Timisoara, 1978. [6] L. A. Fialkow, Which operators are similar to partial isometries? Proc. Amer. Math. Soc. 56, (1976), 140–144. [7] C. Foiaş, Il Bong Jung, E. Ko, C. Pearcy, Complete contractivity of maps associated with the Aluthge and Duggal transforms. Pacific J. Math. 209 (2003), 2, 249–259. [8] J. Guyker, On partial isometries with no isometric part. Pacific J. Math. 62, 2 (1976), 419–433. [9] P. R. Halmos, L. J. Wallen, Powers of partial isometries. J. Math. Mech. 19 (1969/1970), 657–663. [10] C. S. Kubrusly, An introduction to Models and Decompositions in Operator Theory. Birkhäuser, Boston, 1997. [11] M. Mbekhta, Résolvant généralisé et théorie spectrale. J. Operator Theory, 21 (1989), 69–105. [12] M. Mbekhta, Partial isometries and generalized inverses. Acta Sci. Math. (Szeged), 70 (2004), 767–781. [13] S. M. Patel, A note on quasi-isometries. Glasnik Matematicki, 35 (55) (2000), 307– 312. [14] S. M. Patel, A note on quasi-isometries II. Glasnik Matematicki, 38 (58) (2003), 111–120. [15] P. Peticunot, Intrinsic characterizations of some classes of operators similar to power partial isometries. Preprint, 2007, 1–20. [16] L. Suciu, Some invariant subspaces for A-contractions and applications. Extracta Mathematicae, 21 (3) (2006), 221–247. [17] L. Suciu, Maximum subspaces related to A-contractions and quasinormal operators. J. Korean Math. Soc. 45 (2008), No.1, 205–219. [18] B. Sz.-Nagy, On uniformly bounded linear transformations in Hilbert space. Acta Sci. Math. (Szeged), 11 (1947), 152–157. [19] B. Sz.-Nagy, C. Foias, Harmonic analysis of operators on Hilbert space. NorthHolland Publish., Amsterdam-London, 1970. [20] L. R. Williams, Similarity invariants for a class of nilpotent operators. Acta Sci. Math., 38 (1976), 423–428.

590

Mbekhta and Suciu

Mostafa Mbekhta University Lille 1 UFR de Mathematiques UMR-CNRS 8524 59655 Villeneuve d’Ascq France e-mail: [email protected] Laurian Suciu Institut Camille Jordan University Claude Bernard Lyon 1 21 av. Claude Bernard 69622 Villeurbanne cedex France e-mail: [email protected] Submitted: June 14, 2007.

IEOT

Integr. equ. oper. theory 63 (2009), 591–593 c 2009 Birkhäuser Verlag Basel/Switzerland

0378-620X/040591-3, published online February 24, 2009 DOI 10.1007/s00020-009-1664-3

Integral Equations and Operator Theory

Operators of Read’s Type are not Orbit-Reflexive Jean Esterle Abstract. We show that the operators of Read’s type on the Hilbert space constructed by Sophie Grivaux and Maria Roginskaya cannot be orbit-reflexive. Mathematics Subject Classification (2000). Primary 47A16; Secondary 47A15. Keywords. Orbit, reflexivity, projection, operators of Read’s type.

In a recent preprint [1], Sophie Grivaux and Maria Roginskaya adapted to the separable Hilbert space the celebrated construction by Read [5] of a bounded operator on l1 which does not have any proper invariant closed set. They do not reach the same result, but they manage to construct a nonzero operator T ∈ L(H) which possesses the two following properties 1. For every x ∈ L(H), the closed orbit Orb(x, T ) := {T n x}n≥0 of x for T is a closed linear subspace of H. 2. If x, y ∈ H, then either Orb(x, T ) ⊂ Orb(y, T ), or Orb(y, T ) ⊂ Orb(x, T ). Recall that if X is a Banach space, an operator T ∈ L(X) is said to be reflexive if every operator S ∈ L(X) such that {p(U )x}p∈C(ζ) ⊂ {p(T )x}p∈C(ζ) for every x ∈ X (or, equivalently, such that every closed linear subspace of X invariant for T is invariant for U ) belongs to the closure of the set {p(T )}p∈C[ζ] with respect to the strong operator topology (SOT). Similarly an operator T ∈ L(X) is said to be orbit-reflexive [2] if every operator U ∈ L(X) such that Orb(x, U ) ⊂ Orb(x, T ) for every x ∈ X (or equivalently, such that every closed subset of X invariant for T is invariant for U ) belongs to the closure of the set {T n }n≥0 with respect to SOT. Various conditions which insure that an operator is orbit-reflexive are given in [3], and it is not that easy to exhibit operators which are not orbit-reflexive. Grivaux and Roginskaya indicate how a slight modification of their construction gives an operator which is not orbit reflexive. On the other hand it is well known that an operator for which the set of closed invariant linear subspaces is linearly ordered by inclusion is never reflexive [4]. The aim of this paper is to show that nonzero operators T ∈ L(H) satisfying conditions (1) and (2) above are not orbit-reflexive, and so no modification of the

592

Esterle

IEOT

construction of Grivaux and Roginskaya is needed to obtain a non orbit-reflexive operator on the Hilbert space. These operators are neither reflexive nor orbitreflexive. Examples of reflexive operators on some Banach spaces which are not orbit-reflexive are given by Muller and Vrsovsky in [3], but the existence of a reflexive operator on the Hilbert space which is not orbit-reflexive seems to be an open problem (the Volterra operator on L2 ([0, 1]) provides an easy example of a non-reflexive operator which is orbit-reflexive). Other examples of non orbitreflexive operators on the Hilbert space are given in [3].

1. Operators of Read’s type are not orbit-reflexive We introduce the following notion. Definition 1.1. Let X be a Banach space. An operator T ∈ L(X) is said to be of Read’s type if the two following conditions are satisfied: 1. For every x ∈ L(X), the closed orbit Orb(x, T ) of x for T is a closed linear subspace of X. 2. If x, y ∈ X, then either Orb(x, T ) ⊂ Orb(y, T ), or Orb(y, T ) ⊂ Orb(x, T ). Proposition 1.2. Let X be a Banach space, and let T ∈ L(X) be an operator of Read’s type. Assume that P ∈ L(X) is a bounded linear projection from X onto Orb(a, T ) for some a ∈ X. Then Orb(x, P ) ⊂ Orb(x, T ) for every x ∈ X. If, further, {0} = 6 Orb(a, T ) 6= X, then P T 6= T P. Proof. Since P 2 = P, we have Orb(x, P ) = {x, P x} for every x ∈ X. If x ∈ Orb(a, T ), then P x = x, and Orb(x, P ) = {x} ⊂ Orb(x, T ). If x ∈ / Orb(a, T ), then P x ∈ Orb(a, T ) ⊂ Orb(x, T ). Hence Orb(x, P ) ⊂ Orb(x, T ) for every x ∈ X. Now set F = Ker(P ), and assume that {0} = 6 Orb(a, T ) 6= X. Let b ∈ F \{0}. We have P b = 0, hence T n P b = 0 for every n ≥ 0. On the other hand b ∈ / Orb(a, T ) and so a ∈ Orb(b, T ). This means that there exists a sequence (np )p≥1 of positive integers such that a =limp→+∞ T np b. Hence a = P a =limp→+∞ P T np b, and P T np b 6= 0 = T np b when p is sufficiently large, which shows that T P 6= P T. Corollary 1.3. Let H be the separable Hilbert space, and let T ∈ L(H) be an operator of Read’s type. Then T is not orbit-reflexive. Proof. It follows immediately from the definition of SOT that the closure in L(X) of the set {T n }n≥0 is contained in the commutant of T for every T ∈ L(X) if X is a Banach space. So the corollary follows immediately from the proposition, unless Orb(x, T ) = H for every nonzero x ∈ H. But such an operator, which would provide a counterexample to the invariant subspace problem, would be trivially not orbit-reflexive.

Vol. 63 (2009)

Operators of Read’s Type are not Orbit-Reflexive

593

References [1] S. Grivaux and M. Roginskaya, On Read’s type operators on Hilbert spaces, Int. Math. Res. Notices 2008, 2008:rnn083-42. [2] D. Hadwin, E. Nordgren, H. Radjavi, P. Rosenthal, Orbit-reflexive operators, J. London Math. Soc 34 (1986), 111–119. [3] V. Muller and L. Vrsovsky, On orbit reflexive operators, J. London Math. Soc., to appear. [4] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, Berlin, Heidelberg, New-York, 1973. [5] C. J. Read,The invariant subspace problem for a class of Banach spaces II. Hypercyclic operators, Israel J. Math. 63 (1988), 1–40. Jean Esterle Université de Bordeaux IMB, UMR 5252 351, cours de la Libération 33405-Talence-Cedex France e-mail: [email protected] Submitted: October 22, 2008.

Integr. equ. oper. theory 63 (2009), 595–599 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040595-5, published online April 1, 2009 DOI 10.1007/s00020-009-1673-2

Integral Equations and Operator Theory

A Classification of Homogeneous Operators in the Cowen-Douglas Class Adam Kor´anyi and Gadadhar Misra Abstract. A complete list of homogeneous operators in the Cowen-Douglas class Bn (D) is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc. Mathematics Subject Classification (2000). Primary 47B32; Secondary 14F05, 53B35. Keywords. Hermitizable, Cowen-Douglas class, reproducing kernel, homogeneous holomorphic Hermitian vector bundle.

1. Introduction A bounded linear operator T on a complex separable Hilbert space H is said to be homogeneous if its spectrum is contained in the closed unit disc and for every M¨ obius transformation g of the unit disc D, the operator g(T ) defined via the usual holomorphic functional calculus, is unitarily equivalent to T . To every homogeneous irreducible operator T there corresponds an associated projective unitary representation U of the M¨obius group G: Ug∗ T Ug = g(T ),

g ∈ G.

The projective unitary representations of G lift to unitary representations of the ˜ which are quite well-known. We can choose Ug such that k 7→ Uk universal cover G is a representation of the rotation group K ⊆ G. If H(n) = {x ∈ H : Uk x = ei nθ x}, then T : H(n) → H(n + 1) is a block shift. A complete classification of these for ˜ First dim H(n) ≤ 1 was obtained in [1] using the representation theory of G. This research was supported in part by a DST - NSF S&T Cooperation Programme and a PSC-CUNY grant.

596

Kor´ anyi and Misra

IEOT

examples for dim H(n) = 2 appeared in [5]. Recently (cf. [4, 3]), an m-parameter family of examples with dim H(n) = m was constructed. In the present announcement we show that the ideas of [4, 3] lead to a complete classification of the homogeneous operators in the Cowen-Douglas class. A Fredholm operator T on a Hilbert space H is said to be in the CowenDouglas class Bn (Ω) of the domain Ω ⊆ C if its eigenspaces Ew , w ∈ Ω, are of constant finite dimension n. In the paper [2], Cowen and Douglas show that (a) E ⊆ Ω × H with fiber Ew at w ∈ Ω is a holomorphic Hermitian vector bundle over Ω, where the Hermitian structure is given by ksw kw = kιw sw kH ,

s w ∈ Ew ,

and ιw : Ew → H is the inclusion map; (b) isomorphism classes of E correspond to unitary equivalence classes of T ; (c) the holomorphic Hermitian vector bundle E is irreducible if and only if the operator T is irreducible. It can be shown that an operator in the Cowen-Douglas class Bn (D) is ho˜ We mogeneous if and only if the corresponding bundle is homogeneous under G. describe below all irreducible homogeneous holomorphic Hermitian vector bundles over the unit disc and determine which ones of these correspond to homogeneous operators (necessarily irreducible) in the Cowen-Douglas class.

2. Homogeneous holomorphic vector bundles The description of homogeneous vector bundles via holomorphic induction is wellknown. Let t ⊆ gC = sl(2, C) be the algebra Ch + Cy, where 1 1 0 0 0 , y= . h= 1 0 2 0 −1 Linear representations (%, V ) of the algebra t ⊆ gC = sl(2, C), that is, pairs %(h), %(y) of linear transformations satisfying [%(h), %(y)] = −%(y) provide a parametrization of the homogeneous holomorphic vector bundles. ˜ The G-invariant Hermitian structures on the homogeneous holomorphic vector bundle E (making it into a homogeneous holomorphic Hermitian vector bun˜ dle), if they exist, are given by %(K)-invariant inner products on the representation ˜ ˜ space. Here K is the stabilizer of 0 in G. ˜ An inner product can be %(K)-invariant if and only if %(h) is diagonal with real diagonal elements in an appropriate basis. We are interested only in Hermitizable ˜ ˜ bundles, that is, those that admit a G-invariant Hermitian structure. (Since K is not compact, not all bundles are Hermitizable.) So, we will assume without restricting generality, that the representation space of % is Cn and that %(h) is a real diagonal matrix. Since [%(h), %(y)] = −%(y), we have %(y)Vλ ⊆ Vλ−1 , where Vλ = {ξ ∈ Cn : ˜ %(h)ξ = λξ}. Hence (%, Cn ) is a direct sum, orthogonal for every %(K)-invariant

Vol. 63 (2009)

Classification of Homogeneous Operators

597

inner product of “elementary” representations, that is, such that   −ηI0   d .. %(h) =   with Ij = I on V−(η+j) = C j . −(η + m)Im and 

0 Y1   Y := %(y) =   

 0 Y2

0 .. .

..

.

Ym

   ,   0

Yj : V−(η+j−1) → V−(η+j) .

We denote the corresponding elementary Hermitizable bundle by E (η,Y ) . 2.1. The multiplier and Hermitian structures As in [3] we will use a natural trivialization of E (η,Y ) . In this, the sections of the homogeneous holomorphic vector bundle E (η,Y ) are holomorphic functions D ) ˜ action is given by f 7→ J (η,Y f ◦ g −1 with multiplier taking values in Cn . The G g −1 ( p+` 1 p−` 0 (g )(z)η+ 2 Yp · · · Y`+1 if p ≥ `, (η,Y ) (p−`)! (−cg ) Jg (z) p,` = 0 if p < `, ˜ which, for g near e, acting on D by z 7→ where cg is the analytic function on G −1 (az + b)(cz + d) agrees with c. 0

Proposition 2.1. We have E (η,Y ) ≡ E (η ,Y with a block diagonal matrix A.

0

)

if and only if η = η 0 and Y 0 = AY A−1

A Hermitian structure on E (η,Y ) appears as the assignment of an inner product h·, ·iz on Cn for z ∈ D. We can write hζ, ξiz = hH(z)ζ, ξi,

with H(z) 0.

Homogeneity as a Hermitian vector bundle is equivalent to Jg (z)H(g · z)−1 Jg (z)∗ = H(z)−1 ,

g ∈ G, z ∈ D.

The Hermitian structure is then determined by H = H(0) which is a positive block diagonal matrix. We write (E (η,Y ) , H) for the vector bundle E (η,Y ) equipped with −1 the Hermitian structure H. We note that (E (η,Y ) , H) ∼ = (E (η,AY A ) , A∗ −1 HA) for any block diagonal invertible A. Therefore every homogeneous holomorphic Hermitian vector bundle is isomorphic with one of the form (E (η,Y ) , I). If E (η,Y ) has a reproducing kernel K which is the case for bundles corresponding to an operator in the Cowen-Douglas class, then K satisfies K(z, w) = Jg (z)K(gz, gw)Jg (w)∗ and induces a Hermitian structure H given by H(0) = K(0, 0)−1 .

598

Kor´ anyi and Misra

IEOT

3. Construction of the bundles with reproducing kernel For λ > 0, let A(λ) be the Hilbert space of holomorphic functions on the unit disc with reproducing kernel (1 − z w) ¯ −2λ . It is homogeneous under the multiplier λ ˜ ˜ Let A(η) = g (z) for the action of G. This gives a unitary representation of G. m (η+j) dj (η) (η+j) ⊕j=0 A ⊗C . For f in A , we denote by fj the part of f in A ⊗Cdj . We define Γ(η,Y ) f as the Cn -valued holomorphic function whose part in Cd` is given by ` X 1 1 (`−j) Γ(η,Y ) f ` = Y` · · · Yj+1 fj (` − j)! (2η + 2j) `−j j=0 (η,Y )

for ` ≥ j. For invertible block diagonal N on Cn , we also define ΓN := Γ(η,Y ) ◦ (η,Y ) ˜ N . It can be verified that ΓN is a G-equivariant isomorphism of A(η) as a (η,Y ) homogeneous holomorphic vector bundle onto E (η,Y ) . The image KN of the (η) (η,Y ) reproducing kernel of A is then a reproducing kernel for E . A computation (η,Y ) gives that KN (0, 0) is a block diagonal matrix such that its `th block is (η,Y )

KN

(0, 0)`,` =

` X j=0

(η,Y )

(η,Y )

1 1 ∗ Y` · · · Yj+1 Nj Nj∗ Yj+1 · · · Y`∗ . (` − j)! (2η + 2j)`−j −1

(η,Y )

We set HN = KN (0, 0) . We have now constructed a family (E (η,Y ) , HN ) of elementary homogeneous holomorphic vector bundles with a reproducing kernel (η > 0, Y as before, N invertible block diagonal). Theorem 3.1. Every elementary homogeneous holomorphic vector bundle E with a reproducing kernel arises from the construction given above. Sketch of proof. As a homogeneous bundle E is isomorphic to some E (η,Y ) , its ˜ action on the reproducing kernel gives a Hilbert space structure in which the G (η,Y ) (η,Y ) sections of E is a unitary representation U . Now Γ intertwines the unitary ˜ on A(η) with U . The existence of a block diagonal N such that representation of G (η,Y ) ΓN = Γ(η,Y ) ◦ N is a Hilbert space isometry follows from Schur’s Lemma. As remarked before, every homogeneous holomorphic Hermitian vector bundle is isomorphic to an (E (η,Y ) , I); here Y is unique up to conjugation by a block unitary. In this form, it is easy to tell whether the bundle is irreducible: this is the case if and only if Y is not the orthogonal direct sum of two matrices of the same block type as Y . We call such a Y irreducible. Let P be the set of all (η, Y ) such that E (η,Y ) has a reproducing kernel. Using (η,Y ) the formula for KN (0, 0) we can write down explicit systems of inequalities that determine whether (η, Y ) is in P. In particular we have Proposition 3.2. For every Y , there exists an ηY > 0 such that (η, Y ) is in P if and only if η > ηY . Finally, we obtain the announced classification.

Vol. 63 (2009)

Classification of Homogeneous Operators

599

Theorem 3.3. All homogeneous holomorphic Hermitian vector bundles of rank n with a reproducing kernel correspond to homogeneous operators in the CowenDouglas class Bn (D). The irreducible ones are the adjoint of the multiplication operator M on the Hilbert space of sections of (E (η,Y ) , I) for some (η, Y ) in P and irreducible Y . The block matrix Y is determined up to conjugacy by block diagonal unitaries. Sketch of proof. There is a simple orthonormal system for the Hilbert space A(λ) . Hence we can find such a system for A(η) as well. Transplant it using Γ(η,Y ) to E (η,Y ) . The multiplication operator in this basis has a block diagonal form with Mn := M| res H(n) : H(n) → H(n + 1). This description is sufficiently explicit to see Mn ∼ I + 0( n1 ). Hence M is the sum of an ordinary block shift operator and a Hilbert-Schmidt operator. This completes the proof.

References [1] B. Bagchi and G. Misra, The homogeneous shifts, J. Funct. Anal. 204 (2003), 293–319. [2] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), 187–261. [3] A. Kor´ anyi and G. Misra, Multiplicity free homogeneous operators in the CowenDouglas class, Perspectives in Mathematical Sciences, World Scientific Press, to appear, pp. 109–127. , Homogeneous operators on Hilbert spaces of holomorphic functions, J. Func. [4] Anal. 254 (2008), 2419–2436. [5] D. R. Wilkins, Homogeneous vector bundles and Cowen-Douglas operators, Internat. J. Math. 4 (1993), 503–520. Adam Kor´ anyi Lehman College The City University of New York Bronx, NY 10468 USA e-mail: [email protected] Gadadhar Misra Department of Mathematics Indian Institute of Science Bangalore 560 012 India e-mail: [email protected] Submitted: January 8, 2009.