Integral Equations and Operator Theory - Volume 57

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Integr. equ. oper. theory 57 (2007), 1–17 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-17, published online August 8, 2006 DOI 10.1007/s00020-006-1446-0

Integral Equations and Operator Theory

Amenability, Completely Bounded Projections, Dynamical Systems and Smooth Orbits Daniel Beltit¸˘a and Bebe Prunaru Abstract. We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, ﬁxed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representations. Mathematics Subject Classification (2000). Primary 47L25; Secondary 46L07, 43A07. Keywords. Operator space, completely bounded map, amenable semigroup.

1. Introduction If an injective von Neumann algebra is acted on by an amenable group then the corresponding ﬁxed point algebra is in turn injective (Theorem 3.16 in Chapter XV in [28]). This fact turns out to play a key role in several proofs in the theory of operator algebras. In the present paper we investigate what versions that fact might have in the more general framework of operator spaces (see Corollary 3.2 below). Our initial motivation was that it might be useful to have a very general setting where completely bounded projections are associated with actions of semigroups. With the general result at hand (see Theorem 3.1 below) we soon realized that a lot of seemingly unrelated structures in operator theory can now be understood in a unifying manner. Thus such diﬀerent things as dynamical systems, generalized Toeplitz operators or homogeneous spaces of Lie groups can be looked at from a unique point of view. We should point out that the technique of averaging over amenable groups has a long history in functional analysis and related areas. Its applications range from representation theory of ﬁnite and compact groups (see the so-called Weyl’s unitary trick) to ergodic theory (see [17]) and cohomology of von Neumann algebras (see

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the papers [26] and [27]). From this point of view, what we are doing in the present paper is to investigate the relationship between that technique and the idea of completely bounded map. The structure of the paper is as follows: In Section 2 we introduce the notion of operator S-space, which is roughly speaking an operator space X equipped with a semigroup S of completely bounded maps. To each S-invariant subspace Y ⊆ X ∗ and any left invariant mean on the corresponding space of coeﬃcients CX,Y (S) we associate a completely bounded mapping X → Y ∗ and we study some basic properties of that construction. In Section 3 we prove our main result on the existence of completely bounded projections on ﬁxed point subspaces (Theorem 3.1) and then we explore some of the consequences of that theorem (see Theorems 3.4–3.5 and Corollaries 3.6 through 3.10). Preliminaries Our basic references for the theory of operator spaces and completely bounded maps are the monographs [11], [22] and [7]. We shall now recall several basic facts that will be needed in the sequel. If X is a vector space and p, q ≥ 1 then Mp,q (X) is the space of all p by q matrices with entries in X and Mp (X) = Mp,p (X). If X and Y are vector spaces, ϕ : X → Y is a linear mapping and n ≥ 1 then ϕn : Mn (X) → Mn (Y ) is deﬁned by ϕn ([xij ]) = [ϕ(xij )] for every [xij ] ∈ Mn (X). Let H be a complex Hilbert space and B(H) the C ∗ -algebra of all bounded linear operators on H. Then Mn (B(H)) has a unique C ∗ -algebra norm ·n induced by its identiﬁcation with B(H(n) ) where H(n) is the orthogonal sum of n copies of H. An operator space is a complex vector space X endowed with a complete norm · n on every space Mn (X) and with the property that there exists a linear mapping ϕ : X → B(H) for some Hilbert space H such that ϕn : (Mn (X), · n) → (Mn (B(H)), · n ) is isometric for all n ≥ 1. Any closed subspace of B(H) inherits a canonical structure of operator space. In particular this holds true for C ∗ -algebras. More precisely, if A is a C ∗ -algebra and n ≥ 1 then Mn (A) has a unique C ∗ -algebra norm that is induced by an arbitrary faithful representation of A on a Hilbert space. If A and B are C ∗ algebras and ϕ : A → B is linear then ϕ is said to be completely positive if ϕn is a positive map for all n ≥ 1. If X and Y are operator spaces and ϕ : X → Y is linear then ϕ is said def to be completely bounded if ϕcb = sup{ϕn | n ≥ 1} < ∞, and completely contractive if ϕcb ≤ 1. Moreover ϕ is completely isometric if ϕn is isometric for all n ≥ 1. For X and Y operator spaces the space CB(X, Y ) of all completely bounded maps between X and Y is a Banach space when endowed with the norm · cb . Moreover it has an operator space structure given by the isomorphisms Mn (CB(X, Y )) CB(X, Mn (Y )). We shall always denote CB(X, X) = CB(X). When Y = C the space X ∗ = CB(X, C) is called the operator space dual of X. Now let us consider the operator space X ∗∗ = (X ∗ )∗ . Then it can be shown that the canonical injection J : X → X ∗∗ is a complete isometry.

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If X, Y and Z are operator spaces and ϕ : X × Y → Z is a bilinear map then for all p, q ≥ 1 one denotes ϕp;q : Mp (X) × Mq (Y ) → Mpq (Z),

ϕp;q ([uij ], [vkl ]) = [ϕ(uij , vkl )](i,k),(j,l) .

Then ϕ is said to be completely bounded if ϕcb = sup{ϕp;q | p, q ≥ 1} < ∞. As in the case of completely bounded linear maps, the space CB(X × Y, Z) of all completely bounded bilinear maps ϕ : X × Y → Z has an operator space structure; see [11] for details. Y of two We shall also use the operator space projective tensor product X ⊗ operator spaces X and Y (see [11] for the precise deﬁnition). All we need to know is that it is an operator space structure on a certain completion of the algebraic tensor product X ⊗Y so that for any operator space Z there is a canonical complete Y, Z) CB(X × Y, Z). Moreover these spaces are completely isometry CB(X ⊗ isometric to CB(X, CB(Y, Z)). For a given Banach space X there are two distinguished operator space structures on X, the maximal operator space max X = (X, { · max,n }n≥1 ) and the minimal operator space min X = (X, { · min,n }n≥1 ) such that for any other operator space structure (X, { · n }n≥1 ) one has · min,n ≤ · n ≤ · max,n for all n ≥ 1. It can be shown that (max X)∗ min(X ∗ ) and (min X)∗ max(X ∗ ) as operator spaces (see (3.3.13) and (3.3.15) in [11]). An operator algebra is an associative algebra A endowed with an operator space structure so that there exists a completely isometric homomorphism ϕ : A → B(H) for some Hilbert space H. If moreover A is an operator space dual and ϕ can be chosen so that it is, additionally, weak∗ -continuous, then A is said to be a dual operator algebra. (See Chapter 2 in [7] for details.) We now recall a few basic deﬁnitions in diﬀerential geometry that will be needed in Corollary 3.10. A good reference for the diﬀerential geometry of Banach manifolds and homogeneous spaces is [29]. Let M be a Hausdorﬀ topological space. A local chart of M is a homeomorphism ϕ : U → V , where U is an open subset of M and V is an open subset of some real Banach space. A smooth atlas of Uj = M and M is any family of local charts {ϕj : Uj → Vj }j∈J such that j∈J

ϕj ◦ ϕ−1 k : ϕk (Uj ∩ Uk ) → ϕj (Uj ∩ Uk ) is a smooth mapping (between open subsets of Banach spaces) whenever Uj ∩ Uk = ∅. A Banach manifold is a topological space is another Banach manifold with a M equipped with a maximal smooth atlas. If M is smooth smooth atlas {ϕ ej : Uej → Vej }ej∈Je then a continuous mapping f : M → M if ϕ e ◦ f ◦ ϕ−1 : ϕj (f −1 (Ve)) → Vj is smooth whenever f −1 (Ve) ∩ Vj = ∅. j

j

j

j

A Banach-Lie group is a group G which is also a Banach manifold such that the group operations (i.e., multiplication and inversion) are smooth. For instance, if A is a unital associative Banach algebra then its group of invertible elements, denoted by A× , is a Banach-Lie group. Now let G be a Banach-Lie group and H a subgroup of G. We say that H is a Banach-Lie subgroup if there exists a local chart ϕ : U → V of G such that ϕ(U ∩ H) = V ∩ W, where U is an open neighborhood

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of 1 ∈ G, V is an open subset of the Banach space Z, and W is a split subspace of Z (that is, there exists a bounded linear operator E : Z → Z such that E 2 = E and Ran E = W). If this is the case, then G/H with the quotient topology has a structure of Banach manifold such that the natural projection π : G → G/H is smooth and has smooth local cross-sections on a neighborhood of each point of G/H (see e.g., Theorem 8.19 and Corollary 8.3 in [29]). In this case we say that G/H is a homogeneous space of G, and the natural transitive action G × G/H → G/H,

(g1 , g2 H) → g1 g2 H

is a smooth mapping. Now assume that G = A× for some unital associative Banach algebra A and that H is an algebraic subgroup of G (of degree ≤ d) in the sense that there exist an integer d ≥ 1 and a family F of polynomial functions on A × A of degree ≤ d such that H = {g ∈ A× | (∀f ∈ F) f (g, g −1 ) = 0}. Denote L (H) = {a ∈ A | (∀t ∈ R) exp(ta) ∈ H}. Then H with the topology inherited from A is a Banach-Lie group and L (H) (the Lie algebra of H) is a closed subspace of A such that [a, b] := ab − ba ∈ L (H) whenever a, b ∈ L (H). If it happens that L (H) is a split subspace of A, then H is a Banach-Lie subgroup of A× . (See the main theorem in [14] or Theorem 7.14 in [29].)

2. Operator S-spaces We begin this section by introducing some terminology on semitopological semigroups; we refer to [4] and [5] for more details. Definition 2.1. For any semigroup S we denote by Fb (S) the commutative unital C ∗ -algebra of all complex bounded functions on S with the sup norm · ∞ . For each t ∈ S we deﬁne Lt : Fb (S) → Fb (S) and Rt : Fb (S) → Fb (S) by (Lt f )(s) = f (ts) and (Rt f )(s) = f (st) whenever s ∈ S and f ∈ Fb (S). Now assume that the semigroup S is equipped with a topology. We say that S is a right (respectively, left) topological semigroup if for each s ∈ S the mapping S → S, t → ts (respectively, t → st) is continuous. Moreover S is a semitopological semigroup if it is both left and right topological. If the semigroup S is equipped with a topology then we denote by Cb (S) the set of all continuous functions in Fb (S). When S is a right topological semigroup we denote L UCb (S) the set of all left uniformly continuous bounded complex functions on S. That is, f ∈ L UCb (S) if and only if f ∈ Cb (S) and the mapping S → Cb (S), s → Rs f , is continuous. Similarly, when S is a left topological semigroup we deﬁne the set R UCb (S) of all right uniformly continuous bounded complex functions on S by the above condition with Rs replaced by Ls . Moreover, when S is a

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semitopological semigroup we shall need the set UCb (S) = L UCb (S) ∩ R UCb (S) consisting of all uniformly continuous bounded complex functions on S. It is clear that all of the sets L UCb (S), R UCb (S) and UCb (S) are unital C ∗ -subalgebras of Cb (S). Next assume again that S is an arbitrary semigroup and let T be any linear subspace of Fb (S). We say that T is unital if it contains the unit element 1 of Fb (S) (i.e., if each constant function belongs to T ). In this case, a state of T is a linear functional µ : T → C such that µ = µ(1) = 1. Now assume that T is a linear subspace of Fb (S) that is invariant under the operators Lt for each t ∈ S. We say that a linear functional µ : T → C is S-invariant if µ ◦ Lt = µ for all t ∈ S. The unital subspace T of Fb (S) is said to be amenable if it admits an S-invariant state. If the space Fb (S) is amenable, then the semigroup S is said to be amenable. A topological group S is amenable if the space R UCb (S) is amenable. For instance the unitary groups of all injective von Neumann algebras with the strong operator topology, and also the unitary groups of all nuclear unital C ∗ -algebras with the weak topology are amenable groups (see [13] and [21]). It is known that if S is an amenable locally compact group then even the larger space Cb (S) is amenable (see Theorem 2.2.1 in [12]). Definition 2.2. Let S be a semigroup and X an operator space. We say that X is an operator S-space if it is equipped with a mapping α : S × X → X,

(s, x) → α(s, x) = αs (x)

satisfying the following conditions: (i) for all s, t ∈ S we have αst = αs ◦ αt ; (ii) for all s ∈ S the mapping αs : X → X is completely bounded linear, and moreover sup αs cb < ∞. s∈S

We say that X is a dual operator S-space if moreover there exists an operator space X∗ such that X = (X∗ )∗ and (αs )∗ X∗ ⊆ X∗ (⊆ X ∗ ) for all s ∈ S. An equivalent condition is that αs : X → X is weak∗ -continuous for all s ∈ S. In this case, X∗ is said to be a predual of the operator S-space X. Let X be an operator S-space and let Y ⊆ X ∗ be a closed linear subspace such that (αs )∗ Y ⊆ Y for all ∈ S. We denote by CX,Y (S) the smallest unital closed subspace of Fb (S) that contains all the functions fx,ψ := ψ(α(·, x)) for x ∈ X and ψ ∈ Y . We always think of CX,Y (S) as an operator space with the unique operator space structure that makes the inclusion map CX,Y (S) → Fb (S) into a complete isometry. Note that for all s ∈ S, x ∈ X and ψ ∈ Y we have Ls (fx,ψ ) = fx,(αs )∗ ψ . Thus CX,Y (S) is invariant under Ls for all s ∈ S. Lemma 2.3. Let S be a semigroup, X an operator S-space, Y ⊆ X ∗ a closed linear subspace such that (αs )∗ Y ⊆ Y for all s ∈ S, and CX,Y (S) as above. Then the mapping E 0 : Y × X → CX,Y (S), (ψ, x) → (ψ ◦ α)(·, x)

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is a completely bounded bilinear mapping and E 0 cb ≤ sup αs cb . s∈S

Proof. Let p, q be arbitrary integers, denote p = {1, 2, . . . , p} and q = {1, 2, . . . , q}, and consider the bilinear mapping (E 0 )p;q : Mp (Y ) × Mq (X) → Mpq (Fb (S)) deﬁned by

(E 0 )p;q (ψ, x) = ψij (α(·, xkl )) (i,k),(j,l)∈p×q

for ψ = (ψi,j )i,j∈p ∈ Mp (Y ) ⊆ Mp (X ∗ ) CB(X, Mp ) and x = (xkl )k,l∈q ∈ Mq (X). What we have to prove is that the norm of the bilinear mapping (E 0 )p;q is at most sup αs cb . In fact, s∈S

(E 0 )p;q (ψ, x) = sup ψij (α(·, xkl )) (i,k),(j,l)∈p×q s∈S

= sup ψq ((αs )q (x))

(see (1.1.30) in [11])

s∈S

≤ sup ψcb · αs · x s∈S

= sup αs cb · ψ · x

(see (3.2.5) in [11])

s∈S

and we are done.

Definition 2.4. Let S be a semigroup, X an operator S-space with the semigroup action α : S × X → X, and Y ⊆ X ∗ a closed linear subspace such that (αs )∗ Y ⊆ Y for all ∈ S. Consider the bilinear map E 0 : Y × X → CX,Y (S),

(ψ, x) → (ψ ◦ α)(·, x)

from Lemma 2.3. Then for each bounded linear functional µ : CX,Y (S) → C we deﬁne the mapping Eµ : X → Y ∗ ,

(Eµ (x))(ψ) = µ(E(ψ, x)) = µ((ψ ◦ α)(·, x))

for all x ∈ X and ψ ∈ Y . Lemma 2.5. With the notation of Lemma 2.3 and Definition 2.4 the bilinear map → CX,Y (S) ping E 0 gives rise to a completely bounded linear mapping E : Y ⊗X such that E(ψ ⊗ x) = fx,ψ . Its dual is a completely bounded linear mapping E ∗ : CX,Y (S)∗ → CB(X, Y ∗ ) with E ∗ cb ≤ sup αs cb and E ∗ (µ) = Eµ for all µ ∈ CX,Y (S)∗ . In particular, s∈S

for each µ ∈ CX,Y (S)∗ we have Eµ cb ≤ sup αs cb · µ. s∈S

Proof. Denote M := sup αs cb . It follows by the above Lemma 2.3 along with s∈S

Proposition 7.1.2 in [11] that the bilinear mapping E 0 : Y ×X → CX,Y (S) naturally

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→ CX,Y (S) with Ecb ≤ corresponds to a completely bounded mapping E : Y ⊗X M . Consequently, for the mapping dual to E : Y ⊗X → CX,Y (S) we have ∗ = CB(X, Y ∗ ) E ∗ : CX,Y (S)∗ → (Y ⊗X) (the last equality follows by Corollary 7.1.5 in [11]) and E ∗ cb = Ecb ≤ M . Moreover, we have by Corollary 2.2.3 in [11] that any continuous linear functional µ : CX,Y (S) → C is completely bounded and µcb = µ. Consequently Eµ cb = E ∗ (µ)cb ≤ E ∗ cb · µ ≤ M · µ, and the proof is ﬁnished. Lemma 2.6. Let S be a semigroup, X an operator S-space with the semigroup action α : S × X → X, and Y ⊆ X ∗ a closed linear subspace such that (αt )∗ Y ⊆ Y for all t ∈ S. Let E ∗ : CX,Y (S)∗ → CB(X, Y ∗ ) as in Lemma 2.5 and endow the operator space CB(X, Y ∗ ) with the semigroup action γ : S × CB(X, Y ∗ ) → CB(X, Y ∗ ),

(t, θ) → γ(t, θ) = γt (θ) := ((αt )∗ |Y )∗ ◦ θ.

Then for all t ∈ S the diagram γt

CB(X, Y ∗ ) −−−−→ CB(X, Y ∗ )  ∗  E∗ E L∗

t CX,Y (S)∗ −−−− → CX,Y (S)∗

is commutative. In particular, if we have a bounded linear functional µ : CX,Y (S) → C and an element t ∈ S satisfying µ ◦ Lt = µ, then ((αt )∗ |Y )∗ ◦ Eµ = Eµ . Proof. Let x ∈ X and ψ ∈ Y arbitrary. We have ((αt )∗ |Y )∗ (Eµ (x)), ψ = Eµ (x), (αt )∗ (ψ) = Eµ (x), ψ ◦ αt

= µ (ψ ◦ αt )(α(·, x))

= µ ψ(α(t·, x))

= µ Lt ((ψ ◦ α)(·, x))

= L∗t µ (ψ ◦ α)(·, x)

(since αt αs = αts )

= EL∗t µ (x), ψ,

and the proof is complete.

Notation 2.7. Let S be a semigroup and X an operator space such that there is a semigroup action α : S × X → X, (s, x) → α(s, x) = αs (x), with αs ∈ CB(X) for all s ∈ S. Assume that Y ⊆ X ∗ is a closed linear subspace such that (αs )∗ Y ⊆ Y for all s ∈ S. Then there exists a natural semigroup action S × Y ∗ → Y ∗, We denote

(s, z) → ((αs )∗ |Y )∗ (z).

X S := {x ∈ X | (∀s ∈ S) αs (x) = x},

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and similarly (Y ∗ )S = {z ∈ Y ∗ | (∀s ∈ S) ((αs )∗ |Y )∗ (z) = z}.

Also we denote by ιY : X → Y ∗ the mapping deﬁned by ιY (x) (y) = y, x for all x ∈ X and y ∈ Y ⊆ X ∗ . Thus ιY is the composition between the natural embedding X → X ∗∗ and the quotient mapping X ∗∗ → Y ∗ , ψ → ψ|Y . Proposition 2.8. Let S be a semigroup, X an operator S-space with the semigroup action α : S × X → X, and Y ⊆ X ∗ a closed linear subspace such that (αs )∗ Y ⊆ Y for all s ∈ S. Assume that the space CX,Y (S) is amenable and pick an S-invariant state µ ∈ CX,Y (S)∗ . Then (i) for all x ∈ X S we have Eµ (x) = ιY (x), and (ii) ιY (X S ) ⊆ Ran Eµ ⊆ (Y ∗ )S . In particular, if X is a dual operator S-space and Y ∗ = X, then ιY = idX , therefore Eµ (x) = x for all x ∈ X S and Ran Eµ = X S . On the other hand, if Y = X ∗ then ιY coincides with the canonical embedding X → X ∗∗ and, by this identification, it follows again that Eµ (x) = x for all x ∈ X S and X S ⊆ Ran Eµ ⊆ (X ∗∗ )S . Proof. Assertion (i) follows at once in view of the way Eµ was deﬁned (see Deﬁnition 2.4) along with the fact that µ(1) = 1, where 1 ∈ Fb (S) is the function that is constant 1 on S. The ﬁrst inclusion in assertion (ii) follows by (i). The second inclusion follows by S-invariance of µ along with Lemma 2.6. In the following example we show that in the setting of Proposition 2.8 it could happen that X S = Ran Eµ . Example 2.9. Let G be an amenable discrete inﬁnite group, so that Cb (G) = ∞ (G) is the C ∗ -algebra of all bounded complex functions on G, and there exists a G-invariant state µ : ∞ (G) → C. Consider the Banach space of all absolutely summable complex functions on G,

|f (g)| < ∞ , 1 (G) = f : G → C | f := sup F ⊆G g∈F F finite

and for all g ∈ G deﬁne αg : 1 (G) → 1 (G) by (αg f )(h) = f (g −1 h) for h ∈ G and f ∈ 1 (G). Also for each f ∈ 1 (G) consider the convolution operator Cf : ∞ (G) → ∞ (G), (Cf b)(h) = b(g)f (g −1 h) for b ∈ ∞ (G) and h ∈ G. g∈G

Now deﬁne the operator G-space X = max 1 (G) (see Section 3.3 in [11] for the deﬁnition of the functors min and max from Banach spaces to operator spaces) with α : G × X → X, (g, f ) → αg f. Then X ∗ = min ∞ (G) and X ∗∗ = max( ∞ (G))∗ as operator spaces (see (3.3.13) and (3.3.15) in [11]).

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Since G is inﬁnite, it follows that the only absolutely summable constant function on G is 0, hence X G = {0} = (X ∗∗ )G , where (X ∗∗ )G is just the set of all G-invariant continuous linear functionals on the commutative C ∗ -algebra ∞ (G), and this set is diﬀerent from {0} since G is amenable. Moreover, with the notation of Proposition 2.8 we claim that actually X G = {0} = Ran Eµ ⊆ (X ∗∗ )G . In fact it is easy to see that the mapping Eµ : 1 (G) → ( ∞ (G))∗ can be equivalently deﬁned in terms of the convolution operators by (Eµ f )(b) = µ(Cf b) for b ∈ ∞ (G) and f ∈ 1 (G). Hence for f = δ1 (the characteristic function of {1} ⊆ G) we have Eµ δ1 = µ, whence 0 = µ ∈ Ran Eµ , and the above claim is proved.

3. The main results Theorem 3.1. Let S be a semigroup and X an operator S-space with the semigroup action α : S × X → X. Assume that one of the following hypotheses holds: (a) X is a dual operator S-space and S is amenable as a discrete semigroup, or (b) X is a dual operator S-space, S is an amenable locally compact topological group and for each x ∈ X the mapping α(·, x) is continuous with respect to the weak∗ -topology of X, or (c) X is a dual operator S-space, S is an amenable topological group and the natural action of S on the predual of X is strongly continuous, or (d) X is an operator S-space, S is a compact left topological semigroup which is amenable as a discrete semigroup, and for each x ∈ X the mapping α(·, x) is continuous with respect to the the norm topology of X, or (e) X is an operator S-space which is separable as a Banach space, S is a compact left topological semigroup which is amenable as a discrete semigroup, and for each x ∈ X the mapping α(·, x) is weakly continuous, or (f) S is a compact topological group and for each x ∈ X the mapping α(·, x) is weakly continuous. Then there exists a linear map P : X → X with the following properties: (i) P ∈ CB(X) and P cb ≤ sup αs cb, (ii) Ran P = X S , and (iii) P ◦ P = P .

s∈S

Proof. We ﬁrst consider the conditions (a)–(c). Let X∗ be an operator space predual of X as in Deﬁnition 2.2. We are going to make use of Proposition 2.8 for Y = X∗ . To this end we ﬁrst make sure that, if either of the conditions (a)– (c) is satisﬁed, then the function space CX,Y (S) is amenable. In the case (a), this is obvious since the space Fb (S) of all bounded complex functions on S

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is amenable. In the case (b), we have CX,Y (S) ⊆ Cb (S) and the space Cb (S) is amenable since the group S is locally compact. Finally, in the case (c) recall that for all s ∈ S, x ∈ X and ψ ∈ Y we have Ls (fx,ψ ) = fx,(αs )∗ ψ . The hypothesis (c) means that the mapping S → Y , s → (αs )∗ ψ, is continuous for all ψ ∈ Y , hence we get CX,Y (S) ⊆ R UCb (S), while the latter space is amenable since G is an amenable group. Consequently the space CX,Y (S) is amenable in either of the cases (a)–(c). Now pick an S-invariant state µ : CX,Y (S) → C and denote P = Eµ : X → Y ∗ = X. Then we have by Proposition 2.8 that Ran P = X S and P is the identity map on X S , whence the desired properties (ii)–(iii) follow. As for property (i), it is a consequence of Lemma 2.5. We now address the conditions (d)–(f). We are going to apply Proposition 2.8 with Y = X ∗ . Again we ﬁrst need to check that the space CX,Y (S) is amenable. In both cases (d) and (e) this is obvious since the whole space Fb (S) is amenable. In the case (f) note that CX,Y ⊆ Cb (S) and the latter space is amenable. Thus the space CX,Y (S) is amenable under either of the conditions (d)–(f), and then we can pick an S-invariant state µ : CX,Y (S) → C and denote P = Eµ : X → Y ∗ = X ∗∗ . We are going to prove that actually Ran Eµ ⊆ X, and then the desired properties (i)–(iii) will follow just as above, by Proposition 2.8 along with Lemma 2.5. Firstly assume that the condition (d) is satisﬁed and let x ∈ X arbitrary. In order to show that Eµ (x) ∈ X we have to check that Eµ (x) : X ∗ → C is weak∗ continuous. To this end, it is enough to check that Eµ (x) is weak∗ -continuous on the unit ball of X ∗ . (See e.g., Corollary 2 to Theorem 6.2 in Chapter IV of [25].) weak∗

Thus let {ψj }j∈J be a net in X ∗ such that ψj ≤ 1 for all j ∈ J and ψj −→ 0. j∈J

Then ψj −→ 0 uniformly on the compact subsets of X. On the other hand, since S j∈J

is compact, it follows that {α(s, x) | s ∈ S} is a compact subset of X, hence (ψj ◦ weak∗

α)(·, x) −→ 0 uniformly on S. Consequently (Eµ (x))(ψj ) = µ((ψj ◦ α)(·, x)) −→ 0, j∈J

j∈J

and thus Eµ (x) is weak∗ -continuous on the unit ball of X ∗ . In the case (e), ﬁrst recall that µ actually extends to an S-invariant state of Fb (S), and in particular to an S-invariant state µ : Cb (S) → C. Thus µ actually deﬁnes a Radon measure on S. Next, as in the case (d), we let x ∈ X arbitrary and check that Eµ (x) : X → C is weak∗ -continuous on the unit ball of X ∗ . Since X is separable, the weak∗ -topology of the unit ball of X ∗ is metrizable, hence it is weak∗

enough to check that, if {ψj }j≥0 is a sequence in the unit ball of X ∗ with ψj −→ 0 j→∞

then lim Eµ (x) (ψj ) = 0. But this fact follows by Lebesgue’s dominated converj→∞

gence theorem, since Eµ (x) (ψj ) = µ((ψj ◦ α)(·, x)) and (ψj ◦ α)(·, x)∞ ≤ sup αs cb · x for all j ≥ 1. s∈S

In the case (f), since S is a compact group, it follows by Proposition 4.2.2.1 in [30] that for each x ∈ X the mapping α(·, x) is actually continuous, hence the conclusion follows by (d). Alternatively, note that the invariant state µ : Cb (S) → C is deﬁned by a probability Haar measure on S, and use of Proposition 2 and

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Remark 1 in Chapter III, §4, no. 1 in [8] to show that Ran Eµ (x) ∈ X for all x ∈ X. We note that, under the hypothesis (f) of Theorem 3.1, the mapping Eµ : X → X used in the proof shows up in several places in the existing literature. See e.g., Weyl’s unitary trick (that is, the fact that every representation of a compact group is similar to a unitary representation) or, more recently, [10] and [20]. For the ﬁrst corollary of Theorem 3.1 we recall that an operator space Y is said to be injective if for any complete isometry ϕ : X0 → X and every ψ0 ∈ CB(X0 , Y ) there exists ψ ∈ CB(X, Y ) such that ψ ◦ ϕ = ψ0 and ψcb = ψ0 cb . (See [11] for details.) Corollary 3.2. Let X be a dual operator space and S an amenable semigroup of completely contractive, weak∗ -continuous linear mappings on X. If X is an injective operator space, then X S is in turn injective. Proof. It follows by Theorem 3.1 along with condition (a) in Deﬁnition 2.2 that there exists a completely contractive projection P : X → X with Ran P = X S . Now the desired conclusion follows by Proposition 4.1.6 in [11]. It is safe to say that most of the assertions contained in the following two theorems are parts of the folklore of operator algebras. However we would like to show how they follow directly from Theorem 3.1 and to emphasize that the idempotent mappings we construct here are completely bounded. Before going further, we recall that any ∗-homomorphism of C ∗ -algebras is completely contractive and any ∗-automorphism of a von Neumann algebra is weak∗ -continuous. Theorem 3.3. Let A be a C ∗ -algebra, G a topological group, and α : G → Aut (A) a group homomorphism such that for each x ∈ A the map g → αg (x) is continuous with respect to the norm topology of A. (When G is locally compact, the triple (A, G, α) with these properties is called in literature a C ∗ -dynamical system.) Then the following hold true: (a) If G is amenable then there exists a completely bounded idempotent mapping Q : A∗ → A∗ whose range consists of all linear forms φ ∈ A∗ which are αinvariant, i.e., φ(αg (x)) = φ(x) for all g ∈ G and all x ∈ A. Moreover, if A is unital, then Q(S(A)) ⊂ S(A) hence Q maps the set of all states of A onto the set of all α-invariant states. (b) If G is a compact group, then there exists a completely positive and completely contractive idempotent P : A → A with Ran P = {x ∈ A | αg (x) = x for all g ∈ G}. Proof. (a) Let us consider the action β of G on the dual space A∗ deﬁned by β(g, φ)(x) = φ(αg−1 (x)) for all x ∈ A and all φ ∈ A∗ . It easy to see that A∗ becomes, via this action, a dual operator G-space satisfying Theorem 3.1 item (b) for the case when G is

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locally compact or item (c) for the case when G is a topological group. Now the conclusion follows from that theorem. b) Follows immediately from Theorem 3.1(f) with X = A. Remark 3.4. We refer to [23] for general information on C ∗ -dynamical systems. In the case when G is an amenable locally compact group, the item (a) in Theorem 3.3 holds true under the weaker hypothesis that all the functions g → φ(αg (x)) are continuous on G for all x ∈ A and all φ ∈ A∗ . Indeed, in this case we can apply Theorem 3.1 item (b) to the operator G-space A. Theorem 3.5. Let M be a von Neumann algebra, G a topological group and α : G → Aut (M ) a group homomorphism such that for each x ∈ M and each φ ∈ M∗ (the predual of M ) the functions g → φ(αg (x)) are continuous on G. (When G is locally compact, the triple (M, G, α) as above is called a W ∗ -dynamical system.) Then the following hold true: (a) Suppose either G is an amenable locally compact group or G is an amenable (not necessarily locally compact) topological group with the additional hypothesis (in this general case) that for all φ ∈ M∗ the mapping g → α∗g (φ) is continuous with respect to the norm topology of the predual M∗ . Then there exists a completely positive unital (hence completely contractive too) idempotent mapping P : M → M whose range is the fixed point algebra of α. In particular, it follows that if M is injective, then the fixed point algebra is also injective. (b) If G is compact, then there exists a completely contractive idempotent mapping Q : M∗ → M∗ whose range is precisely the set of all α-invariant normal forms on M . Moreover, Q maps the set of all normal states of M onto the set of all normal and α-invariant states of M . The dual map Q∗ : M → M is a faithful completely positive and normal idempotent mapping whose range is the fixed point algebra of α (faithful means that Ker P ∩ M + = {0}). Proof. a) This follows from Theorem 3.1 item (b) for the case when G is locally compact or from item (c) when G is a topological group. b) Let us consider, in a similar way as in the proof of the preceding theorem, the action β : G × M∗ → M∗ deﬁned by β(g, φ)(x) = φ(αg−1 (x)). Then M∗ becomes an operator G-space, and moreover, by Proposition 4.2.2.1 in [30], the action β is also continuous with respect to the norm topology on M∗ . Now the existence and other properties (except faithfulness) of Q∗ follows from item (f) in Theorem 3.1. The expression of Q as an integral with respect to the Haar measure on a compact group shows that Q∗ = P , where P is the one from (a). Thence the asserted properties of Q∗ follow.

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As another consequence of Theorem 3.1 we now get the following version of Theorem 16(b) in [15]. See [16] for more information on generalized Toeplitz operators. Corollary 3.6. Let (S, ·) be an amenable semigroup, H be a complex Hilbert space and ρ : S → B(H) a norm-continuous mapping such that ρ(st) = ρ(s)ρ(t), ρ(1) = idH and ρ(s) ≤ 1 for all s, t ∈ S. Now consider the space of ρ-Toeplitz operators T (ρ) = {C ∈ B(H) | (∀s ∈ S)

ρ(s)Cρ(s)∗ = C}.

Then there exists a completely positive, completely contractive mapping P : B(H) → B(H) with Ran P = T (ρ), P ◦ P = P and P (ADB ∗ ) = AP (D)B ∗ whenever D ∈ B(H) and A and B belong to the commutant of ρ(S). Proof. First note that condition (a) in Deﬁnition 2.2 is satisﬁed for X = B(H) with the structure of dual operator S-space deﬁned by α : S × B(H) → B(H),

α(s, A) = ρ(s)Aρ(s)∗ .

Clearly B(H)S = T (ρ), hence Theorem 3.1 shows that there exists an idempotent completely contractive linear mapping P : B(H) → B(H) with Ran P = T (ρ). Now it follows by the very construction of P that P is completely positive and P (ADB ∗ ) = AP (D)B ∗ whenever D ∈ B(H) and A and B belong to the commutant of ρ(S). Corollary 3.7. Let T (T) be the space of all Toeplitz operators on the Hardy space H 2 (T) associated with the unit disk. Then there exists a completely positive, completely contractive linear mapping P : B(H 2 (T)) → B(H 2 (T)) such that P ◦ P = P and Ran P = T (T). In particular T (T) is an injective operator space. Proof. Let Mz : H 2 (T) → H 2 (T),

(Mz f )(eiθ ) = eiθ f (eiθ ),

the unilateral shift operator. It is well known that T (T) = {C ∈ B(H 2 (T)) | Mz∗ CMz = C}, hence the desired conclusion follows by Corollary 3.6 applied for the Abelian semigroup (S, ·) = (N, +) and ρ : N → B(H 2 (T)), ρ(n)C = (Mz∗ )n C(Mz )n for all n ∈ N. We note that (N, +) is amenable since it is Abelian (see Theorem 1.2.1 in [12]). As another consequence of Theorem 3.1 we now provide an alternative proof of Theorem 2.4(a) in [2]. In the special case when M = B(H), the next corollary shows that the set C= (ϕ) = {X ∈ B(H) | ϕ(X) = X} studied in [24] is an injective operator space provided ϕ : B(H) → B(H) is a weak∗ -continuous, completely positive, completely contractive map.

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Corollary 3.8. Let M be a W ∗ -algebra, α : M → M a weak∗ -continuous completely positive, completely contractive linear mapping, and denote Mα = {x ∈ M | α(x) = x}. Then there exists an idempotent, completely positive, completely contractive, linear mapping P : M → M with Ran P = Mα . Proof. The existence of a completely contractive projection P from M onto Mα follows by Theorem 3.1 for X = M, S = (N, +) and α(n, x) = αn (x) whenever n ∈ N and x ∈ M. To conclude the proof, we only have to remark that the idempotent mapping P given by Theorem 3.1 is completely positive according to its construction, since α : M → M is completely positive (see also the construction of Eµ in Deﬁnition 2.4). We now arrive at a corollary that has interesting consequences in providing certain homogeneous spaces with structures of Banach manifolds. See [3] and also Corollary 3.10 below. Corollary 3.9. Let X be a complex Banach space, S a topological group and α : S → B(X ),

s → αs ,

a norm continuous representation of S by bounded linear operators on X such that α1 = idX and sup αs < ∞. Assume that one of the following hypotheses holds: s∈S

(a) S is an amenable topological group and X is a dual Banach space such that αs : X → X is weak∗ -continuous for all s ∈ S, or (b) S is a compact topological group. Next denote X S = {x ∈ X | (∀s ∈ S) αs (x) = x}. Then there exists a bounded linear operator P ∈ B(X ) such that P ≤ sup αs , Ran P = X S and P 2 = P .

s∈S

Proof. We are going to apply Theorem 3.1 for the operator space X = max X . According to (3.3.9) in [11] we have an isometric identiﬁcation B(X ) CB(max X ), hence it follows at once that max X is an operator S-space. On the other hand, the above identiﬁcation shows that the desired conclusion will follow as soon as we show that each of the present hypotheses implies one of the conditions of Theorem 3.1 for the operator space X = max X . Actually, it is obvious that the present hypothesis (b) implies that condition (f) in Theorem 3.1 is fulﬁlled. As for the present hypothesis (a), note that it implies that the condition (c) in Theorem 3.1 is satisﬁed. In fact, it follows by (3.3.15) in [11] that if Y is a Banach space such that X = Y ∗ then (min Y)∗ = max Y ∗ = max X , hence max X is the dual operator space of min Y, and we are done.

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The next result is a partial extension of Theorem 4.8 in [9] and is also related to Theorems 3.12 and 4.4 in [1]. We note that under hypothesis (a) of this corollary we do not require that the group G should be locally compact, and thus the result holds for inﬁnite-dimensional Lie groups. Corollary 3.10. Let A be a unital operator algebra and denote by A× its group of invertible elements. Consider an amenable topological group G and denote R := {ρ : G → A× | ρ continuous group homomorphism and sup ρ(g) < ∞}. g∈G

Assume that one of the following conditions is satisfied: (i) A is a dual algebra, or (ii) G is compact. Then the orbits of the action A× × R → R,

(a, ρ) → a · ρ(·) · a−1 ,

have natural structures of Banach manifolds that are smoothly acted on by the Banach-Lie group A× . Proof. In the proof we need techniques and ideas from Lie theory that were recalled in the introduction. Fix ρ ∈ R and consider its isotropy group (A× )ρ = {a ∈ A× | (∀g ∈ G)

a · ρ(g) · a−1 = ρ(g)}.

We shall prove that (A× )ρ is a Banach-Lie subgroup of A× , and then the desired conclusion follows by Theorem 8.19 in [29] in view of the natural bijection that exists from A× /(A× )ρ onto the orbit of ρ. To show that (A× )ρ is a Banach-Lie subgroup of A× , we ﬁrst note that it is an algebraic subgroup of A× of degree ≤ 1 in the sense explained in the introduction to the present paper. It then follows that (A× )ρ has a structure of Banach-Lie group with the topology inherited from A× , as a consequence of the main result of [14]. The Lie algebra of A× is L (A× ) = A, while the Lie algebra of (A× )ρ is L ((A× )ρ ) = {a ∈ A | (∀g ∈ G) a · ρ(g) = ρ(g) · a} = ρ(G) , hence it remains to prove that the commutant ρ(G) has a complement in A. To this end, consider the action of G on A deﬁned by α : G × A → A,

α(g, a) = ρ(g)aρ(g)−1 .

This action makes A into an operator G-space, since α(g, ·) is completely bounded on A for all g ∈ G as an easy consequence of Theorem 17.1.2 in [11]. Moreover note that condition (b) in Deﬁnition 2.2 is satisﬁed. In case (i), it follows by Theorem 2.1 in [6] that the multiplication in A is separately weak∗ -continuous, hence A is actually a dual operator G-space. Now we see that in either of the cases (i) and (ii) it follows by Theorem 3.1 that there exists a completely bounded idempotent mapping P : A → A with Ran P = AG = ρ(G) , and we are done.

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We point out that some further smoothness properties of similarity orbits of group representations (in particular existence of complex structures on the unitary orbits) are discussed in [18] and [19].

References [1] E. Andruchow, G. Corach, D. Stojanoﬀ, A geometric characterization of nuclearity and injectivity. J. Funct. Anal. 133 (1995), no. 2, 474–494. [2] A. Arias, A. Gheondea, S. Gudder, Fixed points of quantum operations. J. Math. Phys. 43 (2002), no. 12, 5872–5881. [3] D. Beltit¸a ˘, T.S. Ratiu, Symplectic leaves in real Banach Lie-Poisson spaces. Geom. Funct. Anal. 28 (2005), no. 1, 59-73. [4] J.F. Berglund, K.H. Hofmann, Compact Semitopological Semigroups and Weakly Almost Periodic Functions. Lecture Notes in Mathematics, No. 42, Springer-Verlag, Berlin-New York, 1967. [5] J.F. Berglund, H.D. Junghenn, P. Milnes, Compact Right Topological Semigroups and Generalizations of Almost Periodicity. Lecture Notes in Mathematics, No. 663, Springer-Verlag, Berlin, 1978. [6] D.P. Blecher, Multipliers and dual operator algebras. J. Funct. Anal. 183 (2001), no. 2, 498–525. [7] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules. An Operator Space Approach. London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, Oxford, 2004. ´ ements de Math´ematique. Fasc. XIII. Livre VI: Int´egration, [8] N. Bourbaki, El´ Chapitres 1–4. (Deuxi`eme ´edition revue et augment´ee) Actualit´es Scientiﬁques et Industrielles, No. 1175 Hermann, Paris, 1965. [9] G. Corach, J.E. Gal´e, On amenability and geometry of spaces of bounded representations. J. London Math. Soc. (2) 59 (1999), no. 1, 311–329. [10] S. Doplicher, R. Longo, J.E. Roberts, L. Zsid´ o, A remark on quantum group actions and nuclearity Rev. Math. Phys. 14 (2002), no. 7-8, 787–796. [11] E.G. Eﬀros, Zh.-J. Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York, 2000. [12] F.P. Greenleaf, Invariant Means on Topological Groups and Their Applications. Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New YorkToronto, Ont.-London, 1969. [13] P. de la Harpe, Moyennabilit´e du groupe unitaire et propri´et´e P de Schwartz des alg`ebres de von Neumann. In: “Alg`ebres d’Op´erateurs (S´em., Les Plans-sur-Bex, 1978),” Lecture Notes in Math., 725, Springer Berlin, 1979, pp. 220–227. [14] L.A. Harris, W. Kaup, Linear algebraic groups in inﬁnite dimensions. Illinois J. Math. 21 (1977), no. 3, 666–674. [15] L. K´erchy, Generalized Toeplitz operators. Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 373–400.

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[16] L. K´erchy, Elementary and reﬂexive hyperplanes of generalized Toeplitz operators. J. Operator Theory 51 (2004), no. 2, 387–409. [17] A. L uczak, Invariant states and ergodic dynamical systems on W ∗ -algebras. Math. Proc. Cambridge Philos. Soc. 111 (1992) no. 1, 181–192. [18] M. Martin, Projective representations of compact groups in C ∗ -algebras. In “Linear Operators in Function Spaces (Timi¸soara, 1988),” Oper. Theory Adv. Appl., Birkh¨ auser, 43, Basel, 1990, pp. 237–253. [19] M. Martin, N. Salinas, Diﬀerential geometry of generalized Grassmann manifolds in C ∗ -algebras. In: “Operator Theory and Boundary Eigenvalue Problems (Vienna, 1993),” Oper. Theory Adv. Appl., Birkh¨ auser, 80, Basel, 1995, pp. 206–243. [20] A. Odzijewicz, T.S. Ratiu, Banach Lie-Poisson spaces and reduction. Comm. Math. Phys. 243 (2003), no. 1, 1–54. [21] A.L.T. Paterson, Nuclear C ∗ -algebras have amenable unitary groups. Proc. Amer. Math. Soc. 114 (1992), no. 3, 719–721. [22] V. Paulsen, Completely Bounded Maps and Operator Algebras Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, 2002. [23] G.K. Pedersen, C ∗ -algebras and Their Automorphism Groups. London Mathematical Society Monographs, 14. Academic Press, Inc., London-New York, 1979. [24] G. Popescu, Similarity and ergodic theory of positive linear maps. J. Reine Angew. Math. 561 (2003), 87–129. [25] H.H. Schaefer, Topological Vector Spaces. The Macmillan Co., Collier-Macmillan Ltd., New York-London, 1966. [26] A.M. Sinclair, R.R. Smith, The Hochschild cohomology problem for von Neumann algebras. Proc. Natl. Acad. Sci. USA 95 (1998), no. 7, 3376–3379. [27] A.M. Sinclair, R.R. Smith, A survey of Hochschild cohomology for von Neumann algebras. In “Operator algebras, Quantization, and Noncommutative Geometry,” Contemp. Math., 365, Amer. Math. Soc., Providence, RI, 2004, pp. 383–400. [28] M. Takesaki, Theory of Operator Algebras III. Encyclopaedia of Mathematical Sciences, 127. Operator Algebras and Non-commutative Geometry, 8. Springer-Verlag, Berlin, 2003. [29] H. Upmeier, Symmetric Banach Manifolds and Jordan C ∗ -algebras. North-Holland Mathematics Studies, 104. Notas de Matem´ atica, 96. North-Holland Publishing Co., Amsterdam, 1985. [30] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I. Die Grundlehren der mathematischen Wissenschaften, Band 188. Springer-Verlag, New York-Heidelberg, 1972. Daniel Beltit¸a ˘ and Bebe Prunaru Institute of Mathematics “Simion Stoilow” of the Romanian Academy P.O. Box 1-764, RO-014700 Bucharest Romania e-mail: [email protected] [email protected] Submitted: August 10, 2005

Integr. equ. oper. theory 57 (2007), 19–41 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010019-23, published online August 8, 2006 DOI 10.1007/s00020-006-1445-1

Integral Equations and Operator Theory

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains Jorge Buescu and A. C. Paix˜ao Abstract. We study eigenvalues of positive deﬁnite kernels of L2 integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at inﬁnity along the diagonal. The main result deals at once with all possible orders of diﬀerentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive deﬁnite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter diﬀerential family of inequalities for the kernel which is a consequence of positivity and is a diﬀerential generalization of diagonal dominance. Mathematics Subject Classification (2000). 45C05, 45P05. Keywords. Integral operators, positive deﬁnite kernels, eigenvalues.

1. Introduction and definitions Given an interval I ⊂ R, a linear operator K : L2 (I) → L2 (I) is integral if there exists a measurable function k(x, y) on I × I such that for all φ ∈ L2 (I) (1.1) φ −→ K(φ) = k(x, y) φ(y) dy I

almost everywhere. The function k(x, y) is called the kernel of K. If k(x, y) = k(y, x) a.e. in I 2 then K is self-adjoint. If in addition K satisﬁes the condition k(x, y) φ(y)φ(x) dx dy ≥ 0 (1.2) I

I

for all φ ∈ L2 (I), then it is a positive operator and the corresponding kernel k(x, y) is called a positive deﬁnite kernel . The first author acknowledges partial support by CAMGSD through FCT/POCI/FEDER.

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This paper shall deal exclusively with positive integral operators and the corresponding positive deﬁnite kernels. Its purpose is the study of the asymptotic behavior of eigenvalues of K in the case where I is unbounded. The case where I is a compact interval has been thoroughly studied. We next describe what is known in this case. Integral operators with L2 (I 2 ) kernels are compact. For self-adjoint operators standard spectral methods yield the bilinear expansion for the kernel k(x, y) = λn φn (x) φn (y), (1.3) n≥1

where λn ∈ R are the eigenvalues of K repeated according to multiplicity, ordered non-increasingly by absolute value and accumulating only at 0. The {φn }n≥1 are the corresponding L2 (I)-orthonormal eigenfunctions spanning the range of K and equality is in the L2 (I) sense. If the operator is positive, k(x, y) is a positive deﬁnite kernel and λn ≥ 0, so the eigenvalue sequence {λn }n≥1 is non-increasing. The asymptotic behavior of the eigenvalue sequence {λn }n≥1 is closely related to smoothness properties of the kernel k(x, y). If k is continuous, the classical theorem of Mercer (see e.g. [22]) asserts that eigenfunctions are continuous, convergence of the series (1.3) is absolute and uniform and the operator K is trace class with λn (1.4) tr (K) = k(x, x) dx = I

n≥1

from which the basic eigenvalue estimate λn = o(1/n) is derived. For general (not necessarily positive deﬁnite) kernels it was shown by Weyl [23] that if k(x, y) is C 1 then λn = o(1/n3/2 ). This estimate may be improved when k is a positive deﬁnite kernel, as shown by Reade [16], to λn = o(1/n2 ). It may also be shown that if a positive deﬁnite kernel k, in addition to continuity, satisﬁes a Lipschitz condition of order α, 0 < α ≤ 1, then λn = O(1/n1+α ), and that this estimate is best possible as a power of n. More generally, positive In fact the optimal estimates are deﬁnite C p kernels satisfy λn = o(1/np+1 ) [18]. ∞ slightly sharper: λn = o(1/np+1 ) for odd p and 1 np λn < +∞ for even p; see Ha [10] and Reade [19]. Cochran and Lukas [8] and Chang and Ha [7] derive the corresponding results for the decay rate of eigenvalues when a suitable higher-order derivative is Lipα . Comparatively with the case where I is compact, little is known about eigenvalues of positive deﬁnite kernels in unbounded domains. There are fundamental reasons for this: integral operators in unbounded domains are in general noncompact, so there will be no pure eigenvalue spectrum but in general also a continuous part. The abstract theory of eigenvalue distribution for integral operators uses operator ideals and interpolation between Besov spaces; see e.g. Birman and Solomyak [1], Gohberg-Krein [9], K¨ onig [12], Pietsch [15] and references therein. While this approach allows for more precise estimation of eigenvalue asymptotics (determining Lorentz space summability of {λn }), its results are not directly applicable

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to our context. In fact, most of the results are valid only in bounded domains; for unbounded domains compactness of the operators must be externally forced. Thus Pietsch [15] and Birman and Solomyak [1] achieve this with parametrically weighted kernels. If the weights are suﬃciently strong to ensure that the resulting kernel and its derivatives decay suﬃciently fast at ∞, eigenvalue estimates may be derived. It should however be noted that there are no results in this theory speciﬁcally for positive deﬁnite kernels in unbounded domains. From what has been described for the compact case, it is to be expected that restriction to this class of kernels yields results which improve on the general estimates. Indeed, although a straightforward comparison is much more delicate than in the compact case (see, e.g., Pietsch’s 10-parameter eigenvalue theorem in [15]), this paper shows that one can say much more in this case than follows from the general theory. For instance, we show below that under very mild assumptions (integrability on the diagonal), positivity is suﬃcient to ensure compactness of the operator, thus totally dispensing weight factors.

2. Preliminaries: classes Sn and An The purpose of this paper is the study of the eigenvalue distribution of positive integral operators in the case where I is an unbounded interval in R. For this purpose, it will be essential to restrict to the following classes of kernels. Note ﬁrst of all that if k(x, y) is a continuous positive deﬁnite kernel then ∀x ∈ I k(x, x) ≥ 0 and ∀x, y ∈ I |k(x, y)|2 ≤ k(x, x)k(y, y); we refer to this property as the diagonal dominance inequality for positive deﬁnite kernels. In all this section I ⊂ R is only assumed to be a topologically closed interval; the deﬁnitions and results below apply whether I is bounded or not. Definition 2.1. A function k(x, y) : I 2 → C is said to belong to class A0 (I) if: 1. k(x, y) is continuous in I 2 ; 2. k(x, x) ∈ L1 (I); 3. k(x, x) is uniformly continuous in I. Remark 2.2. If I is compact, a kernel is in A0 (I) if and only if it is continuous in I 2 . Less obviously, if I is unbounded then a positive deﬁnite kernel k ∈ A0 (I) if and only if k(x, y) is continuous in I 2 , k(x, x) ∈ L1 (I) and k(x, x) → 0 as |x| → +∞; see [2]. The following summarizes the essential properties of A0 (I) positive deﬁnite kernels. For compact I these follow from the classical Mercer theorem (see e.g. [22]); for unbounded I they are proved in [2]. If k(x, y) is a positive deﬁnite kernel in class A0 (I), then the associated integral operator K deﬁned by (1.1) is Hilbert-Schmidt, therefore compact, so it has a pure eigenvalue spectrum {λn }n≥1 with λn ≥ 0 forming a non-increasing sequence converging to 0. Eigenfunctions φn associated with nonzero eigenvalues

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are uniformly continuous and so vanish at inﬁnity in I is unbounded. Moreover, Mercer’s theorem holds in this wider context: the bilinear eigenseries (1.3) for the kernel is absolutely and uniformly convergent and the operator K : L2 (I) → L2 (I) is trace class with trace given by (1.4), whence the eigenvalue sequence satisﬁes λn = o(1/n). Class A0 (I) seems to be the most general class of positive deﬁnite kernels (in bounded or unbounded domains) for which Mercer’s theorem holds; see counterexamples in [2] as well as more general results in Novitskii [13]. It is therefore natural to adopt it as the starting point for the study of eigenvalue distribution of positive deﬁnite kernels in unbounded domains. Remark 2.3. Throughout this paper the diagonal {(x, y) ∈ I 2 : y = x} will play a prominent role in determining the behavior of k(x, y). We abbreviate reference to this set simply as “the diagonal”. The following deﬁnitions are useful in the study of properties arising from diﬀerentiability of the kernel k. If x is a boundary point of I, a limit at x will mean the one-sided limit as y → x with x ∈ I. Definition 2.4. Let I ⊂ R be an interval. A function k : I 2 → C is said to be of class Sn (I) if, for every m1 = 0, 1, . . . n and m2 = 0, 1, . . . n, the partial derivatives ∂ m1 +m2 k(x, y) exist and are continuous in I 2 . ∂y m2 ∂xm1 Definition 2.5. Let n ≥ 1 be an integer. A function k : R2 → C is said to belong to class An (I) if k ∈ Sn (I) and k(x, y),

∂2 ∂ 2n k(x, y), . . . n n k(x, y) ∂y∂x ∂y ∂x

are in class A0 (I). Remark 2.6. Observe, in analogy with Remark 2.2, that if I is compact Sn (I) ⊂ An (I). If I is compact the contents of Theorem 2.7 below are essentially proved by Kadota [11]; the extension to unbounded I is proved in [5]. Here H n (I) denotes the Sobolev Hilbert space W n,2 (I). Theorem 2.7. Let k(x, y) be a positive deﬁnite kernel in An (I) with eigenseries expansion (1.3). Then the following statements hold. 1. If λi = 0, φi is in C n (I) ∩ H n (I); 2. each km is a positive deﬁnite kernel in class An−m (I) and ∂ 2m k (m) (m) km (x, y) = (x, y) = λi φi (x)φi (y) ∂y m ∂xm i≥1

uniformly and absolutely in I 2 for each m = 0, 1, . . . , n;

(2.1)

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3. the L2 (I) integral operator Km with kernel km is trace class with tr(Km ) = Km = km (x, x) dx.

23

(2.2)

I

Particular attention will be devoted, in § 4, to positive deﬁnite kernels in class A0 (I) ∩ Sn (I). Observe that, trivially from the deﬁnitions, An (I) ⊂ A0 (I) ∩ Sn (I).

3. Preparatory results We next present methods introduced by Ha [10] and Reade [16]. Although adaptations of these methods have been developed in [6] to the context of integral operators deﬁned in unbounded domains, in this paper we will only need results relative to operators deﬁned in the compact interval [0, L]. Proofs may be found in these papers and will be omitted. 3.1. Best approximations Let k be a continuous positive deﬁnite kernel in [0, L]2 and K be the associated positive integral operator. It follows from the general theory of compact operators N in Hilbert space that, if R is the operator with kernel n=1 λn φn (x) φn (y), then R is the best approximation to K in the operator norm by symmetric operators of rank ≤ N , the minimum distance being K − Rop = λn+1 (see e.g. Gohberg and Krein [9], Theorem III.6.1). Also N n=1 λn φn (x) φn (y) is the best approximation to k(x, t) by L2 ([0, L]) symmetric kernels of rank ≤ N which generate compact ∞ integral operators, the minimum distance being ( n=N +1 λ2n )1/2 (see [22] for a version for integral operators or [9] for linear operators in Hilbert space). N Lemma 3.1. If k(x, y) is continuous in [0, L]2 , then n=1 λn φn (x) φn (y) is the best approximation in the trace norm by L2 ([0, L]) symmetric kernels of rank ≤ N . 3.2. Square roots Any positive operator K in Hilbert space has a unique positive square root S [21]. This fact implies that if K is a positive operator with continuous kernel k satisfying the bilinear eigenfunction expansion (1.3), the corresponding square root operator S is an L2 ([0, L]) positive integral operator. Since K is trace class, standard arguments imply that the positive deﬁnite kernel s(x, y) of S satisﬁes the bilinear expansion s(x, y) = λ1/2 (3.1) n φn (x) φn (y), n≥1

where the last equality is in the sense of L2 convergence. In general, of course, s(x, y) will not be continuous, so the corresponding operator S will not be trace class. However, the following holds.

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Lemma 3.2. If k(x, y) is continuous in [0, L] and s(x, y) is the kernel of the corresponding positive square root operator, then for any f ∈ L2 ([0, L])

L

Sf (x) =

s(x, y)f (y) dy 0

is a continuous function of x. 3.3. A class of finite rank operators We now deﬁne a class of ﬁnite rank operators to be used in the approximation of a positive operator K with continuous kernel k deﬁned in the interval [0, L], L > 0. Let N > 0 be an integer and L > 0 be a positive real number. We deﬁne RN,L to be the L2 ([0, L]) operator with kernel rN,L (u, v) = where ψnN,L(x) =

1 0

N N N,L ψ (u) ψnN,L(v), L n=1 n

L L < x ≤ nN if (n − 1) N otherwise.

Clearly, RN,L is an orthogonal projection in L2 ([0, L]). It is thus a positive operator of rank N with 0 ≤ RN,L ≤ I. Its spectrum is {0, 1}, the eigenvalue 1 having multiplicity N and the corresponding orthogonal (unnormalized) eigenfunctions being the ψnN,L. Given an operator K ∈ L2 ([0, L]) with continuous kernel k and square root S, it follows that 0 ≤ SRN,LS ≤ K; since by Lemma 3.2 ([16], Lemma 3) SRN,LS has a continuous kernel, it is trace class. It then follows (see [16], [6] for details) that n NL L N N nN (k(u, u) − k(u, v)) du dv (3.2) L (n−1) NL (n−1) NL n=1 L n NL N k(u, u) − k(u, v) − k(v, u) + k(v, v) N nN = du dv (3.3) L (n−1) NL (n−1) NL 2 n=1

K − SRN,LStr =

Equations (3.2) and (3.3) will be used in § 4 in the proof of our main results. Remark 3.3. Since self-adjointness of the operator K implies conjugate symmetry of the kernel, it is immediate to conclude that the contribution of the imaginary LL part of k to the integral 0 0 rN,L (u, v)k(v, u) du dv is zero. The same observation obviously applies to integration in any square symmetric with respect to the diagonal. Consequently, we may regard k(u, v) in (3.2) and (3.3) as being real-valued without any loss of generality.

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3.4. Eigenvalues of symmetric derivatives The evaluation of the rate of decay of eigenvalues of the operators Km , m a positive ∂ 2m k(x, y) integer, whose kernels are the symmetric derivatives km (x, y) = ∂y m ∂xm of the class Sm ([0, L]) positive deﬁnite kernel k plays a key role in the study of the eigenvalues of the operator K with kernel k (see e.g. [8], [10]). Recall that, according to Theorem 2.7, km is a continuous positive deﬁnite kernel deﬁned on the compact set [0, L]2 . Conventions and properties, in particular Mercer’s theorem described in § 1, are thus applicable to km . A straightforward adaptation of a result of Ha [10] for positive deﬁnite kernels deﬁned in [0, 1]2 yields the following upper bound for the eigenvalues of an operator K with positive deﬁnite kernel in Sm ([0, L]): Lemma 3.4. Let k : [0, L]2 → C be a positive deﬁnite kernel in class Sm ([0, L]), m ≥ 1. Let {λn (k)}n∈N (resp. {λn (km )}n∈N ) be the sequence of eigenvalues of the integral operator with kernel k (resp. km ). Then λ2n (k) ≤ L

2m

4 π2

λn (km ) (2n − 4m − 1)2m

for every n ≥ 2m + 1. Given a trace class kernel k, we deﬁne its N -tail Tk (N ) = We now state and prove a useful consequence of Lemma 3.4.

∞

n=N +1

λn (k).

Corollary 3.5. Let k : [0, L]2 → C be a positive deﬁnite kernel in class Sm ([0, L]), m ≥ 1. Then there exists N0 ∈ N and a real positive constant C such that Tk (2N + 1) ≤ C

L N

2m Tkm (N )

for N > N0 . Proof. According to Lemma 3.4 we may write, for every n > N0 = 2m + 1, 4 λn (km ) n2m L2m 2 (2n − 4m − 1)2m π n2m 2m L 4 1 ≤ m λn (km ) 4 n π2 2m C L λn (km ), = 2 n

λ2n (k) ≤

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8 where C = m 2 depends only on m. Since λn (k) and λn (km ) are positive non4 π increasing sequences, using the above inequalities we obtain ∞

Tk (2N + 1) =

λn (k)

n=2N +2 ∞

=

λ2n (k) + λ2n+1 (k) ≤ 2

n=N +1 ∞

≤2

n=N +1

=CL

2m

2m

C 2

2m L λn (km ) n

∞

λ2n (k)

n=N +1

∞ λn (km ) n2m

n=N +1 ∞

L λn (km ) N 2m n=N +1 2m L =C Tkm (N ) N ≤C

for N > N0 . This ﬁnishes the proof.

4. Asymptotic distribution of eigenvalues This section is devoted to the proof of our main results. We take without loss of generality I = [0, +∞[ as our model unbounded interval; the adaptations to other types of unbounded intervals are trivial. We mention however that the case I = R is particularly signiﬁcant in view of Fourier transforms, see Remark 4.19 and Corollary 4.20. We begin by establishing some basic lemmas and deﬁnitions. Suppose k is a positive deﬁnite kernel in class A0 ([0, +∞[). Let L > 0 and consider the restriction k L of k to the compact square [0, L]2 . In view of the deﬁnition of class A0 (I), both k and k L are positive deﬁnite kernels associated with trace class operators on the corresponding intervals. Using the notation introduced in 3.4, we have: Lemma 4.1. Let k ∈ A0 ([0, +∞[) and k L be the restriction of k to [0, L]2 . Then there is N0 ∈ N such that, for N > N0 , we have Tk (N ) ≤ TkL (N ) +

∞

k(x, x) dx. L

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Proof. Since both K and K L are trace class operators we have by (1.4) ∞ L ∞ ∞ λn (k) = k(x, x) dx = k(x, x) dx + k(x, x) dx 0

n=1

=

∞

L

λn (k ) +

n=1

0

L

∞

(4.1)

k(x, x) dx. L

∞ Suppose L k(x, x) dx = 0. Since k is continuous and non-negative along the diagonal it follows that k(x, x) ≡ 0 for x ≥ L. Since k is positive deﬁnite, we have by diagonal dominance |k(x, y)|2 ≤ k(x, x)k(y, y), and therefore the support of k is contained in [0, L]2 . Direct calculation then shows that the restriction to [0, L] determines a mapping ϕn → ϕL n from the set of eigenfunctions of k to the set of eigenfunctions of k L which is one-to-one and preserves the associated eigenvalues λn (k). Hence k and k L have the same spectrum (including multiplicities) and Tk (N ) = TkL (N ) for all N . The same conclusion may be derived using the principle of related operators ([12], [14]), since inclusion and truncation in this case act as a relation between the operators K : L2 ([0, ∞[) → L2 ([0, ∞[) and K L : L2 ([0, L]) → L2 ([0, L]), both of which ∞ are compact. ∞ Suppose now L k(x, x) dx > 0. Then (4.1) implies that n=1 λn (k) > ∞ L λ (k ). Thus there exists N ∈ N such that, for N > N , n 0 0 n=1 N n=1

λn (k) ≥

N

λn (k L ).

(4.2)

n=1

From (4.1) and (4.2) we conclude that for N > N0 ∞ Tk (N ) ≤ TkL (N ) + k(x, x) dx, L

ﬁnishing the proof. The following result is a central tool in the study of the decay rate of eigenvalues of a positive deﬁnite kernel k in class A0 ([0, +∞[) ∩ Sn ([0, +∞[). Lemma 4.2. Let β > 1, q > 0, A > 0, B > 0, x0 > 0, δ > 0 be real numbers. Suppose f : [0, +∞[→ [0, +∞[ is a continuous function satisfying the condition ∞ B L f (x) dx ≤ Lβ−1 for L ≥ x0 . Let R(L, N ) be deﬁned by q ∞ L L+ f (x) dx (4.3) R(L, N ) = A N L L (q + 1)β < δ and write γ = . Then there exist N0 ∈ N, D > 0 and an N q+β increasing sequence L(N ) → +∞ such that L(N )/N is decreasing and convergent to zero and such that the inequality whenever

R(L(N ), N ) ≤

D N γ−1

(4.4)

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holds for N ≥ N0 . q . Observe that, since 0 < θ < 1, L(N ) q+β L(N ) is a decreasing sequence in an increasing sequence converging to +∞ and N L(N0 ) converging to 0. Choose N0 such that N0θ = L(N0 ) > x0 and N0θ−1 = < δ. N0 ∞ L(N ) B < δ and L(N ) f (x) dx ≤ Then L(N ) > x0 , whenever N ≥ N0 . N L(N )β−1 Thus, according to (4.3) we may write q ∞ L(N ) L(N ) + f (x) dx (4.5) R(L(N ), N ) = A N L(N ) Proof. Deﬁne L(N ) = N θ with θ =

≤ AN (θ−1)q+θ + B N θ(1−β) .

(4.6)

Performing the corresponding calculations we derive from (4.6) that, for D = A + B, D R(L(N ), N ) ≤ γ−1 N for N ≥ N0 , proving the statement. The optimality of the estimate provided by Lemma 4.2 is the issue of the next result. ∞ Proposition 4.3. Suppose that L f (x) dx ∼ 1/Lβ−1 as L → +∞ while keeping the remaining hypotheses of Lemma 4.2. Then, for any positive sequence L(N ) such that L(N )/N < δ, there exists a subsequence L(N ) and constants N0 , C > 0 such C for N ≥ N0 . In particular, the exponent γ in (4.4) that R(L(N ), N ) ≥ (N )γ−1 cannot be improved for any such sequence L(N ). Proof. The assertion is trivially veriﬁed if L(N ) does not converge to ∞. In this case, L(N ) admits a bounded subsequence L(N ) < L0 for some L0 > 0 and we have q ∞ L(N ) L(N ) + f (x) dx R(L(N ), N ) = A N L(N ) ∞ ≥ f (x) dx ≡ C > 0 (4.7) L0

since f is, by hypothesis, a non-negative continuous function ∞ and C = 0 would imply f (x) = 0 for all x > L0 , contradicting the hypothesis L f (x) dx ∼ 1/Lβ−1 . Since γ > 1, from (4.7) we immediately derive C for N ≥ 1. R(L(N ), N ) ≥ (N )γ−1 Suppose now that L(N ) → +∞. From the hypothesis we derive, in particular, ∞ b that there exist constants b > 0, x0 > 0 such that L f (x) dx ≥ β−1 whenever L

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L ≥ x0 . Since L(N ) → +∞, we have L(N ) > x0 for suﬃciently large N and hence, from (4.3), q ∞ L(N ) R(L(N ), N ) = A L(N ) + f (x) dx N L(N ) q L(N ) b L(N ) + . (4.8) ≥A N L(N )β−1 Recalling the deﬁnition of θ and γ, we rewrite (4.8) in the form q+1 β−1 L(N ) Nθ + b . N γ−1 R(L(N ), N ) ≥ A Nθ L(N )

(4.9)

L(N ) converges either (N )θ γ−1 R(L(N ), N ) → +∞, to 0 or to +∞. Then from (4.9) we conclude that (N ) which implies the assertion of the proposition. On the other hand, if L(N ) does not L(N ) admit such a subsequence it follows that C1 < < C2 for some C1 , C2 > 0 Nθ and all suﬃciently large N . From (4.9) we then derive that for suﬃciently large N , β−1 1 C , which completes say N ≥ N0 , R(L(N ), N ) ≥ γ−1 for C = A C1q+1 +b N C2 the proof. ∞ B Remark 4.4. Notice that the condition L f (x) dx ≤ β−1 for some B > 0 and L B every L ≥ x0 > 0 is implied by the somewhat less general requirement f (x) ≤ β x for all x ≥ x0 with B = B (β − 1). The hypothesis of Lemma 4.2 may thus be seen as a generalized condition on the rate of decay of f (x) as x → +∞. A similar observation applies to the comparison of the hypothesis of Proposition 4.3 with 1 the condition f (x) ∼ β . x The following uniform continuity and Lipschitz conditions will be useful in the sequence. Suppose L(N ) is a subsequence of L(N ) such that

Definition 4.5. Let k : [0, +∞[→ C. We say that k(x, y) is uniformly continuous with respect to y on the diagonal if for every > 0 there is δ > 0 such that |k(x, x) − k(x, y)| < whenever |x − y| < δ. Definition 4.6. Let k : [0, +∞[→ C and α ∈ ]0, 1]. We say that k(x, y) is α-Lipschitz (written Lipα ) with respect to y on the diagonal if there is a positive constant A such that |k(x, x) − k(x, y)| ≤ A|x − y|α for every (x, y) ∈ [0, +∞[2 . Remark 4.7. It is clear that if k is Lipα with respect to y on the diagonal then it is uniformly continuous with respect to y on the diagonal. It is also clear that each of these conditions is implied by their respective counterpart on the plane. More speciﬁcally, uniform continuity on [0, +∞[2 implies the condition in deﬁnition 4.5 and Lipα on [0, +∞[2 implies the condition in deﬁnition 4.6.

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Remark 4.8. If k is a positive deﬁnite kernel in class A0 ([0, +∞[), standard arguments together with diagonal dominance and the fact that k(x, x) → 0 as x → +∞ easily show that k is uniformly continuous on the diagonal [2] and, in particular, satisﬁes the condition in deﬁnition 4.5. Remark 4.9. It is easily seen that functions satisfying conjugate symmetry k(x, y) = k(y, x) on [0, +∞[2 and the conditions of deﬁnitions 4.5 or 4.6 will automatically satisfy the analogues of these with respect to the variable x. Properties of this kind ∂k ∂k , can also be seen to arise from conjugate symfor the partial derivatives ∂x ∂y metry of k. Incidentally, none of these will play any relevant part in the sequence. We are now ready to prove our main results. They describe how, for a kernel in class A0 ([0, +∞[) ∩ Sn ([0, +∞[) — and, in particular, in class An ([0, +∞[) — uniform continuity or Lipα continuity on the diagonal together with the rate of decay of k(x, x) at inﬁnity allow us to control the rate of decay of the eigenvalues. Theorem 4.10. Let m ≥ 0 and suppose k(x, y) is a positive deﬁnite kernel in class A0 ([0, +∞[) ∩ Sm ([0, +∞[). Let {λn }n∈N be the sequence of eigenvalues of the integral operator with kernel k. Then the following statements hold. 1.1 Suppose km (x, y) is uniformly continuous with respect to y on the diagonal. Then: ∞ ∞ i) If β > 1, L k(x, x) dx = O 1/Lβ−1 (resp. L k(x, x)dx = o 1/Lβ−1 ) (2m + 1)β as L → +∞ and γ = , then λn = O (1/nγ ) (resp. λn = 2m + β γ o(1/n ∞ )). ii) If L k(x, x) dx = O 1/Lβ−1 as L → +∞ for all β > 1, then λn = o (1/nγ ) for all γ < 2m + 1. iii) If k(x, x) has compact support, then λn = o 1/n2m+1 . 1.2 Suppose km (x, y) is Lipα with respect to y on the diagonal. Then: ∞ (2m+α+1)β , i) If β > 1, L k(x, x)dx = O 1/Lβ−1 as L → ∞ and γ = 2m + α + β γ then ∞λn = O (1/n ). ii) If L k(x, x) dx = O 1/Lβ−1 as L → +∞ for all β > 1, then λn = o (1/nγ ) for all γ < 2m + α + 1. iii) If k(x, x) has compact support, then λn = O 1/n2m+α+1 . ∂km 2.1 Suppose km (x, y) is continuously diﬀerentiable with respect to x and that ∂x is uniformly continuous with respect to y on thediagonal. Then: ∞ ∞ i) If β > 1, L k(x, x)dx = o 1/Lβ−1 (resp. L k(x, x) dx = o 1/Lβ−1 ) (2m + 2)β , then λn = O (1/nγ ) (resp. λn = as L → +∞ and γ = 2m + 1 + β γ o(1/n ∞ )). ii) If L k(x, x) dx = O 1/Lβ−1 as L → +∞ for all β > 1, then λn = o (1/nγ ) for all γ < 2m + 2. iii) If k(x, x) has compact support, then λn = o 1/n2m+2 .

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2.2 Suppose km (x, y) is continuously diﬀerentiable with respect to x and that

∂km ∂x

is Lipα with respect to y on the diagonal. Then: ∞ (2m+α+2)β i) If β > 1, L k(x, x)dx = O 1/Lβ−1 as L → ∞ and γ = , 2m+α+1+β γ then ∞λn = O (1/n ). ii) If L k(x, x) dx = O 1/Lβ−1 as L → +∞ for all β > 1, then λn = o (1/nγ ) for all γ < 2m + α + 2. iii) If k(x, x) has compact support, then λn = O 1/n2m+α+2 . Proof. Let m ≥ 0 and suppose k is a positive deﬁnite kernel in class A0 ([0, +∞[) ∩ Sm ([0, +∞[). Let k L be the restriction of k to the interval [0, L]2 . By Lemma 4.1 there exists N0 ∈ N such that, for N > N0 , we have ∞ Tk (N ) ≤ TkL (N ) + k(x, x) dx. (4.10) L

In particular, we may write Tk (2N + 1) ≤ TkL (2N + 1) +

∞

k(x, x) dx.

(4.11)

L

According to Corollary 3.5, inequality (4.11) implies, for m ≥ 1, 2m ∞ L Tk (2N + 1) ≤ C Tkm L (N ) + k(x, x) dx N L

(4.12)

for some C > 0 and suﬃciently large N . For m ≥ 0 we now use the results of § 3 in the approximation of the operator L L L with kernel km by ﬁnite rank operators. Let S be the square root of Km . Km N,L Deﬁning R as in § 3.3 and recalling (3.2) and (3.3), we have according to Lemma 3.1 L N,L L (N ) ≤ Km − SR Str Tkm L L N nN N nN L L km (u, u) − km = (u, v) du dv (4.13) L (n−1) NL (n−1) NL n=1 n NL L N L 1 N nN L L L = km (u, u) − km (u, v) − km (v, u) + km (v, v) du dv 2 n=1 L (n−1) NL (n−1) NL (4.14) L where, according to Remark 3.3, we may without loss of generality regard km as being real valued. We are now ready to prove statement 1.1 i). If m = 0 we have by hypothesis γ = 1 and the assertion reduces to the already known fact that, for a kernel k in class A0 , λn (k) = o(1/n). Suppose that m ≥ 1. Since by hypothesis km is in class A0 ([0, +∞[) ∩ Sm ([0, +∞[) and is uniformly continuous with respect to y on the diagonal, for evL L ery > 0 there exists δ > 0, independent of L, such that |km (u, u) − km (u, v)| <

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for all u, v ∈ [0, L] satisfying |v − u| < δ. For every positive L and every positive integer N such that L/N < δ we then have, after performing the relevant calculations from (4.13), 2 N L L N,L L (N ) ≤ K Str ≤ N · = L . (4.15) Tkm m − SR L N Therefore, according to (4.12), we derive for N suﬃciently large that 2m ∞ L Tk (2N + 1) ≤ C L+ k(x, x) dx. N L

(4.16)

We now use the hypothesis on the decayrate of k(x, x) at inﬁnity. Suppose ∞ there exist β > 1, B > 0, x0 > 0 such that L k(x, x) dx ≤ B/Lβ−1 whenever L ≥ x0 . We recall Lemma 4.2 setting q = 2m and A = C and deﬁne a sequence of 2m . Recall, ﬁnite rank operators SRN,LS by setting L = L(N ) = N θ with θ = 2m+β from Lemma 4.2, that L(N ) is an increasing sequence with L(N ) → +∞ and such that L(N )/N is decreasing and convergent to 0. Therefore, there exists N0 such that L(N ) > x0 and L(N )/N < δ for every N > N0 . Therefore we derive from (4.16) and Lemma 4.2 that, for suﬃciently large N , Tk (2N + 1) ≤

C +B D = γ−1 N γ−1 N

(2m + 1)β . 2m ∞+ β Now, if L k(x, x) dx ≤ B/Lβ−1 whenever L ≥ x0 , the above condition is ∞ veriﬁed for some ﬁxed D > 0, which implies that λn = O(1/nγ ). If L k(x, x) dx = o(1/Lβ−1 ) as L → ∞, the same condition holds for arbitrary D > 0 and we derive in this case the stronger conclusion that λn = o(1/nγ ). This ﬁnishes the proof of statement 1.1 i). Consider now the hypothesis of statement 1.2 i). Suppose km is Lipα with respect to y on the diagonal. Choose L > 0 and N ∈ N. Then |u − v| < L/N for L L , nN ]. Since km is Lipα , (4.13) implies all u, v ∈ [(n − 1) N α L L N,L Tkm L (N ) ≤ K Str ≤ A L. (4.17) m − SR N

where γ =

In the case m = 0, we derive from (4.10) and (4.17) that, for suﬃciently large N , ∞ k(x, x) dx Tk (N ) ≤ TkL (N ) + L α ∞ L ≤A L+ k(x, x) dx. (4.18) N L Similarly, if m ≥ 1, we derive from (4.11), (4.12) and (4.17), for suﬃciently large N, 2m+α ∞ L L+ k(x, x) dx. (4.19) Tk (2N + 1) ≤ A C N L

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To use the hypothesis on the decay rate of k(x, x) at inﬁnity we suppose, as in the ∞ previous case, that there exist β > 1, B > 0 and x0 > 0 such that L k(x, x) dx ≤ B/Lβ−1 whenever L ≥ x0 . We recall Lemma 4.2 taking q = 2m + α, δ = ∞, and A replaced with A C in the case m > 1. As in the proof of 1.1 i), deﬁne a sequence of ﬁnite rank operators SRN,LS taking L ≡ L(N ) = N θ , where in this 2m+α case θ = 2m+α+β . According to Lemma 4.2 we then derive from (4.18) and (4.19) that, for suﬃciently large N , Tk (N ) ≤

A+B if m = 0 N γ−1

and Tk (2N + 1) ≤

AC + B if m ≥ 1, N γ−1

(2m + α + 1)β for m ≥ 0. These conditions together imply λn = 2m + α + β γ O(1/n ), completing the proof of statement 1.2 i). To prove statements 2.1. i) and 2.2 i) we start by rewriting the integrand in (4.14) in a more convenient way. Suppose km is continuously diﬀerentiable with respect to x in [0, +∞[2 . We set where γ =

L L L L g(u, v) = km (u, u) − km (u, v) − km (v, u) + km (v, v).

(4.20)

L L (t, v) − km (t, u). Notice that ϕ is in C 1 ([0, L]) For u, v, t ∈ [0, L] deﬁne ϕ(t) = km and that g(u, v) = ϕ(v) − ϕ(u). Hence there exists t0 between u and v such that

g(u, v) = ϕ (t0 )(v − u) L ∂km ∂k L (t0 , v) − m (t0 , u) (v − u) = ∂x ∂x L L ∂km ∂km ∂k L ∂k L (t0 , v) − m (t0 , t0 ) (v − u) + (t0 , t0 ) − m (t0 , u) (v − u) = ∂x ∂x ∂x ∂x (4.21) ∂km We now prove statement 2.1 i). Suppose is uniformly continuous with ∂x respect to y on the diagonal. Then, for every > 0 there exists δ > 0, independent of L, such that

L

L

∂km

∂km

∂x (t0 , v) − ∂x (t0 , t0 ) < for all t0 , v ∈ [0, L] satisfying |v − t0 | < δ. Then, for every positive L and every N ∈ N such that L/N < δ, we derive from (4.21) that if |v − u| < L/N we have g(u, v) ≤ 2 L/N , whence from (4.14) L (N ) ≤ Tkm

L2 . N

(4.22)

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Therefore, it follows from (4.10), (4.11) and (4.22) that ∞ k(x, x) dx Tk (N ) ≤ TkL (N ) + L ∞ L2 k(x, x) dx, ≤ + N L Tk (2N + 1) ≤ C ≤ C

L N

L N

2m

T

2m+1

L km

∞

(N ) +

k(x, x) dx L

∞

L+

(4.23)

(4.24)

k(x, x) dx L

for some C > 0, N suﬃciently large and m ≥ 1. Using the hypothesis on the decay rate of k(x, x) we now proceed as in the proof i). Suppose there exist β > 1, B > 0, x0 > 0 such that ∞ of statement 1.1 β−1 k(x, x) dx ≤ B/L whenever L ≥ x0 . Recall Lemma 4.2 with q = 2m + 1, L A = if m = 0, A = C if m ≥ 1. Deﬁne a sequence of ﬁnite rank operators 2m+1 . According to Lemma 4.2 SRN,LS by setting L = L(N ) = N θ , where θ = 2m+1+β we then derive from (4.23) and (4.24) that, for suﬃciently large N , Tk (N ) ≤

+B D1 = γ−1 if m = 0 N γ−1 N

and Tk (2N + 1) ≤

C +B D2 = γ−1 if m ≥ 1 γ−1 N N

(2m + 2)β . 2m + 1 + β ∞ Finally observe that if L k(x, x) dx = O(1/Lβ−1 ) as L → +∞, the above conditions are satisﬁed for ﬁxed D1 and D2 , which implies λn = O(1/nγ ). If ∞ β−1 ) as L → +∞, the same conditions hold for arbitrary L k(x, x) dx = o(1/L D1 and D2 , implying λn = o(1/nγ ). This ﬁnishes the proof of statement 2.1 i). ∂km is continuous We now focus on the proof of statement 2.2 i). Suppose ∂x α 2 on [0, +∞[ and Lip with respect to y on the diagonal, α ∈ ]0, 1]. Choose L ∈ R+ L L and N ∈ N. Then |u − v| < L/N for all u, v in [(n − 1) N , nN ], n = 1, . . . , N . Since ∂km α is Lip with respect to y on the diagonal we derive from (4.20) and (4.21) ∂x that 1+α L g(u, v) ≤ A N where γ =

and, from (4.14),

Tkm L (N ) ≤ A

L N

1+α L.

(4.25)

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Therefore, according to (4.10), (4.12) and (4.25) we have ∞ k(x, x) dx Tk (N ) ≤ TkL (N ) + ≤A

L N

Tk (2N + 1) ≤ C ≤A

L N

1+α

L N

L

∞

L+

k(x, x) dx,

(4.26)

L

2m

Tkm L (N ) +

2m+α+1

∞

k(x, x) dx L

∞

L+

(4.27)

k(x, x) dx. L

We proceed as in the proof of the previous statements. Using the hipothesis on the decay rate of k(x, x) at inﬁnity and taking q = 2m + α + 1 and replacing A by A C if m ≥ 1, δ = ∞ in Lemma 4.2, we deﬁne the sequence of ﬁnite rank 2m+1+α operators SRN,LS by setting L = L(N ) = N θ , where θ = 2m+1+α+β . Then, from Lemma 4.2, (4.26) and (4.27), it follows that for suﬃciently large N Tk (N ) ≤

A+B if m = 0 N γ−1

and Tk (2N + 1) ≤

AC + B = if m ≥ 1 N γ−1

(2m + 1 + α)β . Both conditions imply λn = O(1/nγ ), completing the 2m + 1 + α + β proof of statement 2.2 i). We now prove statements 1.1 ii), 1.2 ii), 2.1 ii) and 2.2 ii). Notice that the ∞ hypothesis on the decay rate of k(x, x), namely L k(x, x) dx = O(1/Lβ−1 ) as L → +∞ for all β > 1, is common to these four statements. We concentrate on the proof of 1.1 ii), which is based on the contents of statement 1.1 i). Suppose km ∞ is uniformly continuous with respect to y on the diagonal and that L k(x, x) dx = O(1/Lβ−1 ) as L → +∞ for all β > 1. Then, according to 1.1 i), λn = O(1/nγ ) for all γ < 2m+1. This fact actually implies the stronger statement that λn = o(1/nγ ) for every γ < 2m + 1. In fact, if there were γ0 < 2m + 1 such that nγ0 λn → C for some C > 0, then λn would not be O(1/nγ ) for γ0 < γ < 2m+ 1, contradicting the previous result. Thus under this hypothesis λn = o(1/nγ ) for every γ < 2m + 1, proving statement 1.1. ii). The proofs of statements 1.2 ii), 2.1 ii) and 2.2 ii) are derived in the exact same way respectively from 1.2 i), 2.1 i) and 2.2 i), so details are omitted. Finally, we prove statements 1.1 iii), 1.2 iii), 2.1 iii) and 2.2 iii). Notice that the hypothesis that k(x, x) has compact support is common to these four statements. We concentrate on the proof of 1.1 iii). For m = 0 we have γ = 1 and the common assertion reduces to the already known fact that for a positive deﬁnite kernel k in A0 ([0, +∞[), λn (k) = o(1/n). Let then m ≥ 1. Choose L > 0 such that where γ =

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+∞ supp k(x, x) ⊂ [0, L]; with this choice we obviously have L k(x, x) dx = 0. Fix L and proceed as in the proof of 1.1 i), considering (4.11), (4.12), (4.13), (4.15), (4.16). From this last equation we derive in this case that 2m L L Tk (2N + 1) ≤ C N for suﬃciently large N . Since L and C are ﬁxed and > 0 is arbitrary, this implies Tk (2N + 1) = o(1/N 2m ) as N → +∞. For m ≥ 0 it then follows that λn = o(1/n2m+1 ), as asserted. To prove 1.2 iii) we ﬁx L as above and proceed as in the proof of 1.2 i), writing (4.10), (4.11), (4.12), (4.13), (4.17), (4.18) and (4.19). From the last two equations we derive in this case that α L Tk (N ) ≤ A L, N 2m+α L Tk (2N + 1) ≤ A C L N for N suﬃciently large and m ≥ 1. Since A, C and L are ﬁxed, it follows that λn = O(1/n2m+α+1 ) for m ≥ 0, as asserted. The proof of 2.1. iii) follows along the same lines. We ﬁx L as above and proceed as in the proof of 2.1 i), considering (4.10), (4.11), (4.12),(4.14), (4.20), (4.21), (4.22), (4.23) and (4.24). The last two equations yield in this case L2 , N 2m+1 L Tk (2N + 1) ≤ C L N Tk (N )

≤

for N suﬃciently large and m ≥ 1. Since C and L are ﬁxed and is arbitrary this implies that λn = o(1/n2m+α+1 ), for m ≥ 0, as asserted. Finally, to prove 2.2.iii) we ﬁx L as above and proceed as in the proof of 2.2.i), writing (4.10), (4.11), (4.12),(4.14), (4.20), (4.21), (4.25), (4.26) and (4.27). The last two equations yield 1+α L Tk (N ) ≤ A L, N 2m+α+1 L Tk (2N + 1) ≤ A C L N for N suﬃciently large and m ≥ 1. Since A, C and L are ﬁxed, it follows that λn = O(1/n2m+α+2 ) for m ≥ 0, as asserted. This ﬁnishes the proof. Corollary 4.11. Suppose k(x, y) is a positive deﬁnite kernel in A0 ([0, +∞[). Suppose furthermore that k is of class C∞ ([0, ∞[2 ) and that km is uniformly continuous with respect to y on the diagonal for every m ∈ N. Let {λn }n∈N be

Vol. 57 (2007)

Eigenvalues of PDKs on Unbounded Domains

37

the of the integral operator with kernel k. If β > 1 and ∞ sequence of eigenvalues β−1 k(x, x) dx = O(1/L ) as L → +∞, then λn = o (1/nγ ) for all γ < β. L Proof. According to statement 1.1 i) of Theorem 4.10, λn = o (1/nγ ) for every γ (2m + 1)β (2m + 1)β with m ≥ 0. Since, for ﬁxed β, lim = β, of the form γ = m→∞ 2m + β 2m + β an argument similar to the one used in the proof of statement 1.1 ii) implies that λn = o (1/nγ ) for all γ < β. Remark 4.12. Observe that Corollary 4.11 cannot be since its assertion improved, is not true for γ = β even in the weaker form λn = O 1/nβ , as the counterexample λn = log n/nγ shows. A similar observation can be applied to statements 1.1 ii), 1.2 ii), 2.1 ii) and 2.2 ii) of Theorem 4.10. Remark 4.13. As a consequence of Remark 4.8 it is immediate to recognize that the assertion of Corollary 4.11 is valid, in particular, for positive deﬁnite kernels lying in class Am for all m ≥ 0. Corollary 4.14. Let k(x, y) be a positive deﬁnite kernel in class A0 ([0, +∞[). Suppose k is of class C p in [0, +∞[2 and that the partial derivatives up to order p are uniformly continuous with respect to y on the diagonal, and let {λn }n∈N be the of the integral operator with kernel k. If β > 1 and ∞ sequence of eigenvalues ∞ β−1 k(x, x) dx = O(1/L ) (resp. k(x, x) dx = o(1/Lβ−1 )) as L → +∞, then L L (p + 1)β . λn = O (1/nγ ) (resp. λn = o (1/nγ ) ) for γ = p+β Proof. For p even (resp. p odd) the hypotheses are easily seen to imply those of statement 1.1 i) (resp. 1.2.i)) of Theorem 4.10. Setting p = 2m (resp. p = 2m + 1) yields the result. Corollary 4.15. Let k(x, y) be a positive deﬁnite kernel in class A0 ([0, +∞[). Suppose k is of class C p in [0, +∞[2 and that the partial derivatives up to order p are satisfy an α-Lipschitz condition with respect to y on the diagonal and let {λn }n∈N be the sequence of eigenvalues of the integral operator with kernel k. If ∞ β > 1 and L k(x, x) dx = O(1/Lβ−1 ) as L → +∞, then λn = O (1/nγ ) for (p + 1 + α)β . γ= p+α+β Proof. For p even (resp. p odd) the hypotheses are easily seen to imply those of statements 2.1 i) and 2.2 i) in Theorem 4.10. Setting p = 2m (resp. p = 2m + 1) yields the result. Remark 4.16. A few interesting observations can be made from the study of limiting cases in the formulas for the exponent γ given by Theorem 4.10. Suppose m and α are ﬁxed and consider the parameter β which controls the rate of decay of the kernel k along the diagonal for 1 < β < +∞. If β → +∞, the limiting values obtained for γ from formulas in items i) of statements 1.1, 1.2, 2.1 and 2.2 of Theorem 4.10 coincide with the exponent

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determining the bound for the decay rate of eigenvalues given in items ii) and iii) of the corresponding statements. These values ultimately reﬂect the fact that, in the case of kernels decaying rapidly on the diagonal and, in particular, in the case of kernels with compact support, the regularity assumed for k determines the upper bound for the decay rate exponent of eigenvalues. The limiting case m → +∞ is the subject of corollary 4.11 and is, in a way, symmetric to the case above. It shows that operators with indeﬁnitely diﬀerentiable kernels have eigenvalue distributions whose decay rate exponent bound is determined by the decay rate exponent of k along the diagonal. For ﬁxed m and β we ﬁnally observe that, in consonance with the interpretation of α as an index of intermediate diﬀerentiability, the limiting cases α = 0 and α = 1 in statements 1.2 and 2.2 produce the corresponding expected upper bounds for the decay rate exponent γ given by statements 1.1 and 2.1. Remark 4.17. Some observations are relevant to the discussion of the hypothesis of Theorem 4.10. We ﬁrst note that, as indeed in the deﬁnition of class A0 , the essential requirements on the behavior of k in the hypothesis of Theorem 4.10 may be restricted to the diagonal with no consequence on the proofs, a fact which is not apparent in the previous literature. Secondly, observe that, in view of Remark 4.9 and conjugate symmetry of kernel k(x, y), the hypothesis on uniform and Lipschitz continuity with respect to y on the diagonal assumed for partial derivatives of k could have been replaced with similar hypothesis with respect to x on the diagonal for conveniently chosen partial derivatives of k. Stronger conditions not specifying the variable x or y can naturally be imposed but with no advantage to the proofs. Finally we note that uniform continuity requirements, which trivially derive from the Sn condition in the compact domain case, must be explicitly imposed in the case of unbounded domains. Remark 4.18. Clearly, the last assertions (items iii)) of statements 1.1, 1.2, 2.1 and 2.2 in Theorem 4.10 bear a direct connection to the formally identical results known for positive trace class operators deﬁned on a compact interval, namely those which apply to C p or C p+α kernels described in § 1. In fact, if transcribed to the case of kernels deﬁned on compact domains, these assertions require somewhat weaker (yet suﬃcient) versions of the above referred conditions (see Remark 4.17). The ﬁrst part of the proof of Lemma 4.1 clariﬁes this connection by establishing the equivalence of the study of eigenvalues of the operator with compactly supported kernel k deﬁned on [0, ∞[2 and the operator whose kernel is the restriction of k to a square [0, L]2 containing the support of k. The fact that the steps taken in our proofs of the referred assertions collapse into (a combination of) the proofs in [16], [8] and [7] can be seen as a consequence of this equivalence. On the other hand, the same authors show that the results obtained in this case are optimal. This is done by explicitly constructing kernels verifying the required assumptions of diﬀerentiability and Lipschitz continuity whose eigenvalues

Vol. 57 (2007)

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attain the bounds for decay rate established by the theorems. The results in Theorem 4.10 are thus known to be optimal in the limit cases corresponding to compactly supported kernels. These facts and the contents of Proposition 4.3 strongly suggest that the remaining statements in Theorem 4.10 are also optimal. Remark 4.19. Our results are stated and proved for the unbounded interval I = [0, +∞[ for convenience only. They are valid, with the obvious rephrasing, for L2 (I) integral operators K and respective positive deﬁnite kernels k(x, y) for the other types of unbounded intervals in R. Of particular signiﬁcance (see below) is the case where I = R. In this case we are dealing with L2 (R) positive deﬁnite kernels; class A0 (R) kernels are continuous in R2 , with k(x, x) ∈ L1 (R) and k(x, x) → 0 as |x| → +∞. All the results and proofs carry through simply by replacing the required asymptotic behavior of k(x, x) as x → +∞ by the corresponding requirement as |x| → +∞. There is a close connection between positive deﬁniteness of a continuous ˆ 1 , ν2 ). More speciﬁcally, L2 (R) kernel k(x, y) and that of its Fourier transform k(ν it is possible to show that if k is in class A0 (R) as deﬁned in Remark 4.19, then its ˜ 1 , ν2 ) = k(ν ˆ 1 , −ν2 ) is a positive deﬁnite kernel with “rotated” Fourier transform k(ν the same eigenvalues λn as k and whose associated eigenfunctions are the Fourier transforms of the corresponding eigenfunctions of k. Moreover, if k 1/2 (x, x) ∈ ˜ with kernel k˜ is trace class with the L1 (R) then the L2 (R) integral operator K same trace as K; see [3] for details. If k is in class A0 (R), a suﬃcient condition for k 1/2 (x, x) ∈ L1 (R) may be formulated in terms of the asymptotic behavior of k(x, x) as k(x, x) = O(1/xβ ) for some β > 2. The following result is an immediate consequence of Corollaries 4.14 and 4.15 and Remark 4.4. Corollary 4.20. Suppose k(x, y) is a positive deﬁnite kernel in class A0 (R) with k(x, x) = O(1/xβ ) as |x| → +∞ for some β > 1 and let {λn }n∈N be the sequence of eigenvalues of the integral operator with kernel k with associated eigenfunctions ˆ 1 , ν2 ) be the double Fourier transform of k(x, y), k(ν ˜ 1 , ν2 ) = k(ν ˆ 1 , −ν2 ) φn . Let k(ν and φˆn be the Fourier transform of φn . Then the following statements hold. (i) k˜ is a positive deﬁnite kernel with L2 (R) eigenfunction expansion ˜ 1 , ν2 ) = k(ν λn φˆn (ν1 ) φˆn (ν2 ). (4.28) n≥1

(ii) If k is of class C p (R) and the partial derivatives up to order p are uniformly (p + 1)β . continuous on the diagonal, then λn = O(1/nγ ), where γ = p+β p (iii) If k is of class C (R) and the partial derivatives up to order p satisfy an α(p+1+α)β . Lipschitz condition on the diagonal, then λn = O(1/nγ ) with γ = p+α+β If β > 2 we have, in addition, that k˜ is in class A0 (R), the series (4.28) is abso˜ : L2 (R) → L2 (R) with kernel lutely and uniformly convergent and the operator K

40

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k˜ is trace class with ˜ = tr K = tr K

+∞

+∞

k(x, x) dx = −∞

−∞

IEOT

˜ ν) dν = k(ν,

λn .

n≥1

Proof. The hypothesis imply that k is a Mercer-like kernel. Then statements i), ii) and iii) follow, in view of remark 4.4, from corollaries 4.14 and 4.15 and propositions 4.1 and 4.2 in [3]. The hypothesis that k(x, x) = O(1/|x|β ) for some β > 2 implies, in addition, that k 1/2 (x, x) ∈ L1 (R), from which the last statement derives by direct application of theorem 4.4 in [3]. Remark 4.21. It is clearly seen that all conclusions in corollary 4.20 still hold if the hypothesis that k : R2 → C be a continuous positive deﬁnite kernel satisfying k(x, x) = O(1/|x|β ) as |x| → +∞ for some β > 1 is replaced with the assumption ∞ that k is a Mercer-like kernel deﬁned on R2 satisfying L k(x, x) dx = O 1/Lβ−1 for some β > 1 and if condition β > 2 is replaced with the hypothesis that k 1/2 (x, x) ∈ L1 (R) or, more generally, that k(x, y) ∈ L1 (R2 ). Once again the version presented, although somewhat weaker, underlines how the behavior of the kernel on the diagonal controls events.

References [1] M. Birman, M. Solomyak, Estimates of singular numbers of integral operators. Russ. Math. Surv. 32 (1977), 15-89. [2] J. Buescu, Positive integral operators in unbounded domains. J. Math. Anal. Appl. 296 (2004), 244–255. [3] J. Buescu, F. Garcia, I. Lourtie, A. Paix˜ ao, Positive deﬁniteness, integral equations and Fourier transforms. Jour. Int. Eq. Appl. 16, 1 (2004), 33–52. [4] J. Buescu, F. Garcia, I. Lourtie, L2 (R) nonstationary processes and the sampling theorem. IEEE Sign. Proc. Lett. 8, 4 (2001), 117–119. [5] J. Buescu, A. Paix˜ ao, Positive deﬁnite matrices and diﬀerentiable reproducing kernel inequalities. J. Math. Anal. Appl. 320 (2006), 279-292. [6] J. Buescu, A. Paix˜ ao, Eigenvalues of positive deﬁnite integral operators in unbounded intervals. Positivity, to appear. [7] C. Chang, C. Ha, On eigenvalues of diﬀerentiable positive deﬁnite kernels. Integr. Equ. Oper. Theory 33 (1999), 1–7. [8] J. Cochran, M. Lukas, Diﬀerentiable positive deﬁnite kernels and Lipschitz continuity. Math. Proc. Camb. Phil. Soc. 104 (1988), 361–369. [9] I. Gohberg, M. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space. A.M.S., Providence, 1969. [10] C. Ha, Eigenvalues of diﬀerentiable positive deﬁnite kernels. SIAM J. Math. Anal. 17 (1986), 2, 415–419. [11] T. Kadota, Term-by-term diﬀerentiability of Mercer’s expansion. Proc. Amer. Math. Soc. 18 (1967), 69–72.

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[12] H. Konig, Eigenvalue distribution of compact operators. Operator Theory: Advances and Applications, vol. 16. Birkhuser, Basel, 1986. [13] I. M. Novitskii, Representation of kernels of integral operators by bilinear series. Siberian Math. J. 25 (1984), 3, 774–778. Translated form the Russian: Sibirsk. Mat. Zh. 25 (1984), 5, 114–118. [14] A. Pietsch, Zur Fredholmschen Theorie in lokalconvexen R¨ aumen. Stud. Math. 28 (1966/67), 161–179. [15] A. Pietsch, Eigenvalues of integral operators II. Math. Ann. 262 (1983), 343–376. [16] J. Reade, Eigenvalues of positive deﬁnite kernels. SIAM J. Math. Anal. 14 (1983), 1, 152–157. [17] J. Reade, Eigenvalues of Lipschitz kernels. Math. Proc. Camb. Phil. Soc. 93 (1983), 1, 135–140. [18] J. Reade, Eigenvalues of positive deﬁnite kernels II. SIAM J. Math. Anal. 15 (1984), 1, 137–142. [19] J. Reade, Positive deﬁnite C p kernels. SIAM J. Math. Anal. 17 (1986), 2, 420–421. [20] J. Reade, Eigenvalues of smooth positive deﬁnite kernels. Proc. Edimburgh Math. Soc. 35 (1990), 41–45. [21] M. Reed, B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis (revised and enlarged edition). Academic Press, San Diego, 1980. [22] F. Riesz, B. Nagy, Functional Analysis. Ungar, New York, 1952. [23] H. Weyl, Das Asymptotische Verteilunggesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71 (1912), 2, 441–479. Jorge Buescu Departamento de Matem´ atica Instituto Superior T´ecnico Av. Rovisco Pais 1049-001 Lisbon Portugal e-mail: [email protected] A. C. Paix˜ ao Departamento de Mecˆ anica Instituto Superior de Engenharia de Lisboa (ISEL) Lisbon Portugal e-mail: [email protected] Submitted: July 20, 2005

Integr. equ. oper. theory 57 (2007), 43–66 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010043-24, published online August 8, 2006 DOI 10.1007/s00020-006-1444-2

Integral Equations and Operator Theory

Zero Products of Toeplitz Operators with n-Harmonic Symbols Boo Rim Choe, Hyungwoon Koo and Young Joo Lee Abstract. On the Bergman space of the unit polydisk in the complex n-space, we solve the zero-product problem for two Toeplitz operators with n-harmonic symbols that have local continuous extension property up to the distinguished boundary. In the case where symbols have additional Lipschitz continuity up to the whole distinguished boundary, we solve the zero-product problem for products with four factors. We also prove a local version of this result for products with three factors. Mathematics Subject Classification (2000). Primary 47B35; Secondary 32A36. Keywords. Zero product, Toeplitz operator, n-harmonic symbol, Bergman space, polydisk.

1. Introduction Let D be the unit disk in the complex plane. For a ﬁxed positive integer n, the unit polydisk Dn is the cartesian product of n copies of D. Let L2 = L2 (Dn , V ) denote the usual Lebesgue space where V = Vn is the volume measure on Dn normalized to have total mass 1. We let A2 = A2 (Dn ) denote the Bergman space consisting of all holomorphic functions in L2 . Due to the mean value property of holomorphic functions, the Bergman space A2 is a closed subspace of L2 , and thus is a Hilbert space. Since every point evaluation is a bounded linear functional on A2 , there corresponds to every a ∈ Dn a unique function Ka ∈ A2 which has the following reproducing property: f (a) = f (z)Ka (z) dV (z) (1.1) Dn

This research was supported by KOSEF(R01-2003-000-10243-0).

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for f ∈ A2 . The function Ka is the well-known Bergman kernel and its explicit formula is given by Ka (z) =

n j=1

1 , (1 − aj zj )2

z ∈ Dn ;

see, for example, Proposition 1.4.24 of [13]. Here, and elsewhere, zj denotes the j-th component of z. It follows from (1.1) that the Hilbert space orthogonal projection P from L2 onto A2 , called the Bergman projection, is realized as an integral operator ψ(z)Ka (z) dV (z) P ψ(a) = Dn

for ψ ∈ L . For a function u ∈ L∞ , the Toeplitz operator Tu with symbol u is deﬁned by 2

Tu f = P (uf ) for f ∈ A . It is clear that Tu : A → A2 is a bounded linear operator. In this paper we study the zero-product problem of whether the zero-product of several Toeplitz operators has only the trivial solution. More explicitly, the problem we consider is 2

2

Does Tu1 · · · TuN = 0 imply that some uj is identically zero? This problem was ﬁrst studied for Toeplitz operators on the Hardy space. In [5], Brown and Halmos actually studied a more general problem on the Hardy space of the unit disk and proved that Tu Tv = Tϕ if and only if either u¯ or v is holomorphic and ϕ = uv. As an immediate consequence of their result, they easily derived that the zero-product problem with two factors has only the trivial solution. Later, on the same context of Hardy space of the unit disk, the zeroproduct problem has been solved by Guo [12] for products with ﬁve factors and by Gu [11] for products with six factors. Recently, Ding [10] solved this problem for products with two factors on the Hardy space of the polydisk. There has been no progress on higher dimensional balls, as far as we know. For the Bergman space case, the study of the zero-product problem has begun only recently. On the setting of the Bergman space of the unit disk, Ahern ˘ ckovi´c [2] solved the zero-product problem for two factors with harmonic and Cu˘ symbols. More recently, Ahern [1] gave another more general approach to solve the same problem. Recall that bounded measurable functions on the unit circle can be identiﬁed with boundary values of bounded harmonic functions on the unit disk. So, it seems quite natural (to us) to work with harmonic symbols as in [2] as substitutes for general symbols in the Hardy space case. However, the work in [2] shows that the Bergman space case, even with such harmonic symbols, is much more subtle than the Hardy space case. Nevertheless, there have been a couple ˘ ckovi´c [9] obtained some of progresses for symbols other than harmonic ones. Cu˘ ˘ ckovi´c [3] partial results when only one symbol is harmonic. Also, Ahern and Cu˘

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solved the zero-product problem for two factors with radial symbols. Of course, the zero-product problem with general symbols, even for two factors, remains still very far from its solution. The higher dimensional cases have been also studied on the ball and polydisk. Recently, the polydisk case was solved by Choe et al. [8] for two factors with pluriharmonic symbols by extending the method in [2]. More recently, on the setting of the unit ball, the ﬁrst two authors of the present paper [7] used an entirely diﬀerent method to solve the zero-product problem for two factors with harmonic symbols that have local continuous extension property up to the boundary. At the same paper, they also solved the problem for multiple products with number of factors depending on the dimension in the case where symbols have additional (global or local) Lipschitz continuity up to the boundary. In this paper, we study the same problem on the polydisk. Going from the ball to the polydisk, we need to adjust our setting suitable for the polydisk. According to our results below, it turns out that n-harmonic symbols on the polydisk are the right substitutes for harmonic symbols on the ball and the distinguished boundary of the polydisk is the right substitute for the boundary of the ball. Recall that a function u ∈ C 2 (Dn ) is called n-harmonic as in [14] if u is harmonic in each variable separately. More explicitly, u is n-harmonic if ∂j ∂ j u = 0,

j = 1, 2, . . . , n

where ∂j = ∂/∂zj denotes the complex diﬀerentiation with respect to zj . Also, recall that the distinguished boundary of Dn is T n , the cartesian product of n copies of the unit circle T = ∂D. First, we consider n-harmonic symbols with local continuous extension property up to the distinguished boundary T n and solve the zero-product problem for two factors. In what follows, we let h∞ denote the class of all bounded n-harmonic functions on Dn . Also, a “boundary open” set refers to a relatively open subset of T n . The following is our ﬁrst result. In case of the disk, this result coincides with ˘ ckovi´c Theorem 1.1 of [7], which in turn is contained in the work of Ahern and Cu˘ [2] mentioned above. Theorem 1.1. Suppose that u1 , u2 ∈ h∞ are continuous on Dn ∪ W for some boundary open set W . If Tu1 Tu2 = 0, then u1 = 0 or u2 = 0. Next, we consider the case where symbols have additional Lipschitz continuity up to the whole or some part of the distinguished boundary. Such extra Lipschitz continuity enables us to extend our method to products with three or four factors. Given a subset X ⊂ Cn , recall that the Lipschitz class of order ∈ (0, 1], denoted by Lip (X), is the class of all functions f on X such that |f (z)−f (w)| = O(|z−w| ) for z, w ∈ X. Our second result solves the zero-product problem with four factors when symbols are n-harmonic and have global Lipschitz continuity up to the distinn = Dn ∪ T n . guished boundary. We let D

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n ) ∩ h∞ for some > 0 and for some Theorem 1.2. Let u1 , u2 , u3 , u4 ∈ Lip (U ∩ D n open set U containing T . If Tu1 Tu2 Tu3 Tu4 = 0, then uj = 0 for some j. For n-harmonic symbols that have only local Lipschitz continuity up to the distinguished boundary, we can also apply the proof of Theorem 1.2 to solve the zero-product problem, but with only three factors. n ) ∩ h∞ for some > 0 and for some Theorem 1.3. Let u1 , u2 , u3 ∈ Lip (U ∩ D n open set U with U ∩ T = ∅. If Tu1 Tu2 Tu3 = 0, then uj = 0 for some j. While the main idea of our method of proofs is adapted from [7], substantial amount of unexpected analysis is required to overcome some diﬀerent nature of the polydisks being product domains. For example, a uniqueness theorem (Proposition 2.2) and an example that follows reveal more involved nature of the polydisks compared with the balls. The whole part of later arguments is thus necessarily eﬀected in a more complicated direction by such an unexpected uniqueness result. Boundary continuity and n-harmonicity hypotheses in the theorems above play key roles in our arguments of the present paper. First, boundary continuity ensures (Proposition 2.1) that the Berezin transform of products of Toeplitz operators under consideration recovers the products of corresponding symbols at every distinguished boundary point where the symbols have continuous extensions. Also, both hypotheses allow us to use a uniqueness theorem (Proposition 2.2) for n-harmonic functions, which can be viewed as a product version of the local Hopf lemma obtained in [6]. In addition, it provides us quite explicit information on local behavior of symbol functions near a distinguished boundary vanishing point (Lemma 3.1). We do not know whether either boundary regularity or n-harmonicity can be removed in the hypotheses of our theorems above when n ≥ 2. We do not know whether the number of factors are best possible under the given hypotheses, either. In Section 2, we collect a couple of key facts which play the role of the starting point of our proofs. In Section 3, we prove Theorems 1.1, 1.2 and 1.3.

2. Preliminaries In this section we prove two main ingredients of our arguments which might be of independent interests. One is a certain continuous extension property of the Berezin transform and the other is a uniqueness theorem for n-harmonic functions. These two propositions are what led us to require boundary regularity and nharmonicity hypotheses in our theorems. Let a ∈ Dn denote an arbitrary point, unless otherwise speciﬁed. Recall that the well-known Berezin transform of a bounded linear operator L on A2 is a on Dn deﬁned by function L L(a) = Lka (z)ka (z) dV (z) Dn

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where ka is the normalized kernel, namely, n 1 − |aj |2 , ka (z) = (1 − aj zj )2 j=1

47

z ∈ Dn .

is continuous on Dn . In our application L will be It is not hard to see that L the products of Toeplitz operators as in the hypotheses of theorems stated in the Introduction. Such operators L have additional properties useful for our purpose: has the same amount of boundary continuity as symbols and the boundary value L is precisely the product of symbols. In order to prove this, we ﬁrst recall some of L well-known facts. obius We let ϕa (z) = φa1 (z1 ), . . . , φan (zn ) where each φaj is the usual M¨ map on D given by aj − z j , zj ∈ D. φaj (zj ) = 1 − aj z j The map ϕa is an automorphism on Dn such that ϕa ◦ ϕa = id. We deﬁne a linear operator Ua on A2 by Ua f = (f ◦ ϕa )ka n for f ∈ A2 . Since the real Jacobian of ϕa (z) is j=1 |φaj (zj )|2 = |ka (z)|2 , we see that each Ua is an isometry on A2 . Also, since Ua ka = (ka ◦ ϕa )ka = 1, we have Ua Ua = I and thus Ua−1 = Ua . Now, being an invertible linear isometry, Ua is unitary. Also, we have Ua Tu Ua = Tu◦ϕa ; (2.1) this is well known on the disk (see, for example, [4]) and the same proof works on the polydisk. Proposition 2.1. Suppose that functions u1 , . . . , uN ∈ h∞ are continuous on Dn ∪ continuously extends to Dn ∪{ζ} {ζ} for some ζ ∈ T n . Let L = TuN · · · Tu1 . Then L and L(ζ) = (u1 · · · uN )(ζ). The proof below is similar to that of the ball case (Proposition 2.1 of [7]) and is included here for reader’s convenience. Proof. Let a ∈ Dn . Note that ka = Ua 1. Also, by (2.1), we note that Ua LUa = (Ua TuN Ua ) · · · (Ua Tu1 Ua ) = TuN ◦ϕa · · · Tu1 ◦ϕa , because Ua Ua = I. Since Ua∗ = Ua−1 = Ua , it follows that L(a) = Ua LUa 1, 1 = TuN ◦ϕa · · · Tu1 ◦ϕa 1, 1

(2.2)

where , denotes the inner product on L2 . We claim that, as a → ζ, we have TuN ◦ϕa · · · Tu1 ◦ϕa 1 → (uN · · · u1 )(ζ)

in L2

(2.3)

which, together with (2.2), implies the proposition. Now, we prove the claim. Let a → ζ. Then, for a given function u continuous on Dn ∪ {ζ}, we observe that ϕa → ζ pointwise (in fact, uniformly on compact sets) and thus u ◦ ϕa → u(ζ) in L2 by the dominated convergence theorem. In

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particular, we have P (u ◦ ϕa ) → u(ζ) in L2 . So, we see that the claim holds for N = 1. We now proceed by induction on N . Assume that (2.3) holds for some N ≥ 1 and consider the case of N + 1. Having (2.3) as induction hypothesis and denoting the Lp -norm by p , we have TuN +1◦ϕa TuN ◦ϕa · · · Tu1 ◦ϕa 1 − (uN +1 uN · · · u1 )(ζ) 2 ≤ TuN +1◦ϕa [TuN ◦ϕa · · · Tu1 ◦ϕa 1 − (uN · · · u1 )(ζ)] 2 + |(uN · · · u1 )(ζ)| TuN +1 ◦ϕa 1 − uN +1 (ζ) 2 ≤ uN +1 ∞ TuN ◦ϕa · · · Tu1 ◦ϕa 1 − (uN · · · u1 )(ζ)] 2 + |(uN · · · u1 )(ζ)| TuN +1 ◦ϕa 1 − uN +1 (ζ) 2 → 0 so that (2.3) also holds for N + 1. This completes the induction and the proof of the proposition. Now, we turn to a uniqueness property of bounded n-harmonic functions. We ﬁrst recall a certain extension property of bounded n-harmonic functions across the boundary. So, let u ∈ h∞ be an arbitrary function. As is well known, the radial limit u∗ (ζ) = limr→1 u(rζ) exists at almost all points ζ ∈ T n and u is recovered by the Poisson integral of u∗ : n 1 − |zj |2 u(z) = u∗ (ζ) dσ(ζ), z ∈ Dn 2 n |1 − z ζ | T j j j=1 where dσ is the Haar measure on T n . See [14, p. 31] for details. Thus, if u is in addition continuous on Dn ∪ W and vanishes on W for some boundary open set W , then it is easily seen from the Poisson integral formula above that u extends n-harmonically across W . We also need various notation. Working with n-harmonic functions, it seems quite natural and necessary to consider relevant diﬀerential operators componentwise. Recall ∂j = ∂/∂zj . Now, given j = 1, . . . , n, we let Rj = zj ∂j + z j ∂ j √ Tj = −1(zj ∂j − z j ∂ j ) ∆j = 4∂j ∂ j denote the j-th radial diﬀerential operator, the j-th real tangential diﬀerential operator and the j-th Laplacian, respectively. Here, and elsewhere, we abuse the notation zj for the function z → zj . Then it is straightforward to see that Rj Ti = Ti Rj , R2j

+

Tj2

Rj Ri = Ri Rj

= |zj | ∆j 2

∆j Ri = (2δij + Ri )∆j

(2.4) (2.5)

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for each i, j (of course, when applied to suﬃciently smooth functions) where δij is the Kronecker delta. Note that each Rj preserves n-harmonicity by (2.5). We also remark that each Rj has the invariance property Rj (u ◦ L) = (Rj u) ◦ L

(2.6)

for real linear transformations L on Cn of the form L(z) = (L1 (z1 ), . . . , Ln (zn )) where each Li is a real linear transformation on C. In our argument below it seems more convenient to use real notation for diﬀerential operators. So, we let √ Dxj = ∂j + ∂ j , Dyj = −1(∂j − ∂ j ) for j = 1, . . . , n. In terms of these real diﬀerential operators, note that Rj = zj Dxj + zj Dyj ,

Tj = zj Dyj − zj Dxj

for each j. Here, a and a denote the real and imaginary part of a ∈ C, respectively. By a multi-index we mean an n-tuple α = (α1 , . . . , αn ) of nonnegative integers. We use conventional multi-index notation. Given a multi-index α, we let |α| = α1 + · · · + αn and α! = α1 ! · · · αn !. Also, X α = X1α1 · · · Xnαn for X = (X1 , . . . , Xn ). We let Dx = (Dx1 , . . . , Dxn ), Dy = (Dy1 , . . . , Dyn ) and R = (R1 , . . . , Rn ), T = (T1 , . . . , Tn ) for the purpose of multi-index notation. By a componentwise rotation ρ on Cn , we mean that ρ is a rotation of the form ρ(z) = (ζ1 z1 , . . . , ζn zn ), ζ ∈ T n . Note that each Rj is componentwise rotationinvariant by (2.6). The same invariance property for each Tj is also easily veriﬁed. As a consequence, we have componentwise rotation-invariance property for general mixed diﬀerential operators: Rα T β (u ◦ ρ) = (Rα T β u) ◦ ρ

(2.7)

for componentwise rotations ρ and multi-indices α, β. This property will be useful in normalizing our argument later. Finally, we use the notation Λ for the set of all multi-indices α such that each component of α is 0 or 1 and put e = (1, 1, . . . , 1) ∈ T n for simplicity. Proposition 2.2. Suppose that u ∈ h∞ is continuous on Dn ∪W for some boundary open set W . If both u and Rα u vanish on W for all α ∈ Λ, then u = 0 on Dn . Proof. Since u = 0 on W , u extends to an n-harmonic function across W , as mentioned above. We claim Rα u = 0 on W

(2.8)

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for all multi-indices α. In order to see this, we introduce temporary notation. Given an integer m ≥ 1, let Λm be the set of all multi-indices α such that αi ≤ m for each i. Note that we have (2.8) for α ∈ Λ = Λ1 by assumption. We now proceed by induction on m. Suppose that (2.8) holds for all α ∈ Λm . Let α ∈ Λm and ﬁx j. If αj ≤ m − 1, then Rj Rα u vanishes on W by induction hypothesis. Consider αj−1 m−1 αj+1 1 n the case αj = m. Put f = Rα Rj+1 · · · Rα n u for simplicity. Note 1 · · · Rj−1 Rj that f is n-harmonic across W , because u is. Thus, we have by (2.4) Rj Rα u = R2j f = −Tj2 f

on W.

(2.9)

Moreover, Tj2 f = 0 on W , because f = 0 on W by induction hypothesis. So, we see that (2.8) holds for all α ∈ Λm+1 and the induction is complete. Now, let ζ ∈ W be an arbitrary but ﬁxed point. Then it follows from (2.8) that T β Rα u(ζ) = 0 for all multi-indices α and β. Note that we may assume ζ = e by using componentwise rotation-invariance (2.7). So, T β Rα u(e) = 0 for all α and β. On the other hand, we see by routine and straightforward calculations that T β Rα u(e) is of the form T β Rα u(e) = Dxα Dyβ u(e) +

cα β Dxα Dyβ u(e);

(2.10)

lower order

the sum is to be taken over all multi-indices α and β with |α | < |α| or |β | < |β|. Thus, by an inductive argument, we obtain Dxα Dyβ u(e) = 0 for all multi-indices α and β. So, by real-analyticity, we conclude u = 0 in some open subset of Dn and thus on the whole Dn . The proof is complete. Remark. (1) Compared with the ball version (Proposition 4.1 of [7]) of Proposition 2.2, the hypothesis that “ Rα u = 0 for all α ∈ Λ ” might seem too much at a glace. In the same context, one might expect that “ Rj u = 0 for all j ” would be enough. It turns out that such hypothesis requiring all α ∈ Λ is essential, which shows a quite diﬀerent nature caused by the fact that the polydisk is a product domain. To see an example, let g be the harmonic extension on D of a nonzero function continuous on T and vanishing on some open subarc I. Deﬁne u(z) = g(z1 ) · · · g(zn ) for z ∈ Dn and put W = I n . Clearly, we have u = 0, u ∈ h∞ ∩ C(Dn ) and u = 0 on W . Moreover, Rα u = 0 on W for all α ∈ Λ with α = (1, . . . , 1). Thus, just one single missing α ∈ Λ does not guarantee the triviality of u. (2) In the proof of Proposition 2.2, note that the boundedness of given function is used only to ensure the smooth extension across W in each variable sepn for arately. Consider a function u n-harmonic on Dn and continuous on U ∩ D n some open set U with W := U ∩ T = ∅. If u = 0 on W , then one can use the reﬂection principle to see that u extends n-harmonically across W . So, Proposition 2.2 remains valid for such a function u.

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3. Zero products In this section we prove our theorems. We need several technical lemmas. First, we begin with an observation on how Taylor approximations near distinguished boundary points behave when a bounded n-harmonic function vanishes on some boundary open set. Given an integer 0 ≤ k ≤ n, we let Λ(k) be the set of all α ∈ Λ such that |α| = k. Also, for a multi-index β, we say β ∈ Λ∗ (k) if there exists α ∈ Λ(k) such that βi = αi + 1 for some i and βj = αj for all j = i. We let τ (z) = (1 − z1 , . . . , 1 − zn ) η(z) = (|1 − z1 |, . . . , |1 − zn |) for z ∈ C . n

Lemma 3.1. Let u ∈ h∞ ∩C(Dn ∪W ) for some boundary open set W . Suppose that u is not identically 0 on Dn and vanishes on W . Then there exist some ζ ∈ W and an integer 1 ≤ k ≤ n such that   u(ζz) = (−1)|α| Rα u(ζ) τ (z)α + O  η(z)β  β∈Λ∗ (k)

α∈Λ(k)

for z ∈ Dn ∪ ζW near e where ζz = (ζ1 z1 , . . . , ζn zn ). Moreover, some coeﬃcient Rα u(ζ) = 0. Proof. Since u = 0 on W , u extends to an n-harmonic function across W , as mentioned earlier. Since u|W = 0 and u ≡ 0 on Dn by assumption, we have Rα u|W ≡ 0 for some α ∈ Λ by Proposition 2.2. So, the following minimum is well deﬁned and positive: k = min{|α| : Rα u|W ≡ 0, α ∈ Λ}.

(3.1)

Now, choose a point ζ ∈ W such that Rα u(ζ) = 0 for some α ∈ Λ that achieves the above minimum. Note that the function v(z) := u(ζz) is real-analytic near e. Let Dxα Dyβ v(e) τ (z)α Y β (−1)|α| (3.2) v(z) = α!β! α,β

be the Taylor series expansion of v at e. Here, Y = (z1 , . . . , zn ). For the coeﬃcients above, note that Dxα v(e) = Rα v(e) = Rα u(ζ) for α ∈ Λ(k) by (2.7). Thus, we can rewrite the sum in (3.2) as (−1)|α| Rα u(ζ) τ (z)α + Error. (3.3) v(z) = α∈Λ(k)

We now estimate the error term. Consider an arbitrary multi-index α with N (α) ≤ k − 1 where N (α) denotes the number of nonzero components of α. Associated with such α is a multi-index β = (β1 , . . . , βn ) where βi = 0 if αi = 0

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and βi = 1 otherwise. Then, since β ∈ Λ and |β| = N (α) ≤ k − 1, we have Rβ u|W = 0 by minimality of k. Suppose that there is some j such that αj > βj = 1. e Let β = (β1 , . . . , βj−1 , 0, βj+1 , . . . , βn ) and put f = Rβ u. Then we also have e Rβ u|W = 0 by minimality of k. So, we have Rj Rβ u = R2j f = −Tj2 f = 0

on W

by the same argument as in (2.9). Now, repeating the same reasoning as many times as needed, we obtain Rα u|W = 0. In addition, we claim Rα T β u(ζ) = 0

(3.4)

for all multi-indices β. To show this, we assume α = (α1 , . . . , αk−1 , 0, . . . , 0) for notational simplicity . Also, shrinking W if necessary, we may assume W = W1 × · · · × Wn where each Wj is an open arc in T containing ζj . Then it follows from Proposition 2.2 that each function u(·, ξ), ξ ∈ Wk × · · · × Wn , is identically zero on Dk−1 . Therefore, given a multi-index β, we have β

k−1 Rα T1β1 · · · Tk−1 u=0

on Dk−1 × Wk × · · · × Wn

and therefore β

k−1 Tkβk · · · Tnβn Rα T1β1 · · · Tk−1 u=0

on Dk−1 × Wk × · · · × Wn .

This yields (3.4) as a special case, because Ri Tj = Tj Ri for each i, j. Note that we have Rα T β v(e) = 0 by (3.4) and (2.7). So, by (2.10) and (3.4), we have Dxα Dyβ v(e) = 0 for all α with N (α) ≤ k − 1 and for all β. Note that |α| = N (α) = k if and only if α ∈ Λ(k). Therefore, the error term in (3.3) can be decomposed into three pieces Dxα Dyβ v(e) τ (z)α Y β . + + (−1)|α| α!β! |α|>N (α)=k β=0

N (α)=k |β|≥1

N (α)>k |β|≥0

α+β Note that |τ (z)α Y β | ≤ η(z) for all α and β. Hence, it is easily seen that each sum above is dominated by γ∈Λ∗ (k) η(z)γ , as desired. The proof is complete.

Remark. The proof above shows that the point ζ = ζu ∈ W can be chosen arbitrarily, once Rα u(ζ) = 0 for some α ∈ Λ that achieves the minimum (3.1). Thus, given a ﬁnite number of functions u1 , . . . , uN ∈ h∞ ∩ C(Dn ∪ W ) which are not identically 0 on Dn and vanishes on some boundary open set W , one may easily modify the proof above to choose points ζuj ∈ W so that ζu1 = · · · = ζuN . This fact will be used later in the proof of Theorem 1.2 and Theorem 1.3. Before proving our main results, we need some technical estimates which are essentially proved in [7]. First, we need an integral estimate. Given c, s and d real, we let s 1 + log(1 − |a|)(1 − |b|) Φc,s (a, b) = |1 − ab|2+c

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and consider corresponding integral |1 − ξ|d Φc,s (ξ, b) Ic,s,d (a, b) = dV1 (ξ) |1 − aξ|2 D for a, b ∈ D. Lemma 3.2. Given d, s ≥ 0 and c ≥ d − 2, there is a constant C = C(c, s, d) such that if c > d (1 − t)d−c Φ0,s+1 (a, t) Ic,s,d (a, t) ≤ C × Φc−d,s+1 (a, t) if c ≤ d for a ∈ D and 0 < t < 1.

Proof. This follows from Proposition 3.8 of [7].

Next, we need a couple of estimates showing how a certain 1-dimensional Toeplitz operator behaves against test functions. Let Sσ be the 1-dimensional Toeplitz operator with symbol σ where σ(a) = 1 − a for a ∈ D, or more explicitly, Sσ f (a) = D

f (ξ)(1 − ξ) dV1 (ξ), (1 − aξ )2

a∈D

for f ∈ A2 (D). Also, we let µt (a) =

1 , 1 − ta

a∈D

for each 0 < t < 1. Lemma 3.3. Let m > 2 be an integer. Then there exists a polynomial P in two variables with no term of degree less than 2 such that Sσ µm t = for

1 2

1 −1 3 µm−1 + µm t P(1 − t, µt ) + O(|µt | ) m−1 t

< t < 1.

Proof. See the proof of Lemma 4.3 in [7].

Lemma 3.4. Let and m be positive integers such that m > + 1. Then Sσ µm t =

µm− t {1 + Et } (m − 1) · · · (m − )

for 12 < t < 1 where {Et } are some uniformly bounded functions such that Et (t) = o(1) as t → 1. Proof. See the proof of Theorem 1.2 in [7].

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We are now ready to prove our theorems. As is seen in the proofs of Proposition 2.2 and Lemma 3.1, the componentwise rotation-invariance property (2.7) allows us to normalize our argument. We will do the same normalization in the proofs below. This time, however, such normalization eﬀects not only symbol functions but also Toeplitz operators with those symbols. So, we need to make that point clear. Let ρ be a componentwise rotation and let Cρ denote the composition operator f → f ◦ ρ on A2 . Then it is straightforward to verify (for N = 1 and thus for general N ) Cρ (Tu1 · · · TuN )Cρ−1 = Tu1 ◦ρ · · · TuN ◦ρ , which in turn yields Tu1 · · · TuN = 0 ⇐⇒ Tu1 ◦ρ · · · TuN ◦ρ = 0

(3.5)

for any ﬁnite number of Toeplitz operators Tu1 , . . . , TuN . Given t = (t1 , . . . , tn ) where 0 < tj < 1, we let λt (z) = µt1 (z1 ) · · · µtn (zn )

z ∈ Dn ;

this will be the source for our test functions. Note on Constants. For two positive quantities X and Y , we often write X Y or Y X if X is dominated by Y times some inessential positive constant. Also, we write X ≈ Y if X Y X. We ﬁrst prove Theorem 1.1. Proof of Theorem 1.1. Assume Tu1 Tu2 = 0. Then, since u1 and u2 are both continuous on D ∪ W by assumption, we have 0 = (Tu1 Tu2 )= u1 u2

on W

by Proposition 2.1. There are two cases to consider: (i) Both u1 and u2 vanish on W (ii) Either u1 or u2 does not vanish on some boundary open subset of W . In n ) for some open set U with case of (i) we have by Lemma 3.1 u1 , u2 ∈ Lip1 (U ∩ D n U ∩ T = ∅. Thus, the case (i) is contained in Theorem 1.3 to be proved below. So, we may assume (ii). Note that we may further assume that u1 does not vanish on some boundary open set, still denoted by W , because otherwise we can use the adjoint operator (Tu1 Tu2 )∗ = Tu2 Tu1 . We now have u2 = 0 on W . Assume that u2 is not identically 0 on Dn . This will lead us to a contradiction. In the rest of the proof, we let z = (z1 , . . . , zn ) ∈ Dn represent an arbitrary point. Since u2 = 0 on W and u2 is not identically 0 on Dn , we have a point ζ ∈ W and an integer k ≥ 1 provided by Lemma 3.1 (with u2 in place of u). By componentwise rotation-invariance (2.7) and (3.5), we may assume ζ = e without loss of generality. Also, put Cα = Rα u2 (e) for multi-indices α. Then, by Lemma 3.1, we have   Cα τ (z)α + O  η(z)β  u2 (z) = α∈Λ(k)

β∈Λ∗ (k)

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for z ∈ Dn ∪ W near e where some coeﬃcient Cα = 0. Put c1 = u1 (e) = 0 and m2 (z) = Cα τ (z)α α∈Λ(k)

for simplicity. Also, let e1 = u1 − c1 and e2 = u2 − m2 . Then we have 0 = Tu1 Tu2 = Tc1 +e1 Tm2 +e2 = c1 Tm2 + Te1 Tm2 + Tu1 Te2 and thus −c1 Tm2 = Te1 Tm2 + Tu1 Te2 . Now we apply each side of the above to the same test functions and derive a contradiction. Here, we will use test functions λm t where m > 4 is any ﬁxed integer and t = (t1 , . . . , tn ) is chosen as follows. Choice of t: Put

F (x) =

Cα xα

α∈Λ(k)

for x ∈ R . Note that F is a non-zero polynomial on Rn , because some coeﬃcient Cα is not zero. Thus, there is some y ∈ (0, 1)n such that F (y) = 0. Given t ∈ (0, 1), we now choose tj = tj (t) such that tj = 1 − yj (1 − t2 ), j = 1, . . . , n, n

and let t = t(t) = (t1 , . . . , tn ) for the rest of the proof. Note tj ∈ (0, 1) and 1 − t2j = yj (1 − t2 ) for each j so that n

(1 − t2j )αj = (1 − t2 )k y α

(3.6)

j=1

for each α ∈ Λ(k).

Estimate of Tm2 λm t (t): Note that τ (z) = σ(z1 ), . . . , σ(zn ) . Thus, we have n σ αj (wj )µm tj (wj ) Tm2 λm (z) = C dV1 (wj ) α t (1 − zj wj )2 j=1 D α∈Λ(k)

=

Cα

Tm2 λm t (t) =

(3.7)

Sσαj µm tj (zj )

j=1

α∈Λ(k)



so that

n

Cα 

αj =1

α∈Λ(k)

  Sσ µm tj (tj )

  µm tj (tj ) .

αj =0

Note that, by Lemma 3.3, we have Sσ µm t (a) =

(a) µm−1 t {1 + O(|1 − ta|)} m−1

(3.8)

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as t → 1 uniformly in a ∈ D and hence Sσ µm t (t) =

(t) µm−1 t {1 + O(1 − t)} m−1

as t → 1. It follows that Tm2 λm t (t) =

=

1 + o(1) (m − 1)k

 Cα 

 µm−1 (tj )  tj

αj =1

α∈Λ(k)

  µm tj (tj )

αj =0



 + o(1)} Cα  (1 − t2j ) (m − 1)k α =1

λm t (t){1

α∈Λ(k)

j

λm (t){1 + o(1)} (1 − t2 )k F (y) = t (m − 1)k

by (3.6)

and thus m 2 k |Tm2 λm t (t)| λt (t)(1 − t ) |F (y)| =

|F (y)| (1 − t2 )mn−k

as t → 1. Since F (y) = 0, we ﬁnally get |Tm2 λm t (t)|

1 (1 − t2 )mn−k

(3.9)

as t → 1. Estimate of Tu1 Te2 λm t (t): First, recall that

|e2 (z)|

|1 − z1 |β1 · · · |1 − zn |βn .

β∈Λ∗ (k)

Also, for β ∈ Λ∗ (k), note βj ≤ 2 < m − 2 for each j and |β| = k + 1. Thus, by Lemma 3.2, we have |Te2 λm t (z)|

n

β∈Λ∗ (k) j=1

n

β∈Λ∗ (k) j=1

D

|1 − wj |βj dV1 (w) |1 − zj wj |2 |1 − tj w j |m

Φ0,1 (zj , tj ) (1 − tj )m−2−βj

1 (1 − t)mn−2n−k−1

n j=1

Φ0,1 (zj , tj )

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and hence, again by Lemma 3.2, we obtain |Tu1 Te2 λm t (t)|

n

1 (1 −

t)nm−2n−k−1

1 (1 − t)nm−2n−k−1

j=1 n

D

|Φ0,1 (wj , tj )| dV1 (wj ) |1 − tj wj |2

Φ0,2 (tj , tj )

j=1

as t → 1. Note that

2 n 1 + log(1 − tj )2 1 + | log(1 − t)|2n Φ0,2 (tj , tj ) = ≈ (1 − t2j )2 (1 − t)2n j=1 j=1 n

as t → 1. Combining these estimates, we conclude |Tu1 Te2 λm t (t)|

1 + | log(1 − t)|2n o(1) = nm−k−1 (1 − t) (1 − t)nm−k

(3.10)

as t → 1. Estimate of Te1 Tm2 λm t (t): By (3.7), we ﬁrst note    m m m Cα  Sσ µtj (zj )  µtj (zj ) Tm2 λt (z) = αj =1

α∈Λ(k)

and hence by (3.8) |Tm2 λm t (z)|

 |Cα | 

αj =1

α∈Λ(k)

|λm t (z)|

α∈Λ(k)

=

|λm t (z)|

αj =0

 |µm−1 (zj )|O(1)  tj 

|Cα | 



  |µm tj (zj )|

αj =0

|1 − tj zj |

αj =1

|Cα ||1 − t1 z1 |α1 · · · |1 − tn zn |αn

α∈Λ(k)

for z ∈ Dn . Since coeﬃcients Cα are bounded, it follows that n |e1 (w)| m |Te1 Tm2 λt (t)| dV (w). m+2−αj Dn j=1 |1 − tj wj | α∈Λ(k)

Given > 0 small, we let Dj () = {zj ∈ D : |zj − 1| < } for j = 1, . . . , n and put Ω = D1 () × D2 () × · · · × Dn ().

(3.11)

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Corresponding to this region, we now decompose the sum in (3.11) into two pieces as follows: = and = 1

α∈Λ(k)

2

Dn \Ω

α∈Λ(k)

Ω

for convenience.

We ﬁrst estimate 1 . Note that Dn \ Ω is the ﬁnite union of sets of the form J = J1 × J2 × · · · × Jn where each Jj is either Dj () or D \ Dj () and at least one Jj is D \ Dj (). Given such J, let = (J) be an index such that J = D \ D (). Since |1 − t w | ≥ |1 − w | − (1 − t ) ≥ − (1 − t ) − y (1 − t) for w ∈ J as t → 1, we have, for α ∈ Λ(k), dV1 (w ) 1 m+2−α [ − y (1 − t)]m+2−α J |1 − t w | 1 ≤ [ − |y|(1 − t)]m+2 as t → 1. On the other hand, we have dV1 (wj ) 1 ≈ , m+2−αj m−αj |1 − t w | (1 − t j j j) D

j = ;

(3.12)

see Lemma 4.2.2 of [15] for a proof of this well-known estimate. Note that 1 1 ≈ m−α j (1 − tj ) (1 − t)m−αj j=

j=

(1 − t)m−α (1 − t)mn−k (1 − t)m−1 ≤ (1 − t)mn−k =

because |α| = k. It follows that n 1 1 (1 − t)m−1 dV (w) · m+2−α mn−k j (1 − t) [ − |y|(1 − t)]m+2 J j=1 |1 − tj wj | as t → 1. Note that this estimate is uniform in J and α ∈ Λ(k) as t → 1. So, since e1 is bounded, we conclude ν(, t) (3.13) 1 (1 − t)mn−k as t → 1 where ν(, t) = (1 − t)m−1 /[ − |y|(1 − t)]m+2 . √ Next, we estimate 2 . Note that |w − e| < n for w ∈ Ω . Thus, setting √ ω() = sup{|u1 (w) − u1 (e)| : w ∈ Dn ∪ W, |e − w| < n},

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Zero Products of Toeplitz Operators

we have 2

≤ ω()

Dn

α∈Λ(k)

α∈Λ(k)

≈

n

1 dV (w) m+2−αj |1 − t w j j| j=1

n

ω()

59

1 (1 − t )m−αj j j=1

(3.14)

ω() (1 − t)mn−k

where the second inequality holds by (3.12). In summary, with > 0 small and ﬁxed, we have by (3.13) and (3.14) |Te1 Tm2 λm t (t)|

ω() + ν(, t) (1 − t)mn−k

(3.15)

as t → 1 and the estimate is uniform in . Finish of Proof : Setting M = −c1 Tm2 and R = Te1 Tm2 + Tu1 Te2 , we obtain from (3.9), (3.10) and (3.15) that 1=

|Rλm t (t)| ω() + ν(, t) + o(1), |M λm t (t)|

> 0 : ﬁxed

as t → 1 and the estimate is uniform in small. So, now ﬁrst taking the limit t → 1 and then → 0, we have 1 ω() → 0 by continuity of u at e, which is a contradiction. This completes the proof.

Next, we prove Theorem 1.2. Proof of Theorem 1.2. Assume Tu1 Tu2 Tu3 Tu4 = 0. Then, since each uj is continuous up to the distinguished boundary by assumption, we have (Tu1 Tu2 Tu3 Tu4 )= u1 u2 u3 u4 = 0

on T n

by Proposition 2.1. Since u1 u2 u3 u4 is continuous and vanishes everywhere on the distinguished boundary, there exists a boundary open set W ⊂ T n such that either or

uj never vanishes uj = 0 on W

on W,

holds for each j. Since there is nothing to prove if u1 ≡ 0, we may assume that u1 |W never vanishes. Now, assume that each uj is not identically 0 on Dn . We will get a contradiction. Since each uj is not identically 0 on Dn , we may shrink (if necessary) the set W to get a smaller boundary open set, still denoted by W , such that each uj

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satisﬁes (i) uj |W never vanishes,

either

or (ii) uj |W = 0 but uj ≡ 0.

(3.16)

Given j for which the case (ii) holds, we have by Lemma 3.1 a point ζ ∈ W and a positive integer kj such that   uj (ζz) = Rα uj (ζ) τ (z)α + O  η(z)β  (3.17) β∈Λ∗ (kj )

α∈Λ(kj )

for z ∈ Dn ∪ ζW near e where some coeﬃcient Rα uj (ζ) = 0. Note that such a point ζ can be chosen independently of j by the remark mentioned right after Lemma 3.1. So, we may assume ζ = e; this causes no loss of generality by (2.7) and (3.5). Also, we let Cj,α = Rα uj (e) for each j and multi-index α. Using such notation, we rewrite (3.17) as   Cj,α τ (z)α + O  η(z)β  uj (z) = β∈Λ∗ (kj )

α∈Λ(kj )

for z ∈ D ∪ W near e where some coeﬃcient Cj,α = 0. We let kj = 0 if uj (e) = 0. Note k1 = 0, because u1 never vanishes on W . With such convention we have |1 − z1 |α1 · · · |1 − zn |αn (3.18) |uj (z)| n

α∈Λ(kj )

for each j, because Λ(0) consists of the zero multi-index in case kj = 0. Here, and in the rest of the proof, z ∈ Dn represents an arbitrary point. Now, we deﬁne the major part of uj by Cj,α τ (z)α (3.19) mj (z) := α∈Λ(kj )

for each j. More explicitly, we deﬁne  (e)   uj mj (z) = Cj,α τ (z)α  

if uj (e) = 0 if uj (e) = 0

α∈Λ(kj )

for each j. Put ej = uj − mj for each j. Note that ej (e) = 0 for each j. Thus, we have  n    |1 − zi | if uj (e) = 0   i=1 |ej (z)| (3.20)  β1 βn  |1 − z | · · · |1 − z | if u (e) = 0  1 n j   β∈Λ∗ (kj )

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by the Lipschitz continuity hypothesis. We introduce further notation. Put M = Tm1 Tm2 Tm3 Tm4 and R = Tu1 Tu2 Tu3 Tu4 − M for notational convenience. Then, setting R1 = Te1 Tm2 Tm3 Tm4 , R2 = Tu1 Te2 Tm3 Tm4 , R3 = Tu1 Tu2 Te3 Tm4 , R4 = Tu1 Tu2 Tu3 Te4 , we have R = R1 + R2 + R3 + R4 . Note that R = −M , because Tu1 Tu2 Tu3 Tu4 = 0 m by assumption. Now we will estimate the same expression M λm t (t) = −Rλt (t) in two diﬀerent ways and reach a contradiction as in the proof of Theorem 1.1. Here, m > 4 is any ﬁxed integer and t = (t1 , . . . , tn ) is chosen below. Put k = k1 + k2 + k3 + k4 . Note that we have k ≥ 1, because (u1 u2 u3 u4 )(e) = 0. Also, recall that k1 = 0. Choice of t: First, we introduce some notation. Let (α, β, γ) denote an arbitrary triple of α ∈ Λ(k2 ), β ∈ Λ(k3 ) and γ ∈ Λ(k4 ). Given (α, β, γ), we let 1 Cα+β+γ = (m − 1) · · · (m − αi − βi − γi ) αi +βi +γi ≥1

be the set of all multi-indices of the form α + β + γ for simplicity. Also, let Λ where α ∈ Λ(k2 ), β ∈ Λ(k3 ) and γ ∈ Λ(k4 ). Using such notation, we consider two polynomials on Rn given by G(x) : = C2,α C3,β C4,γ xα+β+γ (α,β,γ)

=

 

e h∈Λ

 C2,α C3,β C4,γ  xh

(3.21)

α+β+γ=h

and F (x) : =

(α,β,γ)

=

e h∈Λ

C2,α C3,β C4,γ Cα+β+γ xα+β+γ 

Ch 

 C2,α C3,β C4,γ  xh

(3.22)

α+β+γ=h

for x ∈ Rn . It is clear from (3.21) and (3.22) that G is non-trivial if and only if F is. Note that G

is the product of non-trivial polynomials G2 , G3 and G4 deﬁned by Gj (x) = α∈Λ(kj ) Cj,α xα for j = 2, 3, 4. The polynomial G is therefore non-trivial. So, we conclude that the polynomial F is also non-trivial.

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Now, as in the proof of Theorem 1.1, we choose y ∈ (0, 1)n with F (y) = 0 and deﬁne 0 < t < 1, ti = ti (t) = 1 − yi (1 − t2 ), for i = 1, . . . , n. Let t = t(t) = (t1 , . . . , tn ) for the rest of the proof. Note ti ∈ (0, 1) and 1 − t2i = yi (1 − t2 ) for each i. Also, note that |α + β + γ| = k2 + k3 + k4 = k for all (α, β, γ). So, we have n

(1 − t2i )αi +βi +γi = (1 − t2 )k y α+β+γ

(3.23)

i=1

for each (α, β, γ). Estimate of M λm t (t): Note that M = cTm2 Tm3 Tm4 where c = u1 (e) = 0. Using (3.19) and repeating arguments similar to (3.7), we have n m αi +βi +γi m C2,α C3,β C4,γ Sσ µti (zi ) . Tm2 Tm3 Tm4 λt (z) = i=1

(α,β,γ)

Meanwhile, given (α, β, γ), we deduce from Lemma 3.4 that n

Sσαi +βi +γi µm ti (ti )

i=1

 =





 µm ti (ti )

αi +βi +γi =0

=



 Sσαi +βi +γi µm ti (ti )

αi +βi +γi ≥1

 µm ti (ti )

αi +βi +γi =0



αi +βi +γi ≥1

= Cα+β+γ λm t (t){1 + o(1)}

 i −βi −γi µm−α (t ){1 + o(1)} i ti  (m − 1) · · · (m − αi − βi − γi ) (1 − t2i )αi +βi +γi

αi +βi +γi ≥1

as t → 1 (hence all ti → 1). Also, note that we have

(1 − t2i )αi +βi +γi =

αi +βi +γi ≥1

n

(1 − t2i )αi +βi +γi = (1 − t2 )k y α+β+γ

i=1

by (3.23). Now, combining all the observations in the preceding paragraph, we have m 2 k M λm t (t) = cλt (t){1 + o(1)}(1 − t ) F (y)

and thus m 2 k |M λm t (t)| λt (t)(1 − t ) |cF (y)| =

|cF (y)| (1 − t2 )mn−k

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as t → 1. Since cF (y) = 0, we ﬁnally conclude |M λm t (t)|

1 (1 − t2 )mn−k

(3.24)

as t → 1. Estimate of Rj λm t (t): Recall Rj = Tu1 · · · Tuj−1 Tej Tmj+1 · · · Tm4 for each j. Let j ≤ 3 for a moment. Using the same argument as in the course of the estimate of M λm t (t), we have |Tmj+1 · · · Tm4 λm t (z)|

n (j)

Sσαi +···+γi µm ti (zi )

i=1

(j)

where denote the sum over all α ∈ Λ(kj+1 ), . . . , γ ∈ Λ(k4 ). Thus we have by Lemma 3.4 |Tmj+1 · · · Tm4 λm t (z)|

n (j)

|µti (zi )|m−pi

i=1

where pi = pi (j; α, . . . , γ) = αi + · · · + γi . Hence by (3.20) we have (j) Θj,α,...,γ (z) |Tej Tmj+1 · · · Tm4 λm t (z)| where

 n n  |µti (wi )|m−pi   |1 − w | dV (w)   |1 − zi w i |2 n i=1 =1 D Θj,α,...,γ (z) = n |1 − wi |βi |µti (wi )|m−pi    dV (w)   |1 − z w |2 n β∈Λ∗ (kj )

D

i

i=1

i

if uj (e) = 0 if uj (e) = 0.

Recall 1 − tj ≈ 1 − t as t → 1. So, if uj (e) = 0, then we have by Lemma 3.2   n Φ0,1 (zi , ti ) Φ0,1 (z , t )   Θj,α,...,γ (z) (1 − ti )m−pi −2 (1 − t )m−p −2− =1

≈ =

i=

(1 − t) (1 − t)mn−2n−(p1 +···+pn ) (1 − t)

n

Φ0,1 (zi , ti )

i=1 n

(1 − t)mn−2n−(kj+1 +···+k4 )

Φ0,1 (zi , ti )

i=1

as t → 1. Similarly, if uj (e) = 0, then we have Θj,α,...,γ (z)

1 (1 − t)mn−2n−(kj+1 +···+k4 )−(kj +1)

n i=1

Φ0,1 (zi , ti )

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as t → 1, because |β| = kj + 1 for β ∈ Λ∗ (kj ). Note that these estimates are uniform in α, . . . , γ. Thus, since ≤ 1, it follows that |Tej Tmj+1 · · · Tm4 λm t (z)|

(1 − t) (1 − t)mn−2n−(kj +···+k4 )

n

Φ0,1 (zi , ti )

(3.25)

i=1

as t → 1 and this estimate remains valid even for j = 4. Now, let j ≥ 2 for a moment. Continuing estimates by using (3.25), we deduce from (3.18) and Lemma 3.2 that |Tuj−1 Tej Tmj+1 · · · Tm4 λm t (z)| (1 − t)

(1 − t)mn−2n−(kj +···+k4 ) n |1 − wi |γi Φ0,1 (wi , ti ) × dV1 (wi ) |1 − zi wi |2 γ∈Λ(kj−1 ) i=1 D n (1 − t) Φ−γi ,2 (zi , ti ) (1 − t)mn−2n−(kj +···+k4 ) i=1 γ∈Λ(kj−1 )

because γi = 0, 1 for each i. Note that, for α ∈ Λ(k1 ), . . . , γ ∈ Λ(kj−1 ), we have 0 ≤ αi + · · · + γi ≤ 2

(3.26)

for each i, because α = 0 (recall k1 = 0); it is this step which requires the restriction on the number of factors in the product. Thus, by repeating the same argument using Lemma 3.2, we obtain n (1 − t) m Φ−qi ,j (zi , ti ) |Rj λt (z)| (j) (1 − t)mn−2n−(kj +···+k4 ) i=1

where qi = qi (j; α, . . . , γ) = αi + · · · + γi . Here, the sum (j) is taken over all α ∈ Λ(k1 ), . . . , γ ∈ Λ(kj−1 ). Since 1 − tj ≈ 1 − t as t → 1, each term of the sum above, when evaluated at ti , is estimated as follows: n

Φ−qi ,j (ti , ti ) =

i=1

n 1 + | log(1 − tj )2 |j i=1 n i=1

(1 − t2j )2−qi 1 + | log(1 − t)|4 (1 − t)2−qi

1 + | log(1 − t)|4n (1 − t)2n−(k1 +···+kj−1 )

and this estimate is uniform in α, . . . , γ. Combining these estimates, we obtain |Rj λm t (t)|

(1 − t) (1 + | log(1 − t)|4n ) , (1 − t)mn−k

t→1

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for j ≥ 2 and the same estimate holds for j = 1 by a similar argument using (3.25). We thus conclude o(1) (3.27) |Rλm t (t)| (1 − t)mn−k as t → 1. Finish of Proof : We easily deduce from (3.24) and (3.27) that 1=

|Rλm t (t)| o(1), |M λm t (t)|

which is a contradiction. The proof is complete.

t → 1,

Finally, we prove Theorem 1.3. The proof is exactly the same as that of Theorem 1.2 except at one spot. We only indicate such a diﬀerence. Proof of Theorem 1.3. By a similar argument using Proposition 2.1, we can also have a boundary open set W where u1 u2 u3 vanishes and (3.16) holds. In the proof of Theorem 1.2 we were able to assume that u1 never vanishes on W (and thus k1 = 0) under the global Lipschitz hypothesis because the location of the boundary open set W was of no signiﬁcance. However, we can not do the same under the present local Lipschitz hypothesis. That is, k1 ≥ 1 may well happen and the inequality (3.26) is no longer guaranteed. Namely, αi + βi + γi = 3 might hold for some i and for some α ∈ Λ(k1 ), β ∈ Λ(k2 ), γ ∈ Λ(k3 ) in case k1 , k2 , k3 ≥ 1. What matters here is the number of terms in the sum above which comes from the number of factors. So, if we are given only three factors, all the arguments in the proof of Theorem 1.2 work even in case k1 ≥ 1. This completes the proof.

References [1] P. Ahern, On the range of the Berezin transform, J. Funct. Anal. 215(2004), 206–216. ˇ Cuˇ ˇ ckovi´c, A theorem of Brown-Halmos type for Bergman space [2] P. Ahern and Z. Toeplitz operators, J. Funct. Anal. 187(2001), 200–210. ˇ Cuˇ ˇ ckovi´c, Some examples related to the Brown-Halmos theorem for [3] P. Ahern and Z. the Bergman space, Acta Sci. Math. 70(2004), 373–378. ˇ Cuˇ ˇ ckovi´c, Commuting Toeplitz operators with harmonic symbols, [4] S. Axler and Z. Integr. Equ. Oper. Theory 14(1991), 1–11. [5] A. Brown and P. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213(1964), 89–102. [6] Bouendi and Rothschild, A local Hopf lemma and unique continuation for harmonic functions, International Mathematics Research Notices, 1993, No. 8, 245-251. [7] B. R. Choe and H. Koo, Zero products of Toeplitz operators with harmonic symbols, J. Funct. Anal. 233(2006), No. 2, 307-334.

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[8] B. R. Choe, Y. J. Lee, K. Nam and D. Zheng, Products of Bergman space Toeplitz operators on the polydisk, Math. Ann., to appear. ˇ Cuˇ ˇ ckovi´c, Berezin versus Mellin, J. Math. Anal. Appl. 287(2003), 234–243. [9] Z. [10] X, Ding, Products of Toeplitz operators on the polydisk, Integr. Equ. Oper. Theory 45(2003), 398–403. [11] C. Gu, Products of several Toeplitz operators, J. Funct. Anal. 171(2000), 483–527. [12] K. Guo, A problem on products of Toeplitz operators, Proc. Amer. Math. Soc. 124(1996), 869–871. [13] S. Krantz, Function theory of several complex variables. Second edition., Graduate Studies in Mathematics, 40. Amer. Math. Soc., Providence, RI, 2002. [14] W. Rudin, Function theory in polydiscs, Benjamin, New York, 1969. [15] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York and Basel, 1989. Boo Rim Choe and Hyungwoon Koo Department of Mathematics Korea University Seoul 136–713 Korea e-mail: [email protected] [email protected] Young Joo Lee Department of Mathematics Chonnam National University Gwangju 500-757 Korea e-mail: [email protected] Submitted: May 25, 2005 Revised: June 30, 2006

Integr. equ. oper. theory 57 (2007), 67–81 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010067-15, published online May 3, 2006 DOI 10.1007/s00020-006-1440-6

Integral Equations and Operator Theory

Zeno Product Formula Revisited Pavel Exner, Takashi Ichinose, Hagen Neidhardt and Valentin A. Zagrebnov Abstract. We introduce a new product formula which combines an orthogonal projection with a complex function of a non-negative operator. Under certain assumptions on the complex function the strong convergence of the product formula is shown. Under more restrictive assumptions even operator-norm convergence is veriﬁed. The mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observables in the sense of Davies. Mathematics Subject Classification (2000). Primary 47A55, 47D03, 81Q10; Secondary 47B25. Keywords. Zeno dynamics, product formulæ, resolvents, convergence, generalized observables.

1. Introduction Product formulæ are a traditional tool in various branches of mathematics; their use dates back to the time of Sophus Lie. Such formulæ are often of the form n (1.1) s- lim e−itA/n e−itB/n = e−itC , C := A + B, t ∈ R, n→∞

where A and B are bounded operators on some separable Hilbert space H and s-lim stands for the strong operator topology. A natural generalization to unbounded self-adjoint operators A and B is due to Trotter [23, 24] who showed that the limit exists and is equal to e−itC , t ∈ R, if the operator C, Cf := Af + Bf,

f ∈ dom(C) := dom(A) ∩ dom(B),

is essentially self-adjoint. In [16, 17] Kato focused his interest to products of the type n n s- lim (f (tA/n)g(tB/n)) and s- lim g(tB/n)1/2 f (tA/n)g(tB/n)1/2 (1.2) n→∞

n→∞

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where A, B are non-negative self-adjoint operators and f, g are now so-called Kato functions. Recall that a Borel measurable function f (·) : [0, ∞) −→ R is usually called a Kato function if the conditions 0 ≤ f (x) ≤ 1,

f (0) = 1,

f (+0) = −1,

are satisﬁed. Under these conditions he was able to show that the limits (1.2) exist and are equal to e−tC , t ∈ R, where C is the form sum of A and B. Notice f (x) = e−x is a Kato-functions which yields the well-known Trotter-Kato product formulæ n n s- lim e−tA/n e−tB/n = s- lim e−tB/2n e−tA/n e−tB/2n = e−tC , t ≥ 0. n→∞

n→∞

Both products (1.1) and (1.2) are very useful and admit applications to functional integration, quantum statistical physics and other parts of physics – see, e.g., [8, Chap. V], [25] and references therein. The last decade brought a progress in understanding of the convergence properties of such formulæ in operator-norm and trace-class topology, for a review of these results we refer to the monograph [25]. In the last few years we have witnessed a surge of interest to another type of product formulæ, namely those where the left-hand side is replaced by an expression of the type s- lim

n→∞

P e−itH/n P

n

(1.3)

where H is a self-adjoint operator on some separable Hilbert space H and P is a orthogonal projection on some closed subspace h ⊆ H. Products of such type are motivated by the “quantum Zeno eﬀect” (QZE). We call them therefore Zeno product formulæ. The fact that the limit, if it exists, may be unitary on h is a venerable problem known already to Alan Turing and formulated in the usual decay context of quantum mechanics for the ﬁrst time by Beskow and Nilsson [2]: frequent measurements can slow down a decay of an unstable system, or even fully stop it in the limit of inﬁnite measurement frequency. The eﬀect was analyzed mathematically by Friedman [12] but became popular only after the authors of [21] invented the above stated name. Recent interest is motivated mainly by the fact that now the eﬀect is within experimental reach; an up-to-date bibliography can be found, e.g., in [10] or [22]. The physical interpretation of this formula can be given in the context of particle decay, cf. [8, Chap. II]. The unstable system is characterized by a projection P to a subspace h of the state Hilbert space H of a larger, closed system, the dynamics of which is governed by a self-adjoint Hamiltonian H. Repeating the non-decay measurement experiment with the period t/n, we can describe the time evolution over the interval [0, t] of a state originally in the subspace P H by the interlaced product (P e−itH/n P )n ; the question is how this operator will behave as n → ∞. In [21, Theorem 1] it was shown that if the limit (1.3) exists and there is a conjugation J commuting with P and H, then the Zeno product formula deﬁnes a unitary group on the subspace h. Another simple example shows that this result is not valid generally: the limit (1.3) may exist without deﬁning a unitary group.

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Let H = L2 (R) and H be the momentum operator, i.e H = −i∂x and let P = P[a,b] be the orthogonal projection on some subspace h = L2 ([a, b]), [a, b] ⊆ R. A straightforward calculation shows that P[a,b] e−isH P[a,b] = P[a,b] P[a+s,b+s] e−isH P[a,b] , Therefore, we get

T (t) := s- lim

n→∞

P e−itH/n P

n

= P e−itH h,

s ∈ R. t ∈ R,

(1.4)

which is neither unitary nor it satisﬁes the group property but deﬁnes a contraction semigroup for t ≥ 0. This example is covered by the following more general one. Let H be the minimal self-adjoint dilation of a maximal dissipative operator K deﬁned on the subspace h. Since by deﬁnition of self-adjoint dilations, cf. [11], P e−isH P := e−isK = P e−isH h,

we ﬁnd s- lim

n→∞

P e−itH/n P

n

= e−itK ,

s ≥ 0, t ≥ 0.

From now on the strong convergence in the product formula is considered only on h. Further examples can be found in [20]. However, in all of them the non-unitarity of the limit is related to the fact that H is not semibounded. So we restrict ourself in the following to the case that H is semi-bounded from below; it is clear that without loss of generality we may assume that H is non-negative. It has to be stressed that the last mentioned assumption does not ensure the √ existence of the limit (1.3). Indeed, if dom( H) ∩ h is not dense in h, then it can happen that the left-hand side in the Zeno product formula does not converge, cf. [8, Rem. 2.4.9] or [20]. With these facts in mind we assume in the following that H is a non-negative √ self-adjoint operator such that dom( H)∩h is dense in h. Under these assumptions we claim that a “natural” candidate for the limit of the Zeno product formula (1.3) is the unitary group e−itK , t ∈ R, on h, generated by the non-negative self-adjoint operator K associated with the closed sesquilinear form k, √ √ √ k(f, g) := ( Hf, Hg), f, g ∈ dom(k) = dom( H) ∩ h ⊆ h. (1.5) The claim rests upon the paper [9, Theorem 2.1] where it is shown that T n lim P e−itH/n P f − e−itK f 2 dt = 0, for each f ∈ h and T > 0 (1.6) n→∞

0

holds. This result yields the existence of the limit of the Zeno product formula for almost all t in the strong operator topology, along a subsequence {n } of natural numbers1 . The reason why this result is weaker than the natural conjecture is that the exponential function involved in the interlaced product gives rise to oscillations 1 This

fact that the proof in [9] yields convergence along a subsequence was omitted in the first version of the paper from which the claim was reproduced in the review [22]. A complete proof of this claim is known at present only in the case when P is finite-dimensional, cf. [9].

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which are not easy to deal with. One of the main ingredients in the present paper is a simple observation that one can avoid the mentioned problem when φ(x) = e−ix is replaced by functions with an imaginary part of constant sign. In analogy with the Kato class of the product formula (1.2) it seems to be useful to introduce a class of admissible functions. Defintion 1. We call a Borel measurable function φ(·) : [0, ∞) −→ C admissible if the conditions |φ(x)| ≤ 1,

x ∈ [0, ∞),

φ(0) = 1,

and φ (+0) = −i,

(1.7)

are satisﬁed. Typical examples are φ(x) = (1 + ix/k)−k ,

k = 1, 2, . . . ,

and φ(x) = e−ix ,

x ∈ [0, ∞).

(1.8)

The main goal of this paper is to prove the following result. Theorem 2. Let H be a non-negative self-adjoint operator in H and let h be a closed subspace √ of H such that P : H −→ h is the orthogonal projection from H onto h. If dom( H) ∩ h is dense in h and φ is admissible function which obeys

m (φ(x)) ≤ 0,

x ∈ [0, ∞),

(1.9)

then for any t0 > 0 one has n

s- lim (P φ(tH/n)P ) = e−itK ,

(1.10)

n→∞

uniformly in t ∈ [0, t0 ], where the generator K is defined by (1.5) and the strong convergence is meant on h. One may consider formulæ of type (1.10) as modiﬁed Zeno product formulæ. Examples of admissible functions obeying (1.9) are φ(x) = (1 + ix)−1

and φ(x) = (1 + ix/2)−2 ,

x ∈ [0, ∞).

Unfortunately, not all admissible function do satisfy the condition (1.9). Indeed, the functions φ(x) = (1 + ix/3)−3 and φ(x) = e−ix , x ∈ [0, ∞), are admissible but do not obey (1.9). In particular this yields that the convergence problem for the original Zeno product formula (1.3) is not solved by Theorem 2 and remains open. However, Theorem (2) suggests the following regularizing procedure. We set ∆φ := {x ∈ [0, ∞) : m (φ(x)) ≤ 0}. By (1.7) the set ∆φ contains a neighbourhood of zero. If the subset ∆ ⊆ ∆φ contains also a neighbourhood of zero, then φ∆ (x) := φ(x)χ∆ (x),

x ∈ [0, ∞),

deﬁnes an admissible function obeying m (φ∆ (x)) ≤ 0, x ∈ [0, ∞). By Theorem 2 we obtain that for any t0 > 0 one has n

s- lim (P φ∆ (tH/n)P ) = e−itK n→∞

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uniformly in t ∈ [0, t0 ]. Applying this procedure to φ(x) = e−ix , x ∈ [0, ∞), one has to choose a subset ∆ ⊆ ∆φ := ∪∞ m=0 [2mπ, (2m + 1)π] containing a neighbourhood of zero. From φ(x) = e−ix one can construct a “cutoﬀ” admissible function φ∆ (x) := e−ix χ∆ (x), x ∈ [0, ∞), obeying (1.9). In particular for ∆ = [0, π), the function φ∆ (x) = e−ix χ∆ (x), x ∈ [0, ∞), is admissible and obeys (1.9). This leads immediately to the following corollary. Corollary 3. If the assumptions of Theorem 2 are satisfied, then for any t0 > 0 one has n n s- lim P (I + itH/n)−1 P = s- lim P (I + itH/2n)−2 P = e−itK , (1.11) n→∞

n→∞

and s- lim

n→∞

n P EH ([0, πn/t))e−itH/n P = e−itK

(1.12)

uniformly in t ∈ [0, t0 ] where EH (·) is the spectral measure of H, i.e. H = [0,∞) λ dEH (λ). The ideas to replace the unitary group e−itH by a resolvent, cf. (1.11), or to employ a spectral cut-oﬀ together with e−itH , cf. (1.12), are not new: they were used to derive a modiﬁcation of the unitary Lie-Trotter formula in [14] and [18, 19], respectively, both for the form sum of two non-negative self-adjoint operators. See also [3]. Finally, let us note that formula (1.12) admits a physical interpretation in the context of the Zeno eﬀect. To this end we note that the combination of the energy ﬁltering and non-decay measurement following immediately one after another, see (1.12), can be regarded as a single generalized measurement. In fact, a product of two, in general non-commuting2 projections represents the simplest non-trivial example of generalized observables3 introduced by Davies which are realized as positive maps of the respective space of density matrices [6, Sec. 2.1]. Thus formula (1.12) corresponds to a modiﬁed Zeno situation with such generalized measurements, which depend on n and tend to the standard non-decay yes-no experiment as n → ∞. Let us describe brieﬂy the contents of the paper. Section 2 is completely devoted to the proof of Theorem 2. In Section 3 we handle √ the general case of admissible functions under the stronger assumption h ⊆ dom( H). We show that under this assumption the modiﬁed Zeno product formula converges to e−itK for 2 We are primarily interested, of course, in the nontrivial case when the P does not commute with H, and thus also with the spectral projections EH ([0, πn/t)). 3 Since the spectral projections involved commute with the evolution operator, one can also replace the product P EH ([0, πt/n)) in our formulæ by EH ([0, πt/n))P EH ([0, πt/n)). Such generalized observables represented by symmetrized projection products have been recently studied as almost sharp quantum eﬀects – cf. [1].

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any admissible function not necessary satisfying the additional condition (1.9). In particular, one has n s-lim P e−itH/n P = e−itK , (1.13) uniformly in t ∈ [0, t0 ] for any t0 > 0. Moreover, we shall demonstrate there that under stronger assumptions, unfortunately too restrictive from the viewpoint of physical applications, even the operator-norm convergence can be obtained. We ﬁnish the paper with a conjecture which takes into account the results of [9] and the present paper.

2. Proof of Theorem 2 We set F (τ ) := P φ(τ H)P : h −→ h ,

τ ≥ 0,

(2.1)

and

Ih − F (τ ) : h −→ h , τ > 0 , (2.2) τ where Ih is the identity operator in the subspace h. In the following for an operator X in H we use the notation P XP for the operator P XP := P X h : h −→ h as well as for its extension by zero in h⊥ . Let us assume that √ (2.3) dom(T ) := dom( H) ∩ h S(τ ) :=

is dense in h. We deﬁne a linear operator T : h −→ H by √ (2.4) T f := Hf, f ∈ dom(T ). √ √ Since H is closed and dom( H) ∩ h is dense the operator T is closed and its domain dom(T ) is dense in h. Then T ∗ T : h −→ h is a self-adjoint operator which is identical with K deﬁned by (1.5), i.e. K := T ∗ T : h −→ h

(2.5)

which deﬁnes a non-negative self-adjoint operator in h. Further, let us represent the function φ as φ(x) = ψ(x) − iω(x),

x ∈ [0, ∞),

where ψ, ω : [0, ∞) −→ R are real-valued, Borel measurable functions obeying |ψ(x)| ≤ 1,

ψ (+0) = 0

ψ(0) = 1,

(2.6)

and 0 ≤ ω(x) ≤ 1,

ω(0) = 0,

ω (+0) = 1.

Setting ϕ(x) := 1 − ω(x),

x ∈ [0, ∞),

one has 0 ≤ ϕ(x) ≤ 1,

ϕ(0) = 1,

ϕ (+0) = −1,

(2.7)

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which shows that ϕ is a Kato function. In terms of ψ, ϕ the function φ admits the representation φ(x) = ψ(x) − i(1 − ϕ(x)), x ∈ [0, ∞). We set

1, x = 0, inf (1 − ϕ(s))/s, x > 0, and s∈(0,x] 1, x = 0, p+ (x) := sups∈(0,x] (1 − ϕ(s))/s, x > 0.

p− (x) :=

(2.8)

Both functions are bounded on [0, ∞) and obey 0 ≤ p− (x) ≤ 1 ≤ p+ (x) < ∞,

x ∈ [0, ∞).

(2.9)

The function p− is decreasing, i.e. p− (x) ≥ p− (y), 0 ≤ x ≤ y, and p+ is increasing, i.e. p− (x) ≤ p− (y), 0 ≤ x ≤ y. We deﬁne the sesquilinear forms √ √ √ k− f, g ∈ dom(k− τ ≥ 0, τ (f, g) := (p− (τ H) Hf, Hg), τ ) := dom( H) ∩ h, and

√ √ k+ τ (f, g) := (p+ (τ H) Hf, Hg),

√ f, g ∈ dom(k+ τ ) := dom( H) ∩ h,

τ ≥ 0.

+ Notice that for τ = 0 one has k− 0 = k0 = k where the sesquilinear form k is deﬁned ± by (1.5). Obviously, both forms kτ are non-negative for each τ ≥ 0. Moreover, the form k− τ is closable for each τ > 0 and its closure is a bounded form on h while ± the form k+ τ is already closed for each τ ≥ 0. By Kτ we denote the associated non-negative self-adjoint operators on h. We note that K0± = K. By (2.9) we get + k− τ (f, f ) ≤ k(f, f ) ≤ kτ (f, f ),

+ f ∈ dom(k− τ ) = dom(k) = dom(kτ ),

τ ≥ 0,

which yields Kτ− ≤ K ≤ Kτ+ ,

τ ≥ 0.

{Kτ− }τ ≥0

Since p− is decreasing the family is increasing as τ ↓ 0. Further, from (2.8) one gets that s-limτ →+0 p− (τ H) = IH . Since k− τ ≤ k and √ √ lim k− f, g ∈ dom(k− τ (f, g) = lim (p− (τ H) Hf, Hg) = k(f, g), τ ) = dom(k), τ →+0

τ →+0

we obtain from Theorem VIII.3.13 of [15] that s- lim (Ih + Kτ− )−1 = (Ih + K)−1 . τ →+0

(2.10)

Further, since p+ is increasing the family {Kτ+ }τ ≥0 is decreasing as τ ↓ 0. By s-limτ →+0 p+ (τ H) = IH we ﬁnd √ √ lim k+ f, g ∈ dom(k+ τ (f, g) = lim (p+ (τ H) Hf, Hg) = k(f, g), τ ) = dom(k). τ →+0

τ →+0

Since k is closed we obtain from Theorem VIII.3.11 of [15] that s- lim (Ih + Kτ+ )−1 = (Ih + K)−1 . τ →+0

(2.11)

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Lemma 4. Let {X(τ )}τ >0 , {Y (τ )}τ >0 , and {A(τ )}τ >0 be families of bounded nonnegative self-adjoint operators in h such that the condition 0 ≤ X(τ ) ≤ A(τ ) ≤ Y (τ ) ,

τ > 0,

is satisfied. If s − limτ →0 X(τ ) = s − limτ →0 Y (τ ) = A, where A is a bounded self-adjoint operator in h, then s−limτ →0 A(τ ) = A. Proof. Since for each f ∈ h we have (X(τ )f, f ) ≤ (A(τ )f, f ) ≤ (Y (τ )f, f ) ,

τ > 0,

we get limτ →0 (A(τ )f, f ) = (Af, f ), f ∈ h, or w−limτ →0 A(τ ) = A. Hence w − lim (Y (τ ) − A(τ )) = 0. τ →0

Since Y (τ ) − A(τ ) ≥ 0, τ > 0, we ﬁnd s-lim (Y (τ ) − A(τ ))1/2 = 0 τ →0

which yields s-limτ →0 (Y (τ ) − A(τ )) = 0. Hence s-limτ →0 A(τ ) = A. From (2.2) we obtain 1 1 S(τ ) = P (IH − ψ(τ H))P + i P (IH − ϕ(τ H))P, τ τ

τ > 0.

(2.12)

Let

1 P (IH − ϕ(τ H))P : h −→ h, τ > 0. (2.13) τ Lemma 5. Let H be a non-negative self-adjoint operator in H and let h be a closed √ subspace of H. If dom( H) ∩ h is dense in h and ϕ is a Kato function, then we have s-lim (Ih + L0 (τ ))−1 = (Ih + K)−1 (2.14) L0 (τ ) :=

τ →0

Proof. Since k− τ (f, f )

≤

IH − ϕ(τ H) f, f τ

we ﬁnd Kτ− ≤ P Hence X(τ ) := (Ih + Kτ+ )−1 ≤

≤ k+ τ (f, f ),

√ f ∈ dom( H) ∩ h,

IH − ϕ(τ H) P ≤ Kτ+ , τ

Ih + P

IH − ϕ(τ H) P τ

−1

τ > 0.

≤ (Ih + Kτ− )−1 =: Y (τ ),

τ > 0. Taking into account (2.10),(2.11) and applying Lemma 4 we prove (2.14). We set L(τ ) :=

1 1 P (I − ψ(τ H))P + P (I − ϕ(τ H))P : h −→ h, τ τ

τ > 0.

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Lemma 6. Let H be a non-negative self-adjoint operator in H and let h be a closed √ subspace of H. If dom( H) ∩ h is dense in h, the real-valued Borel measurable function ψ obeys (2.6) and ϕ is a Kato function, then s-lim (Ih + L(τ ))−1 = (Ih + K)−1 . τ →0

(2.15)

Proof. Let ψ(2x) + ϕ(2x) , x ∈ [0, ∞). 2 Notice that ζ is a Kato function. Setting

0 (τ ) := 1 P (IH − ζ(τ H))P, τ > 0, L τ

0 (τ ))−1 = (Ih + K)−1 . By we obtain from Lemma 5 that s-limτ →0 = (Ih + L

L(2τ ) = L0 (τ ) we prove (2.15). ζ(x) :=

We set I − ψ(τ H) P (Ih + L0 (τ ))−1/2 , τ > 0. τ Lemma 7. Let H be a non-negative self-adjoint operator in H and let h be a closed √ subspace of H. If dom( H) ∩ h is dense in h, the real-valued, Borel measurable functions ψ obeys (2.6) and ϕ is a Kato function, then we have M (τ ) := (Ih + L0 (τ ))−1/2 P

s-lim (Ih + M (τ ))

−1

τ →0

= Ih .

(2.16)

Proof. A straightforward computation proves the representation (Ih + L(τ ))−1 = (Ih + L0 (τ ))−1/2 (Ih + M (τ ))

−1

(Ih + L0 (τ ))−1/2 ,

τ > 0.

By (2.14) and (2.15) we get w − lim (Ih + M (τ ))−1 = Ih τ →0

which yields

1/2 −1 s-lim Ih − (Ih + M (τ )) = 0. τ →0

−1 =0 s-lim Ih − (Ih + M (τ ))

Hence

τ →0

which proves (2.16). From (2.16) one gets s-lim (iIh + M (τ )) τ →0

−1

= −iIh .

Hence s-lim (Ih + L0 (τ ))−1/2 (iIh + M (τ )) τ →0

or

−1

(Ih + L0 (τ ))−1/2 = −i(Ih + K)−1 .

−1 1 = (iIh + iK)−1 . s-lim iIh + P (Ih − ψ(τ H))P + iL0 (τ ) τ →0 τ

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Using (2.12) and(2.13) we obtain s-lim (iIh + S(τ ))−1 = (iIh + iK)−1 τ →0

which yields s-lim (Ih + S(τ ))−1 = (Ih + iK)−1 τ →0

We ﬁnish the proof of Theorem 2 applying Chernoﬀ’s theorem [5] or Lemma 3.29 of [7].

3. Arbitrary admissible functions Theorem 2 needs the additional assumption (1.9) and it is unclear whether this assumption can be dropped. In the following √ we are going to show that under stronger assumptions on the domain of H the condition (1.9) is indeed not necessary. Theorem 8. Let H be a non-negative self-adjoint operator on H and let h be a closed subspace of H√such that P : H −→ h is the orthogonal projection from H onto h. If h ⊆ dom( H) and φ is admissible, then s- lim (P φ(tH/n)P )n = e−itK

(3.1)

n→∞

uniformly in t ∈ [0, t0 ] for any t0 > 0 where K is defined by (1.5). √ √ Proof. We note that h ⊆ dom( H) implies that T = HP is a bounded operator, and consequently, K = T ∗ T is also bounded. We may employ the representation √ √ IH − φ(τ H) f, g = p(τ H) Hf, Hg , τ > 0, (3.2) τ √ for f ∈ dom(H) and g ∈ dom( H) where i, x=0 p(x) := . (1 − φ(x))/x, x > 0 Since Cp := supx∈[0,∞) |p(x)| < ∞ by (1.7) one gets p(τ H) B(H) ≤ Cp , τ > 0. √ Hence the equality (3.2) extends to f, g ∈ dom( H), in particular, to f, g ∈ h. This leads to the representation (Ih − F (τ ))f = T ∗ p(τ H)T f ,

τ > 0,

f ∈ h,

(3.3)

or S(τ )f − iKf = T ∗ (p(τ H) − iIH ) T f ,

τ > 0,

f ∈ h.

(3.4)

By assumption (1.7) we ﬁnd s-limτ →0 p(τ H) = iIH which yields s-limτ →0 S(τ ) = iK. In this way we obtain the relation s-lim (Ih + S(τ ))−1 = (Ih + iK)−1 , τ →0

and using Chernoﬀ’s theorem [5] one more time we have proved (3.1).

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It turns out that the convergence (3.1) can be improved to operator-norm convergence under some stronger assumption. Corollary 9. Let the assumptions of Theorem 8 be satisfied. One has n lim (P φ(tH/n)P ) − e−itK B(h) = 0 n→∞

(3.5)

uniformly in t ∈ [0, t0 ] for any t0 > 0 if in addition (i) the operator T is compact or √ (ii) there is α > 0 such that h ⊆ dom( H 1+α ) and Cα := supx∈(0,∞) |pα (x)| < ∞ where 0, x=0 pα (x) := . (p(x) − i)/xα , x > 0 Proof. From (3.4) and the compactness of T we ﬁnd lim S(τ ) − iK B(h) = 0.

(3.6)

τ →0

√ √ If h ⊆ dom( H 1+α ) for some α > 0, then we set Tα := H 1+α P and Kα = Tα∗ Tα . Notice that Tα is a bounded operator. From (3.4) we obtain the representation S(τ ) − iK = τ α Tα∗ pα (τ H) Tα ,

τ > 0.

Hence we ﬁnd the estimate S(τ ) − iK B(h) ≤ τ α Cα Kα B(h) ,

τ > 0,

which yields (3.6). Using the representation t −itK −tS(t/n) e −e = e−(t−s)S(t/n) (S(t/n) − iK)e−isK ds 0

we get the estimate −itK − e−tS(t/n) e

B(h)

Using (3.6) we ﬁnd

≤ t S(t/n) − iK B(h) ,

lim e−tS(t/n) − e−itK

n→∞

B(h)

t ≥ 0.

=0

(3.7)

holds for any t > 0, uniformly in t ∈ [0, t0 ]. We shall combine it with the telescopic estimate F (t/n)n − e−itK ≤ (3.8) B(h) ≤ F (t/n)n − e−tS(t/n) + e−tS(t/n) − e−itK , B(h)

B(h)

where the ﬁrst term can be treated as in Lemma 2 of [4], see also [7, Lemma 3.27], √ (3.9) F (t/n)n − e−tS(t/n) f ≤ n (F (t/n) − Ih ) f , f ∈ h .

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Using the representation (3.3) with τ = t/n, we can estimate the right-hand side of (3.9) by t (F (t/n) − Ih ) f ≤ T ∗ p(tH/n)T f , f ∈ h , t > 0 . n Since p(τ H) B(H) ≤ Cp , τ > 0, we ﬁnd t (F (t/n) − Ih ) f ≤ Cp K B(h) f , f ∈ h . n Inserting this estimate into (3.9) we obtain t ≤ Cp √ K B(h) F (t/n)n − e−tS(t/n) n B(h) which yields lim F (t/n)n − e−tS(t/n) =0 (3.10) n→∞

B(h)

for any t > 0, uniformly in t ∈ [0, t0 ]. Taking into account (3.7), (3.8) and (3.10) we arrive at the sought relation (3.5). Remark 10. Since φ(x) = e−ix , x ∈ [0, ∞), √ is admissible we get from Theorem 8 that under the assumptions h ⊆ dom( H) the original √ Zeno product formula (1.13)√holds and that under the stronger assumptions HP is compact or h ⊆ dom( H 1+α ), α > 0, the original Zeno product formula (1.13) converges in the operator norm. √ Remark 11. Obviously, the conclusion (3.5) is valid if h ⊆ dom( H) and h is a ﬁnite dimensional subspace. Indeed, in this case the operator T is ﬁnite dimensional, and therefore compact. This gives an alternative proof of the result derived in Section 5 of [9] for the case φ(x) = e−ix . Remark 12. In connection with the previous remark let us mention that in the ﬁnite-dimensional case there is one more way to prove the claim suggested by G.M. Graf and A. Guekos [13] for the special case φ(x) = e−ix . The argument is based on the observation that lim t−1 P e−itH P − P e−itK P =0 (3.11) t→0

B(h)

implies (P e−itH/n P )n − e−itK B(h) = n o(t/n) as n → ∞ by means of a natural telescopic estimate. To establish (3.11) one ﬁrst proves that

√ √ t−1 (f, P e−itH P g) − (f, g) − it( HP f, HP g) −→ 0 √ as t → 0 for all f, g from dom( HP ) which coincides in this case with h by assumption. The last expression is equal to −itH √ √ e −I −i HP f, HP g tH and the square bracket tends to zero strongly by the functional calculus, which yields the sought conclusion. We note that the operator in the square brackets is

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well-deﬁned by the functional calculus even if H is not invertible. In the same way we ﬁnd that

√ √ t−1 (f, P e−itK P g) − (f, g) − it( Kf, Kg) −→ 0 √ √ holds as √ t → 0 for any vectors f, g ∈ h. Next we note that ( Kf, Kg) = √ ( HP f, HP g), and consequently, the expression contained in (3.11) tends to zero weakly as t → 0, however, in a ﬁnite dimensional h the weak and operatornorm topologies are equivalent. Conjecture 13. Comparing the results of the present paper with those ones of [9] we conjecture that if we drop the assumption (1.9) in Theorem 2, then at least the convergence T (φ(tH/n))n f − e−itK f 2 dt = 0 lim (3.12) n→∞

0

holds for for each f ∈ h, T > 0 and arbitrary admissible functions φ. The proofs of [9] rely heavily on the analytic properties of the exponential function φ(x) = e−ix . For admissible functions analytic properties are not required which yields the necessity to look for a diﬀerent proof idea.

References [1] A. Arias, S. Gudder, Almost sharp quantum eﬀects J. Math. Phys. 45 (2004), 41964206. [2] J. Beskow, J. Nilsson, The concept of wave function and the irreducible representations of the Poincar´e group, II. Unstable systems and the exponential decay law Arkiv Fys. 34 (1967), 561-569. [3] V. Cachia, On a product formula for unitary groups Bull. London Math. Soc. 37 (2005), no. 4, 621-626. [4] P. R. Chernoﬀ, Note on product formulas for operator semigroups J. Funct. Anal. 2 (1968), 238-242. [5] P. R. Chernoﬀ, Product Formulas, Nonlinear Semigroups and Addition of Unbounded Operators Mem. Amer. Math. Soc. 140, Providence R.I, 1974. [6] E. B. Davies, Quantum Theory of Open Systems Academic Press, London, 1976. [7] E. B. Davies, One-Parameter Semigroups London Mathematical Society Monographs, Academic Press, London-New York, 1980. [8] P. Exner, Open Quantum Systems and Feynman Integrals D. Reidel Publishing Co., Dordrecht, 1985. [9] P. Exner, T. Ichinose, A product formula related to quantum Zeno dynamics Ann. H. Poincar´e 6 (2005), 195-215. [10] P. Facchi, G. Marmo, S. Pascazio, A. Scardicchio, E. C. G. Sudarshan, Zeno’s dynamics and constraints J. Opt. B: Quantum Semiclass. Opt. 6 (2004), S492-S501.

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[11] B. Sz.-Nagy, C. Foias, Harmonic analysis of operators on Hilbert space North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadmiai Kiad, Budapest, 1970. [12] C. Friedman, Semigroup product formulas, compressions, and continual observations in quantum mechanics Indiana Math. J. 21 (1971/72), 1001-1011. [13] G. M. Graf, A. Guekos, private communication. [14] T. Ichinose, A product formula and its application to the Schr¨ odinger equation Publ. RIMS, Kyoto Univ., 15 (1980), 585-600. [15] T. Kato, Perturbation Theory for Linear Operators Springer, Berlin-Heidelberg-New York, 1966. [16] T. Kato, On the Trotter-Lie product formula Proc. Japan Acad. 50 (1974), 694–698. [17] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups Topics in functional analysis, Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New York-London, 1978. [18] M. L. Lapidus, Generalization of the Trotter-Lie formula Int. Eq. Operator Theory 4 (1981), 366-415. [19] M. L. Lapidus, Product formula for imaginary resolvents with application to a modiﬁed Feynman integral J. Funct. Anal. 63 (1985), 261-275. [20] M. Matolcsi, R. Shvidkoy, Trotter’s product formula for projections Arch. Math. 81 (2003), 309-317. [21] B. Misra, E. C. G. Sudarshan, The Zeno’s paradox in quantum theory J. Math. Phys. 18 (1977), 756-763. [22] A. U. Schmidt, Mathematics of the quantum Zeno eﬀect in Mathematical Physics Research on the Leading Edge, Ch. Benton ed., Nova Science Publ., Hauppauge NY, 2004; pp. 113-143. [23] H. F. Trotter, Approximation of semi-groups of operators Paciﬁc J. Math. 8 (1958), 887–919. [24] H. F. Trotter, On the product of semi-groups of operators Proc. Amer. Math. Soc. 10 (1959), 545–551. [25] V. A. Zagrebnov, Topics in the Theory of Gibbs Semigroups Leuven Notes in Mathematical and Theoretical Physics, vol. 10, Series A: Mathematical Physics; Leuven Univ. Press, 2003.

Acknowledgement This work was supported by Czech Academy of Sciences within the project K1010104 and the ASCR-CNRS exchange program, and by Ministry of Education of the Czech Republic within the project LC06002 and the French-Czech CNRS bilateral project. V. Z. and H. N. thanks the Department of Theoretical Physics ˇ z for hospitably and ﬁnancial support. of the Czech Academy of Sciences in Reˇ P. E. and H. N. are grateful to the Universit´e de la M´editerran´ee and Centre de Physique Th´eorique-CNRS-Luminy, Marseille (France), for hospitality extended to them and ﬁnancial support. V. Z. is also thankful the Weierstrass-Institut of

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Berlin for hospitality and ﬁnancial support in 2005 when the paper reached its ﬁnal form. Pavel Exner Department of Theoretical Physics, NPI ˇ z Academy of Sciences, CZ-25068 Reˇ and Doppler Institute, Czech Technical University Bˇrehov´ a 7, CZ-11519 Prague Czech Republic e-mail: [email protected] Takashi Ichinose Department of Mathematics Faculty of Science Kanazawa University Kanazawa 920-1192 Japan e-mail: [email protected] Hagen Neidhardt WIAS Berlin Mohrenstr. 39 D-10117 Berlin Germany e-mail: [email protected] Valentin A. Zagrebnov D´epartment de Physique Universit´e de la M´editerran´ee (Aix-Marseille II) and Centre de Physique Th´eorique Luminy, Case 907 F-13288 Marseille Cedex 9 France e-mail: [email protected] Submitted: September 1, 2005 Revised: February 15, 2006

Integr. equ. oper. theory 57 (2007), 83–99 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010083-17, published online August 8, 2006 DOI 10.1007/s00020-006-1447-z

Integral Equations and Operator Theory

Bivariate Function Spaces and the Embedding of Their Marginal Spaces J.J. Grobler Abstract. For a probability space (X × Y, Σ ⊗ Λ, P) we denote the marginal measures of P, deﬁned on Σ and Λ respectively, by P1 and P2 . If ρ is a function norm deﬁned on L0 (X × Y, Σ ⊗ Λ, P), marginal function norms ρ1 and ρ2 are deﬁned on L0 (X, Σ, P1 ) and L0 (Y, Λ, P2 ). We ﬁnd conditions which guarantee Lρ1 +Lρ2 to be embedded in Lρ as a closed subspace. The problem is encountered in Statistics when estimating a bivariate distribution with known marginals. We ﬁnd a condition which, applied to the binormal distribution in L2 , improves some known conditions. Mathematics Subject Classification (2000). Primary 46E30, 47G10, 47B34, 47B38; Secondary 47B80, 62H12. Keywords. Function spaces, bivariate distribution, marginal measures, marginal distributions, kernel operator, Hilbert-Schmidt operator, Hille-Tamarkin operator.

1. Introduction P.J. Bickel, Y. Ritov and J.A. Wellner [3] and H. Peng, A. Schick [9] considered a probability space (X × Y, Σ ⊗ Λ, P), and studied the question of estimating P from its marginals P1 and P2 , the latter deﬁned by P1 (A) := P(A × Y ) for all A ∈ Σ and P2 (B) := P(X × B) for all B ∈ Λ. The marginal spaces L2 (X, Σ, P1 ) and L2 (Y, Λ, P2 ) embed into L2 (X × Y, Σ ⊗ Λ, P) via the natural embeddings g(x) → g(x)1Y (y) and h(y) → 1X (x)h(y). A question The author thanks B. de Pagter and J. Conradie for discussions about the problem and for their helpful remarks. He is also grateful to G. Buskes for his invitation to spend the fall semester of 2003 at the University of Mississippi at Oxford, USA.

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raised by H. Peng (private communication) is to ﬁnd conditions which guarantee L2 (X, Σ, P1 ) + L2 (Y, Λ, P2 ) to be a closed subset of L2 (X × Y, Σ ⊗ Λ, P). This question is related to a classical problem studied in Hilbert space, namely, to ﬁnd the projection of a Hilbert space H onto the closure of the sum of two closed subspaces Hi (the alternating projection algorithm), see [3]. The basic result there was proven by J. Von Neumann [13] and [14] with contributions by N. Aronszajn [1], H. Nakano [8] and N. Wiener [15]. Our approach will be diﬀerent and will hold in general Banach function spaces (for a deﬁnition and properties of Banach function spaces the reader is referred to A.C. Zaanen [17] and W.A.J. Luxemburg and A.C. Zaanen [7]). We will base our arguments on the fact that the projections onto the closed subspaces are conditional expectations. For a Banach function space of bivariate functions we deﬁne marginal spaces and marginal norms and show that the associate norms of Fatou marginal norms are again marginal if and only if the conditional expectations are contractive projections onto the marginal spaces. We then state a suﬃcient condition for the sum of two marginal spaces to be a closed subspace of the given function space. We derive that the condition used in [3] and in [9], and which is equivalent to the Radon-Nikodym derivative of P1 ⊗ P2 with respect to P to belong to L∞ (P), is also suﬃcient in the general case. We also prove a more general suﬃcient condition which, in the Lp -case, states that the kernel of a composite conditional expectation operator should be Hille-Tamarkin (Hilbert-Schmidt if p = 2). This kernel is a product of Radon-Nikodym derivatives of P1 ⊗ P2 with respect to P and of P with respect to P1 ⊗ P2 . As in [3] and [9] we therefore assume P and P1 ⊗ P2 to be equivalent measures. In the ﬁnal section we consider the case in which the density functions of the measures are known and show that the condition used in [3] and in [9] is never satisﬁed by the binormal distribution whereas the improved condition is satisﬁed for certain values of the correlation coeﬃcient.

2. Preliminaries Let (X, Σ) and (Y, Λ) be measurable spaces and consider the probability space (X × Y, Σ ⊗ Λ, P). The marginal measures P1 and P2 of P are deﬁned on (X, Σ) and (Y, Λ) respectively by P1 (A) := P(A × Y ), A ∈ Σ

and

P2 (B) := P(X × B), B ∈ Λ.

We denote by M + (X×Y, Σ⊗Λ, P) the set of all equivalence classes of P-almost everywhere non-negative P-measurable functions and by L0 (X × Y, Σ ⊗ Λ, P) the vector space of all equivalence classes of P-almost everywhere ﬁnite P-measurable functions. Let ρ be a function norm deﬁned on M + (X × Y, Σ ⊗ Λ, P) and denote the function space Lρ (X × Y, Σ ⊗ Λ, P) by Lρ (see [17, Chapter 15]). If we consider the lattice isomorhisms φ : g(x) → g(x)1Y (y) and ψ : h(y) → 1X (x)h(y) from M + (X, Σ, P1 ) and M + (Y, Λ, P2 ) into M + (X × Y, Σ ⊗ Λ, P), we deﬁne function

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norms on M + (X, Σ, P1 ) and M + (Y, Λ, P2 ) by putting ρ1 (f ) = ρ(φ(f )),

f ∈ M + (X, Σ, P1 )

ρ2 (g) = ρ(ψ(g)),

g ∈ M + (Y, Λ, P2 ).

We call the norms ρi the marginal norms of ρ and the spaces Lρ1 (X, Σ, P1 ) = Lρ1 and Lρ2 (Y, Λ, P2 ) = Lρ2 the marginal spaces of Lρ . Lemma 2.1. If Lρ is a Banach function space, then the marginal spaces Lρ1 (P1 ) and Lρ2 (P2 ) are also Banach function spaces. Proof. It is easy to check that the lattice isomorphisms φ and ψ are order continuous and that the norms ρ1 and ρ2 are function norms. ∞ To check the comsuch that pleteness, let (uk ) be a sequence in L+ ρ k=1 ρ1 (uk ) < ∞. Then 1 ∞ ρ(φ(u )) < ∞ and since L is complete we have by the Riesz-Fischer propk ρ k=1 n erty that φ(s ) = φ(u ) ↑ s ∈ L (see [17, Ch. 15, section 64, Theorem 2]). n k ρ k=1 n But sn := k=1 uk ↑ s in M + (X, Σ, P1 ) and the order continuity of φ implies that φ(sn ) =

n

φ(uk ) ↑ φ(s).

k=1

This shows that s = φ(s) and so ρ1 (s) = ρ(φ(s)) = ρ(s ) < ∞. The function norm ρ1 therefore has the Riesz-Fischer property and so Lρ1 is complete. Similarly, Lρ2 is complete. It follows that φ and ψ are isometric lattice isomorphisms of Lρ1 and Lρ2 ρi onto closed Riesz subspaces of Lρ . We shall denote these isometric images by L for i = 1, 2 and the induced norm on them by ρi respectively. If Lρ does not contain any non-zero function of the form φ(f ) or ψ(g), then Lρi = {0}. However, if L∞ ⊂ Lρ then also L∞ ⊂ Lρi for i = 1, 2. We tacitly assume throughout the paper that L∞ ⊂ Lρ , i.e., Lρ and therefore also the marginal spaces Lρ1 and Lρ2 contain the constant functions. := Σ × Y := We note that P1 is the restriction of P to the sub-σ-algebra Σ {A × Y : A ∈ Σ} of Σ and, similarly, P2 is the restriction of P to the sub-σ-algebra := X × Λ := {X × B : B ∈ Λ} of Σ × Λ. The maps φ and ψ are exactly mapping Λ the Σ-measurable functions onto the Σ-measurable functions and similarly for ψ. We state the following simple remark as a lemma. Lemma 2.2. For the P-integrable, Σ-measurable function f (x)1Y (y) we have f (x)1Y (y) dP = f (x) dP1 . X×Y

X

A similar formula holds for a Λ-measurable function.

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n Then, Proof. Let f (x) = i=1αni 1Ai (x) be a step function. n f (x)1 (y) dP = α P(A × Y ) = α P Y i i=1 i i=1 i 1 (Ai ) = X f (x) dP1 . This X×Y shows that the result holds as stated. Let (Ω, F, P) be an arbitrary probability space and let G be a sub-σ-algebra of F. For f ∈ L1 (Ω, F, P), we denote by EP (f | G) the (P-a.e.) unique G-measurable function with the property that EP (f | G) dP = f dP for all A ∈ G. A

A

The existence of EP (f | G) is a consequence of the Radon-Nikodym theorem. The function EP (f | G) is called the conditional expectation of f with respect to G. If f ∈ L1 (Ω, F, P) and g ∈ L∞ (Ω, G, P), then EP (gf | G) = gEP (f | G). For a proof, and also for properties of EP (· | G) used without proof in this paper, we refer to [4], [12] and [5]. The conditional expectation EP (· | G) can be extended from a mapping of 1 L (Ω, F, P) into itself, to a mapping from M + (Ω, F, P) into itself. However, easy examples show that the extended map does not map L0 (Ω, F, P)+ into itself (see [5]). Hence we deﬁne dom EP (· | G) of EP (· | G) by dom EP (· | G) := {f ∈ L0 (Ω, F, P) : EP (|f | | G) ∈ L0 (Ω, G, P)}. Clearly, dom EP (· | G) is an ideal in L0 (Ω, F, P) which contains L1 (Ω, F, P) and is therefore order dense in L0 (Ω, F, P). For f ∈ dom EP (· | G), we deﬁne: EP (f | G) := EP (f + | G) − EP (f − | G). This deﬁnes a positive linear operator EP (· | G) : dom EP (· | G) → L0 (Ω, G, P) ⊂ L0 (Ω, F, P). Although the conditional expectation does not automatically map L0 (Ω, F, P) into itself, we have the result that if EP (|f | | G) ∈ L0 (Ω, G, P), then f ∈ L0 (Ω, F, P) (see [5, Proposition 2.3]). We need to consider the conditional expectation as a positive operator deﬁned on Lρ = Lρ (X × Y, Σ ⊗ Λ, P) and therefore Lρ has to be contained in dom EP (· , Σ) P A condition which guarantees this is expressed in terms of the and dom E (· , Λ). associate norm of the function norm ρ. The associate norm ρ of a function norm ρ is deﬁned by ρ (g) = sup{ |f g| dP : ρ(f ) ≤ 1}. X×Y

The associate norm is a Fatou function norm (see [17]) and gives rise to the Banach function space Lρ = Lρ which is called the associate space of Lρ . In the case under consideration we note that ρi ≤ (ρ )i . We call the function norm ρ deﬁned on L0 (X × Y, Σ ⊗ Λ, P) fully marginal if ρi = (ρ )i , i = 1, 2.

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Proposition 2.3. Let ρ be a Fatou function norm. Then the conditional expectation is a contractive projection of Lρ onto L ρ1 if and only if for all g ∈ Lρ EP (f |Σ) 1 we have φ(g) ∈ Lρ and ρ1 (g) = ρ (φ(g)). A similar condition is necessary and to be a contractive projection of Lρ onto L ρ2 . In particular, suﬃcient for EP (f |Λ) the function norm ρ is fully marginal if and only if both the conditional expectations above are contractive projections. and Proof. Let f ∈ Lρ , and let ρ (φ(g)) = ρ1 (g). We show that f ∈ dom EP (· | Σ) P −1 P ≤ ρ(f ). For this it is suﬃcient to show that φ E (|f | |Σ) ∈ Lρ1 that ρ(E (|f | | Σ)) −1 P with ρ1 (φ E (|f | |Σ)) ≤ ρ(f ). Since ρ is a Fatou-norm, the same holds for ρ1 as can easily be checked. Hence, by Lemma 2.2, −1 P dP1 = ρ1 (φ E (|f | |Σ)) = sup |gφ−1 EP (|f | |Σ)| ρ1 (g)≤1

= sup

ρ1 (g)≤1

X×Y

X

|φ(g)|EP (|f | |Σ)dP = sup

ρ1 (g)≤1

= sup

ρ1 (g)≤1

X×Y

X×Y

EP (|φ(g)f | |Σ)dP =

|φ(g)f | |dP ≤ sup ρ (φ(g))ρ(f ) ≤ ρ(f ). ρ1 (g)≤1

Hence, if the condition holds, the conditional expectation is a contractive projec ρ1 . tion onto L Conversely, assume that the conditional expectation is a contractive projection deﬁned on Lρ and let g ∈ Lρ1 . Then, for any f ∈ Lρ with ρ(f ) ≤ 1, |φ(g)f |dP = |g(x)f (x, y)|dP = X×Y X×Y dP = dP ≤ = EP (|g(x)f (x, y)| | Σ) |g(x)| EP (|f (x, y)| | Σ) X×Y

X×Y

≤ ρ1 (φ(g))ρ(f ) ≤ ρ1 (φ(g)). ρ1 (E (|f (x, y)| | Σ)) ≤ ρ1 (φ(g))

P

This shows that φ(g) ∈ Lρ and that ρ (φ(g)) ≤ ρ1 (φ(g)). On the other hand, ρ1 (φ(g)) ≤ ρ (φ(g)) follows from the fact that ρ1 is the restriction of ρ to the ρ1 . closed Riesz subspace L For a Σ ⊗ Λ-measurable function f we shall henceforth denote by EP1 (f ) and and EP (f | Λ). Hence, by EP2 (f ) respectively, the conditional expectations EP (f | Σ) P P E1 and E2 are contractive projections onto Lρ1 and Lρ2 respectively. Example 2.4. Let ρ be the Lp -norm. It follows from Lemma 1.2 that φ(f ) pp = |φ(f )|p dP = φ(|f |p ) dP = |f |p P1 = f pp X×Y

X×Y

X

It follows that the marginal norm ρ1 (f ) = f p and in the same way ρ2 (g) = g p . Hence, in this case Lρ1 = Lp (X, Σ, P1 ) and Lρ2 = Lp (Y, Λ, P2 ). The associate norm

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for · p is · q with 1/p + 1/q = 1 and it follows that the conditional expectation p for 1 < p < ∞. It is trivial to see that is a contractive projection of Lp (P) into L it also holds for p = 1, ∞. We say the probability measure P on Σ ⊗ Λ, has a desintegration with respect to each of its marginal measures P1 and P2 if there exist random measures (see [6]) Q1 (x, ·) and Q2 (y, ·) such that for all A ∈ Σ and B ∈ Λ we have P(A × B) =

A

Q1 (x, B) P1 (dx) =

B

Q2 (y, A) P2 (dy).

In that case, we have for every P integrable function f (x, y) f dP = f (x, y)Q1 (x, dy)P1 (dx) = f (x, y)Q2 (y, dx)P2 (dy). X×Y

X

Y

Y

X

Note that for almost every x and y the random measures Q1 (x, ·) and Q2 (y, ·) are probability measures; for example, if A ∈ Σ we have 1(x) P1 (dx) = P1 (A) = P(A × Y ) = Q1 (x, Y )P1 (dx) A

A

and so Q1 (x, Y ) = 1(x) holds P1 -almost everywhere on X and the proof for Q2 is similar. We note the following fact and for the sake of completeness supply the proof. Lemma 2.5. If the measure P has desintegrations Q1 and Q2 then P P E1 (f ) = f (x, y)Q1 (x, dy) and E2 (f ) = f (x, y)Q2 (y, dx). Y

Hence, the operators (see [6]).

X

EPi ,

i = 1, 2, are operators generated by random measures

and f ∈ Lρ (X × Y, Σ ⊗ Λ, P) a direct calculation using the Proof. For A × Y ∈ Σ fact that Q1 (x, Y ) = 1X (x) holds P1 -almost everywhere, shows that f (x, y)Q1 (x, dy) dP = A×Y Y = f (x, y)Q1 (x, dy) Q1 (x, dη)P1 (dx) = A Y Y = f (x, y)Q1 (x, dy) Q1 (x, dη)P1 (dx) = A Y Y = f (x, y)Q1 (x, dy)P1 (dx) = f (x, y) dP. A

Y

A×Y

This proves the ﬁrst assertion and the second one follows similarly.

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Remark. Note that the setting considered is a special case of a more general one. Suppose that (Ω, F , P) is a probability space and suppose that Fi , i = 1, 2, are sub-σ-algebras of F such that the σ-algebra generated by F1 and F2 equals F . If Pi denote the restriction of P to Fi then (Ω, Fi , Pi ) is again a probability space. If ρ is a function norm on L0 (Ω, F , P) then the restrictions of ρ to the spaces L0 (Ω, Fi , Pi ) yield marginal function spaces and the theory developed above can be extended to this more general case.

3. The main result ρ1 + L ρ2 being closed in Lρ , it is necessary to In order to study the problem of L know when a function of two variables can be written as a sum of two functions, each of which is a function of one variable. For the next simple observation, which plays a crucial role in the solution to our problem, the author is indebted to B. de Pagter. Lemma 3.1. Let f : X ×Y → R be a numerical function. Then f (x, y) = g(x)+h(y) with g : X → R and h : Y → R if and only if f (x, y) + f (ξ, η) = f (ξ, y) + f (x, η) for all (x, y),(ξ, η) ∈ X × Y . Proof. Let f (x, y) = g(x) + h(y); then, by rearranging the terms, f (x, y) + f (ξ, η) = g(ξ) + h(y) + g(x) + h(η) = f (ξ, y) + f (x, η). Conversely, If the condition holds, we can write f (x, y) = f (x, η) + f (ξ, y) − f (ξ, η). Taking ξ and η ﬁxed, this is the required decomposition of f.

The immediate result following from this remark is: Corollary 3.2. If the sequence (fn (x, y))∞ n=1 converges pointwise to a function f (x, y) and if, for each n, we have fn (x, y) = gn (x) + hn (y) then there exist functions g(x) and h(y) such that f (x, y) = g(x) + h(y). Our main result depends on having this result available holding P-almost everywhere and this can be proved if P is the product of its marginal measures. Proposition 3.3. Let P = P1 ⊗ P2 be a product measure on X × Y. Let fn , gn , hn be P-almost everywhere ﬁnite functions such that fn (x, y) = gn (x) + hn (y) holds P-almost everywhere. If (fn (x, y))∞ n=1 converges pointwise almost everywhere to f (x, y) ∈ L0 (X × Y, Σ ⊗ Λ, P) then there exist functions g(x) ∈ L0 (X, Σ, P1 ) and h(y) ∈ L0 (Y, Λ, P2 ) such that f (x, y) = g(x) + h(y) holds P-almost everywhere. Proof. By deﬁning fn (x, y), gn (x) and hn (y) to be zero on the common null set we may assume without loss of generality that fn (x, y) = gn (x) + hn (y) holds for all n and for all x and y. Hence, for all x, y, ξ and η, fn (x, y) + fn (ξ, η) = fn (x, η) + fn (ξ, y).

(1)

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Let N0 be the null set such that for all (x, y) ∈ / N0we have fn (x, y) → f (x, y) and |f (x, y)| < ∞. By Fubini’s theorem 0 = P(N0 ) = X P2 (N0 (x)) dP1 and it follows that P2 (N0 (x)) = 0 holds for P1 -almost every x. Let N1 ⊂ X be the exceptional set and let ξ ∈ / N1 . Then N0 (ξ) is a P2 -null set such that fn (ξ, y) → f (ξ, y) and |f (ξ, y)| < ∞ for all y ∈ / N0 (ξ). / N2 . Let Similarly, there is a null set N2 ⊂ Y such that P1 (N0 (y)) = 0 for all y ∈ / N2 , we have that N0 (η) is a P1 -null set such that η∈ / N0 (ξ) ∪ N2 . Then, since η ∈ fn (x, η) → f (x, η) and |f (x, η)| < ∞ for all x ∈ / N0 (η). Since we also have that η ∈ / N0 (ξ), it follows that fn (ξ, η) → f (ξ, η) and |f (ξ, η)| < ∞. Therefore, in equality (1), we have for all (x, y) ∈ / N ∪(N0 (η)×Y )∪(X ×N0 (ξ)) that the left hand side converges to f (x, y) + f (ξ, η) and the right hand side converges to f (x, η) + f (ξ, y) and all the functions in the terms are ﬁnitely valued. Since this union is a P1 ⊗ P2 null set, we have that f (x, y) + f (ξ, η) = f (x, η) + f (ξ, y) P1 ⊗ P2 -almost everywhere and the proof is complete.

Corollary 3.4. Let P be probability measure on Σ ⊗ Λ with marginal measures P1 and P2 and let P be equivalent to P1 ⊗ P2 . Let fn , gn , hn be P-almost everywhere ﬁnite functions such that fn (x, y) = gn (x) + hn (y) holds P-almost everywhere. If 0 (fn (x, y))∞ n=1 converges pointwise almost everywhere to f (x, y) ∈ L (X × Y, Σ ⊗ 0 0 Λ, P) then there exist functions g(x) ∈ L (X, Σ, P1 ) and h(y) ∈ L (Y, Λ, P2 ) such that f (x, y) = g(x) + h(y) holds P-almost everywhere. Proof. Since P and P1 ⊗ P2 are equivalent, a property holds P-almost everywhere if and only if it holds P1 ⊗ P2 -almost everywhere. Proposition 3.5. Let f ∈ L0 (X × Y, Σ ⊗ Λ, P) and suppose that P and P1 ⊗ P2 are equivalent measures. If f has a decomposition f (x, y) = g(x) + h(y) which holds P-almost everywhere, then the decomposition is unique up to a constant. Proof. We may assume that P = P1 ⊗ P2 . If g(x) + h(y) = g (x) + h (y) then γ(x) := g(x) − g (x) = h (y) − h(y) =: η(y) holds P-a.e. Let N be the exceptional / N1 . If set and as before P2 (N (x)) = 0 holds for almost every x, say for all x ∈ x0 ∈ N1 we have (x0 , y) ∈ / N for almost every y, i.e., γ(x0 ) = η(y) for almost every y. Hence, η(y) is constant for almost every y. In a similar way γ(x) is constant for almost every x and this proves the assertion of the proposition. Easy examples can be constructed to show that this is false for general measures. Example 3.6. Consider the case of X = Y = [0, 1]. Consider a rectiﬁable curve of total length 1 in X × Y with equation y = ϕ(x), x ∈ [0, 1]. Let P be a measure deﬁned on the half open intervals R = (a, b]×(c, d] by deﬁning P(R) to be the length of the part of the curve contained in R. Extension to the Borel sets yields a measure

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P which is concentrated on the curve. Let h(y) be any Borel measurable function which is not constant and put g(x) = h(ϕ(x)). Then g(x)1Y (y) = h(y)1X (x) P-almost everywhere and h is not constant. ρ1 + L ρ2 in Lρ in a special case. We now prove the closedness of L ρ1 + L ρ2 is Proposition 3.7. Let P = P1 ⊗ P2 and let ρ be fully marginal. Then L closed in Lρ . ρ1 + L ρ2 which converges to f ∈ Lρ . The Proof. Let fn = gn +hn be a sequence in L sequence then has a subsequence which converges pointwise P-almost everywhere to f and so, by Proposition 3.3 f can be written as g + h with g ∈ L0 (X, Σ, P1 ) = |g|1Y ∈ L0 (X × Y, Σ, P) and so g ∈ and h ∈ L0 (Y, Λ, P2 ). Note that EP (|g| | Σ) P P dom E (· | Σ). Since f ∈ Lρ , we also have f ∈ dom E (· | Σ) which implies that h = It is also easy to check that EP (h | Σ) = ( h(y) dP2 (y))1X . f − g ∈ dom EP (· | Σ). Y We now apply the conditional expectations The same arguments hold for EP (· | Λ). and EP (· | Λ)to both sides of the equation f = g + h and use the fact that EP (· | Σ) it is a contractive projection onto the marginal spaces. This yields ρ1 h(y) dP2 = EP (f ) ∈ L g(x) + Y

and

X

1

ρ2 g(x) dP1 + h(y) = EP2 (f ) ∈ L

and so, using the fact that the marginal spaces contain the constant functions, we ρ2 . ρ1 + L get f = g + h ∈ L For a general measure P the following condition (see [9] and [3]) is used in 2 + L 2 is closed in L2 (X × Y, Σ ⊗ Λ, P) : the L2 -case in order to prove that L For all A ∈ Σ, B ∈ Λ, P(A × B) ≥ γP1 (A)P2 (B), γ > 0.

(CS)

This condition implies that for every positive Σ ⊗ Λ-measurable function f (x, y) we have f (x, y)P(dx, dy) ≥ γ f (x, y)d(P1 ⊗ P2 ). (CS2) X×Y

X×Y

To see this, note that the class of sets for which condition (CS) is true includes the class F of all ﬁnite disjoint unions of sets of the form A × B ∈ Σ ⊗ Λ (which is an algebra) and that it is a monotone class. Hence, by Dynkin’s principle (see [2]) it contains the σ-algebra generated by F , which is Σ ⊗ Λ. It follows that P(C) ≥ γP1 ⊗ P2 (C) for all C ∈ Σ ⊗ Λ. But then (CS2) holds for positive step functions and the claim follows.

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Condition (CS) thus implies that the measure P1 ⊗ P2 is P-absolutely continuous. If r(x, y) is the Radon-Nikodym derivative of P1 ⊗ P2 relative to P we have for all C ∈ Σ ⊗ Λ that P1 ⊗ P2 (C) = and so

r(x, y)dP C

γ C

So, for all C ∈ Σ ⊗ Λ,

r(x, y)dP = γP1 ⊗ P2 (C) ≤

dP C

C

(1 − γr)dP ≥ 0

This shows that condition (CS) implies that the Radon-Nikodym derivative r is an element of L∞ (X × Y, Σ ⊗ Λ, P) and r ∞ ≤ 1/γ. If the function r is strictly positive, the measures P and P1 ⊗P2 are equivalent. Let r (x, y) := 1/r(x, y) if r(x, y) = 0 and r (x, y) := 0 if r(x, y) = 0. Then r (x, y) derivative of P with respect to P1 ⊗ P2 , i.e., C f (x, y)dP = is the Radon-Nikodym f (x, y)r (x, y)d(P ⊗ P2 ). In this case we have, for every Σ ⊗ Λ-measurable set 1 C of the form A × B that r (x, y) dP2 (y) dP1 (x) P(A × B) = A

B

showing that P has the desintegration Q1 (x, B) = B r (x, y) dP2 (y) and by Fubini it also follows that Q2 (y, A) = A r (x, y) dP1 (x). By Lemma 2.5 it therefore follows that = EP (f | Σ) f (x, y)Q1 (x, dy) = f (x, y)r (x, y) dP2 (y) = EP1 ⊗P2 (r f | Σ) Y

and similarly = EP (f | Λ)

Y

X

f (x, y)Q2 (y, dx) =

X

f (x, y)r (x, y) dP1 (x) = EP1 ⊗P2 (r f | Λ).

Letting g = r f, we deduce that, in terms of r we get EP1 1 ⊗P2 (g) = EP1 (rg)

and

EP2 1 ⊗P2 (g) = EP2 (rg)

We use these remarks to prove the following theorem which is known in the L2 -case. Theorem 3.8. Let (X × Y, Σ ⊗ Λ, P) be a probability space with marginal spaces (X, Σ, P1 ) and (Y, Λ, P2 ) and suppose that the measures P and P1 ⊗ P2 are equivalent. Suppose moreover that condition (CS) holds, i.e., that there is a γ > 0 such that for all A ∈ Σ, B ∈ Λ we have P(A × B) ≥ γP1 (A)P2 (B) and that ρ is a fully ρ1 + L ρ2 is closed marginal function norm deﬁned on M + (X × Y, Σ ⊗ Λ, P). Then L in Lρ (X × Y, Σ ⊗ Λ, P).

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(1) (2) ρ1 + L ρ2 and suppose that (fn ) converges to f ∈ Lρ . Proof. Let fn = fn +fn ∈ L The sequence then has a subsequence which converges P-almost everywhere to f and since the measures P and P1 ⊗ P2 are equivalent, the subsequence converges P1 ⊗ P2 -almost everywhere to f. By Corollary 2.4(b) the function f can be decomposed as f = f (1) + f (2) with f (i) almost everywhere ﬁnite. We need to prove that ρi for i = 1, 2. f (i) ∈ L Let r be the Radon-Nikodym derivative of P1 ⊗P2 with respect to P and recall that by condition (CS) we have r ∞ ≤ 1/γ < ∞. This implies that multiplication by r is a continuous operator Mr on Lρ and Mr ≤ 1/γ. By our remarks above we have, for i = 1, 2, that

ρi EPi 1 ⊗P2 (|f |) = EPi (r|f |) = EPi (Mr (|f |)) = (EPi ◦ Mr )(|f |) ∈ L ρi . The argument used in Proposition Hence, f ∈ dom(EPi 1 ⊗P2 ) and EPi 1 ⊗P2 (f ) ∈ L (i) 3.5 shows that it follows from this that f ∈ dom(EPi 1 ⊗P2 ) and so we have that EPi 1 ⊗P2 (f ) = EPi 1 ⊗P2 (f (1) + f (2) ) = f (i) + ci with ci ∈ R. It follows that f (i) ∈ ρi . L

Remark. The Lp -norm is fully marginal for all p, 1 ≤ p ≤ ∞, and so the above theorem holds for all Lp spaces. Still assuming that P and the product of its marginal measures P1 ⊗ P2 are equivalent measures with P = r (P1 ⊗P2 ) and P1 ⊗P2 = rP as before, we investigate which other conditions on these Radon-Nikodym derivatives will guarantee the ρ1 + L ρ2 in Lρ . closedness of L The proof of the preceding theorem provides a clue for a solution of the problem. Essential in the proof above is that EPi 1 ⊗P2 should be deﬁned on Lρ , i.e., that Lρ ⊂ dom EPi 1 ⊗P2 . We have the following result Lemma 3.9. With the Radon-Nikodym derivatives r and r as above, let Q(ξ, η, x, y) =

r(x, y)1Y (η) = r(x, y)r (ξ, y)1Y (η). r(ξ, y)

If the kernel operator with kernel Q maps Lρ into itself, then Lρ ⊂ dom(EP2 1 ⊗P2 ), EP2 1 ⊗P2 (Lρ ) ⊂ dom(EP1 ) and Q is the kernel of the operator EP1 EP2 1 ⊗P2 . Similarly, if R(ξ, η, x, y) =

r(x, y)1X (ξ) = r(x, y)r (x, η)1X (ξ) r(x, η)

is the kernel of an operator which maps Lρ into itself, then Lρ ⊂ dom(EP1 2 ⊗P2 ), EP1 1 ⊗P2 (Lρ ) ⊂ dom(EP2 ) and R is the kernel of the operator EP2 EP1 1 ⊗P2 .

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Proof. Let f ∈ Lρ . Then

EP1 (EP2 1 ⊗P2 (|f |))(ξ) = EP1 ( |f (x, y)| dP1 (x))(ξ) X = ( |f (x, y)| dP1 (x))r (ξ, y) dP2 (y) Y X = r (ξ, y)|f (x, y)| d(P1 ⊗ P2 )(x, y) X×Y r (ξ, y) = |f (x, y)| dP(x, y) r (x, y) X×Y r(x, y) ρ1 . = |f (x, y)| dP(x, y) ∈ L r(ξ, y) X×Y

Hence, EP1 (EP2 1 ⊗P2 (|f |)) ∈ L0 (X × Y, Σ ⊗ Λ, P) which implies by [5, Proposition 2.3] that EP2 1 ⊗P2 (|f |) ∈ L0 (X × Y, Σ ⊗ Λ, P), i.e., f ∈ dom EP2 1 ⊗P2 and that EP2 1 ⊗P2 (f ) ∈ dom EP1 . Thus, the operator EP1 EP2 1 ⊗P2 is well deﬁned and maps Lρ into itself. Exactly the same argument holds in the second case. Theorem 3.10. Let (X × Y, Σ ⊗ Λ, P) be a probability space with marginal spaces (X, Σ, P1 ) and (Y, Λ, P2 ) and suppose that the measures P and P1 ⊗ P2 are equivalent. Let r be the Radon-Nikodym derivative of P1 ⊗ P2 with respect to P and r the Radon-Nikodym derivative of P with respect to P1 ⊗ P2 and let ρ be a fully marginal function norm deﬁned on M + (X × Y, Σ ⊗ Λ, P). If Q(ξ, η, x, y) =

r(x, y)1Y (η) r(ξ, y)

ρ1 + L ρ2 is closed in is the kernel of an operator which maps Lρ into itself, then L Lρ (X × Y, Σ ⊗ Λ, P). The same conclusion holds if R(ξ, η, x, y) =

r(x, y)1X (ξ) = r(x, y)r (x, η)1X (ξ) r(x, η)

is the kernel of an operator which maps Lρ into itself. Proof. We prove the assertion for Q. The assumption implies by the preceding lemma that the operator EP1 EP2 1 ⊗P2 is well deﬁned and maps Lρ into itself and hence into Lρ1 . Again a repitition of the argument used before, yields for a sequence ρ1 + L ρ2 which converges to f ∈ Lρ that f can be written as f (1) (x) + (fn ) ⊂ L f (2) (y) with both terms functions which are almost everywhere ﬁnite and the functions f, f (1) and f (2) are all in dom EP1 . It follows that EP1 (f ) = f (1) + EP1 (f (2) ) ρ1 if and only if EP1 (f (2) ) ∈ L ρ1 . But, since and so, since ρ is fully marginal, f (1) ∈ L ρ2 is the decomposition of an element of Lρ into a sum of elements of Lρ1 and L

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unique up to a constant (Proposition 3.5), we can assume that in the decomposition which we consider, f (2) = EP2 1 ⊗P2 (f ). Hence, EP1 (f (2) ) = EP1 (EP2 1 ⊗P2 (f )) ∈ Lρ1 . So f (1) ∈ Lρ1 and it follows that f (2) = f − f (1) ∈ Lρ2 . This completes the proof. A general condition which guarantees that a kernel operator maps Lρ into itself, is that the kernel should be of ﬁnite double norm (see [16] or [11]). In this case it means that Q(ξ, η, x, y) ∈ Lρ for ﬁxed (ξ, η) and ρ (Q(ξ, η, x, y)) ∈ Lρ . Noting that Q is only a function of ξ, x and y, it follows that the second of the ρ1 . conditions translates to ρ (Q(ξ, η, x, y)) ∈ L We note this as a corollary. Corollary 3.11. Let (X × Y, Σ ⊗ Λ, P) be a probability space with marginal spaces (X, Σ, P1 ) and (Y, Λ, P2 ) and suppose that the measures P and P1 ⊗ P2 are equivalent. Let r be the Radon-Nikodym derivative of P1 ⊗ P2 with respect to P and r the Radon-Nikodym derivative of P with respect to P1 ⊗ P2 and let ρ be a fully marginal function norm deﬁned on M + (X × Y, Σ ⊗ Λ, P). If Q(ξ, η, x, y) =

r(x, y)1Y (η) r (ξ, y)1Y (η) = r (x, y) r(ξ, y)

ρ1 , then satisﬁes Q(ξ, η, x, y) ∈ Lρ for ﬁxed (ξ, η) and ρ (Q(ξ, η, x, y)) ∈ L ρ1 + L ρ2 is closed in Lρ (X × Y, Σ ⊗ Λ, P). L In particular, if for some p, 1 ≤ p ≤ ∞, we have p/q |r(x, y)r (ξ, y)|q dP(x, y) dP1 (ξ) < ∞, p−1 + q −1 = 1. X

X×Y

p (P1 ) + L p (P2 ) is closed in Lp (X × Y, Σ ⊗ Λ, P). Then L Remark. 1. The kernel Q satisfying the condition stated here for the case 1 ≤ p ≤ ∞ is also known as a Hille-Tamarkin kernel and in the case of p = 2 as a Hilbert-Schmidt kernel. 2. As remarked earlier, the theorem also holds under the condition which ρ2 . The condition here will be that the kernel guarantees that EP2 (f (1) )(η) ∈ L r (x, η)1X (ξ) is of ﬁnite double norm and in the case of Lp space R(ξ, η, x, y) = r (x, y) will translate to p/q |r(x, y)r (x, η)|q dP(x, y) dP2 (η) < ∞, p−1 + q −1 = 1. Y

X×Y

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4. Density functions In order to apply the result above we consider the case X = Y = R with the Borel σ-algebra B(R). Suppose that the probability measure P on R2 is deﬁned by the bivariate density function p(x1 , x2 ) with p(x1 , x2 ) > 0 for all x1 and x2 . As is well known and easy to check, the marginal measures Pi , i = 1, 2, then have “marginal densities” ∞ p(x1 , x2 )dxj , j = i. pi (xi ) = −∞

The desintegrations Qi can then be checked to be given by p(x1 , x2 )dx2 Q1 (x1 , B) = B , B ∈ B(R) p1 (x1 ) p(x1 , x2 )dx1 Q2 (x2 , A) = A , A ∈ B(R). p2 (x2 ) From this it follows that the conditional expectations satisfy ∞ f (x1 , x2 )p(x1 , x2 )dxj P Ei (f, B(R)) = −∞ , j = i pi (xi )

and EPi 1 ⊗P2 (f, B(R))

∞

∞

= −∞

−∞

p(ξ, x2 )f (x1 , x2 )dξ dxj , j = i.

Now, since P1 ⊗ P2 has density function p1 (x1 )p2 (x2 ), we have p1 (x1 )p2 (x2 ) dP P1 ⊗ P2 (C) = p1 (x1 )p2 (x2 )dx1 dx2 = p(x1 , x2 ) C C which shows that the Radon-Nikodym derivative r(x1 , x2 ) of P1 ⊗ P2 relative to P satisﬁes p1 (x1 )p2 (x2 ) . r(x1 , x2 ) = p(x1 , x2 ) Thus the kernel Q(ξ, η, x1 , x2 ) satisﬁes Q(ξ, η, x1 , x2 ) =

p1 (x1 )p(ξ, x2 ) r(x1 , x2 ) = . r(ξ, x2 ) p1 (ξ)p(x1 , x2 )

For Q to be a Hilbert-Schmidt kernel it should satisfy ∞ ∞ ∞ p1 (x1 )2 p(ξ, x2 )2 dP(x1 , x2 ) dP1 (ξ) 2 2 −∞ −∞ −∞ p1 (ξ) p(x1 , x2 ) ∞ ∞ ∞ p1 (x1 )2 p(ξ, x2 )2 dx1 , dx2 , dξ < ∞. (4.1) = −∞ −∞ −∞ p1 (ξ)p(x1 , x2 ) Example 4.1. The bivariate normal distribution. We consider the simpliﬁed bivariate normal density for the Hilbert space case.

1 1 (x21 − 2ρx1 x2 + x22 ) exp − p(x1 , x2 ) = 2 2(1 − ρ ) 2π 1 − ρ2

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for which the marginal densities pi (xi ) (i = 1, 2) are x2

1 pi (xi ) = √ exp − i . 2 2π By (4.1) we have to show that the integral of the function 2

ξ2

1 1 2 2 2 κ exp −x21 − + x ξ + − 2ρξx + x − 2ρx x + x 2 1 2 2 2 1 − ρ2 2 2(1 − ρ2 ) 1 with κ :=

(2π)3/2

1 1 − ρ2

is ﬁnite. Using the formula ∞

2

exp(−ax + bx) dx = −∞

π exp a

b2 4a

, a > 0,

we ﬁnd that the required integral equals 1 − ρ2 1 − 5ρ2 and so the Hilbert-Schmidt norm of the operator equals 1/4 1 − ρ2 . 1 − 5ρ2 The conclusion1 is that the sum of the marginal L2 -spaces is closed in L2 (P) if √ | ρ | < 1/ 5. But, the Radon-Nikodym derivative of P1 ⊗ P2 with respect to P satisﬁes 2 2 ρ x1 x22 p1 (x1 )p2 (x2 ) ρ 2 = 1 − ρ exp + r(x1 , x2 ) = x1 x2 − p(x1 , x2 ) 1 − ρ2 2 2 1 − ρ2 and so on the line x2 = x1 , r(x1 , x2 ) = r(x1 , x1 ) =

ρ(ρ − 1) 2 1 − ρ2 exp x 1 − ρ2 1

is bounded if and only if 0 ≤ ρ < 1; however, on the line x2 = −x1 this causes r to be unbounded except if ρ = 0. This means that r ∞ = ∞ except in the case that ρ = 0. Hence, the bivariate normal distribution satisﬁes (CS) only in the trivial case where P = P1 ⊗ P2 . 1 Added

in proof: This was improved by H. Peng and A. Schick [10] who showed that it holds for all |ρ| < 1, a result which is also proved by the author in a fortcoming paper entitled “Bivariate and marginal function spaces”

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Example 4.2. The bivariate normal distribution: The Lp -case. In the sequel 1/p + 1/q = 1 In this case we ﬁrst calculate p1 (x)q p(ξ, y)q dP(x, y) H(ξ) := p1 (ξ)q p(x, y)q and then show that H(ξ)p/q dP1 (ξ) < ∞. It turns out that 1 − ρ2

H(ξ) = exp 1 − ρ2 (q 2 − q + 1)

q(q − 1) ρ2 2 2 1 − ρ (q 2 − q + 1)

ξ2.

Note that the latter integral is convergent (exists) only if 1 − ρ2 (q 2 − q + 1) > 0. This means that it is convergent only if 1 |ρ| < 2 q −q+1 where we also note that q 2 − q + 1 is positive for all q. Hence, H(ξ)p/q dP1 (ξ) ∞ ξ2 1 H(ξ)p/q exp(− ) dξ = √ 2 2π −∞

1/2−p/2q

−1/2 2 p/2q 2 2 = 1−ρ 1 − ρ (q − q + 1) 1 − ρ2 (q 2 + 1) . We note that this is a well deﬁned real number if 1 |ρ| < q2 + 1 The result was derived under the additional condition that 1 |ρ| < q2 − q + 1 which holds automatically if the ﬁrst inequality holds true, because q 2 − q + 1 ≤ ∞ q 2 + 1. The best result is therefore √ obtained in L , for then q 1= 1 and we get a solution for all ρ with | ρ | < 1/ 2. However, in the case of L , with q = ∞ no ρ = 0 satisﬁes this condition and no result is obtained. In the case that p = q = 2, the result holds, as seen before, for 1 |ρ| < √ . 5

References [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.

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[2] R.B. Ash and C.A. Dol´ eans-Dade, Probability and measure theory, Second Edition, Academic Press, San Diego, 2000. [3] P.J. Bickel, Y. Ritov and J.A. Wellner, Eﬃcient estimation of linear functionals of a probability measure P with known marginal distributions Ann. Statist.19 (1991), 1316–1346. [4] J.L. Doob, Measure theory, Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, 1994. [5] J.J. Grobler and B. de Pagter, On operators representable as multiplication conditional expectation operators, J. of Operator theory 48 (2002), 15–40. [6] J.J. Grobler, B. de Pagter and D.T. Rambane, Lattice properties of operators deﬁned by random measures, Quaestiones Mathematicae 26 (2003), 307–319. [7] W.A.J. Luxemburg and A.C. Zaanen Notes on Banach function spaces Indag. Math bf 25 (1963), Note I, 135–147; Note II, 148–153; Note III, 239–250; Note IV, 251–263; Note V, 496–504. [8] H. Nakano, Spectral theory in Hilbert space, Japanese Society for the promotion of science, Tokyo, 1953. [9] H. Peng and A. Schick, On eﬃcient estimation of linear functionals of a bivariate distribution with known marginals, Statist. and Prob. Letters 59 (2002), 83–91. [10] H. Peng and A. Schick, Eﬃcient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the least squares approach. Journal of Multivariate Analysis 95 (2005), 385-409. [11] H.H. Schaefer, Positive operators, Springer Verlag, Berlin, Heidelberg, New York, 1974. [12] K.R. Stromberg, Probability for analysts, Chapman and Hall, New York London, 1994. [13] J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949), 401–485. [14] J. von Neumann, Functional operators, Volume II: The geometry of orthogonal spaces, Annals of Mathematics Studies, Volume 22, Princeton University Press, 1950. [15] N. Wiener, On the factorization of matrices, Comment. Math. Helv. 29 (1955), 97–110. [16] A.C. Zaanen, Linear Analysis, North-Holland, Amsterdam, Noordhoﬀ, Groningen, 1964. [17] A.C. Zaanen, Integration, North-Holland, Amsterdam New York, 1967. J.J. Grobler Unit for Business Mathematics and Informatics North-West University Potchefstroom 2520 South Africa e-mail: [email protected] Submitted: August 11, 2005 Revised: June 15, 2006

Integr. equ. oper. theory 57 (2007), 101–126 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010101-26, published online August 8, 2006 DOI 10.1007/s00020-006-1450-4

Integral Equations and Operator Theory

Shifted Hermite-Biehler Functions and Their Applications Vyacheslav Pivovarchik and Harald Woracek Abstract. We investigate a particular subclass of so-called symmetric indeﬁnite Hermite-Biehler functions and give a characterization of functions of this class in terms of the location of their zeros. For the proof we employ the theory of de Branges Pontryagin spaces of entire functions. We apply our results to obtain information on the eigenvalues of some boundary value problems. Mathematics Subject Classification (2000). Primary 46E22, 46C20, 65L10; Secondary 30D99, 30D25. Keywords. Hermite-Biehler class, Pontryagin space, de Branges space, boundary value problem.

1. Introduction The Hermite-Biehler class is the set of all entire functions E which have no zeros in the open upper half-plane C+ and satisfy |E(z)| ≤ |E(z)|, z ∈ C+ ,

(1.1)

An indeﬁnite generalization of this notion is obtained when these conditions are substituted by the conditions that E(z) and E # (z) := E(z) have no common nonreal zeros and that the kernel i E # (z) E # (w) S(w, z) := 1− (1.2) z−w E(z) E(w) has a ﬁnite number of negative squares. The fact that positive deﬁniteness of the kernel (1.2) coincides with the condition (1.1) is thereby a classical result, cf. [Pi]. Functions of the Hermite-Biehler class appear in several contexts of complex– and functional analysis, see for example [dB], [B] or [L1], and are a classical object of analysis. The origin of this notion goes back to the investigation of polynomials V. Pivovarchik expresses his gratitude to the Vienna University of Technology for hospitality.

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and their zeros, cf. [H]. The various deﬁnitions found in the literature often diﬀer in some unessential details; for a comparison see Remark 2.2. In the present note we deal with two subclasses of indeﬁnite Hermite-Biehler functions. The ﬁrst one is deﬁned by the requirement that the function E satisﬁes the functional equation E(−z) = E # (z), z ∈ C , (1.3) and we speak of symmetric indeﬁnite Hermite-Biehler function, cf. Deﬁnition 2.7. The other one, the class of semibounded indeﬁnite Hermite-Biehler functions, is deﬁned by the requirement that the function A(z) := 12 (E(z) + E # (z)) has only ﬁnitely many zeros oﬀ the positive real axis, cf. Deﬁnition 2.9. These two classes are related via the transformation T : E(z) −→ A(z 2 ) − izB(z 2 ) , i 2 (E(z)

(1.4)

where A(z) := + E (z)) and B(z) := − E (z)), cf. Proposition 2.10. In the main result of this paper we characterize, in terms of the location of their zeros, those symmetric indeﬁnite Hermite-Biehler functions which are Ttransforms of positive deﬁnite semibounded Hermite-Biehler functions. It turns out that all zeros in the upper half-plane must be simple, lie on the imaginary axis, and that their location restricts the freedom of zeros on the negative imaginary axis, cf. Theorem 3.1. Our method of proof relies heavily on the theory of symmetric and semibounded de Branges spaces as developed in [KWW3]. For the particular case of polynomials (note that T maps the set of all polynomials onto the set of all polynomials satisfying (1.3)) an analogous characterization was obtained by diﬀerent methods in [P3]. Our motivation to investigate functions of this particular kind, namely Ttransforms of positive deﬁnite semibounded Hermite-Biehler functions, and to study the location of their zeros stems from two sources. First, the classes of semibounded and symmetric indeﬁnite Hermite-Biehler functions readily appeared in several contexts, where also the connection (1.4) between them played a prominent role. For example in the theory of de Branges spaces of entire functions, cf. [dB, Theorems 47,54], [KW2], [KWW3], and in the study of strings and their indeﬁnite generalizations, cf. [KK], [LW], [KWW2]. It turned out in [KWW2] that the Ttransforms of semibounded positive deﬁnite Hermite-Biehler functions correspond to what is called a generalized string in [LW]. From the viewpoint of complex analysis it is natural to ask for product representations and distribution as well as location of zeros. Secondly, T-transforms of positive deﬁnite semibounded HermiteBiehler functions appear in the study of various boundary value problems and there describe the eigenvalues of the problem. Hence, our results can be employed to describe the location of the eigenvalues of such problems, in particular one obtains information on the eigenvalues lying on the imaginary axis. Knowledge on their location has turned out to be of importance for solving the corresponding inverse problems. In concrete cases results of this type where obtained separately, e.g. in [MP1], [MP2], [Si], [PM] or [MoP]. Let us point out that our aim in the study 1 2 (E(z)

#

#

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of the presently treated boundary value problems was not to obtain new knowledge on their resonances, but to show that the presented general results provide a structural and uniﬁed approach to the study of such questions. Let us summarize the contents of the present note. After this introduction, in Section 2, we set up some notation and provide some preliminary results on the mentioned classes of Hermite-Biehler functions. We deal with their product representations and the structure of the generated de Branges spaces of entire functions. Section 3 is devoted to the formulation and proof of our main result, namely Theorem 3.1. Finally, in Section 4, we apply Theorem 3.1 in four concrete cases; the following boundary value problems are investigated: I. The Regge problem: −y + q(x)y = λ2 y , y(0) = 0 ,

y (a) − iλy(a) = 0 . Here λ is the spectral parameter and the potential q is real-valued and belongs to L2 (0, a). This problem occurs in the theory of scattering when the potential is supposed to have ﬁnite support. It is certainly the most popular among the boundary value problems we deal with and was well-studied by a variety of authors, see e.g. [R1], [R2], [Kr], [Ko], [S], [Hr], [IP], [Si], [KaKo]. II. The generalized Regge problem, cf. [GP], [PM]: −y + q(x)y = λ2 y , y(0) = 0 ,

y (a) − iαλy(a) + βy(a) = 0 . with α > 0 and β ∈ R. III. Vibrations of a damped string. The problem of small transversal vibrations of a damped smooth inhomogeneous string with ﬁxed left endpoint whose right end carries a point mass able to move with damping in the direction orthogonal to the equilibrium position of the string can be reduced to the following spectral problem, cf. [P2], [MP1], [MP2]: −y − ipλy + q(x)y = λ2 y , y(0) = 0 , y (a) + (β − iαλ − mλ2 )y(a) = 0 . Thereby the coeﬃcient p > 0 is proportional to the damping along the length of the string, α > 0 is proportional to the coeﬃcient of damping of the point mass m > 0 at the right end, and β is a real parameter.

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IV. A fourth order problem which describes small transversal vibrations of an elastic beam, cf. [MoP]: y (4) − (g(x)y ) = λ2 y , y(0) = y (0) = 0 , y(a) = 0 ,

y (a) − iαλy (a) = 0 . Hereby g(x) is a continuously diﬀerentiable function describing the distributed stretching or compressing force. The left end of the beam is hinge connected and the right end is hinge connected with damping.

2. Some preliminaries on indefinite Hermite-Biehler functions If Ω ⊆ C is a domain and K(w, z) is a function deﬁned on Ω × Ω, which is analytic in the variables z and w and has the property that K(w, z) = K(z, w), then K is called an analytic symmetric kernel (shortly kernel ) on Ω. Let κ ∈ N ∪ {0}. We say that the kernel K has κ negative squares, if for each choice of n ∈ N and z1 , . . . , zn ∈ Ω the quadratic form QK (ξ1 , . . . , ξn ) :=

n

K(zj , zi )ξi ξj

i,j=1

has at most κ negative squares, and if for some choice of n, z1 , . . . , zn this upper bound is actually attained. Recall that every kernel K with a ﬁnite number κ of negative squares on a domain Ω generates a reproducing kernel Pontryagin space P(K) whose elements are analytic function on Ω, cf. [ADRS]. In fact, P(K) is obtained as the Pontryagin space completion of span{K(w, .) : w ∈ Ω} with respect to the inner product given by [K(w, .), K(w , .)] = K(w, w ). In the present note kernels of a particular form play a crucial role. If Θ is a meromorphic function on the open upper half-plane C+ and Ω denotes its domain of holomorphy, deﬁne SΘ (w, z) := i

1 − Θ(z)Θ(w) , z, w ∈ Ω . z−w

Clearly, SΘ is a kernel on Ω in the above sense. For a function F , we denote by F # the function F # (z) := F (z). We call F real, if F = F # . Let κ ∈ N ∪ {0}. If E is meromorphic on the whole plane C, we write ind− E = κ in order to express the fact that the kernel S E# | has κ E C+ negative squares. 2.1. Definition. Let κ ∈ N ∪ {0}. The set HB κ of Hermite-Biehler functions with κ negative squares is deﬁned to be the set of all entire functions E which satisfy

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ind− E = κ and are such that E and E # have no common nonreal zeros. Moreover, we deﬁne the set of indeﬁnite Hermite-Biehler functions as HB <∞ := HB κ . κ∈N∪{0}

2.2. Remark. The notion of Hermite-Biehler functions appeared frequently in the literature. However, the existing deﬁnitions do not completely coincide with each other. We would like to make this more explicit, and to state the relationship with the present Deﬁnition 2.1. (i) In the book [L1], one ﬁnds the following deﬁnitions: An entire function E belongs to HB, if it has no zeros in the closed lower half-plane C− ∪ R and if |E(z)| < |E # (z)|, Im z > 0 . Moreover, E is said to belong to HB, if E has no zeros in the open lower half-plane C− and |E(z)| ≤ |E # (z)|, Im z > 0. (ii) In [B] (originally in [L2]) a class P is deﬁned as the set of all entire functions E of exponential type which have no zeros in C− and satisfy |E(z)| ≤ |E # (z)|, Im z > 0. (iii) In the book [dB] the class of all entire functions is considered which satisfy |E # (z)| < |E(z)|, Im z > 0. (iv) In the series [KWW1]-[KWW3] as well as in [KW1], [KW2], classes of indeﬁnite Hermite-Biehler functions are deﬁned by requiring the conditions of # Deﬁnition 2.1 and, additionally, that EE is not constant. The relationships among these various, in their essence equivalent but in their details diﬀerent, notions can now be formulated as follows. – An entire function E belongs to P as in (ii) if and only if it belongs to HB, as in (i) and is of exponential type. – An entire function belongs to HB as in (i) if and only if it has no real zeros and the function E # (z) possesses the property stated in (iii). Thus (iii) in comparison to (i) allows real zeros and exchanges the roles of upper and lower half-plane. – We have E ∈ HB 0 as in Deﬁnition 2.1 if and only if E # ∈ HB. This follows from a classical result of G.Pick, cf. [Pi]. # – We have EE = const if and only if there exists a constant λ ∈ C with |λ| = 1 such that λE is real. Hence the deﬁnition mentioned in (iv) diﬀers from Deﬁnition 2.1 only by, in our context, somewhat trivial functions. a. Zeros, product representation and limits In order to study the distribution of zeros of entire functions it is practical to use the language of divisors, see e.g. [R]: Put D := d : C → Z : supp d has no accumulation point in C , where supp d denotes the set of all points w where d assumes a nonzero value. To a function F which is meromorphic in the whole plane, there is associated a

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divisor d(F ) which assigns to each point w the multiplicity of w as a zero of F . For example, for the function F (z) := z + 1z we have   +1 , w = ±i . d(F )(w) = −1 , w = 0   0 , otherwise Clearly, a meromorphic function F is entire if and only d(F ) ≥ 0. In the particular instance of Hermite-Biehler functions, the following subset of D plays a distinguished role: 1 DHB := d ∈ D, d ≥ 0 : #(supp d ∩ C+ ) < ∞, d(w) Im < ∞ . w w=0

2.3. Remark. (i) The following is an immediate corollary of [KL, Satz 6.4] and basic H ∞ theory, see e.g. [RR]: Let E ∈ HB <∞ . Then E # (z) = γe−iaz B(z) · B1 (z) , E(z) where B is the Blaschke product associated with the zeros of E in the open lower half-plane C− , B1 is the (ﬁnite) Blaschke product associated with the zeros of E in C+ , the real number a is the mean type

E# 1 E#

, := lim sup log a = mt E E y→+∞ y and γ = e−2i arg E(0) . Moreover, d(E) ∈ DHB . (ii) We have E ∈ HB <∞ if and only if E can be factorized as E = p · E0 with a function E0 ∈ HB 0 and a polynomial p whose zeros lie in C+ and are diﬀerent from the conjugates of the zeros of E0 . Thereby ind− E is equal to the number of zeros of p counted according to their multiplicities, i.e. ind− E = d(p)(w) = d(E)(w) . w∈C

w∈C+

(iii) If ind− E1 < ∞ and ind− E2 < ∞, then also ind− (E1 E2 ) < ∞. In fact, the inequality ind− (E1 E2 ) ≤ ind− E1 + ind− E2 (2.1) holds. Moreover, if E1 , E2 , E1 E2 ∈ HB <∞ , then in (2.1) actually equality prevails. (iv) Let E be an entire function, let D be meromorphic in the whole plane and # # assume that D = D# . Then (DE) = EE , and hence ind− DE = ind− E. If DE we assume moreover that supp d(D) ⊆ R and d(D)(w)+ d(E)(w) ≥ 0, w ∈ C, then E ∈ HB κ if and only if DE ∈ HB κ . A particular instance of this situation is frequently of good use: Let P be a real Weierstraß product to the real zeros of E and choose D = P −1 above.

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This shows that in the investigation of indeﬁnite Hermite-Biehler functions one can often restrict without loss of generality that no real zeros are present. From these facts we see that Krein’s factorization theorem, cf. [K] or [L1, Lehrsatz VII.3.6], immediately extends to the indeﬁnite case. Its indeﬁnite version can be formulated as follows. Krein’s Factorization Theorem: If E ∈ HB <∞ , then d(E) ∈ DHB and d(E)(w) > 0 ⇒ d(E)(w) = 0, w ∈ C \ R .

(2.2)

The function E admits a locally uniformly convergent product representation of the form E(z) = γD(z)e

−iaz

p(w) n z 1 z d(E)(w) Re n , exp d(E)(w) 1− w n w n=1

(2.3)

w∈R

where D is real, supp d(D) ⊆ R, |γ| = 1, a ≥ 0 and p : supp d(E) \ R → N ∪ {0} satisﬁes d(E)(w) < ∞. |w|p(w)+1 w∈R Conversely, every function of this form belongs to HB <∞ . 2.4. Remark. The above discussion shows that from the viewpoint of the asymptotic distribution of zeros, and hence also from the function theoretic viewpoint of growth etc., the classes of indeﬁnite and positive deﬁnite Hermite-Biehler functions behave the same. However, in questions concerning the exact location of zeros, in the indeﬁnite case essentially new questions and phenomena appear. The set HB 0 is closed with respect to locally uniform convergence. This follows e.g. from the fact that HB 0 = {E # : E ∈ HB}, cf. Remark 2.2, and that, clearly, HB is closed. However, note that none of the sets HB κ for κ > 0, κ≤N HB κ or HB <∞ is closed with respect to locally uniform convergence. In this context we would like to mention the following result: 2.5. Proposition. Let E be an entire function. (i) We have ind− E = κ if and only if there exists a real entire function D such that D−1 E ∈ HB κ . conver(ii) Let HB κ denote the closure of HB κ with respect to locally uniform gence. Assume that E does not vanish identically. Then E ∈ κ∈N∪{0} HB κ if and only if ind− E < ∞, #(supp d(E) ∩ C+ ) < ∞ . Proof. The necessity in (i) follows since for real functions D we have ind− DE = ind− E. To see the suﬃciency assume that ind− E = κ < ∞. Let S := {w ∈ C \ R :

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E(w) = E(w) = 0} and let D be a real entire function with min{d(E)(w), d(E)(w)} , w ∈ S . d(D)(w) = 0 , otherwise E . Then E1 is entire, ind− E1 = ind− E and E1 and E1# have no Put E1 := D common nonreal zeros. Thus E1 ∈ HB κ . We come to the proof of (ii). For suﬃciency assume that E is the locally uniform limit of a sequence (En )n∈N of functions En ∈ HB κ . Then, by the continuity of the kernel S E# , E

ind− E ≤ lim inf ind− En = κ , n→∞

and, by the Theorem of Logarithmic Residues, #(supp d(E) ∩ C+ ) ≤ lim inf #(supp d(En ) ∩ C+ ) = κ . n→∞

To establish necessity in (ii), assume that ind− E < ∞ and #(supp d(E) ∩ C+ ) < ∞. Let p be a polynomial with d(E)(w) , w ∈ C+ d(p)(w) = , 0 , otherwise and put E1 := Ep . Then ind− E1 ≤ ind− E + deg p < ∞. Since, E1 and E1# have no common nonreal zeros, by [KL, Satz 6.4] the number ind− E1 coincides with the number of zeros of E1 in C+ . We see that ind− E1 = 0, and hence E1 ∈ HB 0 . Let S := {w ∈ C− : E1 (w) = p(w) = 0}. Then, for a > 0, z − w + ia E1 (z) ∈ HB 0 . z−w w∈S

Since this function has for suﬃciently small values of a > 0 no common zeros with p# , we conclude that z − w + ia ∈ HB deg p . E a (z) := p(z)E1 (z) z−w w∈S

a

Clearly, lima0 E = E.

b. The de Branges space associated to E ∈ HB <∞ A function E ∈ HB <∞ generates a reproducing kernel Pontryagin space whose elements are entire functions. To this end consider the kernel i E(z)E(w) − E # (z)E # (w) , z, w ∈ C, z = w , 2π z−w (2.4) i ∂E ∂E # # (z)E (z) − E(z) (z) , z ∈ C . K(z, z) := 2π ∂z ∂z The reproducing kernel Pontryagin space generated by this kernel K will be called the de Branges space induced by E and denoted by P(E), cf. [KW1]. K(w, z) :=

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Let E ∈ HB <∞ . A maximal negative subspace of P(E) can be determined explicitly. We do not show this in full generality; we restrict ourselves to the case of simple zeros, since this is enough for what is needed later on. 2.6. Lemma. Let E ∈ HB κ and assume that all zeros of E which lie in the open upper half-plane are simple. Denote these zeros by w1 , . . . , wκ . Then L := span K(wk , .) : k = 1, . . . , κ is a maximal negative subspace of P(E). Proof. Since each wk is a zero of E, we have i E # (z)E(wk ) , k = 1, . . . , κ . 2π z − wk κ It follows that for x, y ∈ L, x = k=1 λk K(wk , .), y = κj=1 µj K(wj , .), K(wk , z) = −

[x, y] =

κ

K(wk , wj )λk µj =

k,j=1

=−

κ i 1 E(wk )λk E(wj )µj . 2π wj − wk k,j=1

Since E(wk ) = 0, k = 1, . . . , κ, and the matrix κ i P := wj − wk j,k=1 is positive deﬁnite (e.g. as the Pick-matrix of the constant function f (z) := i, z ∈ C+ ), it follows that for (λ1 , . . . , λκ ) = 0, ∗    E(w1 )λ1 E(w1 )λ1 κ i 1     .. .. [x, x] = −    < 0.  . . 2π wj − wk j,k=1 E(wκ )λκ E(wκ )λκ We conclude that {K(w1 , .), . . . , K(wκ , .)} is linearly independent and that L is a negative subspace. Since ind− E = κ, it is a maximal negative subspace. c. Symmetric and semibounded Hermite-Biehler functions We are interested in the study of indeﬁnite Hermite-Biehler functions which satisfy a certain functional equation: 2.7. Definition. Let κ ∈ N∪{0}. We deﬁne the set of all symmetric Hermite-Biehler functions with κ negative squares as := E ∈ HB κ : E # (z) = E(−z) . HB sym κ Again let us put HB sym <∞ :=

κ∈N∪{0}

HB sym = E ∈ HB <∞ : E # (z) = E(−z) . κ

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A product representation of symmetric indeﬁnite Hermite-Biehler functions can be deduced from Krein’s factorization theorem. sym Product Representation for HB sym <∞ : If E ∈ HB <∞ , then d(E) ∈ DHB , satisﬁes (2.2) and d(E)(−w) = d(E)(w), w ∈ C .

The function E admits a locally uniformly convergent product representation of the form z d(E)(w) 1− E(z) = γD(z)e−iaz · w w∈iR\{0}

p(w)

[ 2 ] z 2n z 2 d(E)(w) 1 Im w Re 2n , 1 + 2iz 2 − exp 2d(E)(w) · 2 |w | |w| 2n w n=1 Re w>0 w∈R

where D is real and even, i.e. D(z) = D# (z) = D(−z), supp d(D) ⊆ R, γ = ±1, a ≥ 0 and p : {w ∈ supp d(E) : Re w > 0, w ∈ R} → N ∪ {0} satisﬁes

d(E)(w) < ∞. |w|p(w)+1 Re w>0 w∈R

Conversely, every function of this form belongs to HB sym <∞ . Let us recall from [KWW3] that the de Branges space generated by a function E ∈ HB sym <∞ has an important symmetry property. In fact, the map M : F (z) → F (−z) induces an isometric involution of P(E) onto itself. Thus the space P(E) can be decomposed as the direct and orthogonal sum ˙ P(E) = P(E)e [+]P(E) o, where

P(E)e = ker(I − M ) = F ∈ P(E) : F even , P(E)o = ker(I + M ) = F ∈ P(E) : F odd .

The orthogonal projections Pe and Po of P(E) onto P(E)e or P(E)o , respectively, are given by I +M I −M , Po = . Pe = 2 2 For later use, let us recall that indeﬁnite Hermite-Biehler functions are related to generalized Nevanlinna functions. Denote by Nκ the set of all functions q which are meromorphic in C \ R, satisfy q # = q, and are such that the Nevanlinna-kernel Qq (w, z) :=

q(z) − q(w) , z, w ∈ C+ z−w

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has κ negative squares. Moreover, let us put Nκsym := q ∈ Nκ : q(−z) = −q(z) . If E is entire, write E = A − iB with A, B real, i.e. E − E# E + E# , B := i . (2.5) 2 2 Let us make the notational convention that, whenever E is an entire function, we denote by A and B the corresponding functions (2.5). A :=

2.8. Remark. Let E be an entire function. Then the following hold: (i) The common zeros of E and E # correspond to the common zeros of A and B. Since, for real points t, E(t) = 0 if and only if E # (t) = 0, we obtain in particular that real zeros of E correspond to real common zeros of A and B. and Nκsym , respectively) are closely related: (ii) The classes HB κ and Nκ (HB sym κ If E and E # do not have common nonreal zeros, then ind− E = κ if and only # if ind− B have no common A = κ. Hence, E ∈ HB κ if and only if E and E B sym # noreal zeros and A ∈ Nκ . Moreover, we have E (z) = E(−z), z ∈ C, if sym and only if A is even and B is odd. Thus, if E ∈ HB sym , we have B . κ A ∈ Nκ The class of symmetric Hermite-Biehler functions is related to another subclass of indeﬁnite Hermite-Biehler functions as we shall now explain, cf. [KWW3]. 2.9. Definition. Let κ ∈ N ∪ {0} and let E be an entire function. Then E is called a semibounded Hermite-Biehler functions with κ negative squares, if E ∈ HB κ and the meromorphic function B A has only ﬁnitely many poles in C \ [0, ∞). The set of all semibounded Hermite-Biehler functions with κ negative squares will be denoted by HB sb κ , and again we put HB sb HB sb <∞ := κ . κ∈N∪{0}

Let us mention that, in terms of Nevanlinna functions, the corresponding notion is the following, cf. [KWW1]: We write q ∈ Nκep , if q ∈ Nκ and is analytic on C \ [0, ∞) with possible exception of ﬁnitely many poles. Clearly, E ∈ HB sb κ ep means that B A ∈ Nκ . sb The relation between HB sym <∞ and HB <∞ can now be stated as follows. Consider the transformation T which is deﬁned by (TE)(z) := A(z 2 ) − izB(z 2 ) , and deﬁne the sets Bκ , B<∞ , as Bκ := {E ∈ HB sb κ : d(E)|(−∞,0) = 0}, κ ∈ N ∪ {0} , B<∞ := Bκ = {E ∈ HB sb <∞ : d(E)|(−∞,0) = 0} . κ∈N∪{0}

2.10. Proposition. The transformation T induces a bijection of B<∞ onto HB sym <∞ . Thereby ind− TE ≥ ind− E.

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Proof. Assume that E = A − iB ∈ HB sb <∞ , so that q := by [KWW1, Theorem 4.1], the function zq(z 2 ) =

IEOT B A

ep belongs to N<∞ . Then,

zB(z 2 ) A(z 2 )

sym belongs to N<∞ . Assume now moreover that E(t) = 0 for t ∈ (−∞, 0). Then A and B have no common zeros in C\[0, ∞), and hence the functions A1 (z) := A(z 2 ) and B1 (z) := zB(z 2 ) have no common nonreal zeros. Clearly, A1 is even and B1 is odd. We conclude that TE ∈ HB sym <∞ . The map T is clearly injective. To show that T(B<∞ ) = HB sym <∞ , let E = be given. We deﬁne entire functions A− and B− by A − iB ∈ HB sym κ

B(z) . (2.6) z Then A− and B− have no common zeros in C \ [0, ∞) since A and B have no common nonreal zeros. Put E− := A− − iB− , then, clearly, TE− = E. Again appealing to [KWW1, Theorem 4.1] we obtain E− ∈ HB sb ν for some ν ≤ κ. A− (z 2 ) := A(z), B− (z 2 ) :=

2.11. Remark. The transformation T is compatible with multiplication with real functions: Let E be entire and D meromorphic and real, then T(DE)(z) = D(z 2 )(TE)(z) .

(2.7)

This shows that the requirement in Proposition 2.10 that E has no real negative zeros is no essential restriction of generality, since we can always split oﬀ a real Weierstraß product to the real negative zeros of E. #

E 2.12. Remark. A function E ∈ HB sym <∞ satisﬁes E = const if and only if E either satisﬁes E # (z) = E(z) = E(−z) or E # (z) = −E(z) = E(−z), i.e. E# E# = const = E ∈ HB sym ∈ {+1, −1} . {E ∈ HB sym <∞ : <∞ : E E As a short computation shows, we have E# E# ∈ {+1, −1} = E ∈ HB sym ∈ {+1, −1} , T {E ∈ B<∞ : <∞ : E E whereas c−1 TE− (z) = A− (z 2 ) 1 + z c+1

when E− ∈ B<∞ ,

# E− E−

= c, c = ±1.

3. Zeros of symmetric Hermite-Biehler functions In the main result of this paper we give a characterization of the subset T(B0 ) of HB sym <∞ in terms of the location of their zeros. 3.1. Theorem. Let E ∈ HB sym <∞ . Then E ∈ T(B0 ) if and only if its zeros are distributed according to the following two rules:

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(i) All zeros of E in C+ are simple and are located on the imaginary axis. (ii) Let supp d(E) ∩ C+ = {iy1 , . . . , iyκ } with 0 < y1 < . . . < yκ , where κ := ind− E. Then d(E)(w) is odd, k = 2, . . . , κ , w∈[−iyk−1 ,−iyk ]

d(E)(w) is

w∈(0,−iy1 ]

even , d(E)(0) is even odd , d(E)(0) is odd

The proof of this result will be carried out in several steps. The core of the result is to establish that, if E has the described distribution of zeros, it belongs to T(B0 ). Step 1: Removing real zeros. Let us show that without loss of generality we can make the assumption that E has no real zeros with possible exception of a simple zero at the origin. To see this note that, by the symmetry of the zeros of E, there exists an even and real entire function C which satisﬁes   , w ∈ R \ {0} d(E)(w) d(E)(0) . d(C)(w) = 2 · , w=0 2   0 , w ∈ C\R Since C # (z) = C(z) = C(−z), this function assumes real values on R ∪ iR. Deﬁne # an entire function C− by C− (z 2 ) := C(z). Then C− (R) ⊆ R and thus C− = C− . Moreover,  √  , w>0 d(E)( w) d(E)(0) d(C− )(w) = . , w=0 2   0 , w ∈ C \ [0, ∞) The conditions (i) and (ii) on the distribution of zeros clearly hold for E if and ˜ := C −1 E. Also, E belongs to T(B0 ) if and only only if they hold for the function E ˜ ˜− for some E ˜− ∈ B0 . Then C− E˜− if E does: To see this assume ﬁrst that E˜ = TE ˜ ˜ also belongs to B0 , and, by (2.7), T(C− E− ) = C E = E. Conversely, assume that E = TE− for some E− ∈ B0 . Then √   t), t > 0   d(E)( d(E)(0) d(E− )(t) = = d(C− )(t), t ∈ R , ,t=0 2   0, t < 0 −1 −1 E− ∈ B0 . Moreover, again by (2.7), we have T(C− E− ) = and it follows that C− −1 ˜ C E = E. Hence, in all future steps of the proof of Theorem 3.1, we can assume that E has no real zeros with possible exception of a simple zero at the origin.

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Step 2: Proof of necessity, absence of zeros in C+ \ iR. Assume that E = T(E− ) for some E− ∈ B0 . Note that we can write (E− = A− − iB− ) E(z) = A− (z 2 ) − izB− (z 2 ) =

# 2 # 2 E− (z 2 ) − E− (z ) (z ) E− (z 2 ) + E− − iz · i = 2 2 #

E (z 2 ) E− (z 2 ) (1 + z) + − (1 − z) . 2 2 Hence, if z ∈ C is such that Re z > 0 and Im z > 0, so that z 2 ∈ C+ and thus E− (z 2 ) = 0, we have E(z) = 0 if and only if =

# 2 (z ) E− z+1 = . 2 E− (z ) z−1 E#

However, the function E− maps the open upper half-plane into the open unit disk, − whereas the fractional linear transformation z+1 φ(z) := z−1 maps this region as indicated in the below picture:

φ(z) =

z+1 z−1

Thus E cannot have zeros with Re z > 0, Im z > 0, and, by symmetry, therefore also no zeros with Re z < 0, Im z > 0. Step 3: Proof of necessity, zeros on iR. Assume again that E ∈ T(B0 ), that is E(z) = A− (z 2 ) − izB− (z 2 ) with E− := A− − iB− ∈ B0 . We use a geometric idea similar, but more reﬁned, as in the previous step. We show that the functions A− (z) and B− (z) do not have √ common zeros in C: If t ∈ R \ {0} were a common zero of A− and B− , then t ∈ (R ∪ iR) \ {0} would be a common zero of A(z) := A− (z 2 ) and B(z) := zB− (z 2 ), a contradiction. If A− (0) = B− (0) = 0, the function E would have a zero at the origin of order at least 2, again a contradiction. Finally, since E− ∈ HB <∞ , A− and B− cannot have common nonreal zeros.

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The fact that A− and B− have no common zeros shows that the zeros of E, with exception of a possible zero at the origin, coincide with the zeros of the function B− (z 2 ) i + . A− (z 2 ) z

Q(z) :=

We shall deduce the desired distribution of zeros on the imaginary axis from the following statement. 3.2. Lemma. Let q ∈ N0ep be meromorphic in the whole plane and denote by ξn < ξn−1 < . . . < ξ1 < 0 its poles on the negative real axis. Put ξ0 := 0 and ηj :=

|ξj |, j = 0, . . . , n .

Consider the function qˆ(t) := q(−t2 ) + Then we have

1 . t

d(ˆ q )(w) = 1, j = 1, . . . , n ,

(3.1)

w∈(ηj−1 ,ηj )

and

d(ˆ q )(w) =

w∈(ηn ,∞)

0 , limt→−∞ q(t) ≥ 0 1 , limt→−∞ q(t) < 0

(3.2)

Denote by 0 < y1 < . . . < ym the zeros of of qˆ on the positive real axis. Then qˆ(−yj ) = 0, j = 1, . . . , m, and the formulas

d(ˆ q )(w) is odd, yj+1 < ηn ,

(3.3)

w∈[−yj+1 ,−yj ]

and

d(ˆ q )(w) is

w∈[−y1 ,0)

even , d(q)(0) ≥ 0 odd , d(q)(0) < 0

(3.4)

hold true. Proof. The proof of this lemma follows by elementary calculus inspecting the graphs of q(−t2 ) and − 1t :

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−y4

−y2

−y3 −η3

−η2

−y1

y2

y1

−η1

IEOT

η1

0

y3 η2

y4 η3

q(−t2 )

− 1t First of all note that the set of poles of qˆ on R equals {±η1 , . . .±ηn }∪{0}. Thereby d(ˆ q )(±ηj ) = −1, j = 1, . . . , n , d(ˆ q )(0) =

−1 , d(q)(0) ≥ 0 −2 , d(q)(0) < 0

Since q|R is nondecreasing between each pair of consecutive poles, the function q(−t2 ) is nonincreasing on each of the intervals (0, η1 ), (η1 , η2 ), . . . , (ηn−1 , ηn ), (ηn , ∞) . Since 1t is strictly decreasing on R+ , the function qˆ will be strictly decreasing on each of the above intervals. Since it tends to ±∞ at the poles 0, η1 , . . . , ηn , the formula (3.1) follows. Moreover, there is at most one zero on (ηn , ∞). Clearly, such a zero exists if and only if limt→∞ qˆ(t) < 0, which is nothing else than limu→−∞ q(u) < 0. This proves (3.2). For t ∈ [yi , ηi ), i = 1, . . . , n, we have qˆ(t) ≤ 0. Since qˆ(−t) = q(−t2 ) −

2 1 = qˆ(t) − , t t

it follows that qˆ(u) < 0, u ∈ (−ηi , −yi ], i = 1, . . . , n .

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Since limu −ηj qˆ(u) = +∞, j = 1, . . . , n, it follows that (taking into account multiplicities) there is an odd number of zeros on the interval [−yi , −ηi−1 ), i = 2, . . . , n. This shows (3.3). To establish (3.4) it is enough to note that −∞ , d(q)(0) ≥ 0 lim qˆ(u) = u 0 +∞ , d(q)(0) < 0 An application of this lemma with the function q := information on the zeros of E of the form z = it, t ∈ R.

B− A−

now gives the desired

Step 4: Proof of suﬃciency (E(0) = 0), geometric reformulation. Let E ∈ HB sym <∞ and assume that E has no real zeros. Since E(0) ∈ R \ {0} and multiplication with a real constant changes neither of the two conditions in the present theorem, we can assume without loss of generality that E(0) = 1. Let E− ∈ B<∞ be such that TE− = E, i.e. let E− = A− − iB− with A− , B− as in (2.6). Then, by [KWW3, Proposition 4.9], we have P(E)o ∼ = P(E− ). Thus we have to prove that ind− P(E)o = 0. Step 5: Proof of suﬃciency (E(0) = 0), employing the hypothesis. Let E ∈ HB sym <∞ , E(0) = 1, have a zero distribution as described in (i) and (ii). If E(w) = 0, we see from (2.4) that i ∂E # K(w, w) = − E(w) (w) . 2π ∂z # ∂E Since E # (z) = E(−z), we have ∂E ∂z (z) = − ∂z (−z), and hence K(w, w) =

i ∂E E(w) (−w) . 2π ∂z

If w = iy, this formula reads as K(iy, −iy) =

∂E i E(−iy) (iy) . 2π ∂z

Consider the real-analytic function e:

R → R . x → E(ix)

1 e(−y)e (y). Then we can further rewrite K(iy, −iy) = 2π Let 0 < y1 < . . . < yκ be such that wk := iyk , k = 1, . . . , κ, are the zeros of E in the upper half-plane. Since these zeros are all simple and since e(0) = 1 > 0, we must have sgn e (yk ) = (−1)k , k = 1, . . . , κ. Since the (total) number of zeros of e on the interval (−y1 , 0) is even and on each of the intervals (−yk , −yk−1 ), k = 2, . . . , κ, odd, we must have sgn e(−yk ) = (−1)k+1 , k = 1, . . . , κ. Alltogether it follows that K(iyk , −iyk ) < 0, k = 1, . . . , κ .

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IEOT

Step 6: Proof of suﬃciency (E(0) = 0), conclusion. Let w√ 1 , . . . , wκ be the zeros of E in the upper half-plane. Consider the map Φ := 2Pe |L where L := span{K(wk , .) : k = 1, . . . , κ}. Since K(w, −z) = K(−w, z), we have 1 ΦK(wk , .) = √ K(wk , .) + K(−wk , .) , 2 and hence can compute 1 ΦK(wk , .), ΦK(wj , .) = K(wk , .) + K(−wk , .), K(wj , .) + K(−wj , .) = 2 = K(wk , wj ) + K(wk , −wj ) = [K(wk , .), K(wj , .)] + K(wk , −wj ) . Note that, since wj ∈ iR, we have −wj = wj . For j = k, we obtain K(wk , −wj ) = K(wk , wj ) =

i E(wj )E(wk ) − E # (wj )E(wk ) = 0. 2π wj − wk

For j = k, we have proved the previous step that K(wk , −wk ) < 0. It follows in κ that for x ∈ L \ {0}, x = k=1 λk K(wk , .), we have [Φx, Φx] = [x, x] +

κ

|λk |2 K(wk , −wk ) < [x, x] ,

k=1

i.e. that Φ|L is a contraction in the Pontryagin space sense. Since L is a maximal negative subspace, Φ is injective and Φ(L) is again a maximal negative subspace of P(E). We conclude that P(E)e contains a maximal negative subspace, and thus that its orthogonal complement P(E)o is positive deﬁnite. Step 7: Proof of suﬃciency (E(0) = 0). Let E = A − iB ∈ HB sym <∞ have the prescribed distribution of zeros and assume that E(0) = 0. Since, cf. Step 1, E has a simple zero at the origin and since A is even and B is odd, we must have B (0) = 0. Thus we can assume without loss of generality that B (0) = 1. Again deﬁne A− and B− by (2.6), so that A− − iB− ∈ B<∞ and E(z) = (TE− )(z) = A− (z 2 ) − izB− (z 2 ) . Note that A− (0) = 0 and B− (0) = 1. Our aim is to show that ind− E− = 0. Consider the function E+ := iE− = B− + iA− , then E+ ∈ B<∞ and ind− E+ = ind− E− . By [KWW3, Theorem 4.5] the function A− (z 2 ) ˜ E(z) := B− (z 2 ) + i z ∼ ˜ and we have P( E) ). P(E belongs to HB sym = e + <∞ ˜ ˜ has the same distribution of zeros as E Since E(z) = zi E(z), the function E ˜ with the exception that E(0) = 1. The same reasoning as in the above Steps 5 and ˜ e = 0, since this time the number of zeros between 0 6 will show that ind− P(E)

Vol. 57 (2007) Shifted Hermite-Biehler Functions and Their Applications

and −iy1 is odd, hence K(iy, −iy) will be positive and hence the map ˜ o . It follows that be a contraction of L into P(E) ˜ e = 0. ind− E− = ind− E+ = ind− P(E+ ) = ind− P(E) All assertions of Theorem 3.1 are proved.

119

√ 2Po will

✌

4. Application to some boundary value problems In this ﬁnal section we apply Theorem 3.1 in order to describe the location of the pure imaginary eigenvalues of the boundary problems I–IV listed in the introduction. For the problems II–IV their location has been found separately in each case, see [PM] for II, [MP1] for III ([MP2] for a detailed version), and [MoP] for the problem IV. Let us note that in these papers the roles of the upper and lower half-planes were exchanged in comparison to the present paper, cf. the remark we made in the third paragraph of the introduction. I. The Regge problem. This is the problem (4.1) −y + q(x)y = λ2 y , y(0) = 0 , (4.2) y (a) − iλy(a) = 0 , (4.3) where λ is the spectral parameter and the potential q is real-valued and belongs to L2 (0, a). Denote s(ζ, x) the solution of equation −y + q(x)y = ζy,

(4.4)

which satisﬁes the conditions s(ζ, 0) = s (ζ, 0) − 1 = 0. Let us show following [KK] that the function ss(ζ,a) (ζ,a) belongs to the Nevanlinna class and has only a ﬁnite ep number of poles on (−∞, 0), i.e. ss(ζ,a) (ζ,a) ∈ N0 . Let us multiply (4.4) by y and subtract the equation complex-conjugate to (4.4) multiplied by y. Then we obtain

y y − y y = 2i Im ζ|y|2 , or

(y y − y y) = 2i Im ζ|y|2 , Integrating this equation we obtain ! a a y y − y y|0 = 2i Im ζ |y|2 dx, 0

If we take y = s(ζ, x) then we obtain s(ζ, a) Im |s (ζ, a)|2 = 2 Im ζ s (ζ, a)

! 0

a

|s(ζ, x)|2 dx, Im ζ > 0.

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This means that the function ss(ζ,a) (ζ,a) belongs to the Nevanlinna class. It is well known (see e.g. [A]) that the set of zeros of s (ζ, a) is bounded below. That means s(ζ,a) s (ζ,a) is meromorphic and has ﬁnitely many poles in C\[0, ∞). It is also known that s (ζ, a) and s(ζ, a) cannot be zero simultanously. Therefore, the function ϕ(ζ) = s (ζ, a) − is(ζ, a) belongs to B0 . On the other hand, the set of zeros of the function ϕ(λ) = s (λ2 , a) − iλs(λ2 , a) coincides with the spectrum of problem (4.1)–(4.3) and the function ϕ(λ) ˜ belongs to T(B0 ). Therefore, we can apply Theorem 3.1 and obtain that (1) all (if any) eigenvalues of problem (4.1)–(4.3) in the closed upper half-plane are pure imaginary and simple (we denote them {iγj }, j = 1, 2, ..., κ in increasing order: γj < γj+1 ); (2) all the points −iγj , j = 1, 2, ..., κ do not belong to the spectrum of the problem except for iγ1 if γ1 = 0; (3) each interval (−iγj+1 , −iγj ), j = 1, 2, ..., κ − 1 contains an odd number (with account of multiplicities) of the eigenvalues; (4) if γ1 = 0, then the interval (−iγ1 , 0) contains an even number (with account of multiplicities) of the eigenvalues. II. The generalized Regge problem. This is the problem −y + q(x)y = λ2 y , y(0) = 0 ,

y (a) − iαλy(a) + βy(a) = 0 , with α > 0 and β ∈ R. Clearly, multiplying s(ζ, a) by a positive constant α does not change anything αs(ζ,a) (the function α ss(ζ,a) (ζ,a) is also Nevanlinna). To prove that s (ζ,a)+βs(ζ,a) belongs also to the Nevanlinna class, let us consider its imaginary part "# %

−2 $ (ζ, a)

s (ζ, a)

s αs(ζ, a) Im = α

+ β)

Im +β . s (ζ, a) + βs(ζ, a) s(ζ, a) s(ζ, a) (ζ,a) > 0 and consequently If Im ζ > 0, then Im ss(ζ,a) Im

αs(ζ, a) > 0, Im ζ > 0. + βs(ζ, a)

s (ζ, a)

αs(ζ,a) That means that s (ζ,a)+βs(ζ,a) is a Nevanlinna function. The set of zeros of its denominator is bounded below, see [A]. In the same way as for the Regge problem

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121

we obtain that the above statements (1)-(4) are true, i.e. we obtain an alternative proof of [PM, Lemma 2.2, 3)] and [PM, Theorem 3.1, 1)-3)]. III. Vibrations of a damped string. We consider the spectral problem −y − ipλy + q(x)y = λ2 y ,

(4.5)

y(0) = 0 ,

(4.6)

y (a) + (β − iαλ − mλ2 )y(a) = 0 ,

(4.7)

where p > 0, α > 0, m > 0 and β is a real parameter. From the physical motivation of the problem it follows that q(x) and β are such that the spectrum of problem (4.5)–(4.7) lies in the open lower half-plane, see [P2]. Let us change the spectral parameter: z = λ + ip 2 . Then we obtain $ # 2 p − q(x) y = 0, (4.8) y + z 2 y + 4 y(0) = 0, $ # αp mp2 − + β y(a) = 0. y (a) + −mz 2 − i(α − mp)z + 4 2 If we denote by s(ζ, x) the solution of the equation $ # 2 p y + ζy + − q(x) y = 0, 4 which satisﬁes the conditions s(ζ, 0) = s (ζ, 0) − 1 = 0 then we obtain that belongs to N0ep .

(4.9) (4.10)

s(ζ,a) s (ζ,a)

We distinguish three cases: Case 1: mp−α = 0. In this case problem (4.8)–(4.10) is selfadjoint and its spectrum lies on the real axis and on a ﬁnite subinterval of the imaginary axis. The spectrum is symmetric with respect to the real axis and to the imaginary axis. Respectively, the spectrum of the problem (4.5)–(4.7) lies on the axis Im λ = − p2 and on a ﬁnite subinterval of the imaginary axis. The spectrum is symmetric with respect to the imaginary axis and to the axis Im λ = − p2 . is a Nevanlinna function. Let us show Case 2: α−mp > 0. Then again (α−mp)s(ζ,a) s (ζ,a) (α−mp)s(ζ,a) that s (ζ,a)+(β where β1 = β + 1 −mζ)s(ζ,a) First of all we notice that

Im

mp2 4

−

αp 2

is also a Nevanlinna function.

# $−1 s (ζ, a) (α − mp)s(ζ, a) = (α − mp) Im + (β − mζ) = 1 s (ζ, a) + (β1 − mζ)s(ζ, a) s(ζ, a) "# %

−2 $

s (ζ, a)

s (ζ, a)

= + (β1 − mζ) Im + β1 − mζ . s(ζ, a) s(ζ, a)

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If Im ζ > 0, then Im Im

s (ζ,a) s(ζ,a)

IEOT

> 0 and consequently

(α − mp)s(ζ, a) > 0, Im ζ > 0. s (ζ, a) + (β1 − mζ)s(ζ, a)

Let us now consider the denominator s (ζ, a) + (β1 − mζ)s(ζ, a). The set of its zeros coincides with the spectrum of the boundary value problem −y + q(x)y = ζy, y(0) = y (a) + (β1 − mζ)y = 0. The eigenvalues of this problem are real and bounded below, see e.g. [P1]. This means that the function s (z 2 , a) + (β1 − mz 2 )s(z 2 , a) − i(α − mp)zs(z 2, a) belongs to T(B0 ). Taking into account that λ = z − ip 2 and using Theorem 3.1 we obtain that (1) all (if any) eigenvalues of problem (4.5)–(4.7) in the closed half-plane Im λ ≥ − p2 (they all lie in the strip − p2 ≤ Im λ < 0) are pure imaginary and simple (we denote them {iγj }, j = 1, 2, ..., κ in increasing order: γj < γj+1 ); (2) all the points −ip + i|γj |, j = 1, 2, ..., κ do not belong to the spectrum of the problem except of iγ1 if γ1 = − ip 2; (3) each interval (−ip + i|γj+1 |, −ip + i|γj |), j = 1, 2, ..., κ − 1 contains an odd number (with account of multiplicities) of the eigenvalues; ip , then the interval − , −ip + i|γ1 | contains an even number (4) if γ1 = − ip 2 2 (with account of multiplicities) of the eigenvalues. Case 3: mp − α > 0. Then again (mp−α)s(ζ,a) s (ζ,a)+(β1 −mζ)s(ζ,a)

(mp−α)s(ζ,a) is s (ζ,a) mp2 αp 4 − 2 is also

a Nevanlinna function and

where β1 = β + a Nevanlinna function. It means that, if we set ξ = −z, then the function s (ξ 2 , a) + (β1 − mξ 2 )s(ξ 2 , a) − i(mp − α)ξs(ξ 2 , a)

belongs to T(B0 ). Taking into account that λ = −ξ − ip 2 and using Theorem 3.1 we obtain that (1) all (if any) eigenvalues of problem (4.5)–(4.7) in the closed half-plane Im λ ≤ − p2 are pure imaginary and simple (we denote them {iγj }, j = 1, 2, ..., κ in increasing order: |γj | < |γj+1 |); (2) all the points −ip + i|γj |, j = 1, 2, ..., κ do not belong to the spectrum of the problem except of iγ1 if γ1 = − ip 2; (3) each interval (−ip + i|γj |, −ip + i|γj+1 |), j = 1, 2, ..., κ − 1 contains an odd number (with account of multiplicities) of the eigenvalues; ip , then the interval − , −ip + i|γ1 | contains an even number (4) if γ1 = − ip 2 2 (with account of multiplicities) of the eigenvalues. Thus we have deduced [MP1, Theorem 4, 2] and [MP1, Theorem 5, 2’].

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IV. A fourth order problem. We consider the problem y (4) − (g(x)y ) = λ2 y,

(4.11)

y(0) = y (0) = 0,

(4.12)

y(a) = 0,

(4.13)

y (a) − iαλy (a) = 0

(4.14)

where g(x) is a continuously diﬀerentiable function. Denote by sj (λ2 , x) (j = 0, 1, 2, 3) the solution of the equation (4.11) which satisﬁes the conditions (k)

sj (λ2 , 0) = δk,j , j = 0, 1, 2, 3, where δkj is the Kronecker delta. Let us introduce the function ϕ(λ2 , x) = s3 (λ2 , a)s1 (λ2 , x) − s1 (λ2 , a)s3 (λ2 , x) This function automatically satisﬁes the conditions (4.12), (4.13). It satisﬁes (4.14) if ϕ (λ2 a) − iαλϕ (λ2 a) = 0. Let us prove that the function the equation

ϕ (ζ,a) ϕ (ζ,a)

is a Nevanlinna function. To do it we multiply

y (4) − (g(x)y ) = ζy,

(4.15)

by y and subtract the equation complex-conjugate to (4.15) multiplied by y. Then we obtain y (4) y − y(4) y − ((g(x)y ) y − (g(x)y ) y) = 2i Im ζ|y|2 or (y (3) y − y (3) y) − (y (2) y − y(2) y ) − (g(x)y y − g(x)y y) = 2i Im ζ|y|2 . Integrating this equation we obtain

a

a a

(y (3) y − y (3) y) − (y (2) y − y(2) y ) − (g(x)y y − g(x)y y)|0 = 0 0 ! a |y|2 dx . = 2i Im ζ

(4.16)

0

Let us apply this result to ϕ(ζ, x). This solution satisﬁes the conditions (4.12), (4.13), and therefore we obtain from (4.16): ! a −(ϕ(2) (ζ, a)ϕ (ζ, a) − ϕ(2) (ζ, a)ϕ (ζ, a)) = 2i Im ζ |ϕ(ζ, x)|2 dx 0

or

ϕ (ζ, a) (2) Im (2) |ϕ (ζ, a)|2 = 2 Im ζ ϕ (ζ, a)

! 0

a

|ϕ(ζ, x)|2 dx, Im ζ > 0.

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That means that after reducing the ratio ϕϕ(ζ,a) (ζ,a) is a Nevanlinna function. Moreover, it belongs to N0ep because the set of zeros of the denominator ϕ (ζ, a), being nothing else but the spectrum of the problem y (4) − (g(x)y ) = ζy, y(0) = y (0) = y(a) = y (a) = 0 , is real and bounded below. This in turn means that ϕ(2) (λ2 ) − iλϕ (λ2 ) = c(λ2 )ψ(λ) where c(ζ) is a real function bounded below and ψ(λ) ∈ T(B0 ). Therefore, we can apply Theorem 3.1 and obtain that the spectrum of problem (4.11)–(4.14) consists of two subsequences which can intersect. One of them consists of real and pure imaginary eigenvalue which are located symmetrically with respect to the real and to the imaginary axis. The eigenvalues of the second subsequence lie in the open lower half-plane and on the positive imaginary half-axis. The pure imaginary eigenvalues of the second subsequence satisfy the conditions (1)-(4) of I. Thus we have obtained [MoP, Theorem 6.5, 1-6].

References [ADRS] D.Alpay, A.Dijksma, J.Rovnyak, H.de Snoo: Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl. 96, Birkh¨ auser Verlag, Basel 1997. [A] F.V.Atkinson: Discrete and Continuous Boundary Problems, Academic Press, New-York-London, 1964. [B] R.Boas: Entire functions, Academic Press, New York 1954. [dB] L. de Branges: Hilbert spaces of entire functions, Prentice-Hall, London 1968. [GP] G.Gubreev, V.Pivovarchik: Spectral analysis of T. Regge problem with parameters, Funktsional. Anal. i Prilozhen. 31(1) (1997), 70-74. [H] C.Hermite: Oevres I, Paris, 1905. [Hr] S.Hrushev: The Regge problem for a string, unconditionally convergent eigenfunction expansions, and unconditional bases of exponentials in L2 (−T, T ),, J. Operator Theory 14 (1985), 67-85. [IP] S.A.Ivanov, B.S.Pavlov: Carleson resonance series in the Regge problem, Izv. Akad. Nauk SSSR Ser. Mat. 42(1) (1978), 26-55. [KK] I.S.Kac, M.G.Krein: On spectral functions of a string, in F.V.Atkinson, Discrete and Continuous Boundary Problems (Russian translation), Moscow, Mir, 1968, 648-737 (Addition II). I.C.Kac, M.G.Krein, On the Spectral Function of the String. Amer. Math. Soc., Translations, Ser.2, 103 (1974), 19-102. ¨ ck, H.Winkler, H.Woracek: Generalized Nevanlinna func[KWW1] M.Kaltenba tions with essentially positive spectrum, J. Oper. Theory 55(1) (2006), 17-48. ¨ ck, H.Winkler, H.Woracek: Singularities of generalized [KWW2] M.Kaltenba strings, Operator Theory Adv. Appl. 163 (2006), 191-248.

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¨ ck, H.Winkler, H.Woracek: De Branges spaces of entire func[KWW3] M.Kaltenba tions symmetric about the origin, Integral Equations and Operator Theory, to appear, DOI 10.1007/s00020-006-1443-3. [KW1]

¨ ck, H.Woracek: Pontryagin spaces of entire functions I, InteM.Kaltenba gral Equations and Operator Theory 33 (1999), 34-97.

[KW2]

¨ ck, H.Woracek: Hermite-Biehler Functions with zeros close to M.Kaltenba the imaginary axis, Proc. Amer. Math. Soc.133 (2005), 245-255.

[KaKo]

P.Kargaev, E.Korotyaev: Inverse resonance scattering and conformal mappings for Schr¨ odinger operators, unpublished manuscript.

[Ko]

V.M.Kogan: On double completeness of the set of eigen- and associated functions of the Regge problem, Functional. Anal. i Prilozhen. 5(3) (1971), 70-74 (in Russian).

[Kr]

A.O.Kravitskii: On Double Expansion into Series of Eigenfunctions of a Nonself-Adjoint Boundary Problem, Diﬀerentsialnye Uravneniya 4(N1) (1968), 165-177 (in Russian).

[K]

M.G.Krein: On a class of entire and meromorphic functions (Russian), published in: N.I.Achieser, M.G.Krein: Some problems in the theorie of moments (Russian), Charkov, 1938, 231-252. ¨ M.G.Krein, H.Langer: Uber einige Fortsetzungsprobleme, die eng mit der angen. I. Einige Theorie hermitescher Operatoren im Raume Πκ zusammenh¨ Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187-236.

[KL]

[LW]

H.Langer, H.Winkler: Direct and inverse spectral problems for generalized strings, Integral Equations and Operator Theory 30 (1998), 409-431.

[L1]

B.Levin: Nullstellenverteilung ganzer Funktionen, Akademie Verlag, Berlin 1962.

[L2]

B.Levin: On a special class of entire functions and on extremal properties of entire functions of ﬁnite degree, Izvestija Akad. Nauk SSSR Ser. Mat. 14 (1950), 45-84.

[MP1]

C.van der Mee, V.Pivovarchik: A Sturm-Liouville Inverse Problem with Boundary Conditions Depending on the Spectral Parameter, Functional Analysis and Its Applications 36(4) (2002), 315-317; Translated from Funktsional’nyi Analiz i Ego Prilozheniya 36(4) (2002), 74-77.

[MP2]

C.van der Mee, V.Pivovarchik: The Sturm-Liouville Inverse Spectral Problem with Boundary Conditions Depending on the Spectral Parameter, Opuscula Mathematica 25(2) (2005), 243-260.

[MoP]

¨ ller, V.Pivovarchik: Spectral properties of a fourth order diﬀerential M.Mo equation, Zeitschrift f¨ ur Analysis und ihre Anwendungen, to appear.

[P1]

V.Pivovarchik: Inverse problem for a string with concentrated mass at one end, Methods of Functional Analysis and Topology 4(3) (1998), 61-71.

[P2]

V.Pivovarchik: Direct and Inverse Problems for a Damped String, J. Operator Theory 42 (1999), 189-220.

[P3]

V.Pivovarchik: Symmetric Hermite-Biehler polynomials with defect, Operator Theory Adv. Appl., to appear.

126 [PM] [Pi] [R1] [R2] [R] [RR] [S] [Si]

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V.Pivovarchik, C.van der Mee: The inverse generalized Regge problem, Inverse Problems 17 (2001), 1831-1845. ¨ G.Pick: Uber die Beschr¨ ankungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23. T.Regge: Construction of potential from resonances, Nuovo Cimento 9(3) (1958), 491-503. T.Regge: Construction of potential from resonances, Nuovo Cimento 9(5) (1958), 671-679. R.Remmert: Funktionentheorie II, Springer Verlag, Heidelberg 1991. M.Rosenblum, J.Rovnyak: Topics in Hardy classes and univalent functions, Birkh¨ auser Verlag, Basel 1994. A.G.Sergeev: Asymptotics of Jost-functions and eigenvalues of the Regge problem, Diﬀerentsialnye Uravneniya 8(5) (1972), 925-927 (in Russian). B.Simon: Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178(2) (2000), 396-420.

Vyacheslav Pivovarchik Department of Applied Mathematics and Informatics South-Ukrainian State Pedagogical University Staroportofrankovskaya str. 26 UA–65091 Odessa Ukraine e-mail: [email protected] Harald Woracek Institut f¨ ur Analysis und Scientiﬁc Computing Technische Universit¨ at Wien Wiedner Hauptstr. 8–10/101 A–1040 Wien Austria e-mail: [email protected] Submitted: December 23, 2005 Revised: May 15, 2006

Integr. equ. oper. theory 57 (2007), 127–132 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010127-6, published online December 26, 2006 DOI 10.1007/s00020-006-1470-0

Integral Equations and Operator Theory

A Remark on the Essential Spectra of Toeplitz Operators with Bounded Measurable Coeﬃcients Eugene Shargorodsky Abstract. We establish a suﬃcient condition for a point to belong to the essential spectrum of a Toeplitz operator with a bounded measurable coeﬃcient. This condition uses geometric information on the cluster values of the coeﬃcient. Mathematics Subject Classiﬁcation (2000). Primary 47B35; Secondary 45E05. Keywords. Toeplitz operators, cluster values, Fredholm property.

1. Introduction Let T = {ζ ∈ C : |ζ| = 1} be the unit circle. A number c ∈ C is called a (left, right) cluster value of a measurable function a : T → C at a point ζ ∈ T if 1/(a − c) ∈ L∞ (W ) for every neighbourhood (left semi-neighbourhood, right semi-neighbourhood) W ⊂ T of ζ. Cluster values are invariant under changes of the function on measure zero sets. We denote the set of all left (right) cluster values of a at ζ by a(ζ − 0) (by a(ζ + 0)), and use also the following notation a(ζ) = a(ζ − 0) ∪ a(ζ + 0), a(T) = ∪ζ∈C a(ζ). It is easy to see that a(ζ − 0), a(ζ + 0), a(ζ) and a(T) are closed sets. Hence they are all compact if a ∈ L∞ (T). Let K ⊂ C be an arbitrary compact set and λ ∈ C \ K. Then the set w − λ σ(K; λ) = w∈K ⊆T |w − λ| is compact as a continuous image of a compact set. Hence the set ∆λ (K) := T \ σ(K; λ) is open in T. So, ∆λ (K) is the union of an at most countable family of open arcs. Supported by an EPSRC grant GR/R81749/02.

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We call an open arc of T p–large if its length is greater than or equal to p where p = p−1 , 1 < p < ∞. p Let T (a) : H (T) → H p (T) denote the Toeplitz operator generated by a function a ∈ L∞ (T), i.e. T (a)f = P (af ), f ∈ H p (T), where P is the Riesz projection: 1 g(w) 1 dw, ζ ∈ T, P g(ζ) = g(ζ) + 2 2πi T w − ζ and H p (T) := P Lp (T) is the Hardy space, 1 < p < ∞. A well known theorem of I. Gohberg and N. Krupnik describes the spectrum of T (a) with a piecewise continuous a in terms of a(ζ ±0), ζ ∈ T (see [5, 9.3, 9.8] or [1, 5.39]). It is still possible to describe the spectrum of T (a) in terms of a(ζ ± 0), ζ ∈ T if a is not piecewise continuous but a(ζ) consists of at most two points for each ζ ∈ T ([2, 3, 11]). The situation becomes considerably more complicated if a(ζ) is allowed to contain more than two points. Suppose, for example, that a(−1) consists of three points, a(−1 ± 0) = a(T) = {c1 , c2 , c3 } ⊂ C and the closed triangle (c1 , c2 , c3 ) with the vertices c1 , c2 , c3 is non-degenerate . Then the (essential) spectrum of T (a) : H 2 (T) → H 2 (T) is a connected set which contains {c1 , c2 , c3 } and is contained in (c1 , c2 , c3 ) (see, e.g., [1, 2.30, 2.33, 2.35]). It turns out however that this set is not determined solely by c1 , c2 , c3 . A. B¨ottcher has constructed examples where the spectrum of T (a) : H 2 (T) → H 2 (T) (i) does not contain any points of the boundary of the triangle (c1 , c2 , c3 ) other than c1 , c2 , c3 ; (ii) contains a side of (c1 , c2 , c3 ) and no other point of the boundary apart from c 1 , c2 , c3 ; (iii) coincides with the union of two sides of (c1 , c2 , c3 ); (iv) coincides with the boundary of (c1 , c2 , c3 ); (v) coincides with (c1 , c2 , c3 ) (see [1, 4.71–4.78]). These striking examples and the results obtained in [7, 8] imply that if a(ζ) is not required to contain at most two points for every ζ ∈ T, then it is no longer possible to describe the (essential) spectrum of T (a) in terms of the cluster values of a. In other words, it is no longer suﬃcient to know the values of a, it is important to know “how these values are attained” by a. This ﬁeld seems to be wide open at present. Since a complete description of the essential spectrum of T (a) in terms of the cluster values of a ∈ L∞ (T) is out of the question, it is natural to try ﬁnding “good” suﬃcient conditions for a point λ to belong to the essential spectrum. It is well known that a(T) is a subset of the essential spectrum of T (a) : H p (T) → H p (T), 1 < p < ∞ ([10], see also [5, 7.4] or [1, 2.30]). Hence we will assume from now on that λ ∈ a(T). The following result was obtained in [8]. 2π max{p,p } ,

Theorem 1.1. Let 1 < p < ∞, a ∈ L∞ (T), λ ∈ C \ a(T) and suppose that, for some ζ ∈ T, (I) ∆λ (a(ζ − 0)) (or ∆λ (a(ζ + 0)) ) contains at least two p–large arcs, (II) ∆λ (a(ζ)) contains at least one p–large arc.

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Then λ belongs to the essential spectrum of T (a) : H p (T) → H p (T) (i.e. T (a)−λI : H p (T) → H p (T) is not Fredholm ). Note that if a(ζ) consists of two points, then (I) implies (II) because it implies that a(ζ − 0) (or a(ζ + 0) respectively) consist of the same two points. Moreover, (I) is in this case exactly the condition from [2, 3] (see also [1, 5.54-5.58] and, for the matrix-valued case, [11]). It was shown in [8] that condition (I) is optimal in the following sense: for any compact K ⊂ C and λ ∈ C \ K such that ∆λ (K) contains at most one p–large arc there exists a ∈ L∞ (T) such that a(−1 ± 0) = a(T) = K and T (a) − λI : H r (T) → H r (T) is invertible for any r ∈ [min{p, p }, max{p, p }]. It is unclear on the other hand whether condition (II) can be dropped. The aim of this paper is to show that (II) can be relaxed: ∆λ (a(ζ)) can be substituted with a larger set ∆λ (a(ζ + 0)) (or ∆λ (a(ζ − 0)) respectively). It follows from A. B¨ ottcher’s examples mentioned above that in general there is no point outside a(T) which will always belong to the spectrum of T (a) : H 2 (T) → H 2 (T), unless a(ζ − 0) (or a(ζ + 0)) lies on a straight line. Our theorems 1.1 and 2.1 show that this is not the case if p = 2. While the ﬁrst condition in theorems 1.1 and 2.1 means in the case p = 2 that the convex hull of a(ζ − 0) (of a(ζ + 0)) is a straight line segment containing λ, the set of λ’s satisfying the conditions of our theorems increases as p moves away from 2. This phenomenon is well known in the case when a(ζ) and a(ζ − 0) (or a(ζ + 0)) consist of two points (see [2, 3, 11] and [1, 5.54-5.58]).

2. Main Result Theorem 2.1. Let 1 < p < ∞, a ∈ L∞ (T), λ ∈ C \ a(T) and suppose that, for some ζ ∈ T, (i) ∆λ (a(ζ − 0)) (or ∆λ (a(ζ + 0)) ) contains at least two p–large arcs, (ii) ∆λ (a(ζ + 0)) (or ∆λ (a(ζ − 0)) respectively ) contains at least one p–large arc. Then λ belongs to the essential spectrum of T (a) : H p (T) → H p (T). Proof. If p = 2 then (i) means that the convex hull of a(ζ − 0) (of a(ζ + 0)) is a straight line segment containing λ. Then λ belongs to the essential spectrum of T (a) : H 2 (T) → H 2 (T) even if (ii) is not satisﬁed (see, e.g., [8, Theorem 3]). Let us assume that p = 2. Suppose ∆λ (a(ζ − 0)) contains at least two p–large arcs and ∆λ (a(ζ + 0)) contains at least one p–large arc. The case of the opposite choice of “+” and “-” can be treated similarly. The proof is by contradiction and is similar to those from [3, 8]. Suppose λ does not belong to the essential spectrum of T (a) : H p (T) → H p (T), i.e T (a−λ) = T (a) − λI is Fredholm. It can be assumed that |a − λ| = 1 almost everywhere on T (see [5, 8.2] or [1, 2.32]) and that T (a − λ) is invertible (see [5, 7.7]). Further, a − λ can be approximated in the L∞ (T)–norm by a function a0 ∈ L∞ (T) which satisﬁes the following conditions

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• |a0 | = 1 almost everywhere on T and T (a0 ) : H p (T) → H p (T) is invertible, • there exist an open arc γ− ⊂ T with the right end equal toζ and an open arc γ+ with the left end equal to ζ such that ∆0 a0 (ζ − 0) ∪ a0 (γ− ) contains at least two 2π p–large arcs strictly longer than max{p,p and ∆0 a0 (ζ + 0) ∪ a0 (γ+ ) contains }

2π at least one p–large arc strictly longer than max{p,p } . ∞ Then a0 = exp(iψ) and the real-valued ψ ∈ L (T) can be chosen in such a way that π π = , (2.1) ψ − (ϑ± − π) L∞ (γ± ) < π − max {p, p } min {p, p } π π , sup , (2.2) inf ψ(η) − ϑ0 > ψ(η) − ϑ0 < − max {p, p } max {p, p } η∈E η∈γ− \E where eiϑ± is the centre of one of the p–large arcs from ∆0 a0 (ζ ± 0) ∪ a0 (γ± ) mentioned above, eiϑ0 is the centre of another p–large arc from ∆0 a0 (ζ − 0) ∪ a0 (γ− ) , and E is a measurable subset of γ− such that both E ∩V and (γ− \E)∩V have positive measure for any left semi-neighbourhood V ⊂ T of ζ. Since T (a0 ) : H p (T) → H p (T) is invertible, a0 admits a generalized factor−1 p ¯−1 ¯− ∈ H p (T) and ization in Lp (T): a0 = a− a+ , where a+ , a − ∈ H (T), a+ , a “bar” means complex conjugation (see [10] or [5, 8.1, 8.3], [1, 5.4-5.5]). On the other hand, a0 = exp(iψ) = u− u+ , where

u± = exp

˜ i(ψ ± iψ) 2

and ψ˜ denotes the conjugate function (the Hilbert transform) of ψ. Further, r u±1 ¯±1 + ,u − ∈ H (T) for any positive r < π/ ψ ∞ (see, e.g., [4, Ch. III, proof of Corollary 2.6]). Suppose p < 2 (the case p > 2 can be treated similarly or reduced to the case p < 2 by duality). Then min {p, p } = p,

max {p, p } = p .

−1 −1 p The equality a− a+ = u− u+ implies a+ u−1 + = a− u− . Since a+ , a− ∈ L (T), it is not diﬃcult to deduce from (2.1) and Zygmund’s theorem (see [12, Ch. VII, −1 2.11] or [4, Ch. III, Corollary 2.6]) that the function h = a+ u−1 + = a− u− is integrable over each closed subarc of γ± . Hence h admits an analytic continuation to C \ T \ (γ− ∪ γ+ ) (see, e.g., [9, Lemma 4] or [6, Lemma 3]). The point ζ is an isolated singularity of this function and it follows from [9, Theorem 6] that ζ cannot be an essential singularity (cf. [9, proof of Corollary 7]). Since ζ is either a removable singularity or a pole of h, there exists a neighbourhood of ζ in C which does not contain zeros of h apart from, possibly, ζ. We can therefore choose an arc γ ⊆ γ− which intersects both E and γ− \ E in sets of positive measure that are not arcs (modulo sets of measure 0) and on which 1/h is bounded.

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˜ is integrable on γ, which Then u+ = h−1 a+ ∈ Lp (γ), i.e. |u+ |p = exp(−p ψ/2) contradicts (2.2) according to [8, Lemma]. Hence T (a) − λI : H p (T) → H p (T) cannot be Fredholm. Example. Suppose a ∈ L∞ (T), a(−1 − 0) = {±i} and a(−1 + 0) = {±1}. Then ∆0 (a(−1 − 0)) consists of two open arcs of length π and the same is true for ∆0 (a(−1 + 0)). Hence it follows from Theorem 2.1 that 0 belongs to the essential spectrum of T (a) : H p (T) → H p (T) for any p ∈ (1, +∞). On the other hand, ∆0 (a(−1)) consists of four open arcs of length π/2 and Theorem 1.1 implies only that 0 belongs to the essential spectrum of T (a) : H p (T) → H p (T) for p ∈ (1, 4/3) ∪ (4, +∞). Acknowledgement I am grateful to the referee for useful suggestions which have improved the paper.

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Eugene Shargorodsky Department of Mathematics King’s College London Strand, London WC2R 2LS UK e-mail: [email protected] Submitted: February 21, 2006 Revised: October 22, 2006

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Integr. equ. oper. theory 57 (2007), 133–151 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010133-19, published online December 26, 2006 DOI 10.1007/s00020-006-1476-7

Integral Equations and Operator Theory

On the Inverse and Determinant of Certain Truncated Wiener-Hopf Operators Harold Widom Abstract. The solution of a problem arising in integrable systems requires sharp asymptotics for the inverses and determinants of truncated WienerHopf operators, both in the regular case (where the non-truncated WienerHopf operator is invertible) and in singular cases. This paper treats two cases where the symbol of the Wiener-Hopf operator has Fisher-Hartwig singularities, one double zero or two simple zeros. We find formulas for the inverse that hold uniformly throughout the underlying interval with very small error, and formulas for the determinant with very small error. Mathematics Subject Classification (2000). 47B35. Keywords. Wiener-Hopf operator, Wiener-Hopf determinant, Fisher-Hartwig symbol.

1. Introduction and statement of results The solution of a problem arising in integrable systems [10] requires the asymptotics as α → ∞, with very small error, of the inverses and determinants of truncated Wiener-Hopf operators Wα (σ) acting on L2 (0, α), both in the regular case (where the Wiener-Hopf operator W (σ) on L2 (R+ ) is invertible) and in singular cases. This paper treats two cases where σ has simple Fisher-Hartwig singularities, one double zero or two simple zeros. We ﬁrst state a result for the regular case. Assume σ belongs to the Wiener algebra of functions of the form a + kˆ with k ∈ L1 (R),1 and that ∞ σ(ξ) = 0, ∆ arg σ(ξ) = 0. (1) −∞

R ∞ ixξ ˆ Fourier transform we use is k(ξ) = −∞ e k(x) dx. For the Toeplitz case with a weaker assumption see [5, Th. 2.14]. For (2) itself see footnote 5.

1 The

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Then Wα (σ)−1 = Pα W (σ)−1 Pα − Qα H(1/σ − ) H(1/σ+ ) Qα + o(1).

(2)

The notation here is the following. The operators Pα is multiplication by χ(0,α) or extension by zero from (0, α) to (0, ∞), depending on the context. For functions deﬁned on (0, α) we deﬁne (Qα f )(x) = f (α − x), we use H to denote Hankel operators as usual, and for a function v we deﬁne v˜(ξ) = v(−ξ). The term o(1) denotes a family of operators whose operator norms tend to zero as α → ∞. The functions σ± are the Wiener-Hopf factors of σ: their product equals σ, and ±1 ±1 σ− resp. σ+ extend to bounded analytic functions in the lower resp. upper half plane. ∞ If in addition −∞ |x| |k(x)|2 dx < ∞ then the Kac-Achieser formula holds: det Wα (σ) ∼ G(σ)α E(σ),

(3)

∞ 1 G(σ) = exp log σ(ξ) dξ , 2π −∞ ∞ 1 (log σ+ (ξ)) log σ− (ξ) dξ . E(σ) = exp 2πi −∞ In the present paper we assume that

where

σ(ξ) = (ξ 2 − p2 ) τ (ξ), where p is real, and τ is nonzero, has index zero, and its Wiener-Hopf factors τ± are of the order ξ −1 as ξ → ∞. In order to minimize technical details we make a very strong assumption, namely that eδ |x| k(x) ∈ L1 (R) for some δ > 0.2 This will result in an exponentially small error in the approximation. We make a choice of which Wiener-Hopf factor of σ incorporates which zero, and take the factors to be σ− (ξ) = i(ξ − p) τ− (ξ),

σ+ (ξ) = −i(ξ + p) τ+ (ξ).

As functions, we have 1 τ− (p) = + u− (ξ), σ− (ξ) i(ξ − p)

1 τ+ (p) = + u+ (ξ), σ+ (ξ) −i(ξ + p)

where u± are bounded smooth functions. In fact we think of σ± (ξ)−1 as distributions deﬁned by 1 τ− (p) = + u− (ξ), σ− (ξ) 0 + i(ξ − p)

1 τ+ (p) = + u+ (ξ). σ+ (ξ) 0 − i(ξ + p)

Observe that if we set ep (x) = eipx then (0 − i(ξ + p))−1 is the Fourier transform of χ+ ep and (0 + i(ξ − p))−1 is the Fourier transform of χ− ep . (We write χ± for χR± .) 2 We

shall consistently use δ to denote some positive quantity, diﬀerent for each occurrence.

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To state our ﬁrst result for p = 0 we deﬁne ζ = H(1/σ+ ) ep ,

η = H(1/σ − ) ep ,

(here ep is thought of as restricted to R+ ) and A=

2ip . τ− (p) τ+ (−p) e−iαp − τ+ (p) τ− (−p) eiαp

Theorem 1 (p = 0). There is a δ > 0 such that if A = O(eδα ) then Wα (σ) is invertible for large α and Wα (σ)−1 ≡ Pα W (1/σ+ ) W (1/σ− ) Pα − Qα H(1/σ − ) H(1/σ+ ) Qα iαp +A [e τ+ (p) τ− (−p) Pα ζ ⊗ Pα η − τ+ (p) τ+ (−p) Pα ζ ⊗ Qα ζ −τ− (p) τ− (−p) Qα η ⊗ Pα η + eiαp τ+ (p) τ− (−p) Qα η ⊗ Qα ζ]. This requires lots of explanation. First, the sign ≡ between two operators indicates that the diﬀerence is an integral operator whose kernel is uniformly O(e−δα ) for some δ > 0. There are tensor products in the statement. For functions v, w we deﬁne v⊗w to be the operator taking a function f to v (w, f ). (The inner product is the integral of the product—no complex conjugate.) In general we have v1 ⊗ w1 · v2 ⊗ w2 = (w1 , v2 ) v1 ⊗ w2 and (v ⊗ w) · T = v ⊗ T w, where T is the transpose of T . Next, we deﬁne W ((0 − i(ξ + p))−1 ) to be convolution by χ+ ep on (0, ∞). This is deﬁned on any locally integrable function since x ((χ+ ep ) ∗ f )(x) = eipx e−ipy f (y) dy. 0

Our assumption on σ implies that u+ is a constant plus the Fourier transform of an exponentially decaying function,3 so W (u+ ) is deﬁned on any function of at most polynomial growth. This extends the domain of W (1/σ+ ) to any function of at most polynomial growth. Similarly we deﬁne W ((0 + i(ξ − p))−1 ) to be convolution by χ− ep . This is deﬁned only for functions in L1 since ∞ − ipx χ e−ipy f (y) dy. (( ep ) ∗ f )(x) = e x

1

+

Thus W (1/σ− ) acts on L (R ). Finally, there appear Hankel operators whose symbols are distributions. The symbol of H(1/σ+ ) is the Fourier transform of a constant times χ+ ep plus a rapidly decreasing function. Thus the operator has a singular part which can be written as a constant times ep ⊗ ep . Similarly for H(1/σ − ). In the statement of the theorem 3 Our assumption implies that σ (ξ) extends analytically to a strip around the real line and + that σ+ (ξ − iδ)−1 is in the Wiener algebra for suﬃciently small δ > 0. Therefore, since (0 − i(ξ − iδ))−1 = −(δ + iξ)−1 is in the Wiener algebra, so is u+ (ξ − iδ). This implies that u+ is a constant plus the Fourier transform of a function with bound O(e−δx ).

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we see the product of two of these, or one of them acting on ep . We interpret these by going to the Fourier transforms. For example, from the general identity ∞ 1 vˆ(−ξ) w(ξ) ˆ dξ (v, w) = 2π −∞ we obtain

1 1 1 1 dξ = − , 2π 0 + i(ξ − p) 0 − i(ξ + p) 2ip since by going into the upper half-plane we pass the pole of the ﬁrst factor at p + 0i. Similarly we ﬁnd that H((0 − i(ξ + p)−1 ) ep = −ep /2ip. (Everything can be computed by ﬁrst giving p a small positive imaginary part and then passing to the limit.) Although we shall be using distributions throughout they will all be of a simple form: Fourier transforms of linear combinations of e±p (combinations of 1 and x when p = 0) and exponentially decaying functions, and their restrictions to R± . Thus the meaning of the operators v → v± in Fourier transform space is clear: take the inverse Fourier transform, multiply by χ± , and then take the Fourier transform. The statement for p = 0 is simpler since Hankel operators with distribution symbol do not appear. We deﬁne σ+ B =α+i (0), σ− (χ+ ep , χ+ ep ) =

where the factors are chosen so that τ− (0) = τ+ (0).4 Theorem 2 (p = 0). The operator Wα (σ) is invertible for suﬃciently large α and Wα (σ)−1 ≡ Pα W (1/σ+ ) W (1/σ− ) Pα − Qα H(u − ) H(u+ ) Qα −1 −B [Qα H(u − )1 + Pα W (1/σ+ )1] ⊗ [Qα H(u+ )1 + Pα W (1/σ − )1]. (4) There are several results in the literature giving ﬁrst-order asymptotic results for the inverse when the symbol has a single singularity or zero. The method of [9] gives a ﬁrst-order result in terms of convergence in operator norm. Rambour and Seghier [8] considered the Toeplitz case for a symbol with a zero of arbitrary even order and obtained a stronger result in these cases. Ehrhardt [6] obtained results in the Toeplitz case for a general class of symbols with several Fisher-Hartwig singularities, with convergence in the operator norms of weighted 2 spaces, although they do not apply to symbols with two real zeros. In this work, too, distributions played an important role. None of these serves our present purpose, which is to ﬁnd formulas that hold uniformly throughout (0, α) with very small error. An earlier and more involved derivation of (4) than the one we give here is in [11]. The present proof will use a uniformity in the lemma to Theorem 1 and then a passage to the limit. The computation would be tedious if done by hand and we 4 Without

this normalization the second term is to be multiplied by (τ− /τ+ )(0) = −(σ− /σ+ )(0).

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resorted to computer computation. For the skeptical reader we give a direct proof in §7, along the lines of the proof of Theorem 1. For the determinants we assume that in addition log σ ∈ L1 . We use ≡ here to indicate that the diﬀerence of the two sides is O(e−δα ) for some δ > 0. Now we deﬁne ∞ 1 G(σ) = exp (log σ+ + log σ− ) dξ , 2π −∞ ∞ 1 (log σ+ ) log σ− dξ . E(σ) = exp 2πi −∞ In the integrals log σ+ is interpreted as log (ξ + p+ 0i) plus a regular function while log σ− is log (ξ − p − 0i) plus a regular function. These depend on our choice of factors σ± . For p = 0 the result is Theorem 3 (p = 0). We have as α → ∞ det Wα (σ) τ− (p) τ+ (−p) − τ+ (p) τ− (−p) e2iαp E(σ). ≡ α G(σ) τ+ (−p) τ− (p)

(5)

For p = 0 the result is Theorem 4 (p = 0). We have as α → ∞ det Wα (σ) ≡ B E(σ). G(σ)α This may be rewritten det Wα (σ) = α E(σ) + i G(σ)α

σ+ σ−

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(0) E(σ) + O(e−δα ).

Thus it gives the second-order asymptotics with very small error. The ﬁrst-order asymptotics were obtained by Mikaelyan [7]. In the special case σ0 (ξ) = ξ 2 /(1+ξ 2 ) one has Wα (σ0 ) = e−α (1 + α/2) exactly, as was also shown in [7]. Exact formulas for arbitrary rational symbols were obtained by B¨ ottcher [2] and used by him to obtain further asymptotic results in these cases. For the Toeplitz analogue the asymptotics are known for symbols with any number of zeros [3]. (Or see [4, §10.47].) For Wiener-Hopf operators the case of two simple zeros was considered in [1], but the result was obtained only for τ real and positive. In §6 we show that our formula agrees with the one obtained there. Here is how the approximate inverse in Theorem 1 is obtained. Since (2) holds for regular symbols we consider −1 −1 ) W (σ− ) Pα − Qα H(1/σ Pα W (σ+ − ) H(1/σ+ ) Qα

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a possible ﬁrst approximation for singular symbols. So we see what happens when we multiply it by Wα (σ) on the left. We ﬁnd that, with small error, the result is I plus an operator of rank two. The form of this operator suggests the ingredients of a rank two operator that should have been added to (7) to obtain I in the end. A simple computation tells us exactly which operator to take, and the rest is straightforward. For the determinants we introduce a parameter, use the formula for the logarithmic derivative in terms of the inverse, and apply the earlier results.

2. Proof of Theorem 1 Lemma. Deﬁne Tα = Pα W (1/σ+ ) W (1/σ− ) Pα − Qα H(1/σ − ) H(1/σ+ ) Qα +A eiαp τ+ (p) τ− (−p) Pα ζ ⊗ Pα η − τ+ (p) τ+ (−p) Pα ζ ⊗ Qα ζ

−τ− (p) τ− (−p) Qα η ⊗ Pα η + eiαp τ+ (p) τ− (−p) Qα ζ ⊗ Qα η .

(8)

Then if δ is small enough we have, uniformly in p, Wα (σ) Tα = I + O(A e−δα ),

Tα Wα (σ) = I + O(A e−δα ).

Proof. We shall use, innumerable times and without comment, the identities σ2 ) Pα − Qα H( σ1 ) H(σ2 ) Qα , W (σ1 ) Wα (σ2 ) = Wα (σ1 σ2 ) − Pα H(σ1 ) H( and H(σ1 σ2 ) = W (σ1 ) H(σ2 ) + H(σ1 ) W (σ˜2 ), and their familiar consequences. (For the Toeplitz analogues of the stated identities see [4, §2.17 and §7.8].) Let us multiply (7) by Wα (σ) on the left. We have −1 Wα (σ) Pα W (1/σ+ ) W (1/σ− ) Pα = Wα (σ) Wα (1/σ+ ) W (σ− ) Pα = [Wα (σ− ) − Qa H(˜ σ ) H(1/σ+ ) Qα ] W (1/σ− ) Pα = I − Qa H(˜ σ ) H(1/σ+ ) Pα W (1/σ − ) Qα = I − Qa H(˜ σ ) H(1/σ+ ) W (1/σ ) Q − α

+Qa H(˜ σ ) H(1/σ+ ) (I − Pα ) W (σ −

−1

) Qα .

The ﬁrst two terms combine as σ ) H(1/(σ+ σ− )) Qα . I − Qα H(˜ Similarly, σ ) Pα H(1/σ −Wα (σ) Qα H(1/σ − ) H(1/σ+ ) Qα = −Qα W (˜ − ) H(1/σ+ ) Qα = −Qα W (˜ σ ) H(1/σ ) H(1/σ ) Q + Q W (˜ σ ) (I − P ) H(1/ σ − + α α α − ) H(1/σ+ ) Qα . The ﬁrst term here is Qα H(˜ σ ) W (1/σ− ) H(1/σ+ ) Qα = Qα H(˜ σ ) H(1/(σ− σ+ )) Qα ,

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cancelling the same term above. It follows that Wα (σ) times (7) equals I plus 5 Qa H(˜ σ ) H(1/σ+ ) (I−Pα ) W (1/σ σ ) (I−Pα ) H(1/σ − ) Qα +Qα W (˜ − ) H(1/σ+ ) Qα .

For convenience we set c− = τ− (p),

c+ = τ+ (−p).

The Hankel operator H(1/σ+ ) on the left above is the sum of two terms. One is c−1 + ep ⊗ ep while the other is H(u+ ). Consider the contribution of the latter. It has kernel O(e−δ (x+y) ), so H(u+ ) (I − Pα ) has kernel O(e−δ α e−δ (x+y) ). Since the kernels of both H(˜ σ ) and (I − Pα ) W (1/σ − ) Qα are bounded it follows that the contribution of the H(u+ ) summand of H(1/σ+ ) in the ﬁrst term is O(e−δα ). Similarly so is the contribution of the H(u − ) summand of H(1/σ − ) in the second term. All this is uniform in p. Therefore with this error the above may be replaced by 1 1 Qa H(˜ σ )ep ⊗Qα W (1/σ− ) (I−Pα ) ep + Qα W (˜ σ )(I−Pα ) ep ⊗Qα H(1/σ+ ) ep . c+ c− v ) ep , which comes from Using the general fact Qα W (v) (I − Pα ) ep = eiαp Pα H(˜ ∞ ∞ V (α − x − y) eipy dy = V (−x − y) eip(α+y) dy, α

0

we see that the above equals eiαp eiαp Qa H(˜ σ )ep ⊗ Pα H(1/σ Pα H(σ) ep ⊗ Qα H(1/σ+ ) ep − ) ep + c+ c− eiαp eiαp = Qa H(˜ σ )ep ⊗ Pα η + Pα H(σ) ep ⊗ Qα ζ, c+ c− in our notation. This suggests that we add to our ﬁrst guess a linear combination of tensor products with right factors Pα η and Qα ζ. By symmetry (or taking transposes of the above applied to σ ˜ ) the left factors should be Pα ζ and Qα η. So it should be of the form (a Pα ζ + b Qα η) ⊗ η + (c Pα ζ + d Qα η) ⊗ Qα ζ. To see if this works, we compute Wα (σ) Pα ζ and Wα (σ) Qα η. We have Wα (σ) Pα ζ

5 In

= = ≡

Pα W (σ) Pα H(1/σ+ ) ep −1 Pα W (σ) H(1/σ+ ) ep − Pα W (σ) (I − Pα ) H(σ+ ) ep 1 −Pα H(σ) W (1/σ + ) ep + 2ip c+ Pα W (σ) (I − Pα ) ep

=

− σ+1(p) Pα H(σ) ep +

eiαp 2ip c+

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Qα H(˜ σ ) ep .

the regular case with the stated conditions the Hankel operators are all compact, so this operator is o(1). Similarly for the one obtained by multiplying by Wα (σ) on the right. This gives (2) almost immediately. With the stronger assumption on σ this operator is O(e−δα ).

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Similarly σ) η Wα (σ) Qα η = Qα Wα (˜ 1 Qα H(˜ σ ) ep + ≡ − σ− (−p)

eiαp 2ip c−

Pα H(σ) ep .

This will cancel the previous extra terms when we add, if our coeﬃcients are such that in a Pα ζ + b Qα η the coeﬃcient of Pα H(σ) ep is zero while the coeﬃcient of Qα H(˜ σ ) ep is −eiαp /c+ , and in c Pα ζ + d Qα η the coeﬃcient of Qα H(˜ σ ) ep is zero while that of Pα H(σ) ep is −eiαp /c− . We see that this does occur when 2ip σ− (−p) σ+ (p) e2iαp (2ip)3 τ− (−p) τ+ (p) e2iαp = , ∆ ∆

a=

(2ip)2 c− σ− (−p) eiαp (2ip)3 τ− (p) τ− (−p) eiαp =− , ∆ ∆ (2ip)2 c+ σ+ (p) eiαp (2ip)3 τ+ (p) τ+ (−p) eiαp c= =− , ∆ ∆ 2ip σ− (−p) σ+ (p) e2iαp (2ip)3 τ− (−p) τ+ (p) e2iαp d= = , ∆ ∆ b=

where ∆ = (2ip)2 c− c+ − e2iαp σ− (−p) σ+ (p) =

(2ip)3 eiαp . A

Thus a = τ− (−p) τ+ (p) eiαp A, c = −τ+ (p) τ+ (−p) A,

b = −τ− (p) τ− (−p) A,

d = τ− (−p) τ+ (p) eiαp A.

This tells us that when we deﬁne Tα by (8) then the ﬁrst statement of the lemma holds. (The reason for the extra factor A in the statement is that the error inherent in (9) gets multiplied by a factor proportional to A.) The second is obtained by applying the ﬁrst to σ ˜ and taking transposes. This completes the proof of the Lemma. Proof of Theorem 1. The theorem follows from the relations of the lemma by using the easily checked fact that Tα diﬀers from I by an integral operator whose kernel is O(α).

3. Proof of Theorem 2 Since the statement of the lemma to Theorem 1 holds uniformly in p, we need only compute the p → 0 limit of Tα to obtain its analogue when p = 0. The computations simplify if we take τ± (0) = 1, which we may do since the theorem for the general case follows by applying the special case to the symbol σ(ξ)/τ (0). We easily compute that (0) − τ+ (0)))−1 + O(p) = −B −1 + O(p). A = −(α + i(τ−

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For the rest, a tedious but straightforward computation6 shows that all terms of (8) except for the ﬁrst combine as −B −1 times [Pα H(u+ )1−Qα H(u − )1−x+iτ+ (0)]⊗[Pα H(u − )1−Qα H(u+ )1−x−iτ− (0)]+O(p).

(All terms involving p−1 cancel.) Now W (1/σ+ )1 = x + W (u+ )1, while H(u+ )1 has Fourier transform u+ − u+ (0) u+ − u+ (0) =− , 0 + iξ 0 − iξ + + so H(u+ )1 = −W (u+ )1 + u+ (0) = −W (u+ )1 − iτ+ (0). (Observe that τ± (ξ) = 1 ∓ iξ u± (ξ).) Therefore W (1/σ+ )1 = x − H(u+ )1 − iτ+ (0),

and similarly for the right-hand side of the tensor product. Therefore the tensor product equals [Pα W (1/σ+ )1 + Qα H(u − )1] ⊗ [Pα W (1/σ − )1 + Qα H(u+ )1]. Thus if we set Tα0 = Pα W (1/σ+ ) W (1/σ− ) Pα − Qα H(u − ) H(u+ ) Qα −1 −B [Qα H(u − )1 + Pα W (1/σ+ )1] ⊗ [Qα H(u+ )1 + Pα W (1/σ − )1], (10) then we have Wα (σ) Tα0 = I + O(A e−δα ),

Tα0 Wα (σ) = I + O(A e−δα ).

This gives the statement of Theorem 2 when we check that an integral operator whose kernel is O(α).

Tα0

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diﬀers from I by

4. Proof of Theorem 3 Suppose we want to prove (5) for a symbol σ1 . If we take any other σ0 of the same form we can write σ1 = σ0 eρ , and we deﬁne the family σ = σλ = σ0 eλ ρ where λ ∈ [0, 1]. We assume ﬁrst that A satisﬁes the bound in the statement of Theorem 1 for all λ ∈ [0, 1], so that we can use its conclusion and the relation d log det Wα (σ) = tr Wα (σ)−1 Wα (σρ). dλ The operator on the right is Wα (σ)−1 [Wα (σ) Wα (ρ) + Pα H(σ) H(˜ ρ) Pα + Qα H(˜ σ ) H(ρ) Qα ] ρ) Pα + Qα H(˜ σ ) H(ρ) Qα ]. = Wα (ρ) + Wα (σ)−1 [Pα H(σ) H(˜ 6 The

tedious computation was done by Maple.

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We look ﬁrst at

ρ) Pα , tr Wα (σ)−1 Pα H(σ) H(˜ and in particular to the contribution to this of bracketed expression in (8) with its factor A. First, the factors Pα may be removed with error O(e−δα ). Then H(˜ ρ) may be moved around to the left, so the trace of the product ≡ A times that of eiαp τ+ (p) τ− (−p) H(˜ ρ) ζ ⊗ H(σ) η − τ+ (p) τ+ (−p) H(˜ ρ) ζ ⊗ H(σ) Qα ζ iαp −τ− (p) τ− (−p) H(˜ ρ) Qα η ⊗ H(σ) η + e τ+ (p) τ− (−p) H(˜ ρ) Qα ζ ⊗ H(σ) Qα η.

The function H(σ) Qα ζ contains as a summand H(σ) Qα H(u+ ) ep (the rest comes from the ep ⊗ ep part oh H(1/σ+ )), and this is O(e−δα ) since H(σ) and H(u+ ) have exponentially decaying kernels. Similarly for Qα η. Therefore with this error we may make the replacements Qα ζ → −

eiαp eiαp e−p , e−p = − 2ip c+ 2ip τ+ (−p)

eiαp eiαp e−p . e−p = − 2ipc− 2ip τ+ (p) Next, for any plus function v+ the Fourier transform of H(v+ )ep is v+ v+ (p) − v+ . = 0 + i(ξ − p) + 0 − i(ξ − p) Qα η → −

Therefore if we denote the Fourier transform by F we have in particular Fη=

−1 −1 −1 σ (2ip τ− (−p))−1 + σ − σ − (p) − − =− , 0 − i(ξ − p) 0 − i(ξ − p)

and similarly Fζ=

−1 −1 (2ip τ+ (p))−1 + σ+ σ+ (p)−1 − σ+ =− . 0 − i(ξ − p) 0 − i(ξ − p)

If we make these substitutions we ﬁnd that we are left with four tensor products, only two of which have nonzero coeﬃcients. We show them below with their coeﬃcients. e2iαp A−1 , − (13) H(˜ ρ) e−p ⊗ H(σ) e−p , 2ip τ+ (−p) τ− (p) −1 −1 σ+ σ − ⊗ H(σ) F −1 , eiαp τ+ (p) τ− (−p). H(˜ ρ) F −1 0 − i(ξ − p) 0 − i(ξ − p) (14) Next we consider Pα H(σ) H(˜ ρ) Pα multiplied on the left by the −1 . Again, because of the Qα we may Qα H(1/σ − ) H(1/σ+ ) Qα part of Wα (σ) make the replacements −1 iαp e−p ⊗ ep , Qα H(1/σ − ) → c− e

iαp H(1/σ+ ) Qα → c−1 ep ⊗ e−p , + e

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the product of these being −

e2iαp e−p ⊗ e−p , 2ip c− c+

and after bringing the H(˜ ρ) around, the corresponding tensor product is H(˜ ρ) e−p ⊗ H(σ) e−p with the factor e2iαp . − 2ip τ+ (−p) τ− (p) This cancels the term in (13) when we multiply it by A and subtract t his one. To compute the trace of the term in (14) we note that the factor on the right has Fourier transform −1 σ− σ+ σ =− 0 + i(ξ + p) 0 − i(ξ + p) +

since σ+ (−p) = 0. Therefore the inner product of the two factors is −1 σ+ σ+ 1 1 ρ− dξ = − dξ − ρ 2π 0 − i(ξ − p) 0 − i(ξ + p) 2π (0 − i(ξ − p)) (0 − i(ξ + p)) and so the trace with its factor (and the factor A) is A eiαp τ+ (p) τ− (−p)

ρ− (p) − ρ− (−p) . 2ip

(15)

As for the Pα W (1/σ+ ) W (1/σ− ) Pα part of the inverse, this is ≡ the trace of W (1/σ+ ) W (1/σ− ) H(σ) H(˜ ρ) = W (1/σ+ ) H(σ+ ) H(˜ ρ) = −H(1/σ+ ) W (σ ) H(˜ ρ ) = −H(1/σ ) H(˜ ρ σ ). + + + The trace (including the minus sign) equals 1 1 −1 (ρ σ+ ) σ+ ρ− (log σ+ ) dξ, dξ = (16) 2πi 2πi since ρ dξ = 0. So (15)+(16) is the contribution of the Pα H(σ) H(˜ ρ) Pα term in (12) to the σ ) H(ρ) Qα term, observe that trace. For the contribution of the Qα H(˜ σ ) H(ρ) Qα = Qα Wα (˜ σ ) −1 H(˜ σ ) H(ρ) Qα , Wα (σ) −1 Qα H(˜ which has the same trace as σ ) −1 H(˜ σ ) H(ρ) Pα . Wα (˜ Hence we need only replace σ by σ ˜ everywhere in what we derivated above. Then we add two and we get ρ− (p) − ρ− (−p) + ρ+ (−p) − ρ+ (p) A eiαp τ+ (p) τ− (−p) 2ip 1 + [ρ− (log σ+ ) − ρ+ (log σ− ) ] dξ. 2πi

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And ﬁnally we have to add to this tr Wα (ρ) =

α 2π

IEOT

ρ(ξ) dξ.

What is this the derivative of? Since (d/dλ) log σ = ρ, for the last term the answer is α log σ(ξ) dξ. 2π Here we use our chosen factorization of σ to deﬁne the logarithm. For the next to last terms, since (d/dλ) log σ± = ρ± we have d (log σ+ ) log σ− dξ = [ρ+ log σ− + (log σ+ ) ρ− ] dξ, dλ so the middle term is d/dλ of 1 2πi

(log σ+ ) log σ− dξ.

For the ﬁrst term we compute that since dτ± /dλ = ρ± τ± , d 1 d log A = log dλ dλ τ− (p) τ+ (−p) e−iαp − τ+ (p) τ− (−p) eiαp −(ρ− (p) + ρ+ (−p)) τ− (p) τ+ (−p) e−iαp + (ρ+ (p) + ρ− (−p)) τ+ (p) τ− (−p) eiαp = . τ− (p) τ+ (−p) e−iαp − τ+ (p) τ− (−p) eiαp In the numerator we add (ρ− (p) + ρ+ (−p)) τ+ (p) τ− (−p) eiαp to the ﬁrst summand and subtract it from the ﬁrst. Then the quotient becomes −ρ− (p) − ρ+ (−p) − A eiαp τ+ (p) τ− (−p)

ρ− (p) − ρ− (−p) + ρ+ (−p) − ρ+ (p) , 2ip

and so the ﬁrst term is the logarithmic derivative of (τ+ (−p) τ− (p) A)−1 , which is the same as the logarithmic derivative of τ− (p) τ+ (−p) e−iαp − τ+ (p) τ− (−p) eiαp . τ+ (−p) τ− (p) Putting this together gives the logarithmic derivative of det Wα (σ) with error O(e−δα ). Integrating and exponentiating gives 1 det Wα (σ) ≡ C exp ) log σ dξ (log σ + − G(σ)α 2πi (17) −iαp τ− (p) τ+ (−p) e − τ+ (p) τ− (−p) eiαp × , τ+ (−p) τ− (p) where C is a constant independent of λ.

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This was derived under the assumption that, for all λ, A = O(eδα ) for some small δ, which means that 2iαp is not exponentially close to any of the values of τ− (p) τ+ (−p) , (18) log τ+ (p) τ− (−p) in particular if 2iαp is bounded away from these values. Assume that this is so for our original σ1 , and suppose we had taken σ0 (ξ) =

ξ 2 − p2 , (q1 − iξ) (q2 + iξ)

with q1 , q2 in the right half-plane. The corresponding quotient in (18) is q1 − ip q2 − ip . q1 + ip q2 + ip Each factor maps the right half-plane to the lower half-plane, so with proper choice of qi the ratio can be made arbitrarily close to the ratio in (18) coming from σ1 . The values of (18) for σλ depend linearly on λ, so for this choice 2iαp will be bounded away from the values of (18) for all λ ∈ [0, 1]. Therefore (17) holds for some C. We have [2, Prop. 5.2] det Wα (σ0 ) =e

−α(q1 +q2 )

(q1 + ip) (q2 + ip) eiαp − (q1 − ip) (q2 − ip) e−α(p1 +p2 )/2 , 2ip (q1 + q2 )

and a computation shows that for this σ0 (17) holds when C = eiαp . This proves the theorem for our symbol σ1 under the condition that 2iαp is bounded away from the values of (18). To remove the condition, suppose that it is very close to one of these values. Replace σ1 by σ1 eλρ , with ρ now any symbol such that ρ− (p) ρ+ (−p) = 1, ρ+ (p) ρ− (−p) and let λ run over a little (but not too little) circle around zero. Then the corrrespoinding values of (18) will run over circles about the original ones, and so 2iαp will be bounded away from them, uniformly in λ. By analyticity of the two sides in our formula, their values at λ = 0 are obtained by integrating with respect to λ over the circle. We deduce that the diﬀerence of the two sides, which is exponentially small uniformly for λ on the circle, is also exponentially small for λ = 0.

5. Proof of Theorem 4 One can check that the right-hand side of (6) is the limit of the right-hand side of (5) as p → 0. Unfortunately it is not clear from our derivation that (5) holds uniformly in p, although it surely does. Fortunately the derivation of Theorem 4 from Theorem 2 is a lot easier than the derivation of Theorem 3 from Theorem 1, so we proceed with it.

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We start in the same way, with (12). As before, the ﬁrst term has trace α ρ(ξ) dξ, 2π and we proceed to the contribution of tr Wα (σ)−1 Pα H(σ) H(˜ ρ) Pα . −1 −1 ) W (σ− ) Pα part of the inverse is The contribution of the Pα W (σ+ 1 ρ− (log σ+ ) dξ. 2πi

(19)

as before. The contribution of the rest of Wα (σ)−1 is the trace of −Qα H(u − ) H(u+ ) Qα −1

−1 −B [Qα H(u )1] − )1 + Pα W (σ+ )1] ⊗ [Qα H(u+ )1 + Pα W (σ − right-multiplied by Pα H(σ) H(˜ ρ) Pα . In computing the trace we may, as before, move the H(˜ ρ) Pα around to the left. As we saw earlier, any trace involving two Hankel operator with nonsingular symbol with the operator Qα between them will be O(e−δα ). With similar error we may remove the factors Pα . So with this error the trace reduces to that of −1

−1 ρ) W (σ+ )1 ⊗ H(σ) W (σ −B −1 H(˜ −

−1

)1.

The Fourier transform of the second factor is −1 σ− σ+ , = σ 0 + iξ iξ +

while the Fourier transform of the ﬁrst factor is −1 σ + ρ˜ . 0 + iξ +

The ρ˜ may be replaced by ρ˜+ , and the result after replacing ξ by −ξ is −1 σ+ ρ− . 0 − iξ −

Notice that the expression in parenthesis has the singularity (0 − iξ)−2 . When we multiply by the preceding and integrate we may remove the minus subscript, and we see that the trace in question equals ρ− (ξ) 1 dξ = i ρ− (0). 2π (0 − iξ)2 So the contribution of the Pα H(σ) H(˜ ρ) Pα factor is −iB −1 ρ− (0), which is to be added to (19). Similarly (or by applying the preceding to σ ˜ ) we compute the contribution of the Qα H(˜ σ ) H(ρ) Qα term. Adding the two gives 1 1 ρ− (log σ+ ) dξ − ρ+ (log σ− ) dξ + iB −1 (ρ− (0) − ρ+ (0)). 2πi 2πi

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Thus the logarithmic derivative of the determinant equals 1 α 1 ρ(ξ) dξ + ρ− (log σ+ ) dξ + ρ+ log σ− dξ 2π 2πi 2πi +iB −1 (ρ− (0) − ρ+ (0)) + O(e−δα ). The ﬁrst term is the logarithmic derivative of G(σ)α , and the next two terms combine as the logarithmic derivative of 1 (log σ+ ) log σ− dξ , exp 2πi as we saw in the last section. Since (see footnote 4) σ+ B = α − i log + λ (ρ+ − ρ− ) (0), σ− the last term above is the logarithmic derivative of B. This proves the theorem with a constant factor C on the right-hand side. Taking the special case σ0 (ξ) = ξ 2 /(1 + ξ 2 ) shows that C = 1.

6. Comparison with the formula of [1] We show here that (5) agrees with the result of [1]. Take p > 0. We deﬁne two contours, both going from −∞ to ∞. Contour C1 has an indentation above −p and below p, while C2 has an indentation below −p and above p. Each has its own geometric mean Gi (σ) and constant Ei (σ). These are deﬁned by integrals over Ci with integrands involving the associated factorizations. Our factorization σ = σ− σ+ is associated with C1 . Thus G(σ) = G1 (σ), E(σ) = E1 (σ), and if we think of the right-hand side of (5) as a sum then G(σ)α times the ﬁrst summand is exactly G1 (σ)α E1 (σ). For the other terms we ﬁrst express our exponential factor E1 (σ) in terms of E2 (σ). We have 1 log((ξ + p) τ+ ) log((ξ − p)τ− ) dξ log E1 (σ) = 2πi C1 =

1 2πi

∞−iδ

−∞−iδ

log((ξ + p) τ+ ) log((ξ − p)τ− ) dξ − log(−2p τ− (−p)),

1 log E2 (σ) = 2πi =

1 2πi

∞−iδ −∞−iδ

C2

log((ξ − p) τ+ ) log((ξ + p)τ− ) dξ

log((ξ − p) τ+ ) log((ξ + p)τ− ) dξ − log(2p τ− (p)),

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since for the ﬁrst we pass the pole at ξ = −p and for the second the pole at ξ = −p. For the two resulting integrals, both over the contour in the lower half-plane, the ﬁrst one minus the second equals 1 1 ξ−p ξ+p dξ + log((ξ − p) τ− ) dξ (log τ+ ) log log 2πi ξ+p 2πi ξ−p 1 =− 2πi

1 ξ−p ξ+p dξ + log((ξ − p) τ− ) dξ log τ+ log log ξ+p 2πi ξ−p = − log τ+ (p) + log τ+ (−p).

Putting these together shows that E1 (σ) = −

τ− (p) τ+ (−p) E2 (σ). τ− (−p) τ+ (p)

Finally, we compute that G1 (σ)/G2 (σ) = e−2ip . Thus e2iαp G(σ)α = G2 (σ)α , so G(σ)α times the second summand in our formula is exactly G2 (σ)α E2 (σ). Hence our result can be stated det Wα (σ) = G1 (σ)α (E1 (σ) + O(e−δα )) + G2 (σ)α (E2 (σ) + O(e−δα )), which, except for the size of the error terms, agrees with [1].

7. Another proof of Theorem 2 As before, we may assume τ− (0) = τ+ (0). We begin by multiplying Wα (σ) on the right by Tα0 , given by (10). For the product with Pα W (1/σ+ ) W (1/σ− ) Pα we obtain as before Wα (σ) Pα W (1/σ+ ) W (1/σ− ) Pα = I − Qa H(˜ σ ) H(1/σ+ ) Pα W (1/σ − ) Qα , but now we write H(˜ σ ) H(1/σ+ ) Pα W (1/σ −)

−1

σ ) H(u+ ) Pα W (σ = H(˜ σ ) (1 ⊗ 1) Pα W (1/σ − ) + H(˜ − σ ) H(u+ ) W (1/σ ≡ H(˜ σ ) (1 ⊗ 1) Pα W (1/σ − ) + H(˜ −) σ ) H(u+ /σ− ). = H(˜ σ ) (1 ⊗ 1) Pα W (1/σ − ) + H(˜

)

This is to be left- and right-multiplied by Qα . Next, for the product with Qα H(u − ) H(u+ ) Qα we have Wα (σ) Qα H(u σ ) Pα H(u − ) H(u+ ) Qα = Qα W (˜ − ) H(u+ ) Qα ≡ Qα W (˜ σ ) H(u − ) H(u+ ) Qα ,

(20)

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and we compute σ u σ ) W (u− )] H(u+ ) W (˜ σ ) H(u − ) H(u+ ) = [H(˜ − ) − H(˜ σ ) H(u− u+ ) = H(˜ σ u − ) H(u+ ) − H(˜ 1 1 1 1 − σ ) H u+ − =H σ ˜ H(u+ ) − H(˜ σ 0 − iξ σ− 0 + iξ − σ ˜ u+ =H σ) H H(u+ ) + H(˜ − H(˜ σ ) H(u+ /σ− ). 0 + iξ 0 + iξ Here we used the fact that σ ˜ (0) = 0. The last term here cancels the same term appearing in (20). The ﬁrst two terms we may write as 1 1 H(˜ σ) W σ) W H(u+ ) + H(˜ H(u+ ). 0 − iξ 0 + iξ Since

1 1 + = 2πδ = ˆ1, 0 − iξ 0 − iξ

these terms combine as H(˜ σ ) (1 ⊗ 1) H(u+ ). Putting together what we have done gives Wα (σ) [Pα W (1/σ+ ) W (1/σ− ) Pα − Qα H(1/σ − ) H(1/σ+ ) Qα ] σ ) (1 ⊗ 1) Pα W (1/σ σ ) (1 ⊗ 1) H(u+ )] Qα . ≡ I − Qα [H(˜ − ) + H(˜ For the right-hand side of the ﬁrst tensor product we use Qα 1 = Pα 1 to see that (Pα W (1/σ − ) Qα ) 1 = Qα W (1/σ− ) Pα 1 = Pα W (1/σ − ) Qα 1 = Pα W (1/σ − ) Pα 1 = Pα W (1/σ − ) 1.

Thus the right-hand side above equals I − Qα H(˜ σ )1 ⊗ [Pα W (1/σ − )1 + Qα H(u+ )1]. Next, we will not make a guess as we did in the proof of the lemma since we are sure know the answer. So let us just continue by computing −1 −1 Wα (σ) [Qα H(u σ ) H(u − )1+Pα W (σ+ )1] = Qα W (˜ − )1+Pα W (σ) Pα W (σ+ ) Pα 1.

The ﬁrst summand equals 1 1 Qα H σ ˜ − σ ) W (u− )1 1 − Qα H(˜ σ 0 − iξ − σ ˜ σ) 1 = Qα H 1 − u− (0) Qα H(˜ 0 + iξ 1 = Qα H(˜ σ) W σ ) 1, 1 − u− (0) Qα H(˜ 0 − iξ The second summand equals Pα W (σ− ) Pα 1 − α Qα H(˜ σ ) H(1/σ+ ) Qα .

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Using H(1/σ+ ) = 1 ⊗ 1 + H(u+ ) and 1 ⊗ Pα 1 = α 1 ⊗ 1 we see that this is ≡ to σ )1 − Qα H(˜ σ ) H(u+ )1. Pα W (σ− ) Pα 1 − α Qα H(˜ Since H(u+ )1 = −W (u+ )1 + u+ (0) when we combine the terms we get −1 σ )1 + Qα H(˜ σ ) W (σ+ )1 − (u+ (0) + u− (0)) Qα H(˜ σ )1. Pα W (σ− ) Pα 1 − α Qα H(˜

Since, as already observed, u± (0) = ∓i τ± (0), the coeﬃcient in the last term equals −i (τ+ /τ− ) (0) = i (σ+ /σ− ) (0). Therefore the above equals −1 σ ) W (σ+ ) 1 − B Qα H(˜ σ )1. Pα W (σ− ) Pα 1 + Qα H(˜ −1 Let us show that the ﬁrst two terms cancel. We have H(˜ σ ) W (σ+ ) 1 = H(σ − ) 1. Since σ− (0) = 0 this has Fourier transform σ − /iξ, which is minus the Fourier transform of W (σ − ) 1. Therefore the second term above equals

−Qα W (σ − ) 1 = −Qα W (σ − ) Pα 1 = −Pα W (σ− ) Qα 1 = −Pα W (σ− ) Pα 1. We computed above what we obtain when we left-multiply the ﬁrst two terms in (10) by Wα (σ). What we just computed shows that if we do the same with the last term and add, what results is I + O(e−δα ). Similarly for multiplication on the right. (Or apply the preceding to σ ˜ and take the transpose.) This gives (11) and hence the statement of the theorem. Acknowledgements We thank Albrecht B¨ ottcher for very helpful comments. This work was supported by National Science Foundation grant DMS-0552388.

References [1] S. Albeverio and K. A. Makarov, Extension of the Ahiezer-Kac determinnat formula to the case of real-valued symbols with two real zeros, Acta Applic. Math. 62 (2000) 155–186. [2] A. B¨ ottcher, Wiener-Hopf determinants with rational symbols, Math. Nachr. 144 (1989) 39–64. [3] A. B¨ ottcher and B. Silbermann, The asymptotic behavior of Toeplitz determinants for a generating function with zeros of integral order, Math. Nachr. 102 (1981) 79–105. [4] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag Berlin (1989). [5] A. B¨ ottcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag New York (1998). [6] T. Ehrhardt, A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, Oper. Th.: Adv. Appl. 124 (2001) 217–241. [7] L. V. Mikaelyan, Asymptotics of determinants of truncated Wiener-Hopf operators in a singular case, (Russian) Akad. Nauk Armyan. SSR Dokl. 82 (1986) 151–155. [8] P. Rambour and A. Seghier, Exact and asymptotic inverse of the Toeplitz matrix with polynomial singular symbol, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 705–710.

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[9] H. Widom, Extreme eigenvalues of N-dimensional convolution operators, Trans. Amer. Math. Soc. 106 (1963) 391–414. [10] H. Widom, Asymptotics of a class of operator determinants with application to the cylindrical Toda equations, arXiv: nlin.SI/0605021. [11] H. Widom, On the inverse and determinant of a truncated Wiener-Hopf operator, arXiv: math.FA/0605076, v2. Harold Widom Department of Mathematics University of California Santa Cruz, CA 95064 USA e-mail: [email protected] Submitted: October 2, 2006

Integr. equ. oper. theory 57 (2007), 153–166 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020153–152, published online August 8, 2006 DOI 10.1007/s00020-006-1448-y

Integral Equations and Operator Theory

Characteristic Functions for Multicontractions and Automorphisms of the Unit Ball Chafiq Benhida and Dan Timotin Abstract. A multicontraction on a HilbertP space H is an n-tuple of operators ∗ T = (T1 , . . . , Tn ) acting on H, such that n i=1 Ti Ti ≤ 1H . We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball. Keywords. Characteristic function, multicontraction, analytic automorphism.

1. Introduction Let H be a Hilbert space; a multicontraction is an n-tuple of operators T = (T1 , . . . , Tn ) acting on H, such that ni=1 Ti Ti∗ ≤ 1H . A theory of dilation and models for this type of operators has been developed by Gelu Popescu in [9, 10] and a series of subsequent papers. There is no commutativity assumed there, and the isometric dilation obtained is related to the Fock space and to representations of the Cuntz algebra. Starting mainly with [1], interest has developed around the case T is formed by commuting operators. In particular, in the recent paper [5], which is actually the starting point for this note, a notion of characteristic function is introduced for commuting tuples, and in a particular case it is shown that this is a complete unitary invariant. One computes also, in terms of the characteristic function, the curvature introduced by Arveson [2]. Some of the results therein follow from previous work of Popescu in the noncommuting case; moreover, one should note that a signiﬁcantly more general theory (including an extension of the notion of characteristic function) has been developed independently in [12, 13]. However, the direct approach of the commuting case might still be instructive. This note investigates further the characteristic function of a multicontraction. In Section 2 we remind the main deﬁnitions and notations. Section 3 contains some variations around the results in [5]. In Section 4 we investigate a general form

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of fractional transform, of which the characteristic function is a particular case. The main applications are obtained in Section 5, where one obtains the relation of the characteristic function to involutive automorphisms of the ball applied to a multicontraction. Here formulas similar to the Moebius transform of a single contraction are obtained. The case of the general automorphisms are discussed in Section 6; this is connected to the homogeneous operators considered by Misra et al [7, 3]. After this paper was completed, we have learnt that further work on closely related subjects has independently been done by Bhattacharyya, Eschmeier and Sarkar [6], and have also become aware of the recent work of Popescu [12, 13]. The connection to our results will be pointed out when the case appears.

2. Preliminaries and notations If E1 , E2 are two Hilbert spaces, and C : E1 → E2 is a contraction, one deﬁnes the defect operator DC = (1E1 − C ∗ C)1/2 ∈ L(E1 ) and the defect space DC = DC E1 ⊂ E1 . Suppose T = (T1 , . . . , Tn ) ∈ L(H)n is a commuting multicontraction; that is, n Ti Ti∗ ≤ 1H . i=1

This is the same as requiring that the row operator T = (T1 · · · Tn ) : Hn → H is a contraction. (We will currently denote with the same letter T the multioperator and the associated row contraction.) Accordingly, we have the operators DT = (1Hn − T ∗ T )1/2 and DT ∗ = (1H − T T ∗)1/2 , and the spaces DT = DT Hn ⊂ Hn , DT ∗ = DT ∗ H ⊂ H. For further use, for a multiindex α = (α1 , . . . , αn ) ∈ Nn , we shall denote |α| = |α1 | + · · · + |αn |, and T α = T1α1 · · · Tnαn . If, for z ∈ Bn (the unit ball of Cn ), the operator z : Hn → H is given by z = (z1 1H · · · zn 1H ), then z is a strict contraction, and thus 1H − zT ∗ is invertible. We may then deﬁne θT (z) = −T + DT ∗ (1H − zT ∗ )−1 zDT : DT → DT ∗ .

(2.1) n

Thus θT (z) is an analytic contraction valued function deﬁned on B , which is called the characteristic function of T ; it is a multiplier of the corresponding Hardy–Arveson spaces. According to a standard terminology introduced in [16] for the case of a single contraction, we say that two analytic functions Θ : Bn → L(E1 , E2 ), Θ : Bn → L(E1 , E2 ) coincide if there exist unitary operators Ωi : Ei → Ei , i = 1, 2, such that Ω2 Θ(z) = Θ (z)Ω1 for all z ∈ Bn . The Hardy-Arveson space H is equal to the Hilbert space of analytic functions 1 on Bn with reproducing kernel k(z, w) = 1−z,w . The monomials z α (z ∈ Bn , α ∈ n N ) form a complete orthogonal (but not orthonormal) family therein.

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Also, for E a Hilbert space, we denote H(E) = H ⊗ E; thus H = H(C). The standard multishift S = (S1 , . . . , Sn ) on H is deﬁned by Si f = zi f . Again in [1] one shows that S is a commuting multicontraction, and DS ∗ = P0 , where P0 denotes the orthogonal projection onto the constant functions. We will use the same notation S for the corresponding operators (Si ⊗ 1E ) acting on H(E). If A ∈ L(H(E), H(E ∗ )) is an operator that commutes with the standard multishift, then A is uniquely deﬁned by its restriction a to E; we will denote then A = Ma . One can also view a as a function from Bn to L(E, E ∗ ), by writing a(z)(ξ) = a(ξ)(z). If T is an arbitrary multicontraction, we deﬁne a completely positive map ρT : L(H) → L(H) by n Ti XTi∗, (2.2) ρT (X) = i=1

and denote, as in [5], by A∞ (T ) ∈ L(H) the strong limit of the decreasing sequence of positive operators ρkT (1H ). The next theorem appears in [5]; most of its ingredients are already present in [1]. Theorem A. If T is a commuting contractive tuple of operators on H, then there exists a unique bounded linear operator L : H(DT ∗ ) → H satisfying L(f ⊗ ξ) = f (T )DT ∗ ξ

(2.3)

for all f polynomial, ξ ∈ E. The adjoint operator L∗ : H → H(DT ∗ ) is given by (L∗ h)(z) = DT ∗ (1H − zT ∗ )−1 h.

(2.4)

These operators satisfy LSi = Ti L,

Si∗ L∗ = L∗ Ti∗ .

(2.5)

We have the identities LL∗ + A∞ (T ) = 1H , ∗

L L+

MθT Mθ∗T

= 1H(DT ∗ ) .

(2.6) (2.7)

When no confusion is possible, we will usually denote simply A∞ instead of A∞ (T ). Finally, we remind a dilation result from [1]. A spherical operator Z = (Z1 , . . . , Zn ) is a commuting tuple of normal operators such that Z1 Z1∗ + · · · + Zn Zn∗ = 1. Theorem B. If T is a multicontraction, there is a (essentially uniquely deﬁned) standard minimal dilation of T of the form S ⊕ Z, where S is a multishift on H(DT ∗ ), while Z is a spherical operator. We have Z = 0 iﬀ A∞ (T ) = 0. A few words are in order concerning the relation to the noncommuting case considered by Popescu in [9, 10] and subsequent papers. When the contractions Ti

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do not commute, it is necessary to introduce, instead of the Hardy–Arveson space H(E), the Fock space 2

m

Γ(E) = E ⊕ E n ⊕ (E n )⊗ ⊕ · · · ⊕ (E n )⊗ ⊕ · · · . One can identify then H(E) with the subspace of Γ(E) formed by the symmetric tensors; suppose then that πE is the orthonormal projection onto this subspace. The defect spaces of a noncommuting multicontraction are deﬁned in a similar manner, and the characteristic function of T introduced by Popescu in [10] corresponds to an operator µT : Γ(DT ) → Γ(DT ∗ ) which commutes with the creation operators in the Fock space. The relation with the commuting case is then the formula MθT = πDT ∗ µT |H(DT ). Thus most of the results above concerning the space H(E) can be obtained by “compressing” to the symmetric tensors the corresponding facts in the Fock space Γ(E). This approach appears already in [11] (where one ﬁnds, among others, a noncommutative version of Theorem A); it has been pursued systematically (and in greater generality) in the recent papers [12, 13]. Other connections between the commuting and noncommuting cases can be found in [14].

3. Classes of multicontractions By means of the operator A∞ deﬁned above, we can deﬁne some classes of multicontractions, similar to the ones that appear in [10] in the noncommutative case. Definition 3.1. The multicontraction T is called • pure (or C0 ) if A∞ = 0; • C1 if ker A∞ = {0}; • completely noncoisometric (c.n.c) if ker(1 − A∞ ) = {0}. The next result is an immediate consequence of Theorem A. Proposition 3.2. (i) [5] T is pure iﬀ L∗ is an isometry. In this case T is unitarily equivalent to the commuting tuple T = (T1 , . . . , Tn ) on HT = H(DT ∗ ) MθT (H(DT )) deﬁned by Ti = PHT Si |HT . (ii) T is c.n.c. if and only if L∗ is one-to-one. In this case T ∗ is uniquely determined by the second equality in (2.5). The following lemma has been proved for the noncommuting case in [9] (see Remark 2.7 therein). Lemma 3.3. Suppose T is a commuting multicontraction. (i) ker A∞ is invariant with respect to T ∗ , and the compression of T to ker A∞ is pure. (ii) ker(1 − A∞ ) is invariant with respect to T ∗ and, if Tˆ is the compression to T to ker(1H − A∞ ), then DTˆ∗ = {0}. We can also characterize multishifts by means of their characteristic function.

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Proposition 3.4. If T is pure, then T is unitarily equivalent to the multishift S iﬀ θT ≡ 0. Proof. In Theorem A, if S is the multishift, then L in (2.3) becomes the identity. It follows then from (2.7) that MθT = 0, and thus θT ≡ 0. Conversely, if θT ≡ 0, then from Proposition 3.2 it follows that HT = H(DT ∗ ), Ti = Si , and T is unitarily equivalent to T. Definition 3.5. An analytic function Φ : Bn → L(F , E) is called: • inner if MΦ : H(F ) → H(E) is a partial isometry; • outer if MΦ : H(F ) → H(E) has dense range. The following characterizations of classes of commuting multicontractions, by means of their characteristic functions, are similar to those obtained in [16] for one contraction and in [10] for noncommuting contractions. Theorem 3.6. Suppose T is a c.n.c. multicontraction on H. Then: 1. T is pure iﬀ θT is inner. 2. T is of class C1 iﬀ θT is outer. Proof. (1) If T is pure, then L is a coisometry, and thus L∗ L is a projection. From (2.7) it follows that MθT Mθ∗T is also a projection; thus MθT is a partial isometry and θT is inner. Conversely, if MθT is a partial isometry, then (2.7) implies that L is a partial isometry, and then from (2.6) it follows that A∞ is a projection. Since T is c.n.c., we must have ker(1 − A∞ ) = {0}. Thus A∞ = 0, which means that T is pure. (2) Note that ker A∞ = {0}, is equivalent, by (2.6), to ker(1 − LL∗ ) = {0}. But it is easy to see (for any bounded operator L, actually) that this last relation is equivalent to ker(1 − L∗ L) = {0}. By (2.7), this is the same as ker(Mθ∗T ) = {0}, or θT outer. To end the section, let us note that the model provided by Proposition 3.2 (i) for pure contractions can be extended up to a certain point to c.n.c. multicontractions, in a manner similar to [16] (for a single contraction) or to [10] (for noncommuting multicontractions). We give only some indications in this direction, mostly in order to obtain an extension of Proposition 3.4. More details, including an investigation of the model space, can be found in [6], and in a more general context in [13]. Remember that DMθT = (I − Mθ∗T MθT )1/2 H(DT ) ⊂ H(DT ). Consider the space KT = H(DT ∗ ) ⊕ DMθT and the two mappings v : H(DT ) → KT , v∗ : H(DT ∗ ) → KT , deﬁned by v(f ) = MθT f ⊕ DMθT f,

v∗ (f ) = f ⊕ 0.

(3.1)

It is easy to check that v, v∗ are isometries, that KT = v(H(DT )) ∨ v∗ (H(DT ∗ )), and that

v∗∗ v

∗

= MθT ; consequently v v∗ =

Mθ∗T .

(3.2)

Deﬁne HT = KT u(H(DT )).

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If k ∈ HT , and k ⊥ v∗ (H(DT ∗ )), we must have k = 0 by (3.2). Thus the projection PHT onto HT has dense range when restricted to v∗ (H(DT ∗ )). Also, if f ∈ H(DT ∗ ), then v∗ f 2 = PHT v∗ f 2 + v ∗ v∗ f 2 = PHT v∗ f 2 + Mθ∗T f 2 . On the other hand, by (2.7) for any f ∈ H(DT ∗ ) we have Lf 2 +Mθ∗T f 2 = f . Therefore, the map Lf → PHT v∗ f is an isometry. We have just noticed that its range is dense in H; but its domain is also dense in HT by Corollary 3.2. We obtain then a unitary Φ : H → HT , deﬁned by the formula 2

Φ(Lf ) = PHT v∗ f.

(3.3)

∗

Now, since 1K − PHT = vv , we have, using (2.7), v∗∗ (ΦLf ) = v∗∗ PHT v∗ = 1H(DT ∗ ) − v∗∗ (1K − PHT )v∗

= 1H(DT ∗ ) − v∗∗ vv ∗ v∗ = 1H(DT ∗ ) − MθT Mθ∗T = L∗ L.

Again, since the range of L is dense in H, it follows that v∗∗ Φ = L∗ . We may then deﬁne a multioperator T on HT by requiring that Si∗ v∗∗ k. Applying (3.4) and (2.5), we have

(3.4) v∗∗ T∗i k

=

v∗∗ T∗i Φh = Si∗ v∗∗ Φh = Si∗ L∗ h = L∗ Ti∗ h = v∗∗ ΦTi∗ h; since v∗∗ is one-to-one, this shows that T is a multicontraction unitarily equivalent to T . Using this unitary equivalence, one proves in [5] and in a more general context in [13], that the characteristic function is a complete unitary invariant for c.n.c. contractions. Theorem 3.7. Two c.n.c. contractions are unitarily equivalent if and only if their characteristic functions coincide. It followed already from results in [10] that the class of completely noncoisometric contractions is the natural one for which the characteristic function is a complete unitary invariant. As a consequence of Theorem 3.7, we can obtain a generalization of Proposition 3.4. Corollary 3.8. If T is c.n.c., then T is unitarily equivalent to the multishift S iﬀ θT ≡ 0. Remark 3.9. The main drawback of Theorem 3.7 is that not all contractive multipliers coincide with characteristic functions of commuting multicontractions, and, contrary to the noncommuting case of [10], we do not know of a simple way to characterize those that do. A simple example is given by the null characteristic function: θ : Bn → L(E1 , E2 ) deﬁned by θ(z) = 0 for all z coincides with a characteristic function if and only if dim E1 = ∞. Indeed, it is obvious that in this case coincidence is equivalent to equalities of the dimensions of the domain and of the range. On the other hand, Corollary 3.8 implies that, if θ coincides with θT ,

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then T has to be the multishift S on some space H(E). But then dim DS ∗ = dim E (and is thus arbitrary), while (for n ≥ 2) dim DS = ∞. This follows immediately from for instance, all elements in n the fact that dim ker DS = ∞, since itcontains, n H(E) of the form (z f ) ⊕ (−z f ) ⊕ 0 (with f ∈ H(E)). 2 1 i=1 i=3

4. Fractional transforms It is useful to regard characteristic functions of multicontractions in a larger context, namely as a particular case of fractional transforms. This section is a development of some results in [4]. Let A, W ∈ L(E1 , E2 ) be contractions such that the inverse (I + W A∗ )−1 exists; deﬁne the operator ΨA (W ) ∈ L(E1 , E2 ) by the formula ΨA (W ) = A + DA∗ (I + W A∗ )−1 W DA

(4.1)

∗

The inverse exists if for example W A < 1. A related operator is ψA (W ) = ΨA (W )|DA : DA → DA∗ ; ⊥ ΨA (W )|DA

(4.2)

⊥ A|DA ,

note also that = and it maps this subspace unitarily onto DA∗ . We will use repeatedly the formulas Ψ−A (−W ) = −ΨA (W ),

ψ−A (−W ) = −ψA (W ).

(4.3)

Lemma 4.1. We have the relations I − ΨA (W )∗ ΨA (W ) = DA (I + W ∗ A)−1 (I − W ∗ W )(I + A∗ W )−1 DA , I − ΨA (W )ΨA (W )∗ = DA∗ (I + W A∗ )−1 (I − W W ∗ )(I + AW ∗ )−1 DA∗ . In particular, ΨA (W ) ≤ 1, and, if W is an isometry (or coisometry), then ΨA (W ) and ψA (W ) are isometries (or coisometries, respectively). The lemma is proved by straight computation. As a consequence, we can deﬁne isometric operators Ω : DΨA (W ) → DW and Ω∗ : DΨA (W )∗ → DW ∗ by ΩDΨA (W ) x = DW (I + A∗ W )−1 DA x, ∗ −1

Ω∗ DΨA (W )∗ x = DW ∗ (I + AW )

DA∗ x.

(4.4) (4.5)

Remark 4.2. There is an important case when we can strengthen these statements. Suppose that W0 : DA → DA∗ and I + W0 A∗ : DA∗ → DA∗ is invertible. (Note that A(DA ) ⊂ DA∗ and A∗ (DA∗ ) ⊂ DA .) Then, if W ∈ L(E1 , E2 ) is deﬁned by W = W0 PDA + APDA⊥ , then I + W A∗ is invertible, DW ⊂ DA , DW ∗ ⊂ DA∗ , and Ω, Ω∗ are actually unitary. In this case formulas (4.4) and (4.5) yield identiﬁcations of the defect spaces of ΨA (W ). Proposition 4.3. Suppose the operators I + V A∗ , I + W A∗ , I + ΨA (V )W ∗ , I + V ΨA (W )∗ are all invertible. Then, if Ω, Ω∗ are deﬁned by (4.4) and (4.5), we have Ω∗ ψΨA (W ) (V ) = ψW (ΨA (V ))Ω.

(4.6)

It should be noted that Ω and Ω∗ depend only on A and W (and not on V ).

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Proof. Since the two terms of (4.6) act on DΨA (W ) , in order to check it we have to apply them to DΨA (W ) x. The proof is a rather tedious computation for which we give only some indications. One checks ﬁrst the two formulas ΨA (W )DA = DA∗ (I + W A∗ )−1 (W + A) and DW ∗ (I + ΨA (V )W ∗ )−1 DA∗ (I + V A∗ )−1 ∗

= DW ∗ (I + AW ∗ )−1 DA∗ (I + V ΨA (W ) )−1 . Using them, one shows that Ω∗ ψΨA (W ) (V )DΨA (W ) x = DW ∗ (I + ΨA (V )W ∗ )−1 × ΨA (V )DA − DA∗ (I + V A∗ )−1 DA∗ W (I + A∗ W )−1 DA .

(4.7)

On the other hand, writing explicitely the left term in (4.6) yields ψW (ΨA (V ))ΩDΨA (W ) x = W DW (I + A∗ W )−1 DA x + DW ∗ (I + ΨA (V )W ∗ )−1 ΨA (V )(I + A∗ W )−1 DA x − DW ∗ (I + ΨA (V )W ∗ )−1 ΨA (V )W ∗ W (I + A∗ W )−1 DA x. Since DW ∗ (I + ΨA (V )W ∗ )−1 ΨA (V )W ∗ W (I + A∗ W )−1 DA x = −DW ∗ (I + ΨA (V )W ∗ )−1 W (I + A∗ W )−1 DA x + W DW (I + A∗ W )−1 DA x, it follows that ψW (ΨA (V ))ΩDΨA (W ) x = DW ∗ (I + ΨA (V )W ∗ )−1 [ΨA (V )(I + A∗ W )−1 DA x + W (I + A∗ W )−1 DA x] = DW ∗ (I + ΨA (V )W ∗ )−1 [ΨA (V )DA x + (−ΨA (V )A∗ + I)W (I + A∗ W )−1 DA x] = DW ∗ (I + ΨA (V )W ∗ )−1 × [ΨA (V )DA − DA∗ (I + V A∗ )−1 DA∗ W (I + A∗ W )−1 DA ]x. Comparing this last equality with (4.7), one obtains the desired equality.

The relation with the characteristic function is obtained by noting ﬁrst that, for z ∈ Bn , the operator z : Hn → H is a strict contraction, and thus 1H − zT ∗ is invertible. Then, by comparing (4.2) and (2.1), we see that θT (z) = ψ−T (z).

(4.8)

One should note that in the noncommuting case a similar relation has been used in [12].

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5. Involutive automorphisms of the unit ball Formula (4.8) is of interest to us since it allows to make the connection between characteristic functions and automorphisms of the ball applied to multicontractions. The involutive automorphisms of the unit ball Bn are deﬁned [15] by sλ (z − (1 − sλ )Pλ z), φλ (z) = λ − 1 − z, λ where λ ∈ Bn , sλ = (1 − |λ|2 )1/2 , and Pλ is the projection onto the space spanned by λ. Among the properties of these maps, we note that φλ is involutive, that is, φλ ◦ φλ = 1Bn , that it maps the unit ball onto the unit ball, and the unit sphere onto the unit sphere. The relation with the previous section is given by the next proposition, whose proof is a direct computation. Proposition 5.1. If we identify λ, z ∈ Bn with strict contractions in L(Cn , C), then φλ (z) = Ψλ (−z) = ψλ (−z). Denote λ = λ ⊗ 1H : Hn → H; then λ is a strict contraction. If T = (T1 , . . . , Tn ) is a multicontraction, then λ T ∗ is a strict contraction, and we may deﬁne Ψλ (−T ) = ψλ (−T ). In case T is commutative, φλ (T ) is deﬁned by the analytic functional calculus, and it is in turn a commuting row contraction. Moreover φλ (T ) = Ψλ (−T ). There are several properties of the multicontraction T that are also inherited by φλ (T ). The ﬁrst one can be proved directly. Proposition 5.2. With the above notations, ker(1−A∞ (T )) = ker(1−A∞ (φλ (T ))). In particular, T is c.n.c., if and only if φλ (T ) is c.n.c. Proof. Denote φλ (T ) = R = (R1 , . . . , Rn ). Since all Ri are analytic functions in T1 , . . . , Tn , all subspaces of H invariant to T are also invariant to R. But then the relation φλ (R) = T implies that R and T have the same invariant subspaces, and the same is true for T ∗ and R∗ . Suppose then that K = ker(1 − A∞ (T )) is not contained in ker(1 − A∞ (R)). Since the elements of this last subspace are characterized by the fact that ρkR (1) = 1 for all k (ρR as deﬁned by (2.2)), there exists x ∈ ker(1−A∞ (T )) and a ﬁrst index k K ≥ 1 for which ρK R (1)(x) = x. Since ρR (1) ≤ 1 for all k, and K is invariant to ∗ R , it follows that we can ﬁnd y ∈ K, y = 0 (of the form y = Ri∗1 . . . Ri∗K−1 x), such n n that i=1 Ri∗ y2 < y2 . On the other side, y ∈ K implies i=1 Ti∗ y2 = y2 . Consider now the multioperators T , R , which are the compressions of T and R respectively to K. It is easy to see that R = φλ (T ) = Ψλ (−T ). But the deﬁnition of K implies that T , as a row contraction operator, is a coisometry. By Lemma 4.1, R should also be a coisometry, which contradicts the existence of y.

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We have thus proved that ker(1 − A∞ (T )) ⊂ ker(1 − A∞ (R)). But then φλ (R) = T implies that we actually have equality. According to (4.4) and (4.5), we have operators Ω : Dφλ (T ) → DT and Ω∗ : Dφλ (T )∗ → DT ∗ ; moreover, since Remark 4.2 applies, Ω, Ω∗ are actually unitary maps. They provide identiﬁcations of the defect spaces of φλ (T ); what is more interesting, they can be used in order to obtain a formula for the characteristic function of φλ (T ). Theorem 5.3. (i) The operators Ω and Ω∗ are unitaries. (ii) θφλ (T ) coincides with θT (φλ (z)). Proof. (i) follows from the fact that we can apply Remark 4.2 (A = λ is a strict contraction). As for (ii), by Proposition 5.1, formulas (4.8) and (4.3), we have θφλ (T ) (z) = θΨλ (−T ) (z) = ψ−Ψλ (−T ) (z) = −ψΨλ (−T ) (−z). But Proposition 4.3 applied to the case A = λ , V = −z, W = −T says that ψΨλ (−T ) (−z) (and thus also −ψΨλ (−T ) (−z)) coincides with ψ−T (Ψλ (−z)). Note that the unitaries Ω, Ω∗ in Proposition 4.3 do not depend on V , and thus in our case the unitaries implementing the coincidence do not depend on z. Finally, applying again Proposition 5.1 and formula (4.8), we obtain ψ−T (Ψλ (−z)) = ψ−T (φλ (z) ⊗ 1H ) = θT (φλ (z)),

which ends the proof.

Theorem 5.3 is the generalization of the well known formula for the characteristic function of a Moebius transform of a single contraction [16, VI.1.3]. However, the deﬁnition of the automorphism of the ball makes φλ involutive, and thus φ−1 λ = φλ . The change of sign in the usual deﬁnition of the Moebius transforms accounts for the apparition in [16] of a slightly diﬀerent formula. Note also that a weaker result along the lines of Theorem 5.3 appears in [6]. As a ﬁrst application, we obtain a partial extension of the relation between the spectrum and the characteristic function that exists for single contractions [16]. Recall (see, for instance, [8]) that, for a commuting multioperator T = (T1 , . . . , Tn ), one can deﬁne its right spectrum by: n

σr (T ) = {λ ∈ C :

n

(Ti − λi )(Ti − λi )∗ is not invertible}.

i=1 n

Proposition 5.4. If λ ∈ B , then λ ∈ σr (T ) iﬀ θT (−λ) is not surjective. n Proof. Note that, since i=1 Ti Ti∗ is invertible if and only if (T1 · · · Tn ) is surjective, we have σr (T ) = {λ ∈ Cn : (T1 − λ1 ) · · · (Tn − λn ) not surjective}.

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Since T maps unitarily DT⊥ onto DT⊥∗ , it follows that 0 ∈ σr (T ) iﬀ θT (0) is not surjective. Thus the claim is true for λ = 0. For other values of λ, since, by the spectral mapping theorem, σr (φλ (T )) = φλ (σr (T )), we have λ ∈ σr (T ) iﬀ 0 ∈ σr (φλ (T )) (note that φλ (λ) = 0). This is equivalent to θφλ (T ) (0) not surjective. Since, by Theorem 5.3, θφλ (T ) (0) is unitarily equivalent to θT (φ−λ (0)), and φ−λ (0) = −λ, the proposition is proved. Remark 5.5. Naturally, a corresponding result can be proved for the left spectrum σl (T ) := σr (T ), where T = (T1∗ , . . . , Tn∗ ). This would however require the assumption that T is a multicontraction. If one would want to deduce a consequence about the Harte spectrum σH (T ) = σr (T ) ∪ σl (T ), one should assume both T and T multicontractions, which is a rather unnatural hypothesis (for instance, it is not satisﬁed by the multishift for n ≥ 2). The next consequence concerns the multishift. Proposition 5.6. If S is a multishift and λ ∈ Bn , then φλ (S) is also a multishift, of the same multiplicity. Proof. By Proposition 5.2 φλ (S) is a c.n.c. multicontraction, while Theorem 5.3 implies that its characteristic function is identically zero. We may then apply Corollary 3.8 to conclude that φλ (S) is a multishift. The equality of the multiplicities follows from the equality of the defect spaces of S and φλ (S), as given by Theorem 5.3, (i). We can also study the relation with the model spaces. Lemma 5.7. (i) If Z is spherical, then φλ (Z) is also spherical for all λ ∈ Bn . (ii) If the multicontraction V is a minimal dilation for T , then φλ (V ) is a minimal dilation for φλ (T ). Proof. (i) follows immediately from the functional calculus for commuting normal operators. As for (ii), one sees easily that φλ (V ) is a dilation for φλ (T ), and that the space spanned by the powers of φλ (V ) applied to H is contained in the one spanned by powers of V . On the other side, φλ is involutive, which gives the opposite relation. Proposition 5.8. (i) If the standard minimal dilation of T is S ⊕ Z, then the standard minimal dilation of φλ (T ) is φλ (S) ⊕ φλ (Z). (ii) If T is pure, then φλ (T ) is pure. (iii) If T is of class C1 , then φλ (T ) is of class C1 . Proof. (i) and (ii) follow immediately from Proposition 5.6 and Lemma 5.7. As for (iii), we will apply Theorem 3.6. If T is of class C1 , then θT is outer. To show that θφλ (T ) is also outer, note ﬁrst that by Theorem 5.3 it is enough to show that

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θT (φλ (z)) is outer. Denote by Cλ the operator f → f (φλ (z)); it is invertible, since Cλ2 = 1. Then MθT (φλ (z)) Cλ = Cλ MθT , whence it follows that, if MθT has dense range, then MθT (φλ (z)) also has.

6. General automorphisms The general form of an automorphism α of Bn is α = ω ◦ φλ , n

for λ ∈ B , and ω a unitary map of Cn (see, for instance, [15, Theorem 2.2.5]). We may then complete the results of the previous section by taking into account the action of the unitary ω; since we identify Cn with row 1 × n matrices, and regard T as a row operator, it is natural to consider the action of ω as matrix multiplication to the right. Remember that if T is a multicontraction, then the completely positive map ρT : L(H) → L(H) is deﬁned by (2.2). Lemma 6.1. Suppose T = T (1H ⊗ ω). Then ρT = ρT , while θT (z) coincides with θT (zω ∗ ). Proof. We can write ρT (X) = T (X ⊗ 1Cn )T ∗ ; then ρT (X) = T (1H ⊗ ω)(X ⊗ 1Cn )(1H ⊗ ω ∗ )T ∗ = T (X ⊗ 1Cn )T ∗ = ρT (X). As concerns the defect spaces and operators, we have DT ∗ = DT ∗ and DT ∗ = DT ∗ , while DT = (1H ⊗ ω ∗ )DT (1H ⊗ ω) and DT = (1H ⊗ ω ∗ )DT . Consequently θT (z) = −T + DT ∗ (1H − zT ∗ )−1 zDT = −T (1H ⊗ ω) + DT ∗ (1H − z(1H ⊗ ω ∗ )T ∗ )−1 z(1H ⊗ ω ∗ )DT (1H ⊗ ω) = θT (zω ∗ )(1H ⊗ ω), and the lemma is proved.

Corollary 6.2. Suppose T = T (1H ⊗ ω). Then: (i) T is c.n.c. (C1 , C0 ) iﬀ T is c.n.c. (C0 , C1 , respectively). (ii) If T is a multishift, then T is a multishift of the same multiplicity. Proof. The results in (i) are immediate consequences of the equality ρT = ρT , while for (ii) we have to use, besides Lemma 6.1, Corollary 3.8. Gathering the results in Propositions 5.2, 5.6, 5.8, Lemma 6.1, Corollary 6.2, and Theorem 5.3, we obtain a general result concerning the action of an analytic automorphism of the unit ball on a multicontraction. Theorem 6.3. Suppose α : Bn → Bn is an analytic automorphism, while T is a multicontraction. Then: (i) α(T ) is c.n.c. (C1 , C0 ) iﬀ T is c.n.c. (C0 , C1 , respectively).

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(ii) θα(T ) coincides with θT ◦ α−1 . (iii) If T is a multishift, then α(T ) is a multishift of the same multiplicity. Remark 6.4. In [7] the notion of homogeneous n-tuples of operators is introduced in a general context. In our case, a multicontraction T is homogeneous if α(T ) is unitarily equivalent to T for all α automorphism of Bn . Consequently, Theorem 6.3(iii) says that the standard multishift is homogeneous. We thank J. Sarkar for useful discussions, and the referee for bringing to our attention some recent relevant results.

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Benhida and Timotin

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[16] B. Sz.-Nagy, C. Foias: Harmonic analysis of operators on Hilbert space, NorthHolland Publishing Co, 1970. Chafiq Benhida UFR de Math´ematiques Universit´e des Sciences et Technologies de Lille F-59655 Villeneuve D’Ascq Cedex France e-mail: [email protected] Dan Timotin Institute of Mathematics of the Romanian Academy P.O. Box 1-764 RO-014700 Bucharest Romania e-mail: [email protected] Submitted: September 6, 2005 Revised: May 28, 2006

Integr. equ. oper. theory 57 (2007), 167–184 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020167-18, published online August 8, 2006 DOI 10.1007/s00020-006-1453-1

Integral Equations and Operator Theory

Spectrum of the Kerzman-Stein Operator for the Ellipse Michael Bolt Abstract. The skew-hermitian part of the Cauchy operator, deﬁned with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coeﬃcient in the asymptotic expansion of the eigenvalues. Mathematics Subject Classiﬁcation (2000). 45C05, 45E05, 30C40. Keywords. Kerzman-Stein operator, ellipse, eigenvalues.

1. Introduction In their study of Cauchy-Fantappi`e kernels and the Szeg˝ o kernel in higher dimensions, Kerzman and Stein discovered an elegant method for computing the Riemann map in one dimension. See [11, 12]. At the heart of their method is the fact that, for smooth, bounded domains, the Cauchy kernel and Szeg˝o kernel have the same principal singularity at the diagonal. In particular, the skew-hermitian part of the Cauchy operator, called the Kerzman-Stein operator, is compact. In a later article [10], Kerzman posed a number of problems concerning this operator, including the following. Problem (Kerzman, 1979). Relate the spectrum of the Kerzman-Stein operator to the geometry of the domain. In a sense, the spectrum measures the error when the Cauchy kernel is used to approximate the Szeg˝o kernel, and may be useful for estimating the rate of convergence of integral methods solutions to the Riemann map. Here we provide an answer to the problem for the case of an ellipse with small √ eccentricity. In the following theorem, the ellipse has eccentricity 2 ρ/(1 + ρ).

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Theorem. The Kerzman-Stein operator for an ellipse has eigenvalues ±iλl where each ±iλl has multiplicity 2. If the ellipse is parameterized by t → eit + ρe−it with 0 < ρ < 1, and λ1 ≥ λ2 ≥ · · · ≥ 0, then as ρ ↓ 0, λl = βl · ρ2l−1 + o(ρ2l−1 ) where 0 < βl ≤ 1. The coeﬃcients βl can be computed explicitly. Since the operator is compact and skew-hermitian, its spectrum will be discrete and imaginary, except for an accumulation point at zero. It is obvious that the Kerzman-Stein operator vanishes identically for a circle. The ellipse, however, is the ﬁrst simply-connected, smooth, bounded domain to be considered for this problem after the disc. It is known as well that the Kerzman-Stein operator is invariant under M¨ obius transformations of the complex plane. So in a previous article [4] the author computed the spectrum for domains bounded by two circular arcs or two logarithmic spirals—logarithmic spirals are known to have constant inversive curvature. The ellipse, then, is the ﬁrst example for which there is no apparent M¨obius symmetry. See Wilker [19] or Coﬀman and Frantz [6] for more on this topic. Formally, Kerzman’s problem is similar to Schiﬀer’s problem of computing the Fredholm eigenvalues of a plane domain [14, 15] which has been studied extensively. The similarity is explained by Burbea in [5], where he reduced Kerzman’s eigenvalue problem to a problem previously studied by Singh [17]. The connection between Singh’s problem and the Fredholm eigenvalue problem is apparent, for instance, in Bergman’s book [3, p.71]. Loosely, Singh’s problem can be interpreted as a boundary analogue of the Fredholm eigenvalue problem. For an ellipse, the Fredholm eigenvalues are known exactly—they are simply powers of the parameter ρ. For this fact, see [14, p.1195]. For more on the general problem, see also [2, 16]. The author thanks Professor Sidney Webster for supervising the work of his doctoral dissertation, of which this was a part.

2. Preliminaries We follow notation that Bell uses in his book [1]. Suppose Ω ⊂ C is a bounded domain with twice diﬀerentiable boundary, and let T = T (w) be the unit tangent vector at w ∈ ∂Ω, oriented positively with respect to Ω. If ds is arclength measure on ∂Ω, then the Kerzman-Stein operator is the operator deﬁned by T (z) 1 T (w) − f (w) dsw , A(z, w)f (w) dsw = Af (z) = 2πi ∂Ω w − z w−z ∂Ω valid for f ∈ L2 (∂Ω) and z ∈ ∂Ω. In fact, the kernel A(z, w) is bounded at the diagonal—the apparent singularities cancel each other. The space L2 (∂Ω) is deﬁned using the hermitian inner product (f, g)∂Ω = ∂Ω f g ds. Since A(z, w) is bounded and ∂Ω has ﬁnite length, it follows that A is HilbertSchmidt, and is therefore compact on L2 (∂Ω). Since A is also skew-hermitian, the

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spectral theorem says that its spectrum is discrete and bounded, and will consist of imaginary eigenvalues whose only accumulation point is zero. If Ω is unbounded or if ∂Ω has a corner, however, this is not necessarily true. See [4] for speciﬁc examples when A is non-compact. We point out two general facts about A that will be helpful to our study of the ellipse. Let ∆ be the unit disc, ∆ = {z : |z| < 1}. Lemma 1. Suppose h : ∂∆ → ∂Ω is biholomorphic in a neighborhood of ∂∆ and √ h is single-valued near ∂∆. Then the Kerzman-Stein operator on L2 (∂Ω) is unitary equivalent to the operator on L2 (∂∆) with kernel iw h (w) h (z) iz h (w) h (z) 1 − A(z, w) = for w, z ∈ ∂∆. 2πi h(w) − h(z) h(w) − h(z) √ Proof. We ﬁrst establish an isometry L2 (∂∆) ∼ = L2 (∂Ω) given by (f ◦ h) h ← f . Taking f, g ∈ L2 (∂Ω), this follows from √ √ f g ds = (f ◦ h) · (g ◦ h) |h | ds = (f ◦ h) h · (g ◦ h) h ds. ∂Ω

∂∆

∂∆

Next, write for f ∈ L2 (∂Ω), 1 A∂Ω f (z ) = 2πi

∂Ω

and replace w = h(w), T (w ) = z = h(z) and T (z ) =

h (z) |h (z)|

T (w ) T (z ) − w − z w − z

h (w) |h (w)|

f (w ) dsw ,

· iw, and dsw = |h (w)| dsw , and also

· iz. Then,

(A∂Ω f ) ◦ h(z) h (w) iz h (z) iw 1 − (f ◦ h)(w)|h (w)| dsw . = 2πi ∂∆ h(w) − h(z) |h (w)| h(w) − h(z) |h (z)| Multiplying both sides by h (z) gives (A∂Ω f ) ◦ h(z) h (z) iw h (w) h (z) iz h (w) h (z) 1 − = (f ◦ h)(w) h (w) dsw , 2πi ∂∆ h(w) − h(z) h(w) − h(z) and the lemma is proved.

Lemma 2. The Kerzman-Stein operator commutes with the involution Ψ on L2 (∂Ω) given by f → f T ; that is, AΨf = ΨAf . So the (imaginary) spectrum of A is symmetric with respect to 0.

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Proof. Notice that T (w) 1 T (z) T (z) T (w) 1 − = − T (w)T (z) A(z, w) = 2πi w − z w−z 2πi w − z w−z T (w) 1 T (z) − + T (w)T (z) = A(z, w) T (w)T (z), = 2πi w−z w−z so

AΨf = ∂Ω

A(z, w)f (w)T (w) dsw =

∂Ω

A(z, w) f (w) dsw · T (z) = ΨAf.

So, if Af = iλf , then also AΨf = ΨAf = Ψ(iλf ) = −iλΨf , and the lemma is proved.

3. Approximation by Finite Rank Operators on the Unit Circle The theorem is proved as follows. Using Lemma 1, we transform the problem from the ellipse to an equivalent problem on the disc. We express the new kernel using a double Fourier series, and approximate the operator using ﬁnite rank operators with kernels that have rapidly decaying coeﬃcients. We then estimate the eigenvalues of the approximating operators. Finally, we use the Cauchy interlace theorem and a previous estimate of Feldman, Krupnik, and Spitkovsky in order to bound the leading coeﬃcients. 3.1. Pulling the Kerzman-Stein operator back to the unit disk. For the ellipse, take 0 < ρ < 1 and h(z) = z + ρ/z. Then, using the expansions ∞ 1 j ρ 2 (−1)j ρ h (z) = 1 − 2 = z j z 2j j=0 and

ρ ρ ρ ) − (z + ) = (w − z)(1 − ), w z wz and also the identities w = 1/w, z = 1/z, and h(w) − h(z) = (w +

−iw iz = z−w w−z for w, z ∈ ∂∆, we ﬁnd by Lemma 1 that the Kerzman-Stein operator for the ellipse is unitary equivalent to the operator on L2 (∂∆) with kernel 1 1

−2j−l −2k−l 1 w 2 (−1)j+k ρj+k+l 2 z − w2j+l z 2k+l . A(z, w) = w 2π w − z j k j,k,l≥0

The order of summation can be taken symmetrically with respect to j, k, l; that is, the partial sums are sn = j+k+l≤n . We will see that these partial sums are each divisible by w − z in the ring of polynomials.

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Next, identify ∂∆ with the interval √ [0, 2π] and use the standard basis {ψj }j∈Z 2 ijs for L ([0, 2π]) given by ψj (s) = e / 2π. In terms of this basis, the KerzmanStein operator has kernel eis 1 · 2π eis − eit

A(t, s) =

j+k j+k+l

(−1)

ρ

j,k,l≥0

1 1 2 2 e−i(2j+l)s e−i(2k+l)t − ei(2j+l)s ei(2k+l)t , j k

or alternately,

A(t, s) =

aj,k ψj (t) ψk (s)

j,k∈Z

for real coeﬃcients aj,k that satisfy (i) aj,k = 0 if both j, k < 0, if both j, k ≥ 0, or if j +k is odd. −j+k 2

b−j,k if j < 0 ≤ k; the bj,k are determined by 1 1 w 2 w2j+l z 2k+l . bj,k wj z k = (−1)j+k 2 w − z j,k,l≥0 j k

(ii) aj,k = ρ

j>0, k≥0

j+k+l>0

(iii) aj,k = −ak,j for all j, k. (iv) aj,k = a−j−1,−k−1 for all j, k. As an operator, A then behaves as multiplication by the matrix (aj,k )j,k∈Z . Proof of (i)–(iv). To prove (i), rewrite A(z, w) using the expansion 1 1 w 1 1 1 n l l j+k 2 2 w2j z 2k ρ −1 wz (−1) 2π n>0 w2n z 2n w−z j k j,k≥0

1 n 1 1 = ρ −1 2π n>0 w2n z 2n

0≤l≤n

+

0≤l≤n−2 def

j+k=n−l

1 1 − w2 + wz − z 2 wn−1 z n−1 2 2 1 1 2 w2j z 2k . wl z l (−1)j+k 2 j k j,k≥0

w w−z

j+k=n−l

We have assumed n = j +k+l > 0 since the terms for n = 0 cancel; we have also interchanged indices j, k in the term w−2j−l z −2k−l and factored out the quantity (w−2n z −2n − 1). Since 1 1 2 (−1)j+k 2 = 0 j k j,k≥0 j+k=m

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for m ≥ 2, as shown in the next section, we may divide the quantity in brackets by w − z (rather, we divide by w2 − z 2 , but w − z is a factor of this) and obtain 1 1 j −k n aj,k z w = ρ −1 wl z l cp,n−l wp z 2(n−l)−p 2n 2n w z n>0 j,k∈Z 0≤l≤n−1 1≤p≤2(n−l)

−2n+l+p −l−p n = ρ cp,n−l w z − wl+p z 2n−l−p , n>0

0≤l≤n 1≤p≤2(n−l)

for real coeﬃcients cp,m , 1 ≤ p ≤ 2m, that can be given explicitly in terms of the binomial coeﬃcients. The ﬁrst terms on the right-hand side have the form z j w−k for integers j, k satisfying j < 0, k ≥ 0, and j + k even. The second terms have the form z j w−k for integers j, k satisfying j ≥ 0, k < 0, and j + k even. The coeﬃcients aj,k for the remaining j, k must be zero, so (i) is proved. To prove (ii), start with the nth degree terms in the deﬁnition of the bj,k , 1 1 w j k j+k 2 2 w2j+l z 2k+l , bj,k w z = (−1) w − z j k j>0, k≥0 j,k,l≥0 j+k=n

2(j+k+l)=n

then multiply both sides by ρn/2 and sum on n ≥ 1. After substituting w = eis and z = eit , and replacing j → −j on the left-hand side we obtain 1 1 −j+k eis −ijs ikt j+k j+k+l 2 2 ei(2j+l)s ei(2k+l)t 2 ρ b−j,k e e = is (−1) ρ e − eit j,k,l≥0 j k j<0 k≥0

j+k+l>0

=−

k<0, j≥0

aj,k eijt e−iks =

ak,j eijt e−iks .

k<0, j≥0

The third equality uses (iii), which is proved next. Then, interchanging j, k in the −j+k last expression gives aj,k = ρ 2 b−j,k and (ii) is proved. To prove (iii), use the identity eis /(eis − eit ) = e−it /(e−it − e−is ) to verify that A(s, t) = −A(t, s). Then aj,k ψj (t) ψk (s), aj,k ψj (s) ψk (t) = − and after interchanging j, k, it follows that aj,k = −ak,j . To prove (iv), ﬁrst verify that A(t, s) = A(t, s)e−is eit . Then aj,k ψj (t) ψk (s) = aj,k ψj+1 (t) ψk+1 (s) = aj−1,k−1 ψj (t) ψk (s) = a−j−1,−k−1 ψj (t) ψk (s). The second equality uses the replacements j → j − 1, k → k − 1, and the third equality uses the replacements j → −j, k → −k, and the fact that ψ−j = ψj . These properties enable us to show the eigenspaces of A are even-dimensional.

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Lemma 3. If the Kerzman-Stein operator has an eigenvalue iλ that corresponds 1,+ with eigenvector v = v = vj ψj , then it has a second eigenvector v 2,+ = 1,− v−j−1 ψj that corresponds with iλ. There are also eigenvectors v = v j ψj and v 2,− = v −j−1 ψj that correspond with eigenvalue −iλ. 1,+ Proof. Suppose that v 1,+ = = A( j vj ψj ) = j vj ψj so Av j,k aj,k vk ψj and k aj,k vk = iλvj for each j. Then, using (iv), Av 2,+ = aj,k v−k−1 ψj = a−j−1,−k−1 v−k−1 ψj = iλv−j−1 ψj = iλv 2,+ . j,k

j

j,k

A parity argument shows that v another. To see this, decompose

1,+

and v def

v = vodd + veven =

2,+

can be made independent of one

vj ψj +

j odd

vj ψj .

j even

1,+ If vodd = 0, then replace v = v 1,+ = vodd ; else replace v 1,+ = veven . Then, = Av 1,+ 1,+ iλv still holds, since aj,k = 0 for j + k odd. Moreover, v = j vj ψj and v 2,+ = therefore be j v−j−1 ψj will have opposite odd/even parity, and will orthogonal. Finally, we describe the opposite eigenspace. If v 1,− = j v j ψj , we ﬁnd aj,k v k ψj = aj,k v k ψj = −iλv j ψj = −iλv 1,− . Av 1,− = j,k

2,−

j

j,k

= v −j−1 ψj is an eigenvector corresponding with −iλ, since it is Then also v gotten from v 1,− in the same way that v 2,+ is gotten from v 1,+ . From now on, we associate the spectrum of A with a list of values λ1 ≥ λ2 ≥ · · · ≥ 0, so the eigenvalues of A are ±iλj , where each ±iλj has multiplicity 2. 3.2. Approximation by ﬁnite rank operators. We now pass to a set of ﬁnite rank operators An with degenerate kernel An (t, s) =

2n−1

aj,k ψj (t) ψk (s).

j,k=−2n

The rank of An is at most 4n. Furthermore, Lemma 3 also holds for the An since the span of ψ−2n , . . . , ψ2n−1 is unchanged under the transformation ψj → ψ−j−1 . We then associate the spectrum of An with a list of values λ1 ≥ · · · ≥ λn ≥ 0, so that the eigenvalues of An are ±iλj and each ±iλj has multiplicity 2. As an operator, An acts as multiplication by the matrix (aj,k )j,k=−2n,...,2n−1 . To illustrate this, we show the matrix for A3 in Figure 1. The columns and rows are indexed over −6, −5, . . . , +5. Notice that the matrix is skew-hermitian (in fact, real skew-symmetric) and the powers of ρ increase away from the diagonal. This illustrates properties (ii) and (iii) of the aj,k . Properties (i) and (iv) are also apparent. The innermost 4 × 4 and 8 × 8 submatrices correspond with A1 and A2 , respectively.

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0 0 0 0 0 0

            3  ρ  16   0  25ρ4   128   0  81ρ5  128 0

0 0 0 0 0 0 0

3ρ3 16

0 81ρ4 128

0

−81ρ5 128

0 0 0 0 0 0 ρ2 8

0

5ρ3 8

0 −81ρ4 128

0

0 0 0 0 0 0 0

0 0 0 0 0 0 ρ 2

5ρ2 8

0

−ρ 2

0

−ρ2 8

−5ρ2 8

0 −5ρ3 8

−3ρ3 16

0

−25ρ4 128

0 0 0 0 0 0 0

0

0 0

−ρ3 16

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−ρ3 16

0 2

−ρ 8

0 −ρ 2

0 0 0 0 0 0 0

0 −3ρ3 16

0 −5ρ2 8

0 ρ 2

0 0 0 0 0 0

−25ρ4 128

0 3

−5ρ 8

0

5ρ2 8

0 0 0 0 0 0 0

0 −81ρ4 128

0 5ρ3 8

0 ρ2 8

0 0 0 0 0 0

−81ρ5 128

0 4

81ρ 128

0

3ρ3 16

0 0 0 0 0 0 0

0 81ρ5 128

0 25ρ4 128

0 ρ3 16

0 0 0 0 0 0

                       

Figure 1. The matrix associated to A3 The next result assumes that ρ ↓ 0, so we are considering only ellipses with small eccentricity. Lemma 4. As described above, associate the spectrum of An with the list of values λ1 ≥ · · · ≥ λn ≥ 0. Then, for 1 ≤ l ≤ n, λl = βl · ρ2l−1 + o(ρ2l−1 ) for constants βl > 0. The coeﬃcients βl can be computed explicitly. Proof. Consider ﬁrst the characteristic polynomial of the matrix associated to An , p(λ) = det(An − λI) =

2n−1 σ

j=−2n

(−1)σ (aj,σj − λδj,σj ) =

4n

(−1)j αj λ4n−j ,

j=0

where δj,k = 1 if j = k and δj,k = 0 if j = k. The sum is taken over permutations of {−2n, . . . , 2n − 1} and (−1)σ indicates the signature of a permutation σ. The zeros of p(λ) are the eigenvalues of An , and the coeﬃcients αj are the sums of products of eigenvalues taken j at a time. Consider speciﬁcally the coefﬁcient α4l in the expression. A permutation contributes a term to this coeﬃcient only if it acts identically on 4n − 4l entries. (There are zeros along the diagonal of matrix (aj,k ).) Of these permutations, only those that act identically away from 2 {−2l, . . . , 2l − 1} contribute a term with factor ρ4l . The others contribute higher powers of ρ; this uses (ii). In fact, the permutations that act identically away from 2 {−2l, . . . , 2l −1} contribute det Al to the coeﬃcient α4l , and α4l = det Al +o(ρ4l ). Next, since Al is skew-symmetric and zero in its upper-left and lower-right quadrants, its determinant is the square of the determinant of its upper-right

Vol. 57 (2007) Spectrum of the Kerzman-Stein Operator for the Ellipse

quadrant. The entries in the upper-right quadrant are ρ 2 ρ4l (det Bl )2 where Bl is the matrix Bl = (bj,k ) j = 1,...,2l

−j+k 2

175

b−j,k , so det Al =

.

k = 0,...,2l−1

(This also involves interchanging an even number of rows, but this doesn’t aﬀect 2 2 the determinant.) It follows that α4l = ρ4l (det Bl )2 + o(ρ4l ). Proposition 1, stated and proved in the next section, says that det Bl = 0 2 for each l. So then α4l is always comparable to ρ4l . In particular, when l = 1, the product of the four largest eigenvalues is comparable to ρ4 . That is, λ41 = (+iλ1 )(+iλ1 )(−iλ1 )(−iλ1 ) = β1 · ρ4 + o(ρ4 ),

(1)

β1

= 0. In general, the product of the 4l largest eigenvalues is comparable to for 2 ρ4l . That is, 2 2 (2) λ41 λ42 · · · λ4l = βl · ρ4l + o(ρ4l ), 2l−1 2l−1 for βl = 0. From (1) and (2) it follows inductively that λl = βl ρ + o(ρ ) for βl = 0. Next, since λl = O(ρ2l−1 ) for each l, we conclude that α4l and λ41 λ42 · · · λ4l 2 agree to order ρ4l , and 2

2

λ41 λ42 · · · λ4l = ρ4l (det Bl )2 + o(ρ4l ). 1/2

It now follows inductively from (3) that β1 = (det B1 ) (det Bl )1/2 · (det Bl−1 )−1/2 . So the lemma is proved.

(3)

, and for l > 1, βl =

Using a result of Hermann Weyl [18], we recover the same information for A that we now have for An . Weyl proved the result for the case of symmetric kernels; the proof is the same for the case of hermitian or skew-hermitian kernels. See also Porter and Stirling [13, p.146-147]. Lemma 5. Suppose K and K are symmetric kernels in L2 [(a, b) × (a, b)] and denote by χ1 , χ2 , . . ., the eigenvalues for any such kernel, repeated according to multiplicity, and arranged so that |χ1 | ≥ |χ2 | ≥ · · · . Then, |χj+k+1 (K + K )| ≤ |χj+1 (K )| + |χk+1 (K )|. Using the lemma, set K = An , K = A − An , j = l − 1, and k = 0. Then |χl (A)| ≤ |χl (An )| + |χ1 (A − An )|. Also, if K = A and K = An − A then |χl (An )| ≤ |χl (A)| + |χ1 (An − A)|. This means that the eigenvalues of A agree with the eigenvalues of An to within |χ1 (A − An )|. So by (ii), they agree to within O(ρn+1 ), since this is the HilbertSchmidt norm of A − An . Returning to the notation with eigenvalues ±iλl arranged with λ1 ≥ λ2 ≥ · · · , where each ±iλl has multiplicity 2, we now choose n ≥ 2l − 1 and ﬁnd λl (A) = λl (An ) + O(ρ2l ) = βl · ρ2l−1 + o(ρ2l−1 ),

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where the βl = 0 are speciﬁed in the proof of Lemma 4. Apart from Proposition 1, then, we have left to show only that the βl are bounded by 1. 3.3. Bounding the coeﬃcients. We start with a variant of the Cauchy interlace theorem. Following our earlier convention, associate the spectra of Al and Al−1 with values λ1 ≥ λ2 ≥ · · · ≥ λl ≥ 0 and µ1 ≥ µ2 ≥ · · · ≥ µl−1 ≥ 0, respectively. Then, the relevant minimax formula for λj+1 is |(v, Al v)| λj+1 = min max , u1 ,...,uj v (v, v) where the maximum is taken over vectors v orthogonal to vectors u1 , . . . , uj and their images under the transformations vk ψk → v−k−1 ψk , v k ψk , v −k−1 ψk that occur in Lemma 3. Notice that Al is gotten from Al−1 by adding 4 rows and columns, but these rows and columns correspond with a single vector (for instance, v = ψ2l−1 + iψ2l−2 ) and its transformations. Then, following the usual proof of the interlace theorem as given by Franklin [8], for instance, it follows that λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ · · · ≥ µl−1 ≥ λl > 0. Next, we use Feldman, Krupnik, and Spitkovsky’s estimate [7] that says A < 1 for any ellipse, and in particular, λ1 < 1. Then det Al = (λ1 λ2 · · · λl )4 < (1 · λ2 · · · λl )4 ≤ (1 · µ1 · · · µl−1 )4 = det Al−1 . This is true for any ρ, so letting ρ ↑ 1 gives (det Bl )2 = ( det (bj,k ) j = 1,...,2l

k = 0,...,2l−1

)2 ≤ ( det (bj,k ) j = 1,...,2l−2 )2 = (det Bl−1 )2 , k = 0,...,2l−3

and therefore, βl ≤ 1. Numerically, it seems that the coeﬃcients βl increase with an upper limit of 1. Figure 2 illustrates this behavior for 1 ≤ l ≤ 40. The ﬁrst few leading coeﬃcients βl 1.0

0.5

l 10

20

30

40

Figure 2. Plot of the leading coeﬃcients, βl , for 1 ≤ l ≤ 40.

Vol. 57 (2007) Spectrum of the Kerzman-Stein Operator for the Ellipse

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are β1 = 1/2, β2 = 25/32, β3 = 441/512, and β4 = 184041/204800. The author determined these coeﬃcients using Mathematica.

4. Proof of Proposition 1 The deﬁnition of the bj,k in the proposition below is slightly diﬀerent from the previous section—we have multiplied the right-hand side by z. So the coeﬃcients bj,k for 1 ≤ j, k ≤ 2m in this section correspond with the coeﬃcients bj,k for 1 ≤ j ≤ 2m, 0 ≤ k ≤ 2m − 1 in the previous section. The order of summation can again be taken symmetrically with respect to w and z; for instance, use partial sums j+k≤n for the left-hand side, and partial sums j+k+l≤n for the right-hand side. Proposition 1. Let coeﬃcients bj,k be determined by the equation 1 1 wz 2 w2j+l z 2k+l . bj,k wj z k = (−1)j+k 2 w − z j,k,l=0...∞ j k j,k=1...∞

j+k+l>0

If Bm is the matrix (bj,k )j,k=1...2m , then det Bm = 0 for m > 0. The proof is combinatorial in nature, and depends on the fact that the bj,k are dyadic rational; that is, they are either zero or they have the form r = 2n · u for u odd and n ∈ Z. We use the valuation | · |2 deﬁned on all rationals according to −n 2 if r = 0 |r|2 = 0 if r = 0 where in the ﬁrst case r = 2n · u/v for odd u, v. For example, |5/8|2 = 8. The valuation has the following properties (see Jacobson [9, p.211]): (a) |r1 r2 |2 = |r1 |2 · |r2 |2 (b) |r1 + r2 |2 ≤ max(|r1 |2 , |r2 |2 ), with equality only if |r1 |2 = |r2 |2 or r1 = r2 = 0 The proof of Proposition 1 is structured as follows. Let σ range over all permutations of {1, . . . , 2m}. Then, since det Bm = σ j=1...2m (−1)σ bj,σj , we have bj,σj |2 . | det Bm |2 ≤ max | σ

j=1...2m

This will be an equality provided there is a unique permutation σ for which the maximum is attained. We will ﬁnd such a maximizing permutation, then | det Bm |2 = 0 and det Bm = 0. That is, we will show det Bm is nonzero dyadic rational. We begin by introducing intermediate coeﬃcients dj,k deﬁned by 1 1 2 w2j z 2k , dj,k wj z k = wz + (−1)j+k 2 j k j,k=0...∞ j,k=0...∞

j+k>0

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and we claim that if n is ﬁxed, then j+k=n dj,k = 0. For n = 2, it is easy to check that d0,2 + d1,1 + d2,0 = (−1/2) + 1 + (−1/2) = 0. For larger n, notice that dj,k = 0 if either j or k is odd. In particular, if n is odd then j+k=n dj,k = 0. There remains the case when n = 2p for p ≥ 2. We ﬁnd that 1 1 j+k 2 2 , dj,k = (−1) j k j+k=n

j+k=p

and this is the coeﬃcient of xp in the expansion 1 1 √ √ 2 xj+k , (−1)j+k 2 1−x= 1−x· 1−x= j k j,k≥0

which is evidently zero. So the claim is proved. We may then divide by w − z (rather, we divide by w2 − z 2 , but w − z is a factor of this) and obtain coeﬃcients cj,k that satisfy wz cj,k wj z k = dj,k wj z k . w−z j,k=1...∞

j,k=0...∞

The cj,k and dj,k are related by cn,1 = dn,0 and cn−j,j+1 = dn−j,j + cn−j+1,j for 1 ≤ j < n.

(4)

Notice in particular that c2,1 = −1/2 and c1,2 = 1/2. Following Jacobson [9, p.211], deﬁne ν(r) = − log2 |r|2 and ν(0) = ∞. For example, ν(5/8) = − log2 8 = −3. Then the following properties of ν are equivalent to the corresponding properties of | · |2 ; we will use both repeatedly: (a) ν(r1 r2 ) = ν(r1 ) + ν(r2 ) (b) ν(r1 + r2 ) ≥ min(ν(r1 ), ν(r2 )), with equality only if ν(r1 ) = ν(r2 ) or r1 = r2 = 0 Observe that if x denotes the greatest integer less than or equal to x, then p ν( 1/2 p≥0 j/2 for j ≥ 0, since j )= − 1 ( 1 )(− 12 )(− 32 ) · · · ( 12 − j + 1) 2 . = 2 j j! On the right-hand side of this identity, the j multiplicative factors each contribute a factor of 1/2, and the j! contributes j/2 + j/4 + j/8 + · · · factors of 1/2 since j/2k is the number of multiples of 2k in 1, 2, . . . , j. Using this, we can compute the values ν(dj,k ) and obtain a lower estimate for the values ν(cj,k ). Lemma 6. If 1 ≤ j ≤ n, then ν(cj,n+1−j ) ≥ − p>0 n/2p . There is equality when n is a positive power of 2. Proof. If n is odd then dj,n−j ≡ 0, so cj,n+1−j ≡ 0 and ν(cj,n+1−j ) ≡ ∞, and if n = 2 we see directly that ν(c2,1 ) = ν(c1,2 ) = −1. So suppose n ≥ 4 is even, then

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1/2 if j is even. So if j is dj,n−j = 0 if j is odd, and dj,n−j = (−1)n/2 1/2 j/2 (n−j)/2 even, n−j j n ν(dj,n−j ) = − ( 2p + 2p ) ≥ − p , 2 2 2 p>0 p≥0

and since the cj,n+1−j are combinations of the dj,n−j by (4) we have proved the ﬁrst part of the lemma. If n = 2q for q ≥ 1, we show there is equality. First, notice that ν(dj,2q −j ) = ν(0) = ∞ if j is odd. Next, 2q 2q 0 ν(d2q ,0 ) = − ( 2p + p ) = − p , 2 2 2 p>0 p≥0

and for 0 < j < 2q , j even, q j 2 −j 2q 2q ν(d2q −j,j ) = − ( 2p + 2p ) > − 2p = − p . 2 2 2 2 p>0 p≥0

p≥0

Here the inequality is strict because when p = q − 1, then

2q −j 2 2p

+

j 2 2p

= 0+0 < 1=

2q 2 . 2p

Using (4), it then follows for 1 ≤ j ≤ 2q that |cj,2q +1−j |2 = − p>0 2q /2p , and the lemma is proved. Next we show that bj,k = l=0... min(j−1,k−1) cj−l,k−l . This follows from

min(j−1,k−1)

j,k=1...∞

l=0

cj−l,k−l wj z k =

∞

cj−l,k−l wj−l z k−l · wl z l

l=0 j,k=l+1...∞

=

∞

cj,k wj z k · wl z l

l=0 j,k=1...∞ ∞ wz = dj,k wj z k wl z l w−z l=0 j,k=0...∞   1 1 ∞ wz  2 w2j z 2k  wl z l = (−1)j+k 2  w z + w−z j k j,k=0...∞ l=0

j+k>0

=

wz w−z

(−1)j+k

j,k,l=0...∞ j+k+l>0

We now estimate the value of ν on the bj,k . Lemma 7. If 1 ≤ j ≤ n, then ν(bj,n+1−j ) ≥ − n is a positive power of 2.

p>0 n/2

p

1 1 2 2 w2j+l z 2k+l . j k

. There is equality when

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Proof. Using Lemma 6 we compute min(j−1,n−j)

ν(bj,n+1−j ) = ν(

cj−l,n+1−j−l ) ≥

l=0

≥

min

l=0... min(j−1,n−j)

min

l=0... min(j−1,n−j))

−

ν(cj−l,n+1−j−l )

n − 2l n p =− p . 2 2 p>0 p>0

Furthermore, if n = 2q for q ≥ 1, then for l = 0, ν(cj−l,2q +1−j−l ) ≥ −

2q − 2l 2q > − p = ν(cj,2q +1−j ). 2p 2 p>0 p>0

So we ﬁnd that min(j−1,2q −j)

ν(bj,2q +1−j ) = ν(

cj−l,2q +1−j−l )

l=0

=

min

l=0... min(j−1,2q −j)

ν(cj−l,2q +1−j−l ) = ν(cj,2q +1−j ) = −

2q p , 2 p>0

and the lemma is proved.

We come now to the proof of the proposition, and as a matter of notation ∗ set νn∗ = p>0 n/(2p ). Then the previous lemma says ν(bj,k ) ≥ −νj+k−1 for any j, k with equality provided j + k − 1 is a positive power of 2. We claim there is a ∗ unique permutation σ of {1, . . . , 2m} that maximizes 2m ν , and for this j=1 j+σj −1 permutation, j + σj − 1 is a positive power of 2 for all j. The proposition then follows since ∗ ∗ ν(bj,τj ) ≥ − νj+τ >− νj+σ = ν(bj,σj ) j −1 j −1 j=1...2m

j=1...2m

j=1...2m

j=1...2m

for any other permutation τ . In particular, ν(det Bm ) = j=1...2m ν(bj,σj ) = ∞, and det Bm = 0. We describe how σ is chosen. As j +k−1 increases through the even numbers, there is a signiﬁcant drop in the value of ν(bj,k ) when j + k − 1 reaches a power of 2; in fact, the size of the drop increases with successive powers of 2. So working from the lower-right of the matrix, choose the permutation σ so that j + σj − 1 is the largest possible power of 2. This determines σj for the range 2q − 2m < j ≤ 2m, where 2q−1 < 2m ≤ 2q . In particular, σ restricts to a permutation of {2q − 2m + 1, . . . , 2m}. Now repeat the procedure for the remaining (2q − 2m) × uniquely determines (2q − 2m) upper-left submatrix and continue. This procedure a permutation σ and the claim says that σ uniquely minimizes j=1...2m ν(bj,σj ) = ∗ − j=1...2m νj+σ . j −1

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To be precise, if σ 2m is the optimal permutation for {1, . . . , 2m} and if q is chosen so that 2q−1 < 2m ≤ 2q , we will ﬁnd that q 2 − j + 1 if 2q − 2m < j ≤ 2m q σj2m = otherwise. σj2 −2m ∗ As an example, in Figure 3 we show the matrices with entries ν(bj,k ) and νj+k−1 for 1 ≤ j, k ≤ 2m, where m = 3. (See Figure 1 for the related matrix A3 .) The

       

∞ −1 ∞ −1 ∞ −3 ∞ −3 ∞ −3 ∞ −3 ∞ −4 ∞ −4 ∞ −7



 −3 ∞ −4 ∞ −4 ∞   −3 ∞ −7   ∞ −7 ∞   −7 ∞ −7  ∞ −7 ∞

      

0 1 1 3 3 4

1 1 3 3 4 4

1 3 3 4 4 7

3 3 4 4 7 7

3 4 4 7 7 8

4 4 7 7 8 8

       

∗ Figure 3. The matrices (ν(bj,k ))j,k=1,...,6 and (νj+k−1 )j,k=1,...,6

permutation that produces the minimal value for j ν(bj,σj ) is indicated in bold typeface. We ﬁnd that ν(det B6 ) = j ν(bj,σj ) = −30. The claim is proved using induction on m and requires two steps. The ﬁrst step says that the optimal permutation must restrict to a permutation of {2q − 2m + 1, . . . , 2m}, and the second step speciﬁes its values on this set. The second step establishes the base of the induction as a special case; the base of the induction occurs when 2m = 2q , q ≥ 1. q STEP 1: If τj ≤ 2q − 2m for 2 − 2m < j ≤ 2m, then there is a permutation τ ∗ ∗ ∗ such that l=1...2m νl+τ < l=1...2m νl+τ ∗ −1 . l −1 l

Proof of Step 1. The proof of this step is inductive on j, beginning with j = 2m. So suppose ﬁrst that τ2m ≤ 2q − 2m and deﬁne  q  2 − 2m + 1 if l = 2m def τl∗ = τ2m if l = j = τ2−1 q −2m+1  τl otherwise ∗ = τj = 2q − 2m + 1 and τj∗ = τ2m ; otherwise, τ ∗ and τ agree. Then, so that τ2m ∗ + νj∗ +τj −1 = ν2m+τ 2m −1

< =

2m + τ2m − 1 j + (2q − 2m + 1) − 1 + 2p 2p p>0

2q + j + τ2m − 1 2q j + τ2m − 1 = p + p 2 2 2p p>0 p>0

2m + τ ∗ − 1 j + τj∗ − 1 2m ∗ + = ν2m+τ + νj∗ +τ ∗ −1 . ∗ p p 2m −1 j 2 2 p>0

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Here, the ﬁrst and last equalities are by deﬁnition, the strict inequality occurs since both 2m + τ2m − 1 < 2q and j + 2q − 2m < 2q , and the second equality∗ occurs q q since j + τ − 1 ≤ 2m + (2 − 2m) − 1 < 2 . It follows that 2m l=1...2m νl+τl −1 < ∗ ν . l=1...2m l+τ ∗ −1 l

For the inductive step, pick the largest j such that τj ≤ 2q −2m and 2q −2m < j ≤ 2m. Like before, deﬁne τ by  q  2 − j + 1 if l = j def τl = τj if l = j = τ2−1 q −j+1  τl otherwise so that τj = τj = 2q − j + 1 and τj = τj . Consider the two possible cases: ∗ ∗ Case 1: If j > j then l=1...2m νl+τ < l=1...2m νl+τ −1 , since as before, l −1 l

∗ νj+τ + νj∗ +τj −1 = j −1

< =

j + τj − 1 j + (2q − j + 1) − 1 + p 2 2p p>0

(5)

2q + j + τj − 1 2q j + τj − 1 = p + p 2 2 2p p>0 p>0

j + τj − 1 j + τj − 1 ∗ + = νj+τ + νj∗ +τ −1 . p p j −1 j 2 2 p>0

Here, the strict inequality occurs since j + τj − 1 ≤ 2m + (2q − 2m) − 1 < 2q and j + (2q − j + 1) − 1 < j + (2q − j) = 2q , and the second equality occurs since j + τj − 1 ≤ 2m + (2q − 2m) − 1 < 2q . Case 2: If j < j , then the weak since j + (2q − j + 1) − 1 > 2q . inequality∗ in (5) is ∗ Nevertheless, we recover l=1...2m νl+τl −1 ≤ l=1...2m νl+τ for the permutation l −1 q τ for which τj = τj ≤ 2 −2m and j > j. Inductively, then, there is a permutation τ ∗ so that ∗ ∗ ∗ νl+τ ≤ νl+τ νl+τ ∗ −1 < l −1 l −1 l=1...2m

l=1...2m

l

l=1...2m

and the proof of Step 1 is complete.

q q STEP 2: If τ j = 2 − j + 1 for some 2 − 2m ∗< j ≤ 2m then there is a permutation ∗ σ such that l=1...2m νl+τl −1 < l=1...2m νl+σl −1 .

Proof of Step 2. After Step 1 we may assume that 2q − 2m < τj ≤ 2m for all 2q − 2m < j ≤ 2m, and we compute p>0 2q −2m<j≤2m

j + τj − 1 = 2p

p=q 2q −2m<j≤2m

j + τj − 1 + 2p

0

j + τj − 1 , 2p

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since j + τj − 1 < 2q+1 for all j. The ﬁrst sum on the right-hand side gives the number of j for which j + τj − 1 ≥ 2q , and since (j + τj − 1) = 2 · [(2q − 2m + 1) + · · · + (2m)] − [2m − (2q − 2m)] 2q −2m<j≤2m

= (4m − 2q )2q , the second sum is no larger than (4m − 2q )2q

= (4m − 2q ) · 2q−1 + · · · + 2 = (4m − 2q )(2q − 2). p 2 00 2q −2m<j≤2m

This completes the proof of Step 2, so the claim is also proved.

References [1] Steven R. Bell. The Cauchy transform, potential theory, and conformal mapping. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [2] S. Bergman and M. Schiﬀer. Kernel functions and conformal mapping. Compositio Math., 8:205–249, 1951. [3] Stefan Bergman. The kernel function and conformal mapping. American Mathematical Society, Providence, R.I., 1970. [4] Michael Bolt. Spectrum of the Kerzman-Stein operator for model domains. Integral Equations Operator Theory, 50(3):305–315, 2004. [5] Jacob Burbea. The Cauchy and the Szeg˝ o kernels on multiply connected regions. Rend. Circ. Mat. Palermo (2), 31(1):105–118, 1982. [6] Adam Coﬀman and Mark Frantz. M¨ obius transformations and ellipses (preprint). [7] Israel Feldman, Naum Krupnik, and Ilya Spitkovsky. Norms of the singular integral operator with Cauchy kernel along certain contours. Integral Equations Operator Theory, 24(1):68–80, 1996. [8] Joel N. Franklin. Matrix theory. Prentice-Hall Inc., Englewood Cliﬀs, N.J., 1968. [9] Nathan Jacobson. Lectures in abstract algebra. Vol III: Theory of fields and Galois theory. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. [10] N. Kerzman. Singular integrals in complex analysis. In Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, Proc. Sympos. Pure Math., XXXV, Part, pages 3–41. Amer. Math. Soc., Providence, R.I., 1979.

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[11] N. Kerzman and E. M. Stein. The Cauchy kernel, the Szeg¨ o kernel, and the Riemann mapping function. Math. Ann., 236(1):85–93, 1978. [12] N. Kerzman and E. M. Stein. The Szeg˝ o kernel in terms of Cauchy-Fantappi`e kernels. Duke Math. J., 45(2):197–224, 1978. [13] David Porter and David S. G. Stirling. Integral equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1990. [14] M. Schiﬀer. The Fredholm eigen values of plane domains. Pacific J. Math., 7:1187– 1225, 1957. [15] M. Schiﬀer. Fredholm eigen values of multiply-connected domains. Pacific J. Math., 9:211–269, 1959. [16] Menahem Schiﬀer. Fredholm eigenvalues and conformal mappings. Rend. Mat. e Appl. (5), 22:447–468, 1963. [17] Vikramaditya Singh. An integral equation associated with the Szeg¨ o kernel function. Proc. London Math. Soc. (3), 10:376–394, 1960. [18] Hermann Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Diﬀerentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann., 71(4):441–479, 1912. [19] J. B. Wilker. When is the inverse of an ellipse convex? Utilitas Math., 17:45–50, 1980. Michael Bolt Department of Mathematics 1740 Knollcrest Circle SE Calvin College Grand Rapids, Michigan 49546 USA e-mail: [email protected] Submitted: February 21, 2006 Revised: July 13, 2006

Integr. equ. oper. theory 57 (2007), 185–207 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020185-23, published online August 8, 2006 DOI 10.1007/s00020-006-1451-3

Integral Equations and Operator Theory

Calculation of the Defect Numbers of the Generalized Hilbert and Carleman Boundary Value Problems with Linear Fractional Carleman Shift Jorge Ferreira, Gueorgui S. Litvinchuk and Maur´ıcio D. L. Reis Abstract. The defect numbers of the generalized Hilbert and Carleman boundary value problems with a direct or an inverse linear fractional Carleman shift of order 2 (α (α (t)) ≡ t) on the unit circle are computed. The approach followed consists of the reduction of the mentioned problems to singular integral equations with linear fractional Carleman shift and of the factorization of Hermitian matrix functions with negative determinant. Mathematics Subject Classification (2000). Primary 47G10; Secondary 47A68. Keywords. Boundary value problems, singular integral operators, Carleman shift, factorization, Fredholm theory, solvability theory.

1. Introduction As it is well known, the problem of calculation of the defect numbers l, of linearly independent solutions, and ρ, of solvability conditions, for boundary value problems of analytic functions and the description of its defect subspaces are very important and very diﬃcult as well. However, when we consider problems with binomial boundary conditions on a simply connected domain, they can be solved eﬀectively without any complications. For example, we have a scalar Riemann (Riemann-Hilbert) boundary value problem, a Hasemann boundary value problem, a Carleman boundary value problem, etc (see [8]). For these problems the defect pair (l, ρ) is calculated by the Gakhov-Coburn formulas l = max (0, I)

, ρ = max (0, −I)

(1.1)

Gueorgui S. Litvinchuk was supported by European Social Fund (POPRAMIII) through CITMA (Centro de Ciˆencia e Tecnologia da Madeira).

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where I is the index of a Fredholm boundary value problem. For boundary value problems with polynomial boundary conditions, the defect numbers depend not only on topological characteristics but also on conformal and metrical characteristics. As examples, we have the boundary value problems with the shift and the complex conjugation operators, which contain, in the boundary condition, more than two values of unknown functions. One of the most studied problems is the generalized Riemann boundary value problem Φ+ (t) = a (t) Φ− (t) + b (t) Φ− (t) + h (t) (1.2) on a simply connected contour Γ for a piecewise analytic function {Φ+ (z) , Φ− (z)}. This problem appeared in 1946, when the boundary condition (1.2) was considered in the paper of Markushevich [14]. There are a few hundred publications devoted to problem (1.2) and its generalizations and modiﬁcations, which has numerous applications, for example, in the elasticity theory, in the geometry of surfaces, in the distribution of physical ﬁelds, etc. The most complete result about the defect numbers of problem (1.2) on the unit circle was obtained twenty ﬁve years ago by G. Litvinchuk and I. Spitkovsky ([11], [12]), see also ([8],ch.4). The central idea of the papers [11], [12] consists in the establishment of a direct connection between the solvability theory of problem (1.2) and the factorization of the Hermitian matrix function with negative determinant 2 2 |b| − |a| tb , t∈ (1.3) tb 1 and also on ﬁnding an algorithm to compute the partial indices of the matrix (1.3). From the papers [11] and [12] we have l

=

max (0, 2κ) , ρ = max (0, −2κ) , if m ≤ |κ|

l

=

κ + m , ρ = κ − m, if m ≥ |κ|

where κ = IndΓ a (t), (m, −m) are the partial indices of the matrix (1.3). The purpose of this paper is to show that, in the case of a linear fractional Carleman shift on the unit circle, the defect numbers of a generalized Hilbert boundary value problem Re{a(t)Φ+ (t) + b(t)Φ+ (α(t))} = h(t)

(1.4)

and of a generalized Carleman boundary value problem Φ+ (α (t)) = a (t) Φ+ (t) + b (t) Φ+ (t) + h (t)

(1.5)

are also obtained from the factorization of Hermitian matrix functions with negative determinants. We also use the results of the papers [6], [1], on the factorization of singular integral operators with linear fractional Carleman shift. For simplicity we consider problems (1.4) and (1.5) in the H¨older space. However, for any decomposable Banach algebra of continuous functions on , containing the class of rational functions without poles on , the same results are valid.

Vol. 57 (2007)

Generalized Hilbert and Carleman Boundary Problems

187

2. Auxiliary information and notation All the results of this section can be found in [8]. 2.1. Some results on singular integral operators with shift Let Γ be a Lyapunov curve and α : Γ → Γ be a direct or an inverse Carleman shift of order 2 (α (α (t)) = t), such that α (t) = 0 on Γ and α (t) ∈ Hµ (Γ). Let W be the corresponding shift operator. We consider the operator 1 (2.1) K = (aI + bW ) P+ + (cI + dW ) P− , P± = (I ± S) 2 in the H¨ older space Hµ (Γ) (a, b, c, d ∈ Hµ (Γ)), and its companion operator = (aI − bW ) P+ + (cI − dW ) P− . K These operators satisfy the following identity 1 K 0 I I I W = AP+ + BP− + D W −W I −W 0 K 2 where, for a direct shift on Γ (α = α+ (t)), a (t) b (t) c (t) d (t) A= , B= b (α (t)) a (α (t)) d (α (t)) c (α (t)) and, for an inverse shift on Γ (α = α− (t)), a (t) d (t) c (t) b (t) A= , B= b (α (t)) c (α (t)) d (α (t)) a (α (t))

(2.2)

(2.3)

(2.4)

and D is a compact operator given by D = (d (t) (W SW − γS) , c (α (t)) (W SW − γS)) with γ = ±1 if α = α± (t), respectively. The following theorem gives us the Fredholm condition of operator (2.1) and its index formula. Theorem 2.1. The operator (2.1) is a Fredholm operator if and only if ∆1 (t) = c (t) c (α (t)) − d (t) d (α (t)) = 0 ∆2 (t) = a (t) a (α (t)) − b (t) b (α (t)) = 0 when α = α+ (t) is a direct Carleman shift on Γ; and ∆ (t) = a (t) c (α (t)) − d (t) b (α (t)) = 0 when α = α− (t) is an inverse Carleman shift on Γ. The index of the Fredholm operator (2.1) is given by 1 ∆1 (t) if α = α+ (t) indK = arg 4π ∆2 (t) Γ 1 {arg ∆ (t)}Γ if α = α− (t) . indK = − 2π

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2.2. Singular integral operators with linear fractional Carleman shift Now we consider the case of a linear fractional Carleman shift on the unit circle = {t ∈ C : |t| = 1}, t−β α (t) = , t∈ , |β| = 1 (2.5) βt − 1 which is a direct or an inverse shift if |β| < 1 or |β| > 1, respectively. The shift function (2.5) admits the following factorization α (t) = α+ (t) tµ α− (t)

(2.6)

−1 where α+ (t) = λ βt − 1 , α− (t) = λ−1 (t − β) t−1 , λ = 1 − |β|2 , µ = −1 −1 , λ = 1 if |β| < 1 and α+ (t) = (iλ) (t − β) , α− (t) = iλt βt − 1

2 |β| − 1 , µ = −1 if |β| > 1. We also have −1 α± (α (t)) = α± (t) if |β| < 1 α± (α (t))

= α∓ (t) if |β| > 1.

Let us introduce the weighted shift operators (U ϕ) (t) = kγ (t) (W ϕ) (t) , where the functions kγ (t), γ = ±1, are deﬁned by the formulas k1 (t) = k−1 (t) =

−α+ (t) if |β| < 1 α− (t) t−1 if |β| > 1.

Then, kγ (t) kγ (α (t)) ≡ 1, γ = ±1 and U 2 = I , U S = γSU , γ = ±1. It is possible to obtain the defect numbers and also to construct the defect subpaces of the singular integral operator K = (aI + bU ) P+ + (cI + dU ) P− . In particular, it’s defect numbers can be computed using the following theorems, and Theorem 2.1. Theorem 2.2. Let α = α+ (t) be a linear fractional Carleman shift. Let κ1 ≥ κ2 be the partial indices of the matrix C = A−1 B and let ε ∈ {−1, 1}. Then  if κ1 ≤ 0  0 κ1 1−ε if κ1 > 0, κ2 ≤ 0 . + dim ker K =  κ12+κ2 4 = indK if κ2 > 0 2 This theorem can be found in [6]. In the same paper, Proposition 2.2 gives a formula to calculate the number ε, which is uniquely determined by the matrix function C. For the case of an inverse linear fractional Carleman shift we have the following result (see [1], [5]):

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Theorem 2.3. Let α = α− (t) be a linear fractional Carleman shift. Let κ1 ≥ κ2 be the partial indices of the matrix C = A−1 B and let ε ∈ {−1, 1}. Then  0 if κ1 ≤ 0    κ1 if κ1 > 0 and κ2 ≤ 0 and κ1 is even 2 dim ker K = . κ1 −ε if κ1 > 0 and κ2 ≤ 0 and κ1 is odd  2   κ1 +κ2 = indK if κ2 > 0 2 Again, the number ε is uniquely determined by the matrix function C (see [1] or Theorem 1 of section 21.2 of [8]) 2.3. Factorization of Hermitian matrix functions We introduce the Hankel operator H (G) = P− GP+ . Together with this operator we shall consider the operator 1 ∗ H (G) H (G) 2 = |H (G)| ∗

where H (G) is the adjoint operator of H (G). Let s∞ (G) be the right-end of the condensation spectrum (i.e., the set of limit points of the spectrum and eigenvalues of inﬁnite multiplicity) of operator |H (G)|. Let s0 (G) ≥ s1 (G) ≥ · · · be a sequence of eigenvalues of operator H (G), the so called s-numbers of H (G). When the number q of terms in this sequence is ﬁnite, then, by deﬁnition, we put sq (G) = sq+1 (G) = · · · = s∞ (G). We consider the non-singular Hermitian matrix function a (t) b (t) G (t) = b (t) d (t) ), a and d are real functions. Since det G (t) = 0, the function where a, b, d ∈ Hµ ( 2 ∆ (t) = ad − |b| preserves its sign on . If ∆ (t) > 0, then the partial indices of the matrix function G (t) are equal to zero. Let us assume that ∆ (t) < 0 and that one of the diagonal elements of the matrix function G (t) (for example, d) does not vanish on . Then, we have the representations 2 2 , d2 = |d+ | ∆ = − |∆+ | ±1 where ∆±1 + and d+ belong to the class of functions analytic on |z| < 1 and continuous on |z| ≤ 1. Now, we consider the Hankel operator

H (ω) = P− ωP+ b d+ . d ∆+ The following results can be found in [11].

where ω =

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Theorem 2.4. The partial indices of the matrix function G (t), with ∆ (t) < 0, are equal to l and −l (l ≥ 0), where l is the multiplicity of 1 as an s-number of the operator H (ω). In particular, the partial indices are equal to zero if and only if the operator I − H (ω)∗ H (ω) is invertible. The following theorem gives another method for computing the partial indices of the matrix function G (t). Theorem 2.5. Let m be the smallest possible exact (i.e., calculated with multiplicity taken into account) number of coincidences in |z| < 1 of the rational functions r1 and r2 , having no poles in common in |z| < 1and satisfying on the inequalities |b (t) − d (t) r1 (t)| < −∆ (t) < |b (t) − d (t) r2 (t)| . Then partial indices of the matrix function G (t) are equal to ±m.

3. Generalized Hilbert boundary value problem with linear fractional Carleman shift Let Γ be a simple closed Lyapunov curve dividing the complex plane C in two parts D+ and D− , and let α be a direct or inverse shift on Γ, such that α (t) = 0, t ∈ Γ and α ∈ Hµ (Γ). The generalized Hilbert boundary value problem consists of ﬁnding a function Φ+ (z) = u(x, y) + iv(x, y) , z = x + iy , analytic in the domain D+ , so that the limit values of its real and imaginary parts belong to Hµ (Γ) and satisfy on Γ the condition a(t)u(t) + b(t)u(α(t)) + c(t)v(t) + d(t)v(α(t)) = h(t) ,

(3.1)

where a, b, c, d, h ∈ Hµ (Γ) are real functions. The boundary condition (3.1) can also be written in the form Re{A(t)Φ+ (t) + B(t)Φ+ (α(t))} = h(t) ,

(3.2)

with A (t) = a (t) − ic (t) and B (t) = b (t) − id (t). Notice that, if b (t) = d (t) ≡ 0 we obtain the classical Hilbert (RiemannHilbert) boundary value problem (see, e.g., Gakhov [2], Muskhelishvili [15]). Problem (3.1) (or (3.2)) was proposed by E.G. Khasabov and G.S. Litvinchuk [3]. In their papers [3] and [4], they obtained the Fredholm conditions and the index formula of problem (3.1), with a direct or an inverse Carleman shift α of order 2 (α (α (t)) ≡ t) on Γ. In this section we let γ = +1 or γ = −1 if α is a direct or an inverse shift, respectively. We start by introducing some identities. Put ∆ (t) =

A (t) A (α (t)) − B (t) B (α (t)) ,

θ (t) =

A (t)A (α (t)) − B (t) B (α (t)) ,

V (t) =

B (t) A (α (t)) − B (t)A (α (t)) ,

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The following identities can be directly veriﬁed: 2

2

|θ (t)| − |∆ (t)| = V (t) V (α (t)) ,

(3.3)

θ (t) = θ (α (t)) , ∆ (t) = ∆ (α (t)) , V (t) + V (t) = 0 .

(3.4)

Let us note that from the last identity in (3.4) it follows that ReV = 0, consequently V = iV0 where V0 is a real function. We deﬁne Λ (t, γ) = (1 + γ) ∆ (t) + (1 − γ) θ (t) ,

t∈Γ

and κ = indΓ Λ (t, γ) .

The Fredholm condition and the index I of problem (3.2) are given by the the formulas ∆ (t) = 0, if γ = +1 (3.5) θ (t) = 0, if γ = −1 and I = κ + 1, (3.6) respectively. The defect numbers of problem (3.2) were obtained in the paper [3], but only for some particular cases, namely, when θ (t) ≡ 0 (in the case α = α+ ) or V (t) ≡ 0 (in the cases α = α± ). If α is a linear fractional Carleman shift of order 2 on the unit circle , then the defect numbers of problem (3.2) can be computed. 3.1. The case of a direct linear fractional Carleman shift In this subsection we consider α a direct linear fractional Carleman shift of order 2 on . We start by observing that for Γ = , the study of the solvability theory of problem (3.2) can be reduced to the study of the solvability theory of the singular integral operator with shift T = (AI + BW ) P+ − tAI + α (t) BW P− . (3.7) Indeed, we have 2Re AΦ+ + BΦ+ (α) = (AI + BW ) Φ+ + AI + BW CΦ+ , where CΦ+ = Φ+ Now, if we extend the function Φ+ (z) to D− by deﬁning 1 1 Φ− (z) = Φ+ = Φ+ , for z ∈ D− , z z

(3.8)

on we have Φ+ = Φ− . Introducing a new unknown function Φ− 1 (z) deﬁned by 1 − Φ− 1 (z) = Φ (z) , z we obtain (3.9) 2Re AΦ+ + BΦ+ (α) = (AI + BW ) Φ+ + tAI + α (t) BW Φ− 1 . Finally, as Φ− 1 (∞) = 0, we can write (3.9) in the form (3.7).

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Throughout this subsection, we will consider a direct linear fractional Carleman shift α of the form (2.5) and the weighted shift operator, (U ϕ) (t) = u (t) ϕ (α (t)), with u (t) = −α+ (t), introduced in Subsection 2.2. The following conditions hold U 2 = I , U S = SU and u (t) u (α (t)) ≡ 1 . Hence, the singular integral operator (3.7) can be written in the form T = AI + u−1 BU P+ + aI + bU P− , with a (t) = −

α (t) u−1 (t) V (t) and b (t) = . ∆ (t)

tθ (t) ∆ (t)

From the Fredholm condition (3.5) we have ∆ (t) = 0 and, as u (t) u (α (t)) ≡ 1, the functional operator T1 = AI + u−1 BU is continuously invertible. So it remains to study the operator − , with A (t) = 1 T = P+ + AP −tθ (t) I + α (t) u−1 (t) V (t) U . ∆ (t) Now we can apply the results of Subsection 2.1 to the operator T. We note that, in this case, the compact operator D does not appear in (2.2), because we have U S = SU . Thus, the operator in identity (2.2) has the form 1 −tθ (t) α (t) u−1 (t) V (t) M = P+ + BP− , with B = . tu (t) V (α (t)) −α (t) θ (t) ∆ (t) (3.10) ∆(t) and Theorems 2.1 and Note that det B = tα (t) = 0. Therefore, using M ∆(t)

2.2, we obtain the defect numbers of problem (3.7). To this end, we rewrite matrix B in the form 1 −θ (t) u−1 (t) V (t) t B= 0 (t) u (t) V (α (t)) −θ ∆ (t)

0 α (t)

.

(3.11)

Bearing in mind that V = iV0 (where V0 is a real function), we obtain −1 1 iθ (t) u (t) V0 (t) 0 α (t) i 0 B= (3.12) t 0 0 −i iθ (t) −u (t) V0 (α (t)) ∆ (t) Note that −1 u (t) V0 (t) iθ (t)

iθ (t) −u (t) V0 (α (t))

=

u−1 (t) 0 0 1

B0

1 0 0 u (t)

, (3.13)

where

B0 =

V0 (t) iθ (t) iθ (t) −V0 (α (t))

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is an Hermitian matrix function with negative determinant. Indeed, using (3.3), we have 2

2

2

Λ (t) = |B0 | = −V0 (t) V0 (α (t)) − |θ (t)| = V (t) V (α (t)) − |θ (t)| = − |∆ (t)| . Now, supposing V0 (α (t)) = 0 on , we can apply Theorem 2.5 to the matrix B0 . Hence, the partial indices of matrix B0 are ±m, where m is the smallest possible exact (i.e., calculated with multiplicity taken into account) number of and coincidences in |z| < 1 of the rational functions r1 and r2 , without poles on with no common poles in |z| < 1 which satisfy on the inequalities |iθ (t) + V0 (α (t)) r1 (t)| < −Λ (t) < |iθ (t) + V0 (α (t)) r2 (t)| . Then we have the following factorization of matrix B0 m 0 t + B0 = χ χ− , 0 t−m where the matrix functions χ+ and χ− are analytic in |z| < 1 and |z| > 1, respectively, and det χ± = 0. Let us compute the partial indices of the matrix function B. Using (3.12) and (3.13) we obtain −1 1 0 1 t 0 iu (t) 0 B= . B0 u (t) 0 0 α (t) 0 −i ∆ (t) Bearing in mind the factorization α (t) = α+ (t) tα− (t)(see (2.6)) and the identity u (t) = −α+ (t), it follows −1 t 1 0 1 0 0 1 iu (t) 0 B0 B = 0 −i 0 α+ (t) 0 α− (t) −α+ (t) 0 ∆ (t) + −1 t 0 0 1 1 0 α (t) iu (t) 0 = B0 0 −i 0 α+ (t) −1 0 0 α− (t) ∆ (t) −1 + t 0 1 1 0 0 iu (t) 0 α (t) B0 = −1 0 0 α− (t) 0 α+ (t) 0 −i ∆ (t) t R+ B0 R− , = ∆ (t) −i 0 + where the matrix function R = is analytic in |z| < 1 and 0 −iα+ (t) 0 α− (t) det R+ = 0 in |z| < 1 and the matrix function R− = is analytic −1 0 in |z| > 1 and det R− = 0 in |z| > 1. Finally we observe that the function ∆ admits the following factorization, ∆ (t) = ∆+ (t) t−κ ∆− (t) , where κ=

1 arg ∆ (t) = Ind ∆ (t) , 2π

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and the matrix function B can be rewritten in the form 1 1 0 0 tκ 0 t 0 ∆+ (t) ∆− (t) + B= B R R− . 0 1 1 κ 0 t 0 t 0 0 + − ∆ (t) ∆ (t) From this last equality we conclude that the partial indices of B are κ1 = κ + m + 1 and κ2 = κ − m + 1 . Finally, using Theorem 2.2, we obtain the following result about the solvability theory of the generalized Hilbert boundary value problem (3.2) with a direct linear fractional Carleman shift on . Theorem 3.1. The defect numbers l and ρ, of the generalized Hilbert boundary value problem (3.2) with a direct linear fractional Carleman shift of order 2 on are given by: 1. l = max (0, κ + 1) 2. l = dim ker T = even; 3. l =

κ1 2

+

1−ε 4

κ1 2

, =

=

ρ = max (0, −κ − 1) , if κ1 ≤ 0 or κ2 > 0; κ+m+1 2 κ+m 2 κ+m 2

,

ρ=

−κ+m−1 2

if ε = 1

+ 1 if ε = −1 if κ1 > 0, κ2 ≤ 0 and κ1 is odd.

, if κ1 > 0, κ2 ≤ 0 and κ1 is

, ρ=

−κ+m 2 −κ+m 2

− 1 if ε = 1 if ε = −1

,

Now we consider two particular cases of problem (3.2) with α = α+ : the case ∆ (t) = 0 and θ (t) ≡ 0 and the case ∆ (t) = 0 and V (t) ≡ 0. Then, either 1 0 F (t) , with F (t) = α (t) u−1 (t) V (t) , B= F (α (t)) 0 ∆ (t) or 1 −tθ (t) 0 B= . 0 −α (t) θ (t) ∆ (t) From Ind F (t) = 1 and Ind θ (t) = 0 we conclude that in both cases l = max (0, κ + 1) and ρ = max (0, −κ − 1), i.e., the solvability theory is given by the Gakhov-Coburn formulas. We ﬁnish this Subsection with an example of the generalized Hilbert boundary value problem (3.2), whose defect numbers, l and ρ, do not always satisfy the Gakhov-Coburn formulas (1.1). Example 3.2. Let us consider the generalized Hilbert problem Re t−k [u (t) + iv (−t)] = h (t) on , with k ∈ Z .

(3.14)

First of all we note that, for t = eiϕ , with ϕ ∈ [0, 2π[, we have tk = cos (kϕ) + i sin (kϕ). Therefore problem (3.14) can be written in the form Re cos (kϕ) Φ+ (t) − i sin (kϕ) Φ+ (α (t)) = h (t) , t = eiϕ ∈ ,

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where α (t) = −t is a particular case (β = 0) of a direct linear fractional Carleman shift and Φ+ (t) = u (t) + iv (t). Hence, comparing (3.14) and (3.2), we can see that A (t) = cos (kϕ) = Re t−k and B (t) = −i sin (kϕ) = iIm t−k . Then t2k + t−2k t2k − t−2k , V (t) = − (−1)k . 2 2 From this we can see that the Fredholm condition of problem (3.14) is fulﬁlled (∆ (t) = 0) and, as κ = Ind ∆ (t) = 0, its index is I = κ + 1 = 1. We note that, as condition V (t) = 0 is not satisﬁed, we can not apply Theorems 2.4 and 2.5 to obtain the partial indices of matrix B (see (3.11)). However, for this example, the matrix function B takes the form 2k 1 t 0 t + t−2k t2k − t−2k . B=− 0 −t t2k − t−2k t2k + t−2k 2 ∆ (t) = (−1)k , θ (t) = (−1)k

Since the ﬁrst of these matrices is circulant (see e.g. [13], p. 159), we have 2k 2k t + t−2k t2k − t−2k 1 1 t 1 1 0 = . (3.15) 1 −1 1 −1 0 t−2k t2k − t−2k t2k + t−2k Thus, B

1 = − 2 1 = − 2

1 1 1 −1 1 1 1 −1

t2k 0

0

t−2k

t2k+1 0

0 t−2k+1

1 1 t 0 1 1 −1 0 t 0 1 −1 . 1 1

0 −1

(3.16)

Hence, the partial indices of the matrix function B are given by: κ1

= 2k + 1 and κ2 = −2k + 1 ,

if k ≥ 0

κ1

= −2k + 1

if k < 0 .

and κ2 = 2k + 1 ,

Finally we compute the number ε. It can be obtained from the following identity (see Proposition 2.2 of [6]) ε p (3.17) = (Λ+ )−1 (B + )−1 (α) eB + , 0 −ε κ κ where Λ+ = diag (α+ ) 1 , (α+ ) 2 , p is a polynomial of degree at most κ1 − κ2 . For the case k ≥ 0, we have 1 1 1 1 −1 + − B = and B = − , 1 −1 1 1 2 and it follows that ε = −1. Applying Theorem 3.1, we obtain that the number of linearly independent solutions of the generalized Hilbert boundary value problem (3.14) is given by l = k + 1 and this problem is solvable under the fulﬁllment of ρ = k solvability conditions. When k < 0, we need another factorization of matrix B. Observing that, −2k+1 2k+1 0 0 t 0 1 t =e e, with e = , 1 0 0 t−2k+1 0 t2k+1

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from (3.16), we obtain −2k+1 1 1 1 0 1 1 t B=− . 1 −1 −1 1 0 t2k+1 2 1 1 1 1 1 + − , B = −2 , and from (3.17) we Now we have B = −1 1 1 −1 obtain ε = 1. Again as a result of Theorem 3.1, the defect numbers of problem (3.14) are given by l = −k and ρ = −k − 1. Remark 3.3. In the previous example, the Gakhov-Coburn formulas (1.1) hold only if k = 0 or k = −1. Moreover, only in those two cases the non-homogeneous problem (3.14) is unconditionally solvable. We note that problem (3.14) can be reduced to a system of two Schwartz problems, whose solutions are found in the class of even and odd analytic functions (see [4] and also [8], example 23.1). 3.2. The case of an inverse linear fractional Carleman shift Now we will study the solvability theory of the generalized Hilbert boundary value problem (3.2) when Γ = and α is an inverse linear fractional Carleman shift of order 2 (see formula (2.5)). Here the Fredholm condition takes the form θ (t) = 0. Besides, κ = Ind θ (t) and the index of problem (3.2) is given by the formula I = κ + 1. In this subsection we consider the operator U deﬁned by (U ϕ) (t) = u (t) ϕ (α (t)) , with u (t) = α− (t) t−1 , which fulﬁlls the following conditions U 2 = I , U S = −SU and u (t) u (α (t)) ≡ 1 . As we have seen in the previous subsection, the study of the solvability theory of problem (3.2) can be reduced to the study of the solvability theory of operator (3.7), which can be written in the form T = A (t) I + u−1 (t) B (t) U P+ − tA (t)I + α (t) u−1 (t) B (t)U P− . (3.18) Once again, the matricial identity (2.2) does not contain the compact operator D because of the fact that U S = −SU . Then the solvability theory of operator T can be obtained from the solvability theory of the singular integral operator without shift (see (2.2) and (2.4)) A=

M = AP+ + BP− , with u−1 (t) B (t) tA (t) A (t) α (t) u−1 (t) B (t) B= u (t) B (α (t)) α (t) A (α (t)) tu (t) B (α (t)) A (α (t))

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From the Fredholm condition θ (t) = 0 we have det A = α (t) θ (t) = 0, therefore A is an invertible matrix and 1 α (t) A (α (t)) −α (t) u−1 (t) B (t) −1 A = . −u (t) B (α (t)) A (t) α (t) θ (t) Hence, it is enough to study = P+ + CP− , M with −1

C =A

B=

1 α (t) θ (t)

tα (t) ∆ (t) −tu (t) V (α (t))

u−1 (t) α (t) V (t) ∆ (t)

(3.19)

As det C = tθ(t) = 0, according to Theorems 2.1 and 2.3, the defect α(t)θ(t) numbers of the operator T can be obtained from a factorization of the matrix C. We start by rewriting the matrix C in the form 1 α (t) u−1 (t) 0 tu (t) 0 ∆ (t) V (t) C= . 0 1 0 1 −V (α (t)) ∆ (t) α (t) θ (t) Since V = iV0 (where V0 is a real function), we obtain 1 0 1 tu (t) 0 iα (t) u−1 (t) 0 C= C0 , 0 −i 1 0 0 1 α (t) θ (t) where

C0 =

V0 (t) i∆ (t)

i∆ (t) V0 (α (t))

(3.20)

is an Hermitian matrix function with negative determinant. In fact, using (3.3), we obtain |C0 | = V0 (t) V0 (α (t)) − |∆ (t)|2 = −V (t) V (α (t)) − |∆ (t)|2 = − |θ (t)|2 . If we impose the condition V0 (α (t)) = 0, t ∈ and apply Theorem 2.5 to the matrix C0 , arguments similar to those used in Subsection 3.1 (where we have considered matrix B0 ) allow us to conclude that the partial indices of C0 are ±m. Then m 0 t + (3.21) C0 = C0 C0− . 0 t−m Using the factorization α (t) = α+ (t) t−1 α− (t) and the fact that u (t) = α (t) t−1 , from (3.20) we obtain m + 1 0 t 0 1 iα (t) 0 + − C = C0 C0 0 −i 0 t−m α− (t) 0 α (t) θ (t) m 1 0 0 t α(t)θ(t) + = R R− , 1 0 t−m 0 −

α(t)θ(t)

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C0+ is analytic in |z| < 1 with 0 1 − + − det R = 0 in |z| < 1, and the matrix function R = C0 is analytic α− (t) 0 − in |z| > 1 with det R = 0 in |z| > 1. We also observe that the ﬁrst matrix admits the following factorization 1 0 α(t)θ(t) where the matrix function R+ =

0 =t

κ+1

1 α(t)θ(t) 1 α+ (t)θ + (t)

0

0 1 α+ (t)θ + (t)

1 α− (t)θ − (t)

0

0 1 α− (t)θ − (t)

+ κ − where κ = Ind θ (t), and θ (t) = θ (t) t θ (t). Finally we obtain κ+1+m 0 t + C=G G− , 0 tκ+1−m

where

+

G =

1 α+ (t)θ + (t)

0

0

+

−

R ,G =R

1 α+ (t)θ + (t)

−

1 α− (t)θ − (t)

0

(3.22)

0

1 α− (t)θ − (t)

.

From (3.22) we conclude that the partial indices of the matrix function C are κ1 = κ + 1 + m

and κ2 = κ + 1 − m .

Now we can apply Theorem 2.3 to obtain the following result on the defect numbers of the generalized Hilbert boundary value problem (3.2). Theorem 3.4. The defect numbers l and ρ, of the generalized Hilbert boundary value problem (3.2) with an inverse linear fractional Carleman shift of order 2 of the form (2.5) on are given by: 1. l = max (0, κ + 1) 2. l =

κ1 2

=

κ1 −ε 2

3. l = is odd.

κ+1+m 2

=

and and

κ+1+m−ε 2

ρ = max (0, −κ − 1) , if κ1 ≤ 0 or κ2 > 0, ρ= and

−κ−1+m 2

ρ=

, if κ1 > 0 and κ2 ≤ 0 and κ1 is even,

−κ−1+m−ε 2

, if κ1 > 0 and κ2 ≤ 0 and κ1

If V (t) ≡ 0, we have a degenerate case for which the matrix C, introduced in (3.19), takes the form 1 tα (t) ∆ (t) 0 C= . 0 ∆ (t) α (t) θ (t) According to the second identity in (3.4), {arg ∆ (t)}Γ = 0. Therefore, the partial indices of the matrix C are κ1 = κ2 = κ + 1, with κ = Ind θ (t). Hence, as a result of Theorem 2.5, we get l = max (0, κ + 1) and ρ = max (0, −κ − 1).

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In the rest of this subsection we will consider the shift α (t) = 1t . First of all, we note that this is an inverse Carleman shift (α (α (t)) ≡ t) on , but it is not a particular case of a linear fractional Carleman shift of the form (2.5). Together with this shift we consider the so called flip operator U , deﬁned by (U ϕ) (t) = 1t ϕ 1t . It is easy to prove that all the previous results for the case of an inverse linear fractional Carleman shift of the form (2.5) remain valid in these conditions. In fact, it is enough to replace, in the above procedure, u (t) by t−1 and to observe that, since α (t) = t−1 , we have α+ (t) = α− (t) = 1 (see (2.6)). In this case, the defect numbers can be obtained with the aid of Theorem 3.4 and a factorization of the matrix (c.f. (3.19)) t ∆ (t) V (t) C= . (3.23) −V (α (t)) ∆ (t) θ (t) Finally we present an example of the generalized Hilbert boundary value problem with the shift α (t) = 1t , for which the defect numbers l and ρ do not always fulﬁll the Gakhov-Coburn formulas (1.1). Example 3.5. We consider the generalized Hilbert boundary value problem 1 −k Re t u (t) + iv = h (t) on , with k ∈ Z . (3.24) t As in Example 3.2, problem (3.24) can be written in the form (3.2) with α (t) = 1t , A (t) = cos (kϕ) = Re t−k and B (t) = −i sin (kϕ) = iIm t−k , where t = eiϕ , ϕ ∈ [0, 2π[. We have t2k − t−2k t2k + t−2k , V (t) = − . 2 2 In particular, since θ (t) = 1 = 0, the Fredholm condition of problem (3.24) is satisﬁed. Besides, κ = Ind θ (t) = 0 and, consequently, the index of this problem is I = κ + 1 = 1. As (U ϕ) (t) = 1t ϕ 1t , taking into account the remarks made before this example, we obtain (see (3.23)) t t2k + t−2k − t2k − t−2k C = = − t2k − t−2k t2k + t−2k 2 2k 1 −1 0 t 0 −1 0 t + t−2k t2k − t−2k = . 0 1 0 t 0 1 t2k − t−2k t2k + t−2k 2 θ (t) = 1 , ∆ (t) =

From factorization (3.15), we get 2k+1 1 −1 −1 t C= 1 −1 0 2

0

t−2k+1

−1 −1

1 −1

Hence, the partial indices of the matrix function C are: κ1

= 2k + 1 and κ2 = −2k + 1 ,

if k ≥ 0

κ1

= −2k + 1

if k < 0 .

and κ2 = 2k + 1 ,

.

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It remains to ﬁnd the number ε mentioned in Theorem 2.3. It can be obtained from the factorization of matrix C and the identity ε p Λ+ C − (α) eC + = , (3.25) 0 −ε κ1 κ2 κ κ t−β where Λ+ (t) = diag (α+ (t)) 1 , (α+ (t)) 2 = diag , e = , t−β iλ iλ 0 1 , and p is a polynomial of degree at most κ1 − κ2 (see [1] ). 1 0 −1 1 −1 −1 If k ≥ 0, putting C + = 12 and using and C − = −1 −1 1 −1 (3.25) (with α+ (t) = α− (t) = 1) we get ε = −1. Thus, applying Theorem 3.4, we conclude that the homogeneous problem corresponding to problem (3.24) has k +1 linearly independent solutions (l = κ + 1) and the non-homogeneous problem (3.24) is solvable if and only if k solvability conditions are fulﬁlled (ρ = κ). For k < 0, proceeding as in the corresponding case of Example 3.2 we get the following factorization of matrix C −2k+1 1 0 −1 −1 −1 −1 t C= 0 t2k+1 −1 1 −1 1 2 −1 −1 −1 −1 Therefore, C + = 12 and ε = 1. Hence, from , C− = −1 1 −1 1 Theorem 3.4, we obtain that problem (3.24) is solvable if and only if −k − 1 solvability conditions hold (ρ = −k − 1). Moreover, in this case, problem (3.24) has −k linearly independent solutions (l = −k). Remark 3.6. The non-homogeneous problem (3.24) is only unconditionally solvable if k = 0 or k = −1. These are also the only two cases for which the defect numbers l and ρ fulﬁl the Gakhov-Coburn formulas (1.1). Note that problem (3.24) can be reduced to a system of two Schwartz problems, whose solutions are found in the class of functions with real and imaginary coeﬃcients of their expansions in Taylor series (see [4] and also [8], example 23.2).

4. Generalized Carleman boundary value problem with linear fractional shift A generalized Carleman boundary value problem can be formulated as follows (see, for example [8]): Let D+ be a simply connected domain bounded by a Lyapunov curve Γ. Find the function Φ+ (z) analytic in the domain D+ , whose limit values belong to the class Hµ (Γ) and satisfy on Γ the condition Φ+ (α (t)) = a (t) Φ+ (t) + b (t) Φ+ (t)

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(homogeneous problem) or Φ+ (α (t)) = a (t) Φ+ (t) + b (t) Φ+ (t) + h (t)

(4.1)

(non-homogeneous problem). Here α is a direct or an inverse Carleman shift of order 2 on Γ, α (t) = 0; α (t), a (t), b (t), h (t) ∈ Hµ (Γ). With this formulation, problem (4.1) is overdetermined (see, for example [8]), thus we suppose that the functions a, b and h satisfy the conditions a (t) a (α (t)) + b (t) b (α (t)) = 1,

(4.2)

a (t) b (α (t)) + a (α (t))b (t) = 0,

(4.3)

a (t) h (α (t)) + b (t) h (α (t)) + h (t) = 0.

(4.4)

From (4.2) and (4.3) it follows that, for all elements in Γ, the coeﬃcients a, b ∈ Hµ (Γ) satisfy either the inequality |b (t)| > |a (t)| or the inequality |b (t)| < |a (t)|. Thus the function γ (t) = |a (t)| − |b (t)| preserves its sign on Γ. Note that, if f1 and f2 are linearly independent functions belonging to the space Hµ (Γ) such that f1 (t) f2 (t) F (t) = det = 0, f1 (t) f2 (t) then the functions f1 (α (t)) f2 (α (t)) det f1 (t) f2 (t) a (t) = F (t)

det ,

b (t) =

f1 (t) f2 (t) f1 (α (t)) f2 (α (t)) F (t)

,

satisfy conditions (4.2) and (4.3). The Fredholm theory of problem (4.1) was constructed by G. S. Litvinchuk and A. P. Nechaev [9], [10] (see also e.g. [8], Ch. 7). They proved that problem (4.1) is Fredholm only in the following four cases: (1) α is a direct Carleman shift and |b (t)| > |a (t)|; (2) α is a direct Carleman shift and |a (t)| > |b (t)| > 0; (3) α is an inverse Carleman shift and |b (t)| > |a (t)| > 0; (4) α is an inverse Carleman shift and |a (t)| > |b (t)|. For (1) and (2) the index of problem (4.1) is calculated by the formula 1 {arg b (t)}Γ + 1, I= 2π and, in cases (3) and (4) it is given by 1 1 {arg a (t)}Γ + 1 and I = 2 − {arg a (t)}Γ − m− , I=− 2π 2π respectively, where m− is the number of ﬁxed points of α for which a (t) = −1. In [9], [10] it was also established that, in cases (1) and (4), the problem of obtaining the defect numbers, l and ρ, is solved for any Carleman shift α = α+ (t) or α = α− (t). Moreover, the Gakhov-Coburn formulas (1.1) are satisﬁed. For the remaining cases (2) and (3), this problem remained unsolved, until now.

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Considering a direct or an inverse linear fractional Carleman shift on the unit circle , we can obtain the defect numbers for the boundary value problem (4.1) in all cases (1)-(4). 4.1. The case of a direct linear fractional Carleman shift Let α be a direct Carleman shift of the form (2.5). If we introduce a new unknown function Φ− (z), analytic in |z| > 1, by the formula 1 + 1 − Φ (z) = Φ , z z then the generalized Carleman boundary value problem (4.1) is reduced to the boundary value problem Φ+ (α (t)) = a (t) Φ+ (t) + tb (t) Φ− (t) + h (t) ,

(4.5)

for a piecewise analytic function {Φ+ (z) , Φ− (z)} vanishing at inﬁnity. According to [7], problems (4.1) and (4.5) are equivalent, i.e., they are Fredholm boundary value problems simultaneously and they have the same defect numbers over the ﬁelds R and C, respectively. The boundary value problem (4.5) can be reduced to the singular integral operator with shift K ≡ u−1 (t) U P+ − a (t) P+ + tb (t) P− ,

(4.6)

where (U ϕ) (t) = u (t) ϕ (α (t)), u (t) = −α+ (t) and u (t) u (α (t)) = 1. Since U S = SU , the solvability theory of the singular integral operator (4.6) can be obtained by studying the singular integral operator without shift M = AP+ + BP− , where (see formulas (2.2) and (2.3) of Subsection 2.1) −a (t) u−1 (t) tb (t) 0 A= , B= . u−1 (α (t)) −a (α (t)) 0 α (t) b (α (t)) This operator can be rewritten in the form M = A−1 (P+ + CP− ) , with C = A−1 B. Thus, it is enough to consider the operator = P+ + CP− . M Let us determine the matrix function C explicitly. In fact, from conditions (4.2)-(4.4) and using the equality u (t) = u−1 (α (t)), we obtain a (α (t)) =

a (t)

2

2

|a (t)| − |b (t)|

,

b (α (t)) = −

b (t)

2

2

|a (t)| − |b (t)|

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tb (t) 0 0 α (t) b (α (t)) b (t) b (α (t)) −1 1 t 0 a (α (t)) b (t) u (t) b (α (t)) = a (t) b (α (t)) 0 α (t) u−1 (α (t)) b (t) b (t) b (α (t)) u−1 (t) −a (t) −1 t 0 = b . 2 2 0 α (t) a (t) u (t) |b (t)| − |a (t)| 1

C=

a (α (t)) u−1 (t) −1 u (α (t)) a (t)

Hence, the matrix function C = A−1 B can be rewritten in the form −1 −t 0 0 1 |a (t)|2 − |b (t)|2 a (t) . C= b 0 u−1 (t) α (t) u (t) 0 a (t) 1 (4.7) We can see that the matrix function 2 2 |a (t)| − |b (t)| a (t) = C a (t) 1 2

is Hermitian with determinant − |b (t)| < 0. Therefore, applying Theorem 2.5 of are m Subsection 2.3, we obtain that the partial indices of the matrix function C and −m, where m is the smallest possible exact (i.e., calculated with multiplicity taken into account) number of coincidences in |z| < 1 of the rational functions r1 and r2 , having no poles in common in |z| < 1 and satisfying on the inequalities |a (t) − r1 (t)| < |b (t)| < |a (t) − r2 (t)| . Then,

0 tm − . C 0 t−m Using the formulas α (t) = α+ (t) tα− (t), u (t) = −α+ (t) and u−1 (t) = −1 − (α+ (t)) , we obtain m −1 0 1 t t 0 −1 0 0 + − C C C= b . −α+ (t) 0 0 t 0 −α− (t) 0 t−m =C + C

Thus, the partial indices of C are κ1 = κ + m + 1

and κ2 = κ − m + 1,

1 2π

{arg b (t)} where κ = . Now, as a result of Theorem 2.2, we obtain the defect numbers of the generalized Carleman boundary value problem (4.1) with a direct linear fractional Carleman shift. Theorem 4.1. If α is a direct linear fractional Carleman shift of order 2, then, for the defect numbers l and ρ of the generalized Carleman boundary value problem, we have the following possibilities: 1) If κ1 > 0 and κ2 > 0, then l = κ + 1

and

ρ = 0.

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2) If κ1 > 0 and κ2 ≤ 0, then  κ+1+m −κ − 1 + m   and ρ =  l= 2 2    l = κ + 1 + m − ε and ρ = −κ − 1 + m − ε 2 2 where ε ∈ {1, −1}. 3) If κ2 ≤ κ1 ≤ 0, then l = 0

and

, if κ1 is even , , if κ1 is odd

ρ = −κ − 1.

This theorem remains valid if either one of the inequalities |b (t)| > |a (t)| or 0 < |b (t)| < |a (t)| holds for all elements in . Note that when |b (t)| > |a (t)|, the partial indices of 2 2 −1 0 |b (t)| − |a (t)| a (t) = C 0 1 1 −a (t) are zero (see Theorem 1, chapter 4 of [8]), i.e., m = 0. Thus, from Theorem 4.1, we obtain, l = max (0, κ + 1) and ρ = max (0, −κ − 1), which was already known for problem (4.1), with any direct Carleman shift on any Lyapunov curve. 4.2. The case of an inverse linear fractional Carleman shift In this subsection we consider an inverse linear fractional Carleman shift of order 2. Now we have u (t) = α− (t) t−1 , where (U ϕ) (t) = α− (t) t−1 ϕ (α (t)). Proceeding as in the previous subsection, in order to obtain the defect numbers of operator (4.6) we consider the singular integral operator without shift M = AP+ + BP− , where

A=

−a (t) u−1 (α (t))

0 α (t) b (α (t))

, B=

tb (t) u−1 (t) 0 −a (α (t))

.

If a (t) b (t) = 0, then det A (t) = 0 and, as before, we just need to study the solvability theory of = P+ + CP− , M with C = A−1 B. Computing the matrix function C, we obtain 1 tb (t) α (t) b (α (t)) 0 C=− 0 −u−1 (α (t)) −a (t) α (t) a (t) b (α (t)) =

1 a (t)

0 −1 (α (t)) u (t)

−1 0

|b (t)|2 − |a (t)|2 b (t)

b (t) 1

u−1 (t) −a (α (t)) t 0

0

u−1 (t) (4.8)

Again, we can see that the matrix function 2 2 |b (t)| − |a (t)| = C b (t)

b (t) 1

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is Hermitian with negative determinant. Once more, applying Theorem 2.5 we are m and −m (m ≥ 0). Therefore, the matrix obtain that the partial indices of C function C admits the following factorization m 1 0 0 1 1 0 t t 0 + − C C C= −1 −1 0 t−m 0 t 0 0 (α− (t)) (α+ (t)) a (t) and its partial indices are κ1 = −κ + m + 1 and κ2 = −κ − m + 1, 1 {arg a (t)} where κ = . Now, using Theorem 2.3 we have 2π Theorem 4.2. If α is an inverse linear fractional Carleman shift of order 2, then, for the defect numbers l and ρ of the generalized Carleman boundary value problem, we have the following possibilities: 1) If κ1 > 0 and κ2 > 0, then l = −κ + 1

and

ρ = 0.

2) If κ1 > 0 and κ2 ≤ 0, then  −κ + 1 + m κ−1+m   and ρ =  l= 2 2    l = −κ + 1 + m − ε and ρ = κ − 1 + m − ε 2 2 where ε ∈ {1, −1}. 3) If κ2 ≤ κ1 ≤ 0, then l = 0

and

, if κ1 is even , , if κ1 is odd

ρ = κ − 1.

Note that this theorem remains valid, if we consider the flip operator which is given by (U ϕ) (t) = 1t ϕ 1t . In that case the matrix function C admits the following factorization m 0 0 −1 t 0 −1 + t C C C = (a (t)) . − 0 t−m 1 0 0 t We must notice that Theorem 4.2 is valid only if the condition a (t) b (t) = 0 is fulﬁlled. Thus, when |b (t)| > |a (t)| > 0, Theorem 4.2 holds. However, if |a (t)| > |b (t)|, then Theorem 4.2 is not valid. In fact, if we admitted that Theorem 4.2 was satisﬁed, we would have |a (t)| > |b (t)| > 0, which is contradictory. Indeed, if t0 is one of the ﬁxed points of α, it follows from the conditions (4.2) and (4.3) that: (a (t0 ))2 + |b (t0 )|2 = 1, a (t0 ) + a (t0 ) b (t0 ) = 0.

(4.9) (4.10)

Since it is necessary that a (t) b (t) = 0, we have b (t0 ) = 0. Then, from equality (4.10), we get a (t0 )+a (t0 ) = 0, i.e., a (t0 ) = iβ, β = 0. Now, from the equality (4.9) 2 2 it follows that |b (t0 )| = 1−(a (t0 ))2 = 1+β 2 = 1+|a (t0 )| . Thus, |b (t0 )| > |a (t0 )|, which is a contradiction.

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Nevertheless, in the case |a (t)| > |b (t)|, it is already known (see [9], [10], and also [8], Ch. 7) that, for any shift α = α− (t) on any Lyapunov curve Γ, 1 l = max 0, 2 − {arg a (t)}Γ − m− 2π 1 {arg a (t)}Γ + m− , ρ = max 0, −2 + 2π where m− is the number of ﬁxed points of α for which a (t) = −1.

References [1] Drekova, G. V., Kravchenko, V. G., Dimension and structure of the kernel and cokernel of a singular integral operator with linear-fractional Carleman shift and with conjugation. Dokl. Akad. Nauk SSSR, 315(2), 271-274, 1990 (in Russian). English transl.: Soviet Math. Dokl., 42(3), 743-746, 1991. [2] Gakhov, F. D., Boundary value problems. Nauka: Moscow, 1977 (in Russian). English transl.: Perganion Press: Oxford, 1966. [3] Khasabov, E. G., Litvinchuk, G. S., On Hilbert boundary value problem with a shift. Dokl. Akad. Nauk SSSR, 142(6), 274-277, 1962 (in Russian). [4] Khasabov, E. G., Litvinchuk, G. S., On the index of generalized Hilbert boundary value problem. Uspekhi matem. nauk, 20(2), 124-130, 1965 (in Russian). [5] Kovaliova (Drekova), G. V., Kravchenko, V. G., On solutions of singular integral equations with linear fractional Carleman shift and with conjugation. DEP. Odessk. injenerno-stroit. Inst., Odessa, 1-14, 1990 (in Russian). [6] Kravchenko, V. G., Lebre, A. B., Rodr´ıguez, J. S., Factorization of singular integral operators with a Carleman shift and spectral problems. Journal of Integral Equations and Applications, 13(4), 339-383, Winter 2001. [7] Latushkin, Yu. D., Litvinchuk, G. S., Spitkovsky, I. M., To the theory of boundary value problem of N. Vekua, Trudy Tbilissk. Univ., 259, 163-188, 1985 (in Russian). [8] Litvinchuk, G. S., Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer Academic Publishers: Dordrecht-Boston-London, 1-378, 2000. [9] Litvinchuk, G. S., Nechaev, A. P., On the theory of generalized Carleman boundary value problem. Dokl. Akad. Nauk SSSR, 189(1), 38-41, 1969 (in Russian). [10] Litvinchuk, G. S., Nechaev, A. P., Generalized Carleman boundary value problem. Matem. Sb., 82(1), 30-54, 1970 (in Russian). [11] Litvinchuk, G. S., Spitkovsky, I. M., Sharp estimates of the defect numbers of a generalized Riemann boundary value problem. Dokl. Akad. Nauk SSSR, 255(5), 10421046, 1980 (in Russian). English transl.: Soviet Math. Dokl. 22(3), 781-785, 1980. [12] Litvinchuk, G. S., Spitkovsky, I. M., Sharp estimates of the defect numbers of a generalized Riemann boundary value problem, factorization of Hermitian matrixvalued functions and some problems of approximation by meromorphic functions. Matem. Sb., 117(159), No 2, 196-215, 1982 (in Russian). English transl.: Math USSR Sbornik, Vol. 45, No 2, 205-224, 1983.

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[13] Litvinchuk, G. S., Spitkovsky, I. M., Factorization of measurable matrix functions. Academie-Verlag: Berlin and Birkh¨ auser Verlag: Basel-Boston-Stuttgart, 1372, 1987. [14] Markushevich, A. I., On one boundary value problem of the theory of analytic functions. Uchen. Zapiski Moskovsk. Univer., 100, 20-30, 1946 (in Russian). [15] Muskhelishvili, N. I., Singular integral equations. Fizmatgiz: Moscow, 1962 (in Russian). English transl.: Groningen, Noordholf, 1953. Jorge Ferreira Departamento de Matem´ atica e Engenharias - Universidade da Madeira Campus Universit´ ario da Penteada, 9000-390 Funchal Portugal e-mail: [email protected] Gueorgui S. Litvinchuk Departamento de Matem´ atica e Engenharias - Universidade da Madeira Campus Universit´ ario da Penteada, 9000-390 Funchal Portugal and Ukrainian Academy of Sciences Marine Hydrophysical Institute - Hydroacoustic Department Preobrazhenskaya Street 3, 270 100 Odessa Ukraine e-mail: [email protected] Maur´ıcio D. L. Reis Departamento de Matem´ atica e Engenharias - Universidade da Madeira Campus Universit´ ario da Penteada, 9000-390 Funchal Portugal e-mail: m [email protected] Submitted: June 15, 2005 Revised: February 21, 2006

Integr. equ. oper. theory 57 (2007), 209–215 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020209-7, published online August 8, 2006 DOI 10.1007/s00020-006-1452-2

Integral Equations and Operator Theory

On the Range of the Aluthge Transform Guoxing Ji, Yongfeng Pang and Ze Li Abstract. Let B(H) be the algebra of all bounded linear operators on a com1 1 plex separable Hilbert space H. For an operator T ∈ B(H), let Te = |T | 2 U |T | 2 be the Aluthge transform of T and we deﬁne ∆(T ) = Te for all ∈ B(H), where T = U |T | is a polar decomposition of T . In this short note, we consider an elementary property of the range R(∆) = {Te : T ∈ B(H)} of ∆. We prove that R(∆) is neither closed nor dense in B(H). However R(∆) is strongly dense if H is inﬁnite dimensional. Mathematics Subject Classification (2000). Primary 47A15; Secondary 47B20. Keywords. Aluthge transform, polar decomposition, range.

Let H be a complex separable Hilbert space and let B(H) be the algebra of all bounded linear operators on H. Denoted by F (H) and K(H) the ideals of all ﬁnite rank and compact operators in B(H) respectively if H is inﬁnite dimensional. I is the identity operator in B(H). If H = Cp for p < ∞, we identify B(H) with Mp and denote by Ip the identity. Let T = U |T | be a polar decomposition of T 1 with |T | = (T ∗ T ) 2 and U is a partial isometry with the initial space the closure of the range of |T | and the ﬁnal space the closure of the range of T . The Aluthge 1 1 transform T is deﬁned by T=|T | 2 U |T | 2 (cf.[1]). Note that T is independent of the choice of the partial isometry U in the polar decomposition of T . We deﬁne ∆(T ) = T for all T ∈ B(H). Then ∆ is a map deﬁned on B(H). Recently, T and T have been studied by many authors (cf.[2, 3, 4, 5, 6, 7]). However we know very few properties on the range R(∆) = {T : T ∈ B(H)} of ∆. In this short note, we consider an elementary property of R(∆). We prove that R(∆) is neither closed nor dense in B(H). However R(∆) is strongly dense if H is inﬁnite dimensional. We ﬁrst consider ﬁnite dimensional case. Let dim H = p. We identify B(H) with the set of all p × p matrices Mp . Let d1 , d2 , . . . , dp be p complex numbers. We deﬁne diag(d1 , d2 , . . . , dp ) to be the diagonal matrix with diagonal {d1 , d2 , . . . , dp }. This research was supported in part by the National Natural Science Foundation of China (No. 10571114) and the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2005A1).

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Let T ∈ Mp with the polar decomposition T = U |T |. We may assume that U is 1 a unitary matrix. For the positive matrix |T | 2 ∈ Mp , there are a diagonal matrix D = diag(d1 , d2 , . . . , dp ) and a unitary matrix V such that d1 ≥ d2 ≥ · · · ≥ dp and 1 V |T | 2 V ∗ = D. Then we have V TV ∗ = DW D, where W = V U V ∗ is unitary. For a matrix X we denote by rank(X) the rank of X. x y Lemma 1. Let p = 2 and A = , where x and y are any nonzero complex −y 0 numbers. Then A ∈ R(∆). Proof. Note that λR(∆) ⊆ R(∆) for any λ ∈ C. We may assume that y = 1 without loss of generality. Suppose there is a matrix T ∈ M2 such that A = ∆(T ). Let T = U |T | be the polar decomposition of T . Let {e1 , e2 } be the canonical 1 1 1 1 basis of C2 . Then we have (|T | 2 U |T | 2 e2 , e2 ) = (U |T | 2 e2 , |T | 2 e2 ) = 0. We know 1 1 1 that |T | 2 e2 = 0 since A is invertible. Put η = |T | 2 e2 −1 |T | 2 e2 and choose a unit vector ξ such that ξ⊥η. Then we have (U η, η) = 0, which implies that U η = aξ and U ξ = bη for some constants a and b in C such that |a| = |b| = 1 1 since U is unitary. Let |T | 2 e1 = cξ + dη for some constants c, d ∈ C. Then 1 1 1 1 1 (Ae2 , e1 ) = (U |T | 2 e2 , |T | 2 e1 ) = ac|T | 2 e2 = 1, (Ae1 , e2 ) = (U |T | 2 e1 , |T | 2 e2 ) = 1 1 1 bc|T | 2 e2 = −1 and (Ae1 , e1 ) = (U |T | 2 e1 , |T | 2 e1 ) = cbd + acd = dc(a + b) = x = 1 0, that is, c(a + b) = 0. However we have (Ae2 , e1 ) + (Ae1 , e2 ) = |T | 2 e2 c(a + b) = 0, which implies that c(a + b) = 0. This is a contradiction. Thus A ∈ R(∆). The proof is complete. Lemma 2. Assume p ≥ 2 and A ∈ Mp is an idempotent such that rank(A) = p − 1 and R(A) + R(A∗ ) = H. Then A is not in the closure R(∆) of R(∆). Proof. Let A ∈ Mp be an idempotent such that rank(A) = p − 1 and R(A) + R(A∗ ) = H. Then ker A ∩ ker A∗ = {0}. It is known that A ∈ R(∆) by Proposition 1.12 in [2]. Suppose A ∈ R(∆). Then there exists a sequence {Tn : n ∈ N} ⊂ Mp such that lim ∆(Tn ) = A. Without loss of generality, we may choose that Tn is n→∞

invertible for all n. Let Tn = Un |Tn | be the polar decomposition of Tn . Let Dn and 1 Vn be the diagonal and unitary matrices respectively such that Vn |Tn | 2 Vn∗ = Dn , where Dn = diag(d1 (n), d2 (n), . . . , dp (n)) for some positive real numbers di (n), i = n Vn∗ = Dn Wn Dn , where Wn = Vn Un Vn∗ is unitary 1, 2, . . . p and for all n. Then Vn T for all n. By choosing a subsequence if necessary, we may assume that lim Un = U,

n→∞

lim Vn = V and lim di (n) = di ,

n→∞

n→∞

where di is +∞ or a nonnegative real numbers for any i = 1, 2, . . . , p. Then both U and V are unitary and W = lim Wn = V U V ∗ is also unitary. Thus there are n→∞

nonnegative integers k, l and s such that p = k + l + s and lim di (n) = +∞ n→∞

for 1 ≤ i ≤ k, lim di (n) = di = 0 for k < i ≤ k + l and lim di (n) = 0 for n→∞

n→∞

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k + l < i ≤ p. We note that k, s ≥ 1 since A ∈ R(∆). If p = 2, then we have k = s = 1 and l = 0. By assumption, A = lim ∆(Tn ). Then we have n→∞

B = V AV ∗ = lim Vn ∆(Tn )Vn∗ ∈ R(∆) n→∞

and both A and B have the same properties. With the decomposition H = Cp = Ck ⊕ Cl ⊕ Cs , by an elementary calculation, we have   B11 B12 B13 0 . (1) B =  B21 B22 B31 0 0 Note that B 2 = B, then we easily have  2  B11 + B12 B21 + B13 B31 = B11 , B11 B12 + B12 B22 = B12 , B11 B13 = B13 , 2 B21 B11 + B22 B21 = B21 , B21 B12 + B22 = B22 , B21 B13 = 0,   B31 B11 = B31 , B31 B12 = 0, B31 B13 = 0. (2) Note that rank(B) = rank(A) = p − 1. It follows that dim ker(B31 ) ≤ 1. Thus we have that (B12 B13 ) is of at most rank one, which implies that ker(B13 ) is at least s − 1 dimensional. We also have ker(B13 ) is at most 1 dimensional. It follows that s ≤ 2. By considering B ∗ , we similarly have that B31 is at most of rank one. That is, k ≤ 2. Since B = lim Dn Wn Dn , we easily have n→∞



0 W = 0 W31

0 W22 W32

 W13 W23  . W33

(3)

∗ ∗ = Ik and W31 W31 = Ik , that is Note that W is unitary. It follows that W13 W13 ∗ both W13 and W31 are isometries. We thus have 1 ≤ k ≤ s ≤ 2. Case 1. k < s. Then k = 1 and s = 2. In this case, Cp = C ⊕ Cl ⊕ C ⊕ C and   b11 B12 b13 b14  B21 B22 0 0  . B= (4)  b31 0 0 0  b41 0 0 0

It now follows that b31 b13 = b31 b14 = b41 b13 = b41 b14 = 0 by B 2 = B again. If b41 = 0 then b31 = 0. Otherwise rank(B) ≤ p − 2. This contradicts with the assumption that rank(B) = p−1. However we must have b13 = b14 = 0, which also implies that rank(B) ≤ p − 2. This is a contradiction again. Similarly, if b41 = 0, we have b14 = 0 and a contradiction follows.

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∗ Case 2. k = s. Since W31 and W13 are isometries, both W31 and W13 are unitary in this case. It follows that   0 0 W13 W22 0 . (5) W = 0 W31 0 0

(i) k = s = 1. In this case we have Cp = C ⊕ Cl ⊕ C, lim d1 (n) = +∞, n→∞

lim dp (n) = 0,

n→∞



b11 B =  B21 b31

B12 B22 0

 b13 0  0

(6)

and b31 b13 = 0. If b13 = 0, then b31 = 0 since ker B ∩ ker B ∗ = {0}. Let Wn = (wij (n)) for all n. By (5) and (6), we have lim w1p (n) = W13 = 0,

(7)

lim wp1 (n) = W31 = 0

(8)

n→∞

n→∞

and b13 = lim d1 (n)w1p (n)dp (n) = 0,

(9)

b31 = lim d1 (n)wp1 (n)dp (n) = 0.

(10)

n→∞ n→∞

This is a contradiction since we must have lim d1 (n)dp (n) = 0 from (7) and (9) and lim d1 (n)dp (n) = 0 from (8) and (10). n→∞

n→∞

(ii) k = s = 2. Then Cp = C2 ⊕ Cl ⊕ C2 . By (2), we have B21 B12 = 0 since B21 B12 B13 = 0, R(B12 ) = R(B13 ) and B21 B13 = 0. It follows that B31 2 which implies that B12 B21 + B13 B31 is a square zero matrix. Now B11 − B11 = B12 B21 + B13 B31 by (2). It follows that that σ(B11 ) ⊆ {0, 1}. If either B13 or B31 is 0, then rank(B) ≤ p − 2. This is a contradiction. We next assume that both B13 and B31 are not 0. Then they are of rank one. By (2) again we have B11 B12 = B12 since (B12 B13 ) is of rank one and B11 B13 = B13 . It follows that rank(B11 B12 B13 ) = rank(B11 ). If 0 ∈ σ(B11 ), then rank(B11 B12 B13 ) ≤ 1 and therefore rank(B) ≤ p − 2 since rank(B31 ) = 1. This contradicts with the assumption that rank(A) = rank(B) = p − 1. If B11 is invertible, then σ(B11 ) = {1} and B11 = I2 +X, where X is B11 B12 B13 a square zero matrix. We claim that rank = 2. We may B 0 0 31 α1 assume that (B12 B13 ) = , where αi is an 1 × (p − 2) matrix for i = 1, 2 α2 b(p−1)1 b(p−1)2 . and B31 = bp1 bp2

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In fact, if X = 0, that is, B11 = I2 , then by the facts that B11 B12 = 0, B31 B13 = 0 and rank(B12 B13 ) = 1, we have b(p−1)1 α1 +b(p−1)2 α2 = 0 and bp1 α1 + bp2 α2 = 0. It follows that (b(p−1)1 , b(p−1)2 , 0) = b(p−1)1 (1, 0, α1 ) + b(p−1)2 (1, 0,α2 ) B11 B12 B13 and (bp1 , bp2 , 0) = bp1 (1, 0, α1 )+ bp2 (1, 0, α2 ), that is, rank = B31 0 0 2. If X = 0, then there exist a 2 × 2 unitary matrix Y such that Y B11 Y ∗ = 1 x for some x = 0. Thus without loss of generality, we may assume that 0 1 B11 itself has this form. It follows that α2 = 0 and b(p−1)1 = bp1 = 0 since B11 (B12 B13 ) = (B12 B13 ) and B31 B11 = B31 . Then we have   1 x α1  0 B11 B12 B13 1 0   =  0 b(p−1)2 0  . B31 0 0 0 0 bp2 B11 B12 B13 = 2. Then rank B31 0 0 Consequently, we have rank(B) ≤ p−2, which contradicts with that rank(B) = p − 1 again. Thus by the proof above, we have A ∈ R(∆). The proof is complete. Theorem 3. Let H = Cp for p ≥ 2. Then R(∆) is neither closed nor dense in B(H). Proof. We ﬁrstly show that R(∆) is not closed.   √ 1 1 n 0 1 − 2 n n  Case 1. p = 2. Let Dn = , Un =  , 1 1 1 0 − 1 − n2 n n 1 1 ∈ R(∆) by Lemma 1. then An = Dn Un Dn ∈ R(∆) and lim An = −1 0 n→∞   0 1 0 Case 2. p = 3. Let B =  0 0 1 . Then B ∈ R(∆) by Proposition 1.12 0 0 0  1    0 0 0 1 0 n in [2]. Put Pn =  0 n 0  and Vn =  0 0 1 . Then Bn = Pn Vn Pn ∈ 1 0 0 0 0 n1 R(∆) and lim Bn = B, that is B ∈ R(∆). n→∞

Case 3. p > 3. We have B ⊕ Ip−3 ∈ R(∆) by Proposition 1.12 in [2] again. However we have B ⊕ Ip−3 ∈ R(∆) by the proof as above. By Lemma 2 we know that R(∆) is not dense in B(H). The proof is complete. Next we consider inﬁnite dimensional case.

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Lemma 4. Let H be a complex separable infinite dimensional Hilbert space. Then F (H) ⊂ R(∆) but K(H) R(∆). Proof. Let A be a ﬁnite rank operator with A = 1. Put M = R(A) + R(A∗ ). ⊥ Then M is a ﬁnite dimensional subspace of H which reduces A. Let H = M ⊕ M . A1 0 Then we have A = . Let V be an isometry from M into M ⊥ and T = 0 0 0 A1 . Note that T is a partial isometry. By a simply calculation, 1 V (I − A∗1 A1 ) 2 0 we have ∆(T ) = A. On the other hand, let A be a compact operator with nontrivial kernel and dense range. For example, let {en : n ∈ N} be an orthogonal basis of H. Let {an : n ∈ N} be a decreasing sequence of positive numbers satisfying lim an = 0. n→∞

Let A be the backward weighted shift with weights {an : n ∈ N}. That is Ae1 = 0, Aen = an−1 en−1 for n ≥ 2. Then we have A is compact and A ∈ R(∆) by Proposition 1.12 in [2]. The proof is complete. Theorem 5. Let H be a complex separable infinite dimensional Hilbert space. Then R(∆) is neither closed nor dense in norm topology but strongly dense in B(H). Proof. By Lemma 4, we know that R(∆) is not closed but strongly dense in B(H). On the other hand, let Gr be the set of all right invertible but not invertible operators in B(H). Then we know Gr is a nonempty open subset of B(H). We have Gr ∩ R(∆) = ∅ by Proposition 1.12 in [2], that is, R(∆) is not dense in B(H). The proof is complete. Acknowledgments. The authors would like to thank the referee for his/her suggestions and comments.

References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations and Operator Theory, 13(1990), 307-315. [2] I. B. Jung, E. Ko, and C. Pearcy, Aluthge transforms of operators, Integral Equations and Operator Theory, 37(2000), 437-448. [3] I. B. Jung, E. Ko, and C. Pearcy, Spectral pictures of Aluthge Tranforms of operators, Integral Equations and Operator Theory, (2001), 52-60. [4] X. Liu and G. Ji, Some properties of generalized Aluthge transform, Nihonkai Math. J., 15(2004), 101-107. [5] P. Y. Wu, Numerical range of Aluthge transform of operator, Linear Alg. Appl., 357(2002), 295-298. [6] T. Yamazaki, Parallelisms between Aluthge transformtion and powers of operators, Acta Sci. Math. (Szeged), 67(2001), 809-820.

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[7] T. Yamazaki, On numerical range of the Aluthge transformtion, Linear Alg. Appl., 341(2002), 111-117. Guoxing Ji, Yongfeng Pang and Ze Li College of Mathematics and Information Science Shaanxi Normal University Xian, 710062 People’s Republic of China e-mail: [email protected] Submitted: January 12, 2006

Integr. equ. oper. theory 57 (2007), 217–228 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020217-12, published online December 26, 2006 DOI 10.1007/s00020-006-1479-4

Integral Equations and Operator Theory

A Remark on Two Duality Relations Emanuel Milman Abstract. We remark that an easy combination of two known results yields a positive answer, up to log(n) terms, to a duality conjecture that goes back to Pietsch. In particular, we show that for any two symmetric convex bodies K, T in Rn , denoting by N (K, T ) the minimal number of translates of T needed to cover K, one has: N (K, T ) ≤ N (T ◦ , (C log(1 + n))−1 K ◦ )C log(1+n) log log(2+n) , where K ◦ , T ◦ are the polar bodies to K, T , respectively, and C ≥ 1 is a universal constant. As a corollary, we observe a new duality result (up to log(n) terms) for Talagrand’s γp functionals. Mathematics Subject Classification (2000). 46A, 46B, 47B. Keywords. Entropy, covering numbers, duality, convex bodies, compact operators.

1. Introduction Let K and T denote two convex bodies in Rn (i.e. convex compact sets with nonempty interior). Throughout this note we assume that all bodies in question are centrally symmetric w.r.t. to the origin (e.g. K = −K). For a convex body L, we denote by L◦ its polar body, deﬁned as L◦ = {x ∈ Rn ; x, y ≤ 1 ∀y ∈ L}. The covering number of K by T , denoted N (K, T ), is deﬁned as the minimal number of translates of T needed to cover K, i.e.:     (xi + T ) . N (K, T ) = min N ; ∃x1 , . . . , xN ∈ Rn , K ⊂   1≤i≤N

In this note, we address the following conjecture of Pietsch ([12, p. 38]) from 1972, originally formulated in operator-theoretic notations: Supported in part by BSF and ISF.

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Duality Conjecture for Covering Numbers. Do there exist numerical constants a, b ≥ 1 such that for any dimension n and for any two symmetric convex bodies K, T in Rn one has: b−1 log N (T ◦ , aK ◦ ) ≤ log N (K, T ) ≤ b log N (T ◦ , a−1 K ◦ ) ?

(1.1)

This problem may be equivalently formulated using the notion of entropy numbers. For a real number k ≥ 0, denote the k’th entropy number of K w.r.t. T as: ek (K, T ) = inf{ε > 0; N (K, εT ) ≤ 2k }. Then the duality conjecture may be equivalently formulated with (1.1) replaced by: (1.2) a−1 ebk (T ◦ , K ◦ ) ≤ ek (K, T ) ≤ aeb−1 k (T ◦ , K ◦ ) for all k ≥ 0 (and there is no loss in generality if we assume that k is an integer). As already mentioned, the duality conjecture originated from operator theory, where entropy numbers are used to quantify the compactness of an operator u : X → Y between two Banach spaces. Leaving the ﬁnite dimensional setting for a brief moment, if K = u(B(X)) and T = B(Y ), where B(Z) denotes the unit-ball of a Banach space Z, then it is easy to see that ek (K, T ) → 0 as k → ∞ iﬀ the operator u is compact. Since u is compact iﬀ its adjoint u∗ : Y ∗ → X ∗ is too, and since u∗ (B(Y ∗ )) = u∗ (T ◦ ) and B(X ∗ ) = u∗ (K ◦ ), it follows that ek (K, T ) → 0 iﬀ ek (T ◦ , K ◦ ) → 0. Hence, it is natural to conjecture that the rate of convergence to 0 is asymptotically similar in both cases. A strong interpretation of this similarity is given by (1.2). We will mention other weaker interpretations below. Although the general problem is still not completely settled, there has been substantial progress in recent years, and the answer is known to be positive for a wide class of bodies. We begin by describing some results in this direction. We comment here that when the result imposes the same restrictions on K and T , it is obviously enough to specify only one side of the inequalities in (1.1) or (1.2). When both K and T are ellipsoids, it is easy to see that in fact N (K, T ) = N (T ◦ , K ◦ ). Other special cases were settled in [19],[5],[6], [9],[18]. In [7], it was shown that: C −n N (T ◦ , K ◦ ) ≤ N (K, T ) ≤ C n N (T ◦ , K ◦ ), for some universal constant C > 1. This implies that the tail behaviour of the entropy numbers satisﬁes the duality problem, i.e. eλk (K, T ) ≤ 2ek (T ◦ , K ◦ ) for some universal constant λ > 0 and all k ≥ n. This was subsequently generalized in [16]. Another variant of the problem, is to consider not the individual entropy numbers, but rather the entire sequences {ek (K, T )} and {ek (T ◦ , K ◦ )}. Then one may ask whether: C −1 {ek (T ◦ , K ◦ )} ≤ {ek (K, T )} ≤ C {ek (T ◦ , K ◦ )}

(1.3)

for some universal constant C > 1 and any symmetric (i.e. invariant to permutations) norm ·. When one of the bodies is an ellipsoid, this was positively settled in [25]. Later, in [4], this was extended to the case when one of the bodies is

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uniformly convex or more generally K-convex (see [4] and [17] for deﬁnitions), in which case the constant C in (1.3) depends only on the K-convexity constant. The technique developed in [4] played a crucial role in some of the subsequent results on this problem, and one particular remark will play an essential role in this note. Returning to the duality problem of individual entropy numbers, it was shown in [11] that there exist universal constants a, b ≥ 1 such that when T = D is an ellipsoid: ebk (D◦ , K ◦ ) ≤ a(1 + log k)3 ek (K, D), for all k ≥ 0. In addition, the authors of [11] observed a connection between (one side of) the duality conjecture with T = D and a certain geometric lemma. Later, the case when one of the bodies is an ellipsoid was completely settled in [1], by showing that: b−1 log N (D◦ , aK ◦ ) ≤ log N (K, D) ≤ b log N (D◦ , a−1 K ◦ ). The main new tool developed in [1] was the so called “Reduction Lemma”, which roughly reduces the problem (1.1) for all K, T to the case K ⊂ 4T . This will be the second important tool in this note. Finally, in [2], the Reduction Lemma was combined with the techniques developed in [4], to transfer the results obtained there for the sequence of entropy numbers, to the individual ones. Thus, when one of the bodies K or T is Kconvex, (1.1) was shown to hold with the constants a, b depending solely on the K-convexity constant. The key ideological step in [2] was to separate the question of “complexity” from the question of duality, by explicitly introducing a new notion of convexified packing number, which was implicitly used in [4]. We will later refer to this new notion as well. Our ﬁrst new observation in this note is in fact an immediate consequence of Theorem 6 in [4] and the Reduction Lemma in [1]. It settles the duality problem (1.1) (and (1.2)) up to log(n) terms, and in fact strengthens and generalizes all previously known results into a single statement. Because of the symmetry between K and T (as will be explained below), we formulate this as a one sided inequality: Theorem 1.1. Let K, T be two symmetric convex bodies in Rn . Then: log N (K, T ) ≤ V log(V ) log N (T ◦ , V −1 K ◦ ),

(1.4)

where V = min(V (K), V (T )) and V (L) is defined as: V (L) := inf {log(CdBM (L, B))f (K(XB )); B is a convex body in Rn } ,

(1.5)

where C > 0 is a universal constant, and f is a function depending solely on K(XB ), the K-convexity constant of the Banach space XB whose unit ball is B. For more information on the function f see Remark 2.3 below. Recall that the Banach-Mazur distance dBM (L, B) of two symmetric convex bodies L, B is deﬁned as: dBM (L, B) := inf {γ ≥ 1; B ⊂ T (L) ⊂ γB} ,

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where the inﬁmum runs over all linear transformations T . Since V (L) = V (L◦ ) because dBM (L, B) = dBM (L◦ , B ◦ ) and K(XB ◦ ) = K(XB ), applying the Theorem to K = T ◦ and T = K ◦ gives the opposite inequality (with the same V ): (V log(V ))−1 log N (T ◦ , V K ◦ ) ≤ log N (K, T ).

(1.6)

In addition, since by John’s Theorem, the Banach-Mazur distance √ of any symmetric convex body in Rn from the Euclidean ball D is at most n, and since K(D) = 1, we immediately have: Corollary 1.2. With the same notations as in Theorem 1.1: log N (K, T ) ≤ C log(1 + n) log log(2 + n) log N (T ◦ , (C log(1 + n))−1 K ◦ ), where C > 0 is a universal constant. This should be compared with the previously known best estimate (to the best of our knowledge) for general symmetric convex bodies K, T : log N (K, T ) ≤ C log N (T ◦ , (Cn)−1/2 K ◦ ), which is derived by comparing K with its John ellipsoid and using the duality result of [1] for ellipsoids. The novelty here in comparison to the results of [2] lies in the logarithmic dependence in the Banach-Mazur distance. Although there has been much progress in recent years towards a positive answer to the duality conjecture, it is still not clear that a positive answer should hold in full generality. In view of the Corollary 1.2, and Pisier’s well known estimate K(XB ) ≤ C log(1 + n) for any symmetric convex body B in Rn , we conjecture a weaker form of the duality problem: Weak Duality Conjecture for Covering Numbers. Does there exist a numerical constant C ≥ 1 such that for any dimension n and for any two symmetric convex bodies K, T in Rn one has: log N (K, T ) ≤ V log N (T ◦ , V −1 K ◦ ), where V = C min(K(XK ), K(XT )) ? We present the proof of Theorem 1.1 and several other connections to previously mentioned notions in Section 2. In Section 3, we give an application of Corollary 1.2 for Talagrand’s celebrated γp functionals, which was in fact our motivation for seeking a result in the spirit of Corollary 1.2. Recall that for a metric space (M, d) and p > 0, γp (M, d) is deﬁned as: 2j/p d(x, Mj ) γp (M, d) := inf sup x∈M

j≥0

where the inﬁmum runs over all admissible sets {Mj }, meaning that Mj ⊂ M j and |Mj | = 22 (we refer to [24, Theorem 1.3.5] and [23] for the connection to equivalent deﬁnitions). For two symmetric convex bodies K, T , let us denote γp (K, T ) := γp (K, dT ), where dT is the metric corresponding to the norm induced by T . The γ2 (·, D) functional, when D is an ellipsoid, was introduced to study

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the boundedness of Gaussian processes (see [24] for an historical account on this topic). It was shown by Talagrand in his celebrated “Majorizing Measures Theorem” ([21], see also [22],[23]), that in fact γ2 (K, D) and E supx∈K x, G, where G is a Gaussian r.v. (with covariance corresponding to D in an appropriate manner), are equivalent to within universal constants. This was later extended to various other classes of stochastic processes, where the naturally arising metric d is not the l2 norm (again we refer to [24] for an account). Our second observation in this note is the following duality relation for the γp functionals: Theorem 1.3. Let K, T be two symmetric convex bodies in Rn . Then for any p > 0: γp (K, T ) ≤ Cp log(1 + n)2+1/p log log(2 + n)1/p γp (T ◦ , K ◦ ), where Cp > 0 depends solely on p. Although we strongly feel that this is unlikely, one could conjecture that the log(n) terms are not required in the last Theorem (at least for some values of p). In that case, as will be evident from the proof, we mention that such a conjecture is independent of the duality conjecture for covering numbers, in the sense that neither one implies the other. Acknowledgments. I would like to sincerely thank my supervisor Prof. Gideon Schechtman for motivating me to prove Proposition 3.3, sharing his knowledge, and for reading this manuscript.

2. Duality of Entropy As emphasized in the Introduction, the proof of Theorem 1.1 is immediate once we recall two previously known results. The ﬁrst is the recently observed “Reduction Lemma” ([1, Proposition 12]), which uses a clever iteration procedure to telescopically expand and reduce the appearing terms. We carefully formulate it below: Theorem 2.1 ([1]). Let T be a convex symmetric body in a Euclidean space such that, for some constants a, b ≥ 1, for any convex symmetric body K ⊂ 4T , one has: log N (K, T ) ≤ b log N (T ◦ , a−1 K ◦ ). Then for any convex symmetric body K: log N (K, T ) ≤ b log2 (48a) log N (T ◦ , (8a)−1 K ◦ ). Dually, if K is fixed and the hypothesis holds for all T verifying K ⊂ 4T , then the conclusion holds for any T . The second known result goes back to the work of [4]. It uses the so called Maurey’s Lemma, which (roughly speaking) estimates the covering number of the convex hull of m points by the unit-ball of a K-convex space. We combine Theorem 6 and the subsequent remark from [4] into the following:

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Theorem 2.2 ([4]). Let K, T be two symmetric convex bodies in Rn , such that K ⊂ 4T . Then: log N (K, T ) ≤ V log N (T ◦ , V −1 K ◦ ), where V = min(V (K), V (T )) and V (L) is given by (1.5). Combining these two results, we immediately deduce Theorem 1.1. Note that if V = V (T ) in Theorem 1.1, we proceed by ﬁxing T , applying Theorem 2.2 for all K satisfying K ⊂ 4T and use the ﬁrst part of the Reduction Lemma to deduce (1.4) for all K; if V = V (K), we ﬁx K and repeat the argument by interchanging the roles of K and T and using the second part of the Reduction Lemma. Remark 2.3. The proof of Theorem 2.2 in fact gives an explicit expression for V , rather than the implicit one used in (1.5): V := C1 inf {log(C2 γ)(10Tp (XB ))q ; K ⊂ γB, B ⊂ 4T, γ ≥ 1} ,

(2.1)

where the inﬁmum runs over all symmetric convex bodies B in Rn , Tp (XB ) is the type p (1 < p ≤ 2) constant of the Banach space XB whose unit-ball is B, q = p∗ = p/(p − 1) and C1 , C2 ≥ 1 are two universal constants (see [10] for the deﬁnition of type). Theorem 1.1 was formulated using an implicit function f of K(XB ), since by several important results of Pisier ([13],[14]), an inﬁnite dimensional Banach Space is K-convex iﬀ it has some non-trivial type p > 1. We comment that in [14], an explicit formula bounding K(XB ) as a function of Tp (XB ) and q was obtained. It is possible to obtain an explicit reverse bound using the results in [15], but it is much easier to use an abstract argument which infers the existence of a p > 1, depending solely on K(XB ), such that Tp (XB ) depends solely on K(XB ) (see, e.g. [8, Lemma 4.2]). The advantage of using the K-convexity constant K(XB ) (instead of Tp (XB ) and q), lies in the fact that we may use duality and deduce the other side of the duality inequality (1.6) with the same V , as explained in the Introduction. We also remark that once V in (2.1) is expressed using K(XB ), it is clear that V ≤ min(V (K), V (T )) where V (L) is given by (1.5). We need this “separable” estimate on V , so that we may apply the Reduction Lemma (where the estimate on one of the bodies must be ﬁxed). It is important to note that the proof of Theorem 2.2 actually connects the notions of covering and convexiﬁed packing, mentioned in the Introduction. For two symmetric convex bodies K and T , the convexiﬁed packing number, or convex separation number, was deﬁned in [2] as:

∃x1 , . . . , xN ∈ K such that ˆ (K, T ) = max N ; M . (xj + intT ) ∩ conv {xi ; i < j} = ∅ ˆ (K, T ) ≤ Here int(T ) denotes the interior of the set T . Note that we always have M N (K, T /2) by a standard argument (see [2]). Then the proof actually shows: Theorem 2.4 ([4]). Under the same conditions as in Theorem 2.2: ˆ (K, V −1 T ). log N (K, T ) ≤ V log M

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Using John’s Theorem as in Corollary 1.2, we have: Corollary 2.5 ([4]). Let K, T be two symmetric convex bodies in Rn , such that K ⊂ 4T . Then: ˆ (K, (C log(1 + n))−1 T ). log N (K, T ) ≤ C log(1 + n) log M We mention these variants of Theorem 1.1 and Corollary 1.2 here, because the framework developed in [2] suggests that this is the correct way to understand the duality problem. The cost of transition from covering to convex separation, as given by Theorem 2.4 and Corollary 2.5, is a certain measure of the complexity of the bodies involved. Once the transition is achieved, the duality framework developed in [1] and [2] ﬁnishes the job. Indeed, it was shown in [2] that the convex separation numbers always satisfy a duality relation, for any pair of symmetric convex bodies K, T : ˆ (K, T ) ≤ M ˆ (T ◦ , K ◦ /2)2 . M Using Theorem 2.4, we conclude that when K ⊂ 4T : ˆ (K, V −1 T ) ≤ 2V log M ˆ (T ◦ , (2V )−1 K ◦ ) log N (K, T ) ≤ V log M ≤ 2V log N (T ◦ , (4V )−1 K ◦ ). The Reduction Lemma now immediately gives Theorem 1.1. To conclude this section, we mention that Theorem 2.4 is already stronger than all of the results in [2] connecting the covering and the convex separation numbers. The technique involving the use of Maurey’s Lemma, which was also used in [2] (see also [3]), is optimally exploited in the proof of Theorem 2.4 (Theorem 6 in [4]), by using a clever iteration procedure, producing the log factor in the various deﬁnitions (1.5) and (2.1) of V . All previous general results (with no restriction on K and T ) pay a linear penalty in the√Banach-Mazur distance from “low-complexity” bodies, which may be as large as n.

3. Duality of Talagrand’s γp Functionals Given Corollary 1.2, proving Theorem 1.3 is rather elementary, although we will need to collect several elementary observations which we have not been able to ﬁnd a reference for. Before proceeding, we remark that for our purposes, it is totally immaterial whether the points {xi } in the deﬁnition of N (K, T ) are chosen to lie inside K or not. Indeed, denoting by N (K, T ) the variant where the points are required to lie inside K, it is elementary to check that: N (K, 2T ) ≤ N (K, T ) ≤ N (K, T ). Since throughout this note we allow the insertion of homothety constants in all expressions, or multiplying the entropy numbers by universal constants, this lack of distinction is well justiﬁed.

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First, recall that by Dudley’s entropy bound ([24]): γp (K, T ) ≤ Cp k 1/p−1 ek (K, T ),

(3.1)

k≥1

where Cp > 0 is some constant depending on p. The argument is elementary: γp (K, T ) := inf sup 2j/p dT (x, Kj ) ≤ inf 2j/p sup dT (x, Kj ). x∈K j≥0

x∈K

j≥0 j

Choosing Kj to be the set of 22 points (inside K) attaining the minimum in the deﬁnition of N (K, e2j (K, T )T ), we see that: γp (K, T ) ≤ 2j/p e2j (K, T ). j≥0

It is elementary to verify that for p ≥ 1 and j ≥ 1: 2

j/p

≤ Cp

j −1 2

k 1/p−1 ,

k=2j−1

where Cp = (p(1 − 2

−1/p

−1

))

. Since ek is a non-increasing sequence, we have: j

γp (K, T ) ≤ e1 (K, T ) + Cp ≤ e1 (K, T ) + Cp

−1 2

k 1/p−1 e2j (K, T )

j≥1 k=2j−1

k 1/p−1 ek (K, T ) ≤ (1 + Cp )

k≥1

k 1/p−1 ek (K, T ).

k≥1

A similar argument works for 0 < p < 1. Dudley’s entropy upper bound appears naturally when studying the supremum of Gaussian processes on a set K, e.g. E supx∈K x, G where G is a Gaussian r.v. As mentioned in the Introduction, a deep theorem of Talagrand asserts that the latter expectancy is in fact equivalent (to within universal constants) to γ2 (K, D) where D is an ellipsoid corresponding to the covariance of G. The corresponding lower bound on E supx∈K x, G is due to Sudakov ([20]): E sup x, G ≥ c sup k 1/2 ek (K, D). x∈K

k≥1

When the body T is not an ellipsoid or when p = 2, there is no direct connection between Gaussian processes and γp (K, T ). Nevertheless, we note that the analogue to Sudakov’s Minoration bound holds in full generality: Lemma 3.1.

γp (K, T ) ≥ 2−1/p sup k 1/p ek (K, T ). k≥1

Proof. Let k ≥ 1 be given, and let j ≥ 0 be such that 2j ≤ k < 2j+1 . Then: γp (K, T ) := inf sup 2l/p dT (x, Kl ) ≥ inf sup 2j/p dT (x, Kj ). x∈K

l≥0

x∈K

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j

Since for any admissible set Kj we have |Kj | = 22 ≤ 2k , it follows by deﬁnition that supx∈K dT (x, Kj ) ≥ ek (K, T ). Hence: γp (K, T ) ≥ 2j/p ek (K, T ) ≥ 2−1/p k 1/p ek (K, T ). Since k ≥ 1 was arbitrary, the assertion follows.

We will need one last lemma for the proof of Theorem 1.3: Lemma 3.2. For all k ≥ 3n: ek (K, T ) ≤ 2en (K, T ) exp(−ck/n), where c > 0 is some universal constant. Proof. Denote ek = ek (K, T ) and en = en (K, T ) for short. W.l.o.g. we assume that N (K, ek T ) = 2k and N (K, en T ) = 2n . Obviously we have: N (K, ek T ) ≤ N (K, en T )N (en T, ek T ).

(3.2)

Also N (en T, ek T ) = N (T, eenk T ), and by a standard volume estimation argument, n n we can ﬁnd an ek /en T -net of T with cardinality no greater than (1 + 2e ek ) . Plugging everything into (3.2), we see that: n 2en k n 2 ≤2 1+ , ek or equivalently:

k−n 2en exp log(2) . −1≤ n ek

Since k ≥ 3n, we can ﬁnd a universal constant c > 0 such that: k−n k exp log(2) − 1 ≥ exp c . n n

The assertion now readily follows.

We can now deduce the following equivalence, up to a log(n) term, of the γp functional, Sudakov’s lower bound and Dudley’s upper bound. Although this is surely known, we did not ﬁnd a reference for it, so we include a proof for completeness. Proposition 3.3. Let K, T denote two symmetric convex bodies in Rn , and denote ek = ek (K, T ) and γp = γp (K, T ) for short. Then for any p > 0: k 1/p−1 ek ≤ log(1 + n)Cp sup k 1/p ek , 2−1/p sup k 1/p ek ≤ γp ≤ Cp k≥1

k≥1

k≥1

where Cp , Cp > 0 are universal constants depending solely on p.

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Proof. The ﬁrst inequality is Sudakov’s lower bound (Lemma 3.1) and the second one is Dudley’s upper bound (3.1). We will show the third inequality. Let us split

the sum k≥1 k 1/p−1 ek into two parts, up to and from k = 3n. For the ﬁrst part, we obviously have: 3n−1

k 1/p−1 ek ≤

3n−1

k=1

k=1

1 sup k 1/p ek ≤ C log(1 + n) sup k 1/p ek . k k≥1 k≥1

We use Lemma 3.2 to evaluate the second sum: k 1/p−1 ek ≤ 2en k 1/p−1 exp(−ck/n). k≥3n

k≥3n

For p ≥ 1, k 1/p−1 is non-increasing, so we use: k 1/p−1 exp(−ck/n) ≤ (3n)1/p−1 exp(−ck/n) k≥3n

= (3n)1/p−1

k≥3n

exp(−3c) n ≤ (3n)1/p−1 exp(−3c) ≤ Cn1/p . 1 − exp(−c/n) c

For 0 < p < 1, we evaluate the sum with an integral (although the series may not be monotone, is has at most one extremal point, and this can be handled by a loose estimate): ∞ k 1/p−1 exp(−ck/n) ≤ 3 x1/p−1 exp(−cx/n)dx 3n−1

k≥3n

n 1/p

∞

x1/p−1 exp(−x)dx = 3c−1/p Γ(1/p)n1/p . c 0 We conclude that in both cases: k 1/p−1 ek ≤ Cp n1/p en ≤ Cp sup k 1/p ek . ≤3

k≥1

k≥3n

Summing the two parts together, we conclude the proof. Using Corollary 1.2, the proof of Theorem 1.3 is now clear: Proof of Theorem 1.3. Corollary 1.2 implies that: ek (K, T ) ≤ C log(1 + n)ek/(C log(1+n) log log(2+n)) (T ◦ , K ◦ ),

for some universal constant C ≥ 1 and all k ≥ 0. Using Proposition 3.3 twice, we conclude: γp (K, T ) ≤ Cp log(1 + n) sup k 1/p ek (K, T ) k≥1

≤

Cp

2

log(1 + n) (log(1 + n) log log(2 + n))1/p sup k 1/p ek (T ◦ , K ◦ ) k≥1

2+1/p

≤ Cp log(1 + n)

1/p

log log(2 + n)

◦

γp (T , K ◦ )

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References [1] S. Artstein, V. Milman, and S. J. Szarek, Duality of metric entropy, Ann. of Math. (2) 159 (2004), no. 3, 1313–1328. [2] S. Artstein, V. Milman, S. Szarek, and N. Tomczak-Jaegermann, On convexified packing and entropy duality, Geom. Funct. Anal. 14 (2004), no. 5, 1134–1141. [3] S. Artstein, Entropy methods, Ph.D. thesis, Tel-Aviv University, 2004. [4] J. Bourgain, A. Pajor, S. J. Szarek, and N. Tomczak-Jaegermann, On the duality problem for entropy numbers of operators, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 50–63. [5] B. Carl, Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spaces, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 3, 79–118. [6] Y. Gordon, H. K¨ onig, and C. Sch¨ utt, Geometric and probabilistic estimates for entropy and approximation numbers of operators, J. Approx. Theory 49 (1987), no. 3, 219–239. [7] H. K¨ onig and V. D. Milman, On the covering numbers of convex bodies, Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math., vol. 1267, Springer, Berlin, 1987, pp. 82–95. [8] B. Klartag and E. Milman, On volume distribution in 2-convex bodies, to appear in Israel Journal of Mathematics, www.arxiv.org/math.FA/0604594, 2006. [9] H. K¨ onig, V. D. Milman, and N. Tomczak-Jaegermann, Entropy numbers and duality for operators with values in a Hilbert space, Probability in Banach spaces 6 (Sandbjerg, 1986), Progr. Probab., vol. 20, Birkh¨ auser Boston, Boston, MA, 1990, pp. 219–233. [10] V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer, Berlin, 1986. [11] V. D. Milman and S. J. Szarek, A geometric lemma and duality of entropy numbers, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 191–222. [12] A. Pietsch, Theorie der Operatorenideale (Zusammenfassung), Friedrich-SchillerUniversit¨ at, Jena, 1972. [13] G. Pisier, Sur les espaces de Banach qui ne contiennent pas uniform´ement de ln1 , C. R. Acad. Sci. Paris S´er. A-B 277 (1973), A991–A994. [14]

, Holomorphic semigroups and the geometry of Banach spaces, Ann. of Math. (2) 115 (1982), no. 2, 375–392.

[15]

, On the dimension of the lpn -subspaces of Banach spaces, for 1 ≤ p < 2, Trans. Amer. Math. Soc. 276 (1983), no. 1, 201–211.

[16]

, A new approach to several results of V. Milman, J. Reine Angew. Math. 393 (1989), 115–131.

[17]

, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989.

[18] A. Pajor and N. Tomczak-Jaegermann, Volume ratio and other s-numbers of operators related to local properties of Banach spaces, J. Funct. Anal. 87 (1989), no. 2, 273–293.

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[19] C. Sch¨ utt, Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory 40 (1984), no. 2, 121–128. [20] V. N. Sudakov, Gaussian random processes, and measures of solid angles in Hilbert space, Dokl. Akad. Nauk SSSR 197 (1971), 43–45. [21] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), no. 1-2, 99–149. , Majorizing measures: the generic chaining, Ann. Probab. 24 (1996), no. 3, [22] 1049–1103. , Majorizing measures without measures, Ann. Probab. 29 (2001), no. 1, 411– [23] 417. , The generic chaining, Springer-Verlag, Berlin, 2005, Upper and lower [24] bounds of stochastic processes. [25] N. Tomczak-Jaegermann, Dualit´e des nombres d’entropie pour des op´erateurs ` a valeurs dans un espace de Hilbert, C. R. Acad. Sci. Paris S´er. I Math. 305 (1987), no. 7, 299–301. Emanuel Milman Department of Mathematics The Weizmann Institute of Science Rehovot 76100 Israel e-mail: [email protected] Submitted: October 6, 2006 Revised: November 8, 2006

Integr. equ. oper. theory 57 (2007), 229–234 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020229-6, published online December 26, 2006 DOI 10.1007/s00020-006-1467-8

Integral Equations and Operator Theory

On Smooth Local Resolvents Vladim´ır M¨ uller Abstract. We exhibit an example of a bounded linear operator on a Banach space which admits an everywhere defined local resolvent with continuous derivatives of all orders. Mathematics Subject Classification (2000). Primary 47A10; Secondary 47A11. Keywords. Local resolvent, C ∞ -functions.

Let T be a bounded linear operator acting on a complex Banach space X. It is well known that the resolvent z → (T − z)−1 deﬁned on the complement of the spectrum σ(T ) is unbounded. More precisely, (T − z)−1 → ∞ whenever z approaches the spectrum σ(T ). Let x ∈ X be a nonzero vector. By a local resolvent of T at x we mean a function f : U → X deﬁned on a set U ⊃ C \ σ(T ) such that (T − z)f (z) = x (z ∈ U ). Clearly the local resolvent is uniquely determined for all z ∈ C \ σ(T ) and f (z) = (T − z)−1 x. Thus any local resolvent is analytic on the complement of the spectrum. It was observed in [3] that a local resolvent can be bounded. Bounded local resolvents were further studied in [1], [4], [2] and it was shown that they are rather frequent. In [2], an example of a continuous everywhere deﬁned local resolvent was given (such a local resolvent is necessarily bounded since each local resolvent vanishes for z → ∞). The aim of this note is to exhibit an example of an everywhere deﬁned C ∞ local resolvent. Note that by a basic result of local spectral theory there are no analytic everywhere deﬁned local resolvents. Denote by C, R and Z+ the sets of all complex numbers, real numbers and nonnegative integers, respectively. Let X be a complex Banach space and f : C → X a function. As usually we identify C with R2 and consider f to be a function of two real variables x and ∂ k+l f y. We say that f is a C ∞ -function if it has continuous derivatives ∂x k ∂y l of all orders. ˇ The research was supported by the grant No. 201/06/0128 of GA CR..

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Theorem 1. There exist a Banach space X, an operator T ∈ B(X), a nonzero vector x ∈ X and a C ∞ -function f : C → X such that (T − z)f (z) = x

(z ∈ C).

Proof. Let ϕ : C → 0, 1 be a C ∞ -function such that ϕ(z) = 1 for |z| < 1/3 and f (z) = 0 for |z| > 2/3. (z = 0), g(0) = 0. Clearly g is a Let g : C → C be deﬁned by g(z) = ϕ(z)−1 z ∞ C -function and g(z) = − z1 for |z| > 2/3. k+l

∂ g Write z = x + iy and let gkl = ∂x k ∂y l ; formally we write g00 = g. It is easy to show by induction that −1 k+l+1 gkl (z) = (k + l)! il z for all z, |z| > 2/3 and k, l ∈ Z+ . In particular, all derivatives gkl are bounded functions. For n = 0, 1, . . . choose positive constants Kn such that Kn ≥ nKn−1 (n ≥ 1) and max{|gkl (z)| : z ∈ C} ≤ Kk+l for all k, l ≥ 0. Clearly K0 ≥ 1 and Kn ≥ n! for all n. Denote by D = {z ∈ C : |z| < 1} the open unit disc in the complex plane.

ukl α

Let X0 be the vector space with a Hamel basis formed by the vectors u and (k, l ∈ Z+ , α ∈ D). k+l+1 − η1 (k + l)!il u. Note that ukl For |η| = 1 write for short ukl η = η =

gkl (η)u. Let M ⊂ X0 be the subset formed by the following elements: u, 1 ukl (α ∈ D, k, l ∈ Z+ ), Kk+l α 1 (αu00 (α ∈ D, 2/3 < |α|), α + u) K2 (1 − |α|)2 kl 1 k+1,l uα+t − ukl (α ∈ D, k, l ∈ Z+ , t ∈ R, |t| < 1/3, α + t ∈ D), α − tuα 2 t Kk+l+2 kl 1 k,l+1 u (α ∈ D, k, l ∈ Z+ , − ukl α − suα s2 Kk+l+2 α+is s ∈ R, |s| < 1/3, α + is ∈ D). Let U be the absolutely convex hull of M . Clearly U is absorbing. Let · be the Minkowski seminorm determined by U , i.e., for v ∈ X0 we have |γm | : v = γm m , v = inf m∈M

m∈M

where the coeﬃcients γm are complex and only a ﬁnite number of them are nonzero.

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We show ﬁrst that u = 1. Clearly u ≤ 1. Deﬁne the linear functional h : X0 → C by h(u) = 1 and h(ukl α ) = gkl (α) (α ∈ D, k, l ∈ Z+ ). We show that |h(v)| ≤ v for all v ∈ X0 . To this end, it is suﬃcient to show that |h(m)| ≤ 1 for all m ∈ M . For k, l ∈ Z+ , α ∈ D we have |h(ukl α )| = |gkl (α)| ≤ Kk+l , and for α ∈ D, |α| > 2/3, |h(αu00 α + u)| = αg(α) + 1 = 0. Further, for k, l ∈ Z+ , α ∈ D, t, s ∈ R, |t|, |s| < 1/3, α + t, α + is ∈ D we have kl k+1,l h(ukl ) = gkl (α + t) − gkl (α) − tgk+1,l (α) α+t − uα − tuα ≤ t2 max{|gk+2,l (z)| : z ∈ C} ≤ t2 Kk+l+2 and similarly, kl k,l+1 h(ukl ) = gkl (α + is) − gkl (α) − sgk,l+1 (α) ≤ s2 Kk+l+2 . α+is − uα − suα Thus |h(m)| ≤ 1 for all m ∈ M and consequently, |h(v)| ≤ v for all v ∈ X0 . In particular, u ≥ |h(u)| = 1, and so u = 1. Deﬁne now the linear mapping T0 : X0 → X0 by T0 u T0 u00 α T0 ukl α

= 0, = αu00 (α ∈ D), α +u k−1,l = αukl + ku + iluk,l−1 α α α

(α ∈ D, k, l ∈ Z+ , k + l ≥ 1)

(here we set formally ukl α = 0 if either k < 0 or l < 0). We show that T0 v ≤ 4v for all v ∈ X0 . To this end, it is again suﬃcient to show that T0 m ≤ 4 for all m ∈ M . We have 00 T0 u00 α = αuα + u ≤ |α|K0 + 1 ≤ 2K0 and, for k + l ≥ 1, kl k−1,l + iluk,l−1 ≤ |α|Kk+l + kKk+l−1 + lKk+l−1 ≤ 2Kk+l . T0 ukl α = αuα + kuα α

For 2/3 < |α| < 1 we have 00 00 T0 (αu00 α + u) = |α| · T0 uα ≤ αuα + u.

Let k, l ∈ Z+ , α ∈ D, t, s ∈ R, |t|, |s| < 1/3, α + t ∈ D, α + is ∈ D. Then kl kl kl k+1,l + |t| · uk+1,l ≤ t2 Kk+l+2 + |t| · Kk+l+1 (1) ukl α+t − uα ≤ uα+t − uα − tuα α and similarly, kl 2 ukl α+is − uα ≤ s Kk+l+2 + |s|Kk+l+1 .

For α ∈ D, α + t ∈ D and α + is ∈ D we have 00 1,0 00 1,0 00 T0 (u00 (α + t)u00 − u − tu ) = − αu − αtu − tu α+t α α α+t α α α 00 1,0 00 ≤ |α| · u00 + |t| · u00 α+t − uα − tuα α+t − uα ≤

t2 K2 + |t|3 K2 + t2 K1 ≤ 3t2 K2 .

(2)

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Similarly,

IEOT

00 0,1 2 T0 (u00 α+is − uα − suα ) ≤ 3s K2 .

For |η| = 1, |t| < 1/3, |η + t| < 1 we have 00 1,0 00 00 2 2 T0 (u00 η+t − uη − tuη ) = T0 uη+t = (η + t)uη+t + u ≤ (1 − |η + t|) K2 ≤ t K2 00 0,1 and analogous estimate holds for T0 (u00 η+is − uα − suα ) . Let k + l ≥ 1, α, α + t, α + is ∈ D. We have T0 (ukl − ukl − tuk+1,l ) α+t α α k−1,l k,l−1 kl k−1,l = (α + t)ukl − iluk,l−1 α+t + kuα+t + iluα+t − αuα − kuα α k+1,l k,l k+1,l−1 −tαuα − (k + 1)tuα − itluα kl k+1,l k−1,l + k uk−1,l ≤ |α| · ukl − tuk,l α+t − uα α+t − uα − tuα α k,l−1 k,l−1 k+1,l−1 kl + |t| · uα+t − ukl +l uα+t − uα − tuα α ≤ t2 Kk+l+2 + (k + l)t2 Kk+l+1 + |t|3 Kk+l+2 + t2 Kk+l+1 ≤ 4t2 Kk+l+2 .

Similarly, T0 (ukl − ukl − suk,l+1 ) α+is α α kl k−1,l k−1,l+1 ≤ |α| · ukl − suk,l+1 + k uk−1,l α+is − uα α+is − uα − suα k,l−1α kl + |s| · ukl ≤ 4s2 Kk+l+2 . +l uα+is − uk,l−1 − suk,l α α α+is − uα Let k + l ≥ 1, |η| = 1, t ∈ R, |t| < 1/3 and η + t ∈ D. We have kl kl k+1,l u ) ≤ t2 Kk+l+2 . η+t − (uη + tuη k+l+1 (k + l)!il u. Hence Recall that uη = − η1 T0 (ukl − ukl − tuk+1,l ) = T0 ukl η+t η η η+t 2 k−1,l k,l−1 kl = (η + t)uη+t + kuη+t + iluη+t ≤ 3t Kk+l+2 + k+1,l k,l−1 (η + t)ukl + kuk−1,l + ktukl + iltuk+1,l−1 η + (η + t)tuη η η + iluη η −1 k+l ≤ 3t2 Kk+l+2 + (k + l − 1)!il−1 u η −i (η + t)ti lti kti · (η + t) (k + l) + (k + l) + il − (k + l) (k + l)(k + l + 1) + ki − η η2 η η −1 k+l+1 (η + t)t = 3t2 Kk+l+2 + (k + l + 1) − η + kt + lt (k + l)!il u · η + t − η η t = 3t2 Kk+l+2 + (k + l)!|t| · 1 − (k + l + 1) − (k + l + 1) + k + l η = 32 Kk+l+2 + (k + l + 1)!t2 ≤ 4t2 Kk+l+2 . Similarly, for |η| = 1, s ∈ R, |s| < 1/3 and η + is ∈ D we have kl k,l+1 T0 (ukl ) ≤ 4s2 Kk+l+2 . η+is − uη − suη Hence T0 m ≤ 4 for all m ∈ M and consequently, T0 v ≤ 4v for all v ∈ X0 .

Vol. 57 (2007)

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233

Let X1 = {v ∈ X0 : v = 0}. Then T0 X1 ⊂ X1 . Let X be the completion of X0 /X1 . Then T0 induces the operator T : X → X and T ≤ 4. Deﬁne the function f : C → X by f (z) = u00 z + X1 for |z| < 1 and f (z) = − uz + X1 for |z| ≥ 1. Clearly (T − z)f (z) = u + X1 for all z ∈ C and u + X1 = 0. It remains to show that f is inﬁnitely diﬀerentiable. Clearly f is even analytic for |z| > 1. For |z| < 1 we can show by induction ∂ k+l f kl that ∂x k ∂y l (z) = uz + X1 . Indeed, this follows from the estimates uk,l − uk,l z − uk+1,l lim z+t ≤ lim |t|Kk+l+2 = 0 z t→0 t→0 t and similarly,

uk,l − uk,l z − uk,l+1 lim z+is =0 z s→0 s

Finally, for |z| = 1 we show by induction that t→z,

∂ k+l f ∂xk ∂y l (z)

= ukl z + X1 . Clearly

kl ukl gkl (z + t) − gkl (z) z+t − uz = lim u = gk+1,l (z)u = uk+1,l z |z+t|>1 t t→z, |z+t|>1 t

lim

and t→z,

ukl − ukl z+t z − uk+1,l = 0. z t |z+t|<1

lim

Similar statements hold for derivatives in the imaginary direction y. Thus for all ∂ k+l f kl k, l ∈ Z+ and |z| = 1 we have ∂x k ∂y l (z) = uz + X1 . By (1) and (2), all the derivatives z → ukl z are continuous. Remark 2. In [2], there was constructed an example of an operator T ∈ B(X) and x = 0 such that int σ(T ) = ∅ and there is a continuous local resolvent f : C → X satisfying (T − z)f (z) = x (z ∈ C). It was raised a question whether there is a similar example with smooth local resolvent. As it was observed by J. Kol´aˇr, such an example with C1 -local resolvent cannot exist. Indeed, each local resolvent is analytic on the complement of the spectrum. Therefore it satisﬁes the Cauchy-Riemann conditions on C \ σ(T ). If the local resolvent is C1 and int σ(T ) = ∅, then the local resolvent satisﬁes the CauchyRiemann conditions everywhere, and so it is an entire function. It is well known (and an easy consequence of the Liouville theorem) that such a local resolvent cannot exist.

References [1] T. Bem´ udez, M. Gonz´ alez, On the boundedness of the local resolvent function, Int. Eq. Operator Theory 34 (1999), 1–8. [2] J. Braˇciˇc, V. M¨ uller, On bounded local resolvents, Int. Eq. Operator theory, to appear.

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[3] M. Gonz´ alez, An example of a bounded local resolvent, Operator Theory, Operator algebras and related topics, Proc. 16th Int. Conf. on Operator Theory at Timisoira. The Theta Foundation (1997), 159–162. [4] M.M. Neumann, On local spectral properties of operators on Banach spaces, Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 15–25. Vladim´ır M¨ uller Mathematical Institute Czech Academy of Sciences Zitna 25 115 67 Prague 1 Czech Republic e-mail: muller@@math.cas.cz Submitted: January 15, 2006

Integr. equ. oper. theory 57 (2007), 235–246 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020235-12, published online August 8, 2006 DOI 10.1007/s00020-006-1455-z

Integral Equations and Operator Theory

Semisimplicity of Some Class of Operator Algebras on Banach Space H. S. Mustafayev Abstract. Let G be a locally compact abelian group and let T = {T (g)}g∈G be a representation of G by means of isometries on a Banach space. We define weak operator topology of the set W n T as the closure owith respect to the R fˆ (T) : f ∈ L1 (G) , where fˆ (T) = f (g)T (g) dg is the Fourier transform G

of f ∈ L1 (G) with respect to the group T. Then WT is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp (T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra WT is semisimple. Some related problems are also discussed. Mathematics Subject Classification (2000). 47Dxx, 46J05. Keywords. Representation (semi)group, Banach algebra, radical, semisimplicity.

1. Introduction Let A be a complex commutative Banach algebra. The radical of A, denoted by RadA is the set of all quasi-nilpotent elements in A. If RadA = {0}, then A is said to be semisimple. Let X be a complex Banach space and B (X) the algebra of all bounded, linear operators on X. As usual, we denote by σ (T ) the spectrum of + T ∈ B (X). If T in B (X), we let A+ T (resp. WT ) denote the closure in the uniform operator topology (resp. in the weak operator topology) of all polynomials in T . If T is invertible, then AT (resp. WT ) denotes the closure in the uniform operator topology (resp. in the weak operator topology) of all trigonometric polynomials + in T . The algebras A+ T and AT (resp. WT and WT ) are commutative Banach algebras. This research was supported by the YYUBAP Project No: 2006-FED-B12.

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Let A be a closed commutative subalgebra of B(X). It follows from the spectral radius formula that RadA = {R ∈ A : σ (R) = {0}}. Hence, the algebra A is semisimple if and only if it does not contain a non-zero operator with zero spectrum. For example, if T is a bounded self-adjoint operator on a Hilbert space, then the algebra A+ T is isometrically isomorphic to the algebra of all continuous functions on σ (T ) and therefore, A+ T is semisimple. Recall that an operator T ∈ B (X) is said to be hermitian if exp (itT ) = 1 for all t ∈ R. As is well known, hermitians in a Hilbert space are exactly bounded self-adjoint operators. In contrast with self-adjoint operators on a Hilbert space there is a hermitian operator that generates a non-semisimple algebra [16, Corollary 3.4]. Sinclair [16, Theorem 3.1] proved that if T is a hermitian operator with countable spectrum, then WT+ is semisimple. In [6] Feldman showed that if T is an invertible isometry on a Banach space with countable spectrum, then AT is semisimple. Note that if T is a hermitian operator (resp. invertible isometry) on a Banach space X, then T = {exp(itT )}t∈R (resp. T = {T n }n∈Z ) is a one-parameter (resp. discrete) group of isometries on X. Thus this raises the natural question of whether the similar results are valid for the algebras generated by the representations of an abelian (semi)groups. In this paper we generalize both Sinclair’s and Feldman’s results for the representation of a locally compact abelian group using the concept of spectrum of a representation introduced by Arveson in [2].

2. Arveson spectrum and semisimplicity ˆ A map T : G → Let G be a locally compact abelian group with dual group G. B (X) is a representation of G on a Banach space X, if T is a homomorphism of G into the group of invertible isometries in B(X) which is continuous, i.e. limg→0 T (g) x − x = 0 for all x ∈ X. ˆ the Let T = {T (g)}g∈G be a representation of G on X. For each χ ∈ G, representation Tχ = {Tχ (g)}g∈G will be deﬁned by Tχ (g) = χ (g)T (g). T∗ will denote the set {T (g)∗ }g∈G . Let L1 (G) be the group algebra of G. As usual, fˆ will denote the Fourier transform of f ∈ L1 (G): ˆ fˆ (χ) = f (g)χ (g)dg, χ ∈ G. G

If T = {T (g)}g∈G is a representation of G on X, then for each f ∈ L1 (G) we can deﬁne fˆ (T) ∈ B (X), by fˆ (T) = f (g)T (g) dg. G

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The map f → fˆ (T) is a continuous homomorphism from L1 (G) into B (X). We deﬁne AT (resp. WT ) as the closure with respect to the uniform operator topology 1 ˆ (resp. weak operator topology) of the set f (T) : f ∈ L (G) . Then AT and WT are commutative Banach algebras. Let {U } be a net of compact neighborhoods of 0 contracting to {0} and eU , the non-negative function supported by U and satisfying eU (g) dg = 1. It is easy U

to verify that eˆU (T) → 1X -in the strong operator topology, as U → {0}; 1X is the identity operator on X. This shows that thealgebra WT has the unit element 1X . ˆ let J 0 = f ∈ L1 (G) : suppfˆ ∩ S =Ø }. As is For a closed subset S of G, S 1 0 known [10, p.182-183], JS := JS is the smallest closed ideal in L (G) whose hull is S; IS = f ∈ L1 (G) : fˆ (S) = {0} is the largest closed ideal in L1 (G) whose hull is S (in the case when S = {0}, instead of JS and IS we will write J0 and I0 , respectively). S is a set of synthesis if and only if IS = JS . It is a famous theorem ˆ contains a subset S of of Malliavin [10, p.199] that if G is not compact, then G non-synthesis. Example 2.1. Let G be a locally compact but non-compact abelian group. For a 1 ˆ let T = {T (g)} closed subset S of G, g∈G be the translation group on L (G) /JS ; T (g) : f + JS = fg + JS where, fg (s) = f (s − g). Then T is a representation of G on L1 (G)/JS and ˆ (T) (f + JS ) = h ∗ f + JS for every h ∈ L1 (G). Since L1 (G)/JS has a bounded h ˆ (T) = h + JS . It follows that AT is by one approximate identity, we have h isometrically isomorphic to L1 (G)/JS . Since RadL1 (G)/JS = IS /JS , in case S fails to be of spectral synthesis AT (resp. WT ) is not semisimple. Let T = {T (g)}g∈G be a representation of G on X. The set IX = f ∈ L1 (G) : fˆ (T) = 0 is a closed ideal in L1 (G). The Arveson spectrum sp (T) of T is deﬁned as the hull (IX ), i.e. ˆ : fˆ (χ) = 0, f ∈ IX . sp (T) = χ ∈ G ˆ and We see that sp (T) is a closed subset of G Jsp(T) ⊆ IX ⊆ Isp(T) .

(2.1)

As proved in [2, p.215, Corollary 1], sp (T) =Ø if and only if X = {0}. In [2, p.223, Corollary 2], it is also shown that sp (T) can be identiﬁed with the maximal ideal space of AT . Recall that a Hausdorﬀ topological space S is said to be scattered if S does not contain a nonempty perfect subset. For example, the scattered subsets of the real line (or complex plane) are countable (Cantor-Bendixson Theorem). Note also

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that if S is a scattered set, then every nonempty closed subset of S contains an isolated point. As is known [10, Corollary 8.5.1], scattered subsets of the dual ˆ of a a locally compact abelian group G are sets of synthesis. group G The main result of this section is the following theorem. Theorem 2.2. Let T = {T (g)}g∈G be a representation of a locally compact abelian group G on a Banach space X. If sp (T) is a scattered set, then the algebra WT is semisimple. For the proof of this theorem we need some preliminary results with which we now proceed. Let T = T (g)g∈G be a representation of G on X. Lemma 2.3. If sp(T) is a set of synthesis, then fˆ(T) = 0 if and only if fˆ = 0 on sp (T) . Proof. Since sp (T) is a set of synthesis, it follows from (2.1) that IX = Isp(T) . We deﬁne the kernel of T as the set KerT = {x ∈ X : T (g) x = x, g ∈ G}. The range of T is deﬁned as RanT = span {T (g) x − x : x ∈ X, g ∈ G}. As is known [7, Theorem 3.8.4], KerT ∩ RanT = {0} . It can be deduced even more. Proposition 2.4. If T = {T (g)}g∈G is a representation of G on X, then for every x ∈ KerT, x = dist x, RanT . Proof. We put E = KerT and F = RanT. It is enough to show that x ≤ dist (x, F ) for every x ∈ E. We have F ⊥ = {ϕ ∈ X ∗ : T (g)∗ ϕ = ϕ, g ∈ G} . Let Cb (G) be the space of all bounded continuous functions on G. It is well known that G is amenable, namely there exists a Φ ∈ Cb (G)∗ such that: i) Φ (1) = 1, where 1 is the constant one function on G; ii) Φ (f ) ≥ 0 for every f ≥ 0; iii) Φ (f s ) = Φ (f ), where f s (g) = f (g + s). Let ϕ ∈ X ∗ be ﬁxed. For x ∈ X, we deﬁne a function fϕ,x on G by fϕ,x (g) =< ϕ, T (g) x >. Clearly, fϕ,x ∈ Cb (G). Then we can deﬁne a ψ ∈ X ∗ by < ψ, x >= Φ(fϕ,x ). Since s ∗ < T (s) ψ, x >=< ψ, T (s) x >= Φ fϕ,x = Φ (fϕ,x ) =< ψ, x > (s ∈ G), the map ϕ → ψ is a projection from X ∗ onto F ⊥ . Taking into occount Φ = 1, we have ψ ≤ ϕ. Now, let x ∈ E be given. Since < ϕ, x >=< ψ, x >, we obtain x = sup {|ϕ (x)| : ϕ ∈ X ∗ : ϕ ≤ 1} ≤ sup |ψ (x)| : ψ ∈ F ⊥ , ψ ≤ 1 = dist (x, F ) .

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Let T = {T (g)}g∈G be a representation of G on X ant let E be a closed subspace of X invariant under T, i.e. T (g)E ⊆ E for all g ∈ G. Then we may deﬁne a subrepresentation T|E = {T |E (g)}g∈G of G on E by T |E (g) = T (g)|E , where T (g)|E is the restriction of T (g) to E. In this case IE = f ∈ L1 (G) : fˆ (T) x = 0, x ∈ E . Since IX ⊆ IE , we have hull(IE ) ⊆ hull(IX ), so that sp(T|E ) ⊆ sp(T). Lemma 2.5. Let T = {T (g)}g∈G be a representation of G on X and let E be a closed subspace of X invariant under T. If sp(T|E ) = {0}, then E ⊆ KerT. Proof. Since hull (IE ) = {0} and {0} is a set of synthesis, we have IE = I0 . It follows that f (g)ϕ (T (g) x) dg = 0, G

for all f ∈ I0 , ϕ ∈ X ∗ and all x ∈ E. Thus ϕ (T (g) x) = const and hence T (g) x = x for all x ∈ E and all g ∈ G. Recall that x ∈ X\ {0} (resp. ϕ ∈ X ∗ \ {0}) is said to be an eigenvector of ˆ and T (resp. of T∗ ) if T (g) x = χ (g) x (resp. T (g)∗ ϕ = χ (g)ϕ ) for some χ ∈ G for all g ∈ G. Lemma 2.6. Let T = {T (g)}g∈G be a representation of G on a Banach space X which satisfies one of the following conditions: a) The linear span of the set of eigenvectors of T∗ is weak ∗ dense in X ∗ . b) The linear span of the set of eigenvectors of T is weakly dense in X. Then the algebra WT is semisimple. Proof. a) Let ϕ be an eigenvector of T∗ and let R ∈ RadWT . It is suﬃcient to ∗ ˆ and for all g ∈ G. show that R∗ ϕ = 0. We have T (g) ϕ = χ (g)ϕ for some χ ∈ G ∗ 1 ˆ ˆ It follows that f (T) ϕ = f (−χ) ϕ for all f ∈ L (G). Since R ∈ WT , there exists a net (fλ )λ∈Λ in L1 (G) such that < ϕ, fˆλ (T) x >→< ϕ, Rx >, x ∈ X. If x ∈ Kerϕ, in particular we have ∗ 0 =< fˆλ (−χ) ϕ, x >=< fˆλ (T) ϕ, x >

=< ϕ, fˆλ (T) x >→< ϕ, Rx >=< R∗ ϕ, x > . Thus we obtain Kerϕ ⊆ KerR∗ ϕ, which implies that R∗ ϕ = λϕ for some λ ∈ C. Since σ (R) = {0}, we have λ = 0 and hence R∗ ϕ = 0. The proof of b) is similar to that of a).

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It is not clear whether the condition b) is satisﬁed for the representations with scattered spectrum. However, the proof of Theorem 2.2 below, shows that for the representation T with scattered spectrum the set of eigenvectors of T∗ is weak ∗ dense in X ∗ . Proof of Theorem 2.2. Let M be the weak ∗ closure of the linear span of the set of ˆ eigenvectors of T∗ . Since (RanTχ )⊥ ⊂ M , we have ⊥ M ⊂ RanTχ for all χ ∈ G. Consequently, ⊥ M ⊂ E, where E = ∩χ∈sp(T) RanTχ . By Lemma 2.6 a) it is suﬃcient to prove that E = {0}. Suppose on the contrary that E = {0}. We see that E is a closed subspace of X invariant under T. Then sp (T |E ) is a nonempty closed subset of sp (T). Since sp (T) is a scattered set, sp (T |E ) has an isolated point or single point. We may assume that this isolated or single point is {0}. Assume that {0} is an isolated point of sp (T |E ). Let e ∈ L1 (G) be such that eˆ (0) = 1 and eˆ (χ) = 0 for all χ ∈ sp (T |E ) \ {0}. Since the Fourier 2 transform of e∗e−e vanishes on sp (T |E ), by Lemma 2.3 we have eˆ (T) x = eˆ (T) x for all x ∈ E. Hence eˆ (T) is a projection on E. We put F = eˆ (T) E. Let us show that sp (T |F ) = {0}. For this it is enough to show that IF = I0 , where IF = f ∈ L1 (G) : fˆ (T) eˆ (T) x = 0, x ∈ E . If f ∈ IF , then it follows from Lemma 2.3 that the Fourier transform of f ∗ e vanishes on sp (T |E ). Since {0} ∈ sp (T |E ) and eˆ (0) = 1 this implies fˆ (0) = 0, so that f ∈ I0 . If f ∈ I0 , then the Fourier transform of f ∗ e vanishes on sp (T |E ). By Lemma 2.3 we have fˆ (T) eˆ (T) x = 0 for all x ∈ E, so that f ∈ IF . Now, if {0} is an isolated point of sp (T |E ) we let Z = F , and if sp (T |E ) = {0} we let Z = E. Then Z is a non-null closed subspace of X and sp (T |Z ) = {0}. By Lemma 2.5 we have Z ⊆ KerT. On the other hand, Z ⊆ RanT. Since KerT ∩ RanT = {0}, we obtain Z = {0}. This is a contradiction. We do not know whether Theorem 2.2 true if sp(T) is a synthesis set. Let T = {T (g)}g∈G be a representation of G on X. A straightforward application of a separation theorem shows that the union of the ranges of the operators fˆ (T), f ∈ L1 (G) is dense in X. Theorem 2.2 can be reformulated as follows: Theorem 2.7. Let h : L1 (G) → B (X) be a continuous homomorphism such that the union of the ranges of the operators h(f ), f ∈ L1 (G) is dense in X. If W OT

hull(kerh) is a scattered set, then the algebra h (L1 (G))

is semisimple.

Proof. Under the hypotheses of the theorem there exists a representation T = {T (g)}g∈G of G on X such that h (f ) = f (g)T (g) dg G

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[11, Theorem 32C]. It remains to apply Theorem 2.2.

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3. C0 -semigroup and semisimplicity A family T = {T (t)}t≥0 in B (X) is called a C0 -semigroup if the following properties are satisﬁed: 1) T (0) = 1X ; 2) T (t + s) = T (t) T (s), for every t, s ≥ 0; 3) limt→0+ T (t) x − x = 0, for all x ∈ X. The generator of the C0 -semigroup T = {T (t)}t≥0 is the linear operator A with domain D (A) deﬁned by Ax = lim

t→0+

1 (T (t) x − x) , t

x ∈ D (A) ;

1 D (A) = x ∈ X : lim (T (t) x − x) exists . t→0+ t The generator is always a closed, densely deﬁned operator. The C0 -groups are deﬁned analogously to C0 -semigroups, the only diﬀerence being that the role of the index family t ≥ 0 is replaced by t ∈ R. The generator of a C0 -group T = {T (t)}t∈R is deﬁned as the generator of the associated C0 -semigroup. A C0 -semigroup T = {T (t)}t≥0 will be said to be bounded if supt≥0 T (t) < ∞. If T = {T (t)}t≥0 is a bounded C0 -semigroup with generator A, then the spectrum σ (A) of A belongs to the closed left half-plane. σ (A) ∩ iR is called the unitary spectrum of A. If T = {T (t)}t≥0 is a C0 -semigroup of contractions i.e., T (t) ≤ 1 (t ≥ 0) on a Banach space X, then for every x ∈ X the limit limt→∞ T (t) x exists and is equal to inft≥0 T (t) x. Note also that if T is a bounded C0 -semigroup on X, then |x| = supt≥0 T (t)x is an equivalent norm on X with respect to which T becomes a C0 -semigroup of contractions. Similarly, if T is a bounded C0 -group on X, then |x| = supt∈R T (t)x is an equivalent norm on X with respect to which T becomes a C0 -group of isometries. Let L1 (R+ ) be the space of all absolutely integrable measurable complex functions on the half-line R+ . L1 (R+ ) is a commutative Banach algebra when convolution is taken as the multiplication. L1 (R+ ) can be considered (in the natural way) as a subalgebra of L1 (R). For a function f ∈ L1 (R+ ), we put ∞ fˆ (T) =

f (t) T (t)dt. 0

Then the map f → fˆ (T) is a continuous homomorphism from L1 (R+ ) into B (X). + We deﬁne A+ T (resp. WT ) as the closure with respect to the uniform operator

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topology (resp. weak operator topology) of the set

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fˆ (T) : f ∈ L1 (R+ ) . Then

+ A+ T and WT are commutative Banach algebras. Note also that if T is a bounded + C0 -group then A+ T (resp. WT ) is a subalgebra of AT (resp. WT ). Let en (t) = 2nχ[0,1/n] (t) (n ∈ N), where χ[0,1/n] (t) is the characteristic function of the interval [0, 1/n]. It is easy to verify that eˆn (T) → 1X -in the strong operator topology, as n → ∞. This shows that WT+ has the unit element 1X . In [5] it is shown that if T is a bounded C0 -group with generator A, then sp (T) = σ (iA). As an immediate corollary of Theorem 2.2 we have the next result.

Corollary 3.1. Let T = {T (t)}t∈R be a bounded C0 -group on a Banach space with generator A. If σ (A) is at most countable, then WT (resp. WT+ ) is semisimple. Let T = {T (t)}t≥0 be a C0 -semigroup of contractions on X. Arendt-BattyLyubich-Vu (ABLV) theorem [13, Theorem 5.1.5] states that if the unitary spectrum of the generator of T is countable and contains no residual spectrum, then T is stable, i.e. limt→∞ T (t)x = 0 for all x ∈ X. The proof of the ABLV theorem uses essentially the following construction [13, Theorem 5.1.2]. Lemma 3.2. Let T = {T (t)}t≥0 be a C0 -semigroup of contractions on a Banach space X, with generator A. If inft≥0 T (t) x > 0 for some x ∈ X\ {0}, then there exist a Banach space Y = {0}, a bounded linear operator J : X → Y with dense range and a C0 -semigroup of isometries V = {V (t)}t≥0 on Y with generator B such that: i) Jx = limt→∞ T (t) x for all x ∈ X; ii) V (t) J = JT (t) for all t ≥ 0; iii) σ (B) ⊆ σ (A) . The triple (Y, J, V) will be called the isometric limit semigroup associated to T. As a consequence of Lemma 3.2 and Theorem 2.2 we have the following: Theorem 3.3. Let T = {T (t)}t≥0 be a C0 -semigroup of contractions on a Banach space X with generator A. If the unitary spectrum of A is at most countable, then for every R ∈ RadWT+ and x ∈ X, lim T (t) Rx = 0.

t→∞

Proof . If limt→∞ T (t) x = 0 for every x ∈ X\ {0}, then there is nothing to prove. Hence we may assume that inft≥0 T (t) x > 0 for some x ∈ X\ {0}. Let (Y, J, V) be the isometric limit semigroup associated to T. Any operator S ∈ WT+ generates an operator S0 on JX by S0 Jx = JSx, x ∈ X. Then we have S0 Jx = JSx = lim T (t) Sx = lim ST (t) x t→∞

t→∞

≤ S lim T (t) x = S Jx . t→∞

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Since J has dense range, S0 can be extended continuously to whole Y . We will ˆ ˆ denote this extension by S. Then we have SJ = JS and Sˆ ≤ S for all S ∈ WT+ . Now, let us show that Sˆ ∈ WV+ . Since S ∈ WT+ , there exists a net (fλ )λ∈Λ in L1 (R+ ) such that < ϕ, fˆλ (T) x, >→< ϕ, Sx >, for all ϕ ∈ X ∗ and x ∈ X. By Lemma 3.2 ii) since fˆλ (V) J = J fˆλ (T) (λ ∈ Λ), we have < ψ, fˆλ (V) Jx >=< ψ, J fˆλ (T) x >=< J ∗ ψ, fˆλ (T) x >→ ˆ >, < J ∗ ψ, Sx >=< ψ, JSx >=< ψ, SJx for all ψ ∈ Y ∗ and x ∈ X. Since J has dense range we obtain that ˆ >, < ψ, fˆλ (V) y >→< ψ, Sy for all ψ ∈ Y ∗ and y ∈ Y . The mapping h : S → Sˆ is a contractive algebra-homomorphism from WT+ into WV+ . Since h is a unital algebra-homomorphism, we obtain the spectral incluˆ ⊆ σ(S) for all S ∈ W + . It follows that if σ(S) = 0 then σ(S) ˆ = 0. sion σ(S) T + We claim that the algebra WV is semisimple. As is known [13, Lemma 5.1.1], if V is an arbitrary C0 -semigroup of isometries with generator B, then either σ(B) = {z ∈ C : Rez ≤ 0} or σ(B) ⊆ iR. Since σ(A) ∩ iR is countable, by Lemma 3.2 iii) σ(B) ∩ iR is also countable. Consequently, σ(B) is a proper subset of iR. By [13, Corollary 5.1.3], V extends to a C0 -group of isometries U = {U (t)}t∈R on X with generator B. Since σ(B) is at most countable, by Corollary 3.1 WU is semisimple. Also since WV+ = WU+ , the algebra WV+ is semisimple. ˆ = {0}. Since the Now, let R ∈ WT+ be such that σ(R) = {0}. Then σ(R) + ˆ algebra WV is semisimple we have R = 0, so that JR = 0. By Lemma 3.2 i) this means that limt→∞ T (t) Rx = 0 for every x ∈ X. A C0 -semigroup T = {T (t)}t≥0 on a Banach space X is bounded away from zero if inft≥0 T (t)x > 0 for all x ∈ X\ {0} [13, p.180]. Corollary 3.4. Let T = {T (t)}t≥0 be a bounded C0 -semigroup on a Banach space which is bounded away from zero. If the unitary spectrum of the generator A is at most countable, then WT+ is semisimple. Example 3.5. Let V be the Volterra operator on the Hilbert space L2 [0, 1], deﬁned t by (V f )(t) = f (s)ds and let T = e−tV t≥0 . Notice that the exponential formula 0 [14, Theorem 1.8.3] yields e−tV = 1 for all t ≥ 0. On the other hand, V is a + + non-zero quasinilpotent operator and V ∈ A+ V = AT . This shows that AT (resp. + WT ) is not semisimple.

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4. C1 -contractions and semisimplicity An invertible operator T on a Banach space X is called double power bounded (d.p.b.) if supn∈Z T n < ∞. It is easy to see that if T is a d.p.b. operator, then σ (T ) ⊆ Γ; Γ = {z ∈ C : |z| = 1}. Every d.p.b. operator deﬁnes a bounded representation T = {T n }n∈Z of Z on X. In this case for an arbitrary f = {f (n)}n∈Z ∈ L1 (Z), fˆ (T) is deﬁned by f (n) T n . fˆ (T) = n∈Z

Hence AT (resp. WT ) coincides with the AT (resp. WT ). Note that [10, Theorem −1 the joint 4.5.1] the maximal ideal space of AT can be identiﬁed with σAT T, T , −1 −1 with respect to A →ξ . Note also that the map ξ, ξ spectrum of T, T T homeomorphically identiﬁes σAT T, T −1 with σAT (T ), the spectrum of T with respect to AT . On the other hand, since σAT (T ) ⊆ Γ, by the Shilov’s theorem [10, Corollary 2.3.2] we have σAT (T ) = σ (T ). Hence in this case sp(T) = σ(T ). The next corollary is an immediate consequence of Theorem 2.2. Corollary 4.1. If T is a d.p.b. operator on a Banach space with countable spectrum, then WT is semisimple. An operator T in B (X) is called power-bounded if supn∈N T n < ∞. Note that if T is a power-bounded operator then σ (T ) ⊆ D; D = {z ∈ C : |z| < 1}. σ (T ) ∩ Γ is called the unitary spectrum of T . Note also that if T is a contraction i.e., T ≤ 1, then for every x ∈ X the limit limn→∞ T n x exists and is equal to infn∈N T n x. The main result of this section is the following theorem. Theorem 4.2. If T is a contraction on a Banach space X with countable unitary spectrum, then for every R ∈ RadWT+ and x ∈ X, lim T n Rx = 0.

n→∞

For the proof we need the following lemma [15, Lemma 2.1]. Lemma 4.3. Let T be a contraction on a Banach space X. If infn∈N T n x > 0 for some x ∈ X \ {0}, then there exist a Banach space Y = {0}, a bounded linear operator J : X → Y with dense range and an isometry V on Y such that: i) Jx = limn→∞ T n x for all x ∈ X; ii) V J = JT ; iii) σ (V ) ⊆ σ (T ) . The triple (Y, J, V ) will be called the limit isometry of T . Proof of Theorem 4.2. If limn→∞ T n x = 0 for every x ∈ X, then there is nothing to prove. Hence we may assume that infn∈N T n x > 0 for some x ∈ X\{0}.

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Let (Y, J, V ) be the limit isometry of T . Any operator S ∈ WT+ generates an operator S0 on JX by S0 Jx = JSx, x ∈ X. Then we have S0 Jx = JSx = lim T n Sx = lim ST n x ≤ S Jx . n→∞

n→∞

Since J has dense range, S0 can be extended continuously to whole Y . We will ˆ Then we have SJ ˆ = JS and denote this extension by S. Sˆ ≤ S for all S ∈ WT+ . Now, let us show that Sˆ ∈ WV+ . Since S ∈ WT+ , there exists a net of polynomials (Pλ )λ∈Λ such that < ϕ, Pλ (T ) x >→< ϕ, Sx >, ∗ for every ϕ ∈ X and x ∈ X. By Lemma 4.3 ii) since Pλ (V ) J = JPλ (T ) (λ ∈ Λ), we have < ψ, Pλ (V ) Jx >=< ψ, JPλ (T ) x >=< J ∗ ψ, Pλ (T ) x >→ for every ψ ∈ Y ∗

ˆ < J ∗ ψ, Sx >=< ψ, JSx >=< ψ, SJx >, and x ∈ X. Since J has dense range we obtain that ˆ >, < ψ, Pλ (V ) y >→< ψ, Sy

for every ψ ∈ Y ∗ and y ∈ Y . The mapping h : S → Sˆ is a contractive algebra-homomorphism from WT+ into WV+ . Since h is a unital algebra-homomorphism, we obtain the spectral incluˆ ⊆ σ(S) for all S ∈ W + . It follows that if σ(S) = 0 then σ(S) ˆ = 0. sion σ(S) T + Now, we claim that the algebra WV is semisimple. As is known [3, p.27, Proposition 1.15] if V is an arbitrary isometry on a Banach space, then either σ(V ) = D or σ(V ) ⊆ Γ. Since σ(T ) ∩ Γ is countable, by Lemma 4.3 iii) σ(V ) ∩ Γ is also countable, so that σ(V ) ⊂ Γ. Hence V is an invertible isometry with countable spectrum. By Corollary 4.1, WV is semisimple. It follows that the algebra WV+ is semisimple. ˆ = {0}. Since the alLet R ∈ WT+ be such that σ(R) = {0}. Then σ(R) + ˆ = 0 and hence JR = 0. This means that gebra WV is semisimple we have R limn→∞ T n Rx = 0 for every x ∈ X. An operator T in B(X) is said to be of class C1 if infn∈N T n x > 0 for all x ∈ X\{0} [3, p.249]. As a corollary of Theorem 4.2 we have the next result. Corollary 4.4. Let T be a power-bounded operator of class C1 on a Banach space. If the unitary spectrum of T is at most countable, then WT+ is semisimple. Example 4.5. Let V be the Volterra operator on H = L2 [0, 1] and let T = −1 (1H + V ) . Then T n = 1 for all n ∈ N [8, Problem 150]. We see that σ (T ) = + {1} but T = 1H . Hence A+ T (resp. WT ) is not semisimple. Acknowledgment The author is grateful to the referee for helpful comments and suggestions.

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References [1] W. Arendt and C.J.K. Batty, Tauberian theorems for one-parameter semigroups, Trans. Amer. Math. Soc., 306(1988), 837-852. [2] W. Arveson, The harmonic analysis of automorphism groups of operator algebras, Proc. Symp. Pure Math., 38(1982), 199-269. [3] A. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, NorthHolland, Amsterdam, 1988. [4] J. Esterle, E. Strouse and F. Zouakia, Stabilite asymptotique de cartains semigroupes d’operateurs et ideaux primaries, J. Operator Theory, 28(1992), 203-228. [5] D. E. Evans, On the spectrum of a one-parameter strongly continuous representation, Math. Scand., 39(1976), 80-82. [6] G. M. Feldman, The semisimplicity of an algebra generated by isometric operators, Funk. Anal. Prilozhen., 8(1974), 93-94 (Russian). [7] F. Greenleaf, Invariant Means on Topological Groups and their Application, Van Nostrand, New York, 1969. [8] P. R. Halmos, A Hilbert Space Problem Book, Von Nostrand, Princeton, 1967. [9] E. Hewitt and K. Ross, Abstract Harmonic Analysis-II., Springer-Verlag, New York, 1973. [10] R. Larsen, Banach Algebras, Marcel-Dekker Inc., New York 1973. [11] L. H. Loomis An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York, 1953. [12] Yu. I. Lyubich and V. Q. Phong, Asymptotic stability of linear diﬀerential equations on Banach spaces, Studia Math., 88(1988), 37-42. [13] J. M. A. M. van Neerven, The Asymptotic Behavior of Semigroups of Linear Operators, Birkh¨ auser Verlag, 1996. [14] A. Pazy, Semigroup of Linear Operators and Applications to Partial Diﬀerential Equations, Springer-Verlag, New York, 1983. [15] V. Q. Phong, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal., 103(1992), 74-84. [16] A. M. Sinclair, The Banach algebra generated by a hermitian operator, Proc. London Math. Soc., 24(1972), 681-691. H. S. Mustafayev Yuzuncu Yil University, Faculty of Arts and Sciences, Department of Mathematics 65080 Van, Turkey e-mail: [email protected] Submitted: February 3, 2006 Revised: April 8, 2006

Integr. equ. oper. theory 57 (2007), 247–261 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020247-15, published online December 26, 2006 DOI 10.1007/s00020-006-1468-7

Integral Equations and Operator Theory

Non- (Quantum) Diﬀerentiable C 1-Functions in the Spaces with Trivial Boyd Indices Denis Potapov and Fyodor Sukochev Abstract. If E is a separable symmetric sequence space with trivial Boyd indices and SE is the corresponding ideal of compact operators, then there exists a C 1 -function fE , a self-adjoint element W ∈ SE and a densely deﬁned / closed symmetric derivation δ on SE such that W ∈ Dom δ, but fE (W ) ∈ Dom δ. Mathematics Subject Classiﬁcation (2000). Primary 47A55; Secondary 47L20. Keywords. Commutator estimates, derivations.

1. Introduction This paper studies properties of inﬁnitesimal generator δ S of a strongly continuous group α = {αt }t∈R in Banach algebras S ⊆ B(H), given by αt (y) = eitX ye−itX , y ∈ S, where X is an unbounded self-adjoint operator in the Hilbert space H. The generator δ S is a densely deﬁned closed symmetric derivation on S and we are concerned with the question when its domain Dom δ S satisﬁes the following condition x = x∗ ∈ Dom δ S ⇒ f (x) ∈ Dom δ S , for every C 1 -function f : R → C? In the recent paper [3], it is shown that there are C ∗ -algebras S and operators X for which the implication above fails (see also [9]). In this paper, we consider the case when the Banach algebra S is a symmetrically normed ideal of compact operators on H. (see e.g. [4] and Section 3 below). It is immediately clear that for every self-adjoint operator X, the group α acts isometrically on such an ideal S and, in fact, is a C0 -group on S, provided that S is separable (see e.g. [2]). It is an interesting problem to determine the class of ideals S in which Dom δ S is closed with respect to the C 1 -functional calculus. Note, that the class of such ideals is non-empty. For example it contains the Hilbert-Schmidt ideal. The proof of the latter claim may be found in [10]. On the other hand, it is unclear whether this class contains the Schatten-von

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Neumann ideals Sp when 1 < p < ∞, p = 2. In this paper, we however show that the class of all symmetrically normed ideals S whose Boyd indices are trivial fail the implication above. For various geometric characterizations of the latter class we refer to [1] (see also Section 3 below). Our methods are built upon and extend those of [3, 9]. Our results also contribute to the study of commutator bounded operator-functions initiated in [5–7].

2. Schur multipliers Let Mn (C) be the C ∗ -algebra of all n × n complex matrices, let B ∈ Mn (C) be a diagonal matrix diag{λ1 , λ2 , . . . , λn }. The Schur multiplier Mf (B) associated with the diagonal matrix B and the function f : R → C is deﬁned as follows. For every matrix X = {ξjk }nj,k=1 ∈ Mn (C), the matrix Mf (B)(X) ∈ Mn (C) has (j, k) entry given by [Mf (B)(X)]jk = ψf (λj , λk )ξjk , 1 ≤ j, k ≤ n, where

  f (λ) − f (µ) , λ = µ, λ−µ ψf (λ, µ) =  0, λ = µ.

Alternatively, if {Pj }nj=1 is the collection of one-dimensional spectral projections of the matrix B then B = nj=1 λj Pj , and Mf (B)X = ψf (λj , λk )Pj XPk . (2.1) 1≤j,k≤n

For every matrix X ∈ Mn (C) the following equation outlines the interplay between the Schur multiplier Mf (B) and the commutator [B, X] = BX − XB Mf (B)([B, X]) = [f (B), X]. Indeed, Mf (B)([B, X]) =

ψf (λj , λk )Pj

=

λs Ps , X Pk

s=1

1≤j,k≤n

n

ψf (λj , λk )(λj − λk )Pj XPk

1≤j,k≤n

=

(f (λj ) − f (λk ))Pj XPk

1≤j,k≤n

=

1≤j,k≤n

Pj

n s=1

f (λs )Ps , X Pk = [f (B), X].

(2.2)

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3. Symmetric spaces with trivial Boyd indices Let E = E(0, ∞) be a symmetric Banach function space, i.e. E = E(0, ∞) is a rearrangement invariant Banach function space on (0, ∞) (see [8]) with the additional property that f, g ∈ E and g ≺≺ f imply that g E ≤ f E . Here g ≺≺ f denotes submajorization in the sense of Hardy, Littlewood and Polya, i.e. t t g ∗ (s) ds ≤ f ∗ (s) ds, t > 0, 0

0

where f ∗ (respectively, g ∗ ) stands for the decreasing rearrangement of the function f (respectively, g). Let us consider the group of dilations {στ }τ >0 deﬁned on the space S = S(0, ∞) of all Lebesgue measurable functions on (0, ∞). The operator στ , τ > 0 is given by (στ f )(t) = f (τ −1 t), t > ∞. If E is a symmetric Banach function space, then the lower (respectively, upper) Boyd index αE (respectively, βE ) of the space E is deﬁned by

log στ E→E log στ E→E αE := lim respectively, βE := lim . τ →+∞ τ →+0 log τ log τ We say that the space E has the trivial lower (resp. upper) Boyd index when αE = 0 (respectively, βE = 1). It is known that, if αE = 0 (respectively, βE = 1), then the space E is not an interpolation space in the pair (L1 , Lp ) for every p < ∞ (respectively, (Lq , L∞ ) for every 1 < q), [8, Section 2.b]. Proposition 3.1. ( [8, Proposition 2.b.7]) If E be a symmetric sequence space and αE = 0 (respectively, βE = 1), then for every ε > 0 and every n ∈ N there exist n disjointly supported vectors {xj }nj=1 in E, having the same distribution, such that for every scalars {aj }nj=1 the following holds n max |aj | ≤ aj xj ≤ (1 + ε) max |aj | 1≤j≤n

j=1

1≤j≤n

E

n n n respectively, (1 − ε) |aj | ≤ aj xj ≤ |aj | . j=1

j=1

E

j=1

(3.1) If E is separable then xj can be chosen ﬁnitely supported. SE denotes the corresponding symmetric ideal of compact operators on the Hilbert space 2 = 2 (N), i.e. the space of all compact operators x such that s(x) ∈ E, where s(x) is the step function such that s(x)(t) = sk , k < t ≤ k + 1, k ≥ 0 and {sk }k≥0 the sequence of singular numbers (counted with multiplicities) of the operator x (see e.g. [4]). The norm in the space SE is given by x SE := s(x) E . In particular, if E = Lp , then the ideal Sp = SLp , 1 ≤ p < ∞ stands for the

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Schatten-von Neumann ideals of compact operators and S∞ stands for the ideal of all compact operators equipped with the operator norm, see [4]. Let n2 be the subspace in 2 spanned by the ﬁrst n standard unit vector basis. If an element B ∈ SE is such that B = B|n2 , then we identify B with its matrix from Mn (C). Proposition 3.2. Let E be a separable symmetric function space and αE = 0 (respectively, βE = 1). For every scalar ε > 0 and every positive integer n ∈ N there exist linear operators Φn and Ψn such that (i) Φn , Ψn : Mn (C) → Mkn (C), where {kn }n≥1 is a sequence of positive integers; (ii) the operators Φn , Ψn map diagonal (respectively, self-adjoint) matrices to diagonal (respectively, self-adjoint) matrices; (iii) if Mf (B), Mf (Φn (B)) are the Schur multiplier associated with the diagonal matrices B ∈ Mn (C), Φn (B) ∈ Mkn (C) and the function f , then Ψn (Mf (B)X) = Mf (Φn (B))Ψn (X) for every matrix X ∈ Mn (C); (iv) X S∞ ≤ Ψn (X) SE ≤ (1 + ε) X S∞ (respectively, (1 − ε) X S1 ≤

Ψn (X) SE ≤ X S1 ) for every matrix X ∈ Mn (C). Proof. Let n be a ﬁxed positive integer and ε > 0 be a ﬁxed positive scalar. Let {xj }nj=1 be a sequence of ﬁnitely and disjointly supported vectors, having the same distribution such that (3.1) holds. Let X0 be the matrix given by X0 = diag{x∗1 (k)}k≥1 , i.e. X0 is the ﬁnite diagonal matrix in SE that corresponds to the decreasing rearrangement x∗1 in E. Let I be the identity matrix of the same size as X0 . We deﬁne the linear operators Φn and Ψn by Φn (X) := X ⊗ I, and Ψn (X) := X ⊗ X0 , X ∈ Mn (C). The claims (i), (ii), now, follow immediately from the deﬁnition of Φn and Ψn and the claim (iii) follows from (2.1). Let us prove (iv). For every matrix X ∈ Mn (C) there exist unitary matrices U, V such that U XV = diag{s1 , s2 , . . . , sn }. Now, it follows from elementary properties of tensors, that Φn (U )Ψn (X)Φn (V ) = (U ⊗ I)(X ⊗ X0 )(V ⊗ I) = (U XV ) ⊗ X0 = Ψn (U XV ) = diag{sj X0 }nj=1 , and so

Ψn (X) SE = Φn (U )Ψn (X)Φn (V ) SE n = diag{sj X0 }nj=1 SE = sj xj . j=1

E

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Now, the claim in (iv) for αE = 0 (respectively, βE = 1) follows from combining the equality above with the ﬁrst estimate in (3.1) (respectively, the second estimate in (3.1)). The operators Φn , Ψn are very similar to those, constructed in the proof of [1, Theorem 4.1].

4. Commutator estimates From now on let h : R → R be a function with the following properties (a) h(t) ∈ C 1 (R \ {0}); (b) h(t) = h(−t) when t = 0, h(0) ≥ 0; (c) h(·) is increasing function on (0, ∞); (d) h(±∞) = +∞; (e) 0 ≤ h (t)/h(t) ≤ 1 when t ∈ (0, ∞). Proposition 4.1. Let h(t) be a function that satisfies the conditions (a)–(e) above. If f is a function defined as follows |t|(h(log |t|))−1 , if |t| < 1, t = 0, f (t) = (4.1) 0, if t = 0. then f (t) ∈ C 1 (−1, 1) and f (t) ≥ 0 for every t ∈ (0, 1). Proof. The function given in (4.1) is even so it is suﬃcient to consider only the case t ≥ 0. It follows from the deﬁnition of the function f that for every t ∈ (0, 1) function f is continuously diﬀerentiable. To calculate the derivative at zero, we use the deﬁnition (d) f (t) − f (0) = lim (h(log t))−1 = 0. f (0) = lim t→0 t→0 t−0 In order to verify that f (t) → 0 when t → +0, we note ﬁrst that

h (log t) −1 f (t) = (h(log t)) 1− , 0 < t < 1. h(log t) Since h(t) ≥ 0 for every t ∈ R, together with the property (e), it now follows that for every t ∈ (0, 1) 0 ≤ f (t) ≤ 2(h(log t))−1 → 0, as t → +0.

Let matrices D, V ∈ Mm (C) and A, B ∈ M2m (C) be deﬁned as follows D = diag{e−1 , e−2 , . . . , e−m }, V = {vjk }m j,k=1 , (k − j)−1 (e−j + e−k )−1 , vjk = 0,

if j = k, , if j = k. (4.2)

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and

A=

0 −V

V , 0

B=

D 0

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0 . −D

(4.3)

The following proposition provides commutator estimates in the norm of the ideal of compact operators which are very similar to those established in [3] and [9]. Proposition 4.2. For any function f : R → R given by (4.1), there exists an absolute constant K0 such that for every m ≥ 3 and for every scalar 0 < p ≤ 1 the following estimates hold (i) [B, A] S∞ ≤ π, (ii) [f (pB), A] S∞ ≥

m pK0 log . h(m − log p) 2

Proof. The proof of the ﬁrst claim is based on the norm estimates of the Hilbert matrix, see [3, the proof of Lemma 3.6]. Hence, we need to establish only the second one. Let us ﬁrst note, since the function f is even, it follows from deﬁnition of matrices A, B that

0 f (pD)V − V f (pD) f (pB)A − Af (pB) = , f (pD)V − V f (pD) 0 so

[f (pB), A] S∞ = [f (pD), V ] S∞ . If S = {sjk }m j,k=1 = f (pD)V − V f (pD) ∈ Mm (C), then skj = sjk =

f (pe−j ) − f (pe−k ) ≥ 0, 1 ≤ j, k ≤ m. (e−j + e−k )(k − j)

If 1 ≤ j < k ≤ m, then, since functions h(t) and et are monotone, we have

pe−j pe−k − sjk = (e−j + e−k )−1 (k − j)−1 h(j − log p) h(k − log p) p(e−j − e−k ) −j (2e (k − j))−1 h(k − log p) p(1 − e−1 ) p(1 − e−1 ) ≥ . ≥ 2h(k − log p)(k − j) 2h(m − log p)(k − j) ≥

Now, using m j=1

k−1

1 j=1 j

sjk ≥

≥ log k, we have

k−1 j=1

k−1

sjk ≥

p(1 − e−1 ) p(1 − e−1 ) 1 ≥ log k. 2h(m − log p) j=1 k − j 2h(m − log p)

(4.4)

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Finally, letting x = (m−1/2 , m−1/2 , . . . , m−1/2 ) ∈ Cm , we obtain

S S∞ ≥ Sx, x = ≥ ≥

m m 1 1 p(1 − e−1 ) sjk ≥ log k m m 2 h(m − log p)

−1

j,k=1 m

1 p(1 − e ) m 2 h(m − log p)

k=1

log k

k=[m/2]

1 p(1 − e−1 ) m m log . m 2 h(m − log p) 2 2

Setting K0 = (1 − e−1 )/4, we have

[f (pD), V ] S∞ = S S∞ ≥

m pK0 log . h(m − log p) 2

which, together with (4.4), completes the proof.

Together with (2.2), Proposition 4.2 provides a lower estimate for the operator norm of Schur multiplier associated with the function f , given by (4.1), and diagonal matrix pB given by (4.2) and (4.3) for every scalar 0 < p ≤ 1 and every integer m ≥ 3. Now we extend that lower estimate to a larger class of ideals. Proposition 4.3. Let E be a separable symmetric function space with trivial Boyd indices. For every m ≥ 3, let Am , Bm ∈ M2m (C) be given by (4.2) and (4.3), Φ2m , Ψ2m be the operators from the Proposition 3.2 for the ε = 1/2. There exists an absolute constant K1 such that for every scalar sequence 0 < pm ≤ 1, and for the sequence of the diagonal matrices Wm = Φ2m (pm Bm ) ∈ Mkm (C) the following estimate holds m K1 log , m ≥ 3,

Mf (Wm ) SE →SE ≥ h(m − log pm ) 2 where f : R → R is an arbitrary function given by (4.1). Proof. Letting ∞ Xm = [pm Bm ,

1 Am ], m ≥ 3, pm

we infer from Proposition 4.2 and from (2.2) that for every m ≥ 3 ∞

S∞ ≤ π,

Xm

1 Am ] S∞ pm m K0 log . ≥ h(m − log pm ) 2

∞ ) S∞ = [f (pm Bm ),

Mf (pm Bm )(Xm

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It follows from the deﬁnition of Schur multiplication and duality that

Mf (pm Bm ) S1 →S1 = Mf (pm Bm ) S∞ →S∞ m K0 log , m ≥ 3. ≥ πh(m − log pm ) 2 1 The last estimate implies that there exists a sequence of Xm ∈ M2m (C) such that 1 m ) S1 K0

Mf (pm Bm )(Xm log , m ≥ 3. ≥ 1

Xm S1 2π h(m − log pm ) 2 ∞ Suppose now, that αE = 0 and set Xm = Ψ2m (Xm ) for every m ≥ 3. It follows from Proposition 3.2 that, for every m ≥ 3, Wm is a ﬁnite diagonal self-adjoint matrix such that ∞ )} SE

Mf (Wm )(Xm ) SE

Ψ2m {Mf (pm Bm )(Xm = ∞

Xm SE

Ψ2m (Xm ) SE ∞ m ) S∞ 2 Mf (pm Bm )(Xm 2K0 log . ≥ ≥ ∞ 3 Xm S∞ 3πh(m − log pm ) 2

Mf (Wm ) SE →SE ≥

If we put K1 = 2K0 /(3π), that completes the proof of the case αE = 0. The only 1 ∞ instead of Xm diﬀerence in treating the case βE = 1 is that we need to use Xm in the above estimates. The following proposition proves that if a function f : R → R is given by (4.1) and the multipliers Mf (Wm ) are not uniformly bounded in SE , then this function is not commutator bounded in the sense of [6]. Proposition 4.4. Let E be a separable symmetric function space. If f is a C 1 function and Wm ∈ Mkm (C) is a sequence of diagonal matrices (m ≥ 3) such that

Mf (Wm ) SE →SE → ∞,

(4.5)

then there exist self-adjoint operators W , X, acting on 2 , such that [W, X] ∈ SE ,

[f (W ), X] ∈ / SE .

If, in addition, the norms Wm S∞ are uniformly bounded, then W (Dom X) ⊆ Dom X, and if the following series converges

Wm SE , m≥3

then operator W belongs to S

E

and

W SE ≤

m≥3

Wm SE .

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Proof. It follows from (4.5) that there exists a subsequence of positive integers mr (r ≥ 1) and a sequence of self-adjoint matrices Xr(1) ∈ Mkr (C) such that

Mf (Wr )(Xr(1) ) SE ≥ 2r3 Xr(1) SE , r ≥ 1, where we let, for brevity, kr {λj }j=1

kr

= kmr and

Wr

(4.6)

= Wmr ∈ Mkr (C). Let r ≥ 1 be ﬁxed, k

r let be the sequence of eigenvalues of the matrix Wr , and let {Pj }j=1 be the collection of corresponding one-dimensional spectral projections. For λ ∈ R, we set Qλ = Pj .

1≤j≤kr λj =λ

There are only a ﬁnite number of non-zero projections among {Qλ }λ∈R , let us denote them as {Qj }sj=1 , 1 ≤ s ≤ kr and the corresponding sequence of eigenvalues as {λj }sj=1 , the scalars λj are mutually distinct. We consider the self-adjoint matrices s ˆr = ˆr . X Qj Xr(1) Qj , and Xr(2) = Xr(1) − X j=1

It follows from (2.1) that (recall that ψf (λ, λ) = 0) ˆ r Pl ˆr ) = ψf (λj , λl )Pj X Mf (Wr )(X 1≤j,l≤kr

=

=

s

t=1

1≤j,l≤kr

s

ψf (λj , λl )Pj Qt Xr(1) Qt Pl

ψf (λt , λt )Qt Xr(1) Qt = 0,

t=1 1≤j,l≤kr λj =λl =λt

and so Mf (Wr )(Xr(2) ) = Mf (Wr )(Xr(1) ).

(4.7)

ˆ r SE ≤ X SE (see [4, Theorem III.4.2]) and, hence Now, noting that X r

Xr(2) SE ≤ 2 Xr(1) SE , Xr(2) , we infer from (4.7) and (4.6) (1)

Mf (Wr )(Xr(2) ) SE ≥ r3 Xr(2) SE . We set Xr(3) =

λjl Pj Xr(2) Pl ,

1≤j,l≤kr

where

 0, λjl =

λj = λl , −i  , λj = λl . λj − λl

(4.8)

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The matrix Xr(3) is self-adjoint and Pj Xr(2) Pl = i Xr(2) = 1≤j,l≤kr

λjl (λj − λl )Pj Xr(2) Pl

1≤j,l≤kr

=i

IEOT

λjl Pj (Wr Xr(2) − Xr(2) Wr )Pl

1≤j,l≤kr

= i Wr ,

λjl Pj Xr Pl = i[Wr , Xr(3) ]. (2)

1≤j,l≤kr

(4.9) Finally, we let X (3) . Xr = r−2 Xr(2) −1 SE r

(4.10)

For every r ≥ 1 we have constructed so far the ﬁnite self-adjoint matrices Wr , Xr such that

[Wr , Xr ] S∞ ≤ [Wr , Xr ] SE

(4.10)

(3) E = r−2 Xr(2) −1 SE [Wr , Xr ] S

(4.9)

(2) E = r−2 Xr(2) −1 SE Xr S =

1 r2 (4.11)

and

[f (Wr ), Xr ] SE

(4.10)

= r−2 Xr(2) −1

[f (Wr ), Xr(3) ] SE SE

(2.2)

= r−2 Xr(2) −1

Mf (Wr )([Wr , Xr(3) ]) SE SE

(4.9)

= r−2 Xr(2) −1

Mf (Wr )(Xr(2) ) SE SE

(4.8)

≥ r Xr(2) −1

Xr(2) SE ≥ r. SE (4.12)

kr

Now, we set H = r≥1 C , X = the deﬁnition we have

r≥1

Xr and W =

H = {{ξr }r≥1 : ξr ∈ Ckr ,

r≥1 Wr .

Recall, that by

ξr 2 < ∞},

r≥1

Dom X = {ξ = {ξr }r≥1 ∈ H : X(ξ) = {Xr (ξr )}r≥1 ∈ H}, Dom W = {ξ = {ξr }r≥1 ∈ H : W (ξ) = {Wr (ξr )}r≥1 ∈ H}. W , X are self-adjoint operators, acting on the separable Hilbert space H and

[W, X] SE ≤

r≥1

[Wr , Xr ] SE

(4.11) 1

≤

r≥1

r2

< ∞,

Non- (Quantum) Diﬀerentiable C 1 -Functions

Vol. 57 (2007)

[f (W ), X] SE ≥ max [f (Wr ), Xr ] SE

(4.12)

r≥1

If we assume that

257

= ∞.

Wm SE < ∞,

m≥3

then

W SE ≤

Wr SE ≤

r≥1

Wm SE < ∞.

m≥3

If we assume that supm≥1 Wm S∞ ≤ M < ∞, then, by (4.11), for every ξ = [ξr ]r≥1 ∈ Dom X,

12

12

Xr (Wr (ξr )) 2 =

Wr (Xr (ξr )) − [Wr , Xr ](ξr ) 2 r≥1

r≥1

≤M

Xr (ξr ) 2

r≥1

+ sup [Wr , Xr ] S∞ r≥1

12

ξr 2

12

< ∞.

r≥1

Hence W (ξ) ∈ Dom X. The claim is proved.

It follows from Propositions 4.3 and 4.4 that any function f : R → R given log(m/2) by (4.1) with the function h satisfying the condition h(m−log pm ) → ∞, as m → ∞ (here {pm }m≥0 is some scalar sequence satisfying 0 < pm ≤ 1) is not commutator bounded in any separable symmetrically normed ideal with trivial Boyd indices. In other words, there exist self-adjoint operators W , X, acting on a separable Hilbert / SE . We shall now show how space H, such that [W, X] ∈ SE but [f (W ), X] ∈ further adjustments to the choice of the function h and the sequence {pm }m≥0 can be made in order to guarantee that the operator W above belongs to SE . First, we need the following auxiliary results. Proposition 4.5. For every ε > 0 there exists a function χε such that (i) χε ∈ C 1 (R), (ii) χε (t) = 0, if t ≤ 0, (iii) χε (t) = 1, if t ≥ 1, (iv) 0 ≤ χε ≤ 1 + ε. Proof. Let ξε (t) be the continuous function such that ξ (t) = 0, if t ≤ 0 or t ≥ 1, ξ (t) = 1 + , if /(1 + ) ≤ t ≤ 1/(1 + ) and linear elsewhere. It then follows, that the function t

χε (t) = satisﬁes the assertion.

−∞

ξε (τ ) dτ, t ∈ R,

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Proposition 4.6. Let sm , qm (m ≥ 0) be two increasing sequences such that (i) sm → +∞, s0 = 0, (ii) qm → +∞, q0 = 1, log qm − log qm−1 (iii) α = sup < 1. sm − sm−1 m≥1 Then there exists a function h that satisfies the conditions (a)–(e) (preceding Proposition 4.1) and such that h(sm ) = qm for every m ≥ 0. Proof. Let ε = 1/α − 1, and χε be the function from Proposition 4.5. For every t ≥ 0 we deﬁne H(t) =

χε

m≥1

t−s m−1 (log qm − log qm−1 ). sm − sm−1

(4.13)

We have that x ≥ sm−1 (respectively, x < sm ) if and only if x − sm−1 x − sm−1 ≥ 0 respectively, <1 . sm − sm−1 sm − sm−1 Now, it follows from above that for every ﬁxed t ≥ 0 the sum (4.13) is ﬁnite, H(s0 ) = H(0) = 0 = log q0 , and for every k ≥ 1 H(sk ) =

χε

m≥1

=

k

s −s k m−1 (log qm − log qm−1 ) sm − sm−1

(log qm − log qm−1 ) = log qk .

m=1

We set h(t) := exp(H(t)) for every t ≥ 0 and h(t) = h(−t) for every t < 0, then h(sm ) = qm for every m ≥ 0. Let us check the conditions (a)–(e). (a) The function H is a C 1 -function as a ﬁnite sum of C 1 -functions, so h is a C 1 -function for every t = 0; (b) this item holds by the deﬁnition of h(t), and h(0) = exp(H(0)) = 1; (c) the function H(t) is increasing, for every t ≥ 0, as it is the sum of increasing functions, so the function h is increasing also; (d) since the sequence h(sm ) = qm tends to inﬁnity and since h(·) is an increasing even function, we have h(±∞) = +∞;

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(e) for every t ≥ 0 there exists an integer k ≥ 1 such that sk−1 ≤ t < sk , thus it follows from Proposition 4.5, that t − sm−1 log qm − log qm−1 H (t) = χε , sm − sm−1 sm − sm−1 m≥1 t−s log q − log q k−1 k k−1 = χε sk − sk−1 sk − sk−1 ≤ α(1 + ε) = 1, and so 0 ≤ H (t) =

h (t) ≤ 1. h(t)

Now, we are in a position to prove our main result. Theorem 4.7. For every separable symmetric function space E with trivial Boyd indices, there exists a C 1 -function fE , self-adjoint operators W , X, acting on a separable Hilbert space H such that / SE . W ∈ SE , [W, X] ∈ SE , W (Dom X) ⊆ Dom X, [fE (W ), X] ∈ Proof. Let m ≥ 0, qm := (log(m + e))1/2 , Bm be the diagonal matrices, given by (4.2) and (4.3), Φ2m , Ψ2m be the operators from Proposition 3.2 for ε = 1/2. Let {pm }m≥0 be a sequence that satisﬁes the following ﬁve conditions (i) pm is decreasing to zero; (ii) 0 < pm ≤ 1; (iii) p0 = 1; 1 ≥ m2 Φ2m (Bm ) SE , m ≥ 1; (iv) pm 1 q2 1 (v) ≥ 2m · , m ≥ 1. pm eqm−1 pm−1 We construct such a sequence by induction. If the numbers p0 , p1 , . . . , pm−1 satisfy the conditions above, then pm can be taken to be any positive number for which 2 1 1 qm 1 2 ≥ max 1, , m Φ2m (Bm ) SE , 2 · . pm pm−1 eqm−1 pm−1 It follows from (v) above, that q2 epm−1 ≥ 2m , pm qm−1 and, taking logarithms, 1 1 − log ≥ 2(log qm − log qm−1 ). pm pm−1 = m − log pm , we have log qm − log qm−1 1 0≤ ≤ , m ≥ 0. sm − sm−1 2 1 + log

Putting sm

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Thus, we have veriﬁed that the sequences {qm }m≥0 and {sm }m≥0 satisfy the conditions of Proposition 4.6, and so there exists a function hE (t) such that hE (m − log pm ) = hE (sm ) = qm = (log(e + m))1/2 . If fE is the function, given by (4.1), with the above choice of hE , Wm:= Φ2m (pm Bm ) ∈ Mkm (C), where Bm given by (4.2) and (4.3), is the ﬁnite diagonal matrix then it follows from Proposition 4.3 m log (m/2) K1 log = K1 → ∞.

Mf (Wm ) SE →SE ≥ h(m − log pm ) 2 (log(m + e))1/2 On the other hand, by our choice of pm (see ((iv)) above) we have that 1

Wm S∞ ≤ Wm SE = pm Φ2m (Bm ) SE ≤ 2 , m ≥ 3. m Now the assertion of Theorem 4.7 follows from Proposition 4.4.

5. Domains of generators of automorphism ﬂows Let E be a separable symmetric function space with trivial Boyd indices, let SE be the corresponding symmetrically normed ideal and let X be a self-adjoint operator (may be unbounded), acting on a separable Hilbert space H. We consider a group α = {αt }t∈R of automorphisms on SE given by α(t)T = eitX T e−itX , T ∈ SE , t ∈ R. It follows from separability of SE that α is a C0 -group, [2, Corollary 4.3]. The inﬁnitesimal generator δ of α(t) is deﬁned by α(t)T − T E Dom δ := T ∈ S : there exists · SE − lim , t→0 t α(t)T − T for every T ∈ Dom δ. δ(T ) := · SE − lim t→0 t δ is a closed densely deﬁned symmetric derivation on SE , i.e. densely deﬁned closed linear operator such that δ(T ∗ ) = δ(T )∗ and δ(T S) = δ(T ) S + T δ(S), for all T, S ∈ Dom δ, see e.g. [2, Proposition 4.5]. It is proved in [3, Proposition 2.2], that Dom δ = {T ∈ SE : T (Dom X) ⊆ Dom X, [T, X] ∈ SE } and δ(T ) = i[T, X], T ∈ Dom δ. Now, Theorem 4.7 yields Corollary 5.1. For every separable symmetric function space E with trivial Boyd indices, there exists a C 1 -function fE , a self-adjoint operator W ∈ SE and closed densely defined symmetric derivation δ on SE , such that W ∈ Dom δ but / Dom δ. fE (W ) ∈

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References [1] J. Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces, Integral Equations Operator Theory 1 (1978), no. 4, 453–495. [2] B. de Pagter and F. A. Sukochev, Commutator estimates and R-ﬂows in noncommutative operator spaces, Proc. Edinb. Math. Soc. (to appear). [3] B. de Pagter, F. A. Sukochev, and W. van Ackooij, Domains of inﬁnitesimal generators of automorphism ﬂows, J. Funct. Anal. 218 (2005), no. 2, 409–424. [4] I. C. Gohberg and M. G. Kre˘ın, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Izdat. “Nauka”, Moscow, 1965. [5] E. Kissin and V. S. Shulman, Classes of operator-smooth functions. II. Operatordiﬀerentiable functions, Integral Equations Operator Theory 49 (2004), no. 2, 165– 210. , Classes of operator-smooth functions. I. Operator-Lipschitz functions, Proc. [6] Edinb. Math. Soc. (2) 48 (2005), no. 1, 151–173. , Classes of operator-smooth functions. III. Stable functions and Fuglede [7] ideals, Proc. Edinb. Math. Soc. (2) 48 (2005), no. 1, 175–197. [8] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin, 1979. [9] A. McIntosh, Functions and derivations of C ∗ -algebras, J. Funct. Anal. 30 (1978), no. 2, 264–275. [10] D. S. Potapov and F. A. Sukochev, Lipschitz and commutator estimates in symmetric operator spaces, to appear in J. Operator Theory. Denis Potapov and Fyodor Sukochev School of Informatics and Engineering Faculty of Science and Engineering Flinders University of SA Bedford Park, 5042 Adelaide, SA Australia e-mail: [email protected] [email protected] Submitted: February 6, 2006

Integr. equ. oper. theory 57 (2007), 263–281 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020263-19, published online August 8, 2006 DOI 10.1007/s00020-006-1456-y

Integral Equations and Operator Theory

The Fredholm Index of Locally Compact Band-dominated Operators on Lp(R) Vladimir S. Rabinovich and Steﬀen Roch Abstract. We establish a necessary and suﬃcient criterion for the Fredholmness of a general locally compact band-dominated operator A on Lp (R) and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of A. The results are applied to operators of convolution type with almost periodic symbol. Mathematics Subject Classification (2000). Primary 47A53; Secondary 47G10, 45E10. Keywords. Fredholm index, limit operators, band-dominated operators, Wiener-Hopf operators.

1. Introduction Throughout this paper, let 1 < p < ∞, and for each Banach space X, let L(X) stand for the Banach algebra of all bounded linear operators on X, K(X) for the closed ideal of the compact operators, BX for the closed unit ball of X, and X ∗ for the Banach dual space of X. For each function ϕ ∈ BU C, the algebra of the bounded and uniformly continuous functions on the real line R, and for each t > 0, set ϕt (x) := ϕ(tx) and write ϕI for the operator on Lp (R) of multiplication by ϕ. An operator A ∈ L(Lp (R)) is called band-dominated if lim Aϕt I − ϕt A = 0

t→0

for each function ϕ ∈ BU C. The set Bp of all band-dominated operators forms a closed subalgebra of L(Lp (R)). In this paper we will exclusively deal with banddominated operators of the form I + K where I is the identity operator and K is locally compact (which means that ϕA and AϕI are compact for each function ϕ ∈ BU C with bounded support). We write Lp for the set of all locally compact band-dominated operators on Lp (R).

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The announced Fredholm criterion and the index formula will be formulated in terms of limit operators. To introduce this notion, we will need the shift operators Uk : Lp (R) → Lp (R), (Uk f )(x) := f (x − k) where k ∈ Z. Let h : N → Z be a sequence which tends to inﬁnity in the sense that |h(n)| → ∞ as n → ∞. An operator Ah ∈ L(Lp (R)) is called a limit operator of A ∈ L(Lp (R)) with respect to the sequence h if lim (U−h(m) AUh(m) − Ah )ϕI = 0

m→∞

and lim ϕ(U−h(m) AUh(m) − Ah ) = 0

m→∞

for each function ϕ ∈ BU C with bounded support. If a limit operator Ah of A with respect to h exists, then it is uniquely determined. The set of all limit operators of a given operator A ∈ L(Lp (R)) is called the operator spectrum of A and denoted by σop (A). Clearly, the existence of Ah implies that Ah exists and that Ah = Ah for each subsequence h of h. Since each sequence h with |h(n)| → ∞ contains a subsequence h converging to +∞ or −∞, it follows that the operator spectrum of A splits into two components σ+ (A) ∪ σ− (A) which collect the limit operators of A with respect to sequences h tending to +∞ and to −∞, respectively. An operator A ∈ L(Lp (R)) is said to be rich or to possess a rich operator spectrum if every sequence h tending to inﬁnity possesses a subsequence g for which the limit operator Ag exists. The sets of all rich operators in Bp and Lp will be denoted by Bp$ and L$p . Let χ+ and χ− stand for the characteristic functions of the sets R+ and R− of the non-negative and negative real numbers, respectively. The operators χ+ Kχ− I and χ− Kχ+ I are compact for each operator K ∈ Lp . Indeed, let ε > 0 be arbitrarily given. Since K is band-dominated, there is a continuous function f which is 1 on [0, ∞) and 0 on (−∞, −nε ] with suﬃciently large nε such that f K − Kf I < ε. Thus, χ+ Kχ− I − χ+ Kf χ− I = χ+ (f K − Kf )χ− I < ε. The operator χ+ Kf χ− I is compact since f χ− has a bounded support and K is locally compact. Since further ε can be chosen arbitrarily small, the compactness of χ+ Kχ− I follows. The compactness of χ− Kχ+ I can be checked analogously. This simple observation implies that, for a Fredholm operator of the form A = I + K with K ∈ Lp , the operators χ+ Aχ+ I and χ− Aχ− I, considered as acting on Lp (R+ ) and Lp (R− ), are Fredholm operators again. We call ind+ A := ind (χ+ Aχ+ I) and ind− A := ind (χ− Aχ− I) the plus- and the minus-index of A. Clearly, ind A = ind+ A + ind− A.

Vol. 57 (2007)

Index of Locally Compact BDO

265

Recall in this connection that a bounded linear operator A on a Banach space X is said to be Fredholm if its kernel ker A and its cokernel coker A := X/im A are linear spaces of ﬁnite dimension, and that in this case the integer ind A := dim ker A − dim coker A is called the Fredholm index of A. Here is the main result of the present paper. Theorem 1. Let A = I + K with K ∈ L$p . (a) The operator A is Fredholm on Lp (R) if and only if all limit operators of A are invertible and if the norms of their inverses are uniformly bounded. (b) If A is Fredholm, then for arbitrary operators B+ ∈ σ+ (A) and B− ∈ σ− (A), ind+ B+ = ind+ A

and

ind− B− = ind− A

(1)

and, consequently, ind A = ind+ B+ + ind− B− .

(2)

This result has a series of predecessors. One of the simplest classes of banddominated and locally compact operators on Lp (R) is constituted by the operators of convolution by L1 (R)-functions and by the restrictions of these operators to the half line, the classical Wiener-Hopf operators. The theory of the convolution type operators on the half line originates from the fundamental papers by Krein and Gohberg/Krein [8, 5] where the Fredholm theory for these operators is established and an index formula is derived. See also the monograph [4] by Gohberg/Feldman for an axiomatic approach to this circle of questions. For convolution type operators with variable coeﬃcients which stabilize at inﬁnite, a Fredholm criterion and an index formula have been obtained by Karapetiants/Samko in [6]; see also their monograph [7]. In [13, 14], there is developed the limit operator approach to study Fredholm properties of general band-dominated operators on spaces lp of vector-valued sequences. In [11] we demonstrated that this approach also applies to operators of convolution type acting on Lp spaces if a suitable discretization reducing Lp - to lp -spaces is performed. (To be precisely: If the sequences in lp take their values in an inﬁnite dimensional Banach space, then we derived in [14] a criterion for a generalized form of Fredholmness, called P-Fredholmness; see below. But the results of [11] refer to common Fredholmness.) The long standing problem to determine the Fredholm index of a band-dominated operator in terms of its limit operators, too, has been ﬁnally solved in [12] for band-dominated operators on the space l2 with scalar-valued sequences. All mentioned results can be also found in the monograph [15]. The index formula has been generalized to lp -spaces in [16]. In the present paper we will undertake a further generalization to banddominated operators with compact entries acting on lp -spaces of vector-valued functions. Thereby these results will get the right form to become applicable to locally compact band-dominated operators on Lp -spaces (and thus, to prove assertion (b) of the theorem).

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The paper is organized as follows. We start with recalling some basic facts on sequences of compact operators. For the reader’s convenience, the proofs are included. The main work will be done in Section 3 where we will derive the Fredholm criterion and the index formula for band-dominated operators on lp with compact entries. In Section 4, these results will be applied to locally compact band-dominated operators on Lp which mainly requires to construct a suitable discretization mapping. Some applications will be discussed in the ﬁnal section. This work was supported by the CONACYT project 43432. The authors are grateful for this support.

2. Sequences of compact operators Let X be a complex Banach space which enjoys the following symmetric approximation property (sap): There is a sequence (ΠN )N ≥1 of projections (= idempotents) ΠN ∈ L(X) of ﬁnite rank such that ΠN → I and Π∗N → I ∗ strongly as N → ∞. Evident examples of Banach spaces with sap are the separable Hilbert spaces, the spaces lp (ZK ) and the spaces Lp [a, b]. It is also clear that if X is a reﬂexive Banach space with sap, then X ∗ has sap, too, and the corresponding projections can be chosen as Π∗N . Definition 2. A sequence (Kn ) of operators in L(X) is said to be (a) relatively compact if the norm closure of {Kn : n ∈ N} is compact in L(X); (b) collectively compact if the set ∪n∈N Kn BX is relatively compact in X; (c) uniformly left (right, two-sided) approximable if, for each ε > 0 there is an N0 such that, for each n ∈ N and each N ≥ N0 , Kn − ΠN Kn < ε

(Kn − Kn ΠN < ε,

Kn − ΠN Kn ΠN < ε).

Note that the uniform left approximability of (Kn ) is equivalent to lim sup Kn − ΠN Kn = 0.

N →∞ n∈N

Proposition 3. Let X be a Banach space with sap. The following conditions are equivalent for a sequence (Kn ) of compact operators on X : (a) (Kn ) is relatively compact; (b) (Kn ) and (Kn∗ ) are collectively compact; (c) (Kn ) is uniformly left and uniformly right approximable; (d) (Kn ) is uniformly two-sided approximable. Proof. (a) ⇒ (b): Let (xn ) be a sequence in ∪n Kn BX . For each n ∈ N, choose r(n) ∈ N and yn ∈ BX such that xn = Kr(n) yn . By hypothesis (a), the sequence (Kr(n) ) has a convergent subsequence (Kr(nk ) ). Let K denote the limit of that subsequence. Then xnk − Kynk = Kr(nk ) ynk − Kynk ≤ Kr(nk ) − K → 0.

(3)

Since K is compact and ynk ≤ 1, the sequence (Kr(nk ) ) has a convergent subsequence. From (3) we conclude that then the sequence (xnk ) (hence, the sequence

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(xn )) has a convergent subsequence, too. This yields the collective compactness of the sequence (Kn ). Since (Kn∗ ) is relatively compact whenever (Kn ) is relatively compact, the collective compactness of (Kn∗ ) follows in the same way. (b) ⇒ (c): We will show that the collective compactness of (Kn ) implies the uniform left approximability of that sequence. We will not make use of the strong convergence of ΠN to I ∗ in this part of the proof. So is becomes evident that then also the collective compactness of (Kn∗ ) implies the uniform left approximability of (Kn∗ ) with respect to the sequence (Π∗N ) which is equivalent to the uniform right approximability of (Kn ). Contrary to what we want to show, assume that (Kn ) is not uniformly left approximable. Then there are an ε > 0, a monotonically increasing sequence (N (r))r≥1 and operators Kn(r) ∈ {Kn : n ∈ N} such that (I − ΠN (r) )Kn(r) ≥ ε for all r ∈ N. Choose xn(r) ∈ BX such that (I − ΠN (r) )Kn(r) xn(r) ≥ ε/2 for all r ∈ N.

(4)

By hypothesis (b), the sequence (Kn(r) xn(r) ) has a convergent subsequence. Let x0 denote its limit. We conclude from (4) that (I − ΠN (r) )x0 ≥ ε/4 for all suﬃciently large r. Letting r go to inﬁnity, we arrive at a contradiction. (c) ⇒ (d): This implication follows immediately from Kn − ΠN Kn ΠN ≤ ≤

Kn − ΠN Kn + ΠN Kn − ΠN Kn ΠN Kn − ΠN Kn + ΠN Kn − Kn ΠN

and from the uniform boundedness of the projections ΠN due to the BanachSteinhaus theorem. (d) ⇒ (a): We consider a subsequence of (Kn ) which we write as (Kn )n∈N0 with an inﬁnite subset N0 of N. Since the projections ΠN have ﬁnite rank, there are an inﬁnite subset N1 of N0 such that the sequence (Π1 Kn Π1 )n∈N1 converges, an inﬁnite subset N2 of N1 such that the sequence (Π2 Kn Π2 )n∈N2 converges, etc. Thus, for each N ≥ 1, one ﬁnds an inﬁnite subset NN of NN −1 such that the sequence (ΠN Kn ΠN )n∈NN converges. Let k(n) denote the nth number in NN (ordered with ˆ n )n≥1 is a subsequence ˆ n := Kk(n) . Clearly, (K respect to the relation <) and set K of each of the sequences (Kn )n∈NN up to ﬁnitely many entries. Thus, for each ˆ n ΠN )n≥1 converges. Now we have N ∈ N, the sequence (ΠN K ˆ n ΠN ) − (K ˆ m ΠN ) + ΠN (K ˆn − K ˆ m = (K ˆ n − ΠN K ˆ m − ΠN K ˆn − K ˆ m )ΠN . K Let ε > 0. By hypothesis (d), there is an N such that ˆ n ΠN < ε/3 ˆ n − ΠN K K for all n ∈ N. Fix N , and choose n0 such that ˆn − K ˆ m )ΠN < ε/3 ΠN (K

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ˆ n ΠN )n≥1 converges). for all m, n ≥ n0 (which can be done since the sequence (ΠN K ˆ ˆ Hence, Kn − Km < ε for all m, n ≥ n0 . This implies the convergence of the ˆ n ) and, thus, the relative compactness of (Kn ). sequence (K

3. The Fredholm index of discrete band-dominated operators with compact entries Let X be a complex Banach space with sap. By E := lp (Z, X) we denote the Banach space of all sequences x : Z → X with xn pX < ∞. xpE := n∈Z

For k ∈ Z, let Vk : l (Z, X) → l (Z, X) stand for the shift operator (Vk x)n := xn−k . In what follows, we will have to consider shift operators on diﬀerent spaces lp (Z, X). In order to indicate the underlying space we will sometimes also write Vk, X for the shift operator Vk on lp (Z, X). Further, for each non-negative integer n, let the projection operators Pn : lp (Z, X) → lp (Z, X) be deﬁned by xk if |k| ≤ n (Pn x)k := 0 if |k| > n, p

p

and set Qn := I − Pn and P := (Pn )n≥0 . Sometimes we will also write Pn, X in place of Pn in order to indicate the underlying space. Each operator A ∈ L(E) can be represented in the obvious way by a twosided inﬁnite matrix with entries in L(X) (in analogy with the representation of an operator on lp (Z) := lp (Z, C) with respect to the standard basis). The operator A ∈ L(E) is called a band operator if its matrix representation (Aij ) is a band matrix, i.e., if there is a k ∈ N such that Aij = 0 if |i − j| ≥ k. The closure of the set of all band operators on E is a closed subalgebra of L(E) which we denote by AE . The elements of AE will be called band-dominated operators. By CE we denote the closed ideal of AE which consists of all band-dominated operators which have only compact entries in their matrix representation. Following the terminology introduced in [15], an operator K ∈ L(E) is called P-compact if lim KQn = lim Qn K = 0. n→∞

n→∞

We denote the set of all P-compact operators by K(E, P), and we write L(E, P) for the set of all operators A ∈ L(E) for which both AK and KA are P-compact whenever K is P-compact. Then L(E, P) is a closed subalgebra of L(E) which contains K(E, P) as a closed ideal. Definition 4. An operator A ∈ L(E, P) is called P-Fredholm if the coset A + K(E, P) is invertible in the quotient algebra L(E, P)/K(E, P), i.e., if there exist an operator B ∈ L(E, P) and operators K, L ∈ K(E, P) such that BA = I + K and AB = I + L.

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This deﬁnition is equivalent to the following one: An operator A ∈ L(E, P) is P-Fredholm if and only if there exist an m ∈ N and operators Lm , Rm ∈ L(E, P) such that Lm AQm = Qm ARm = Qm . Thus, P-Fredholmness is often referred to as local invertibility at infinity. If X has ﬁnite dimension, then the notions P-Fredholmness and Fredholmness are synonymous. All band-dominated operators belong to L(E, P). This can be easily checked for the two basic types of band-dominated operators: the shift operators and the operators of multiplication by a function in l∞ (Z, L(X)), and it follows for general band-dominated operators since L(E, P) is a closed algebra. Hence, it makes sense to speak about their P-Fredholmness. A criterion for the P-Fredholmness of a band-dominated operator A can be given in terms of the limit operators of A. These are, in analogy with the notions from Section 1, deﬁned as follows. Let A ∈ L(E), and let h : N → Z be a sequence which tends to inﬁnity. An operator Ah ∈ L(E) is called a limit operator of A with respect to the sequence h if lim Pk (V−h(n) AVh(n) − Ah ) = lim (V−h(n) AVh(n) − Ah )Pk = 0

n→∞

n→∞

for every k ∈ N. The set of all limit operators of A will be denoted by σop (A) and is called the operator spectrum of A again. An operator A ∈ L(E) is said to be rich or to possess a rich operator spectrum if each sequence h which tends to inﬁnity possesses a subsequence g for which the limit operator Ag exists. We refer to the $ for rich operators in AE as rich band-dominated operators and write A$E and CE the Banach algebra of the rich band-dominated operators and for its closed ideal consisting of the rich operators in CE . The following is the main result on P-Fredholmness of rich band-dominated operators. Its proof is in [15], Theorem 2.2.1. Theorem 5. An operator A ∈ A$E is P-Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded, i.e., sup{(Ah )−1 : Ah ∈ σop (A)} < ∞. In case X = C, P-Fredholmness coincides with common Fredholmness. In this case one can also express the Fredholm index of a Fredholm band-dominated operator in terms of the (local) indices of its limit operators. To cite these results from [12, 16], let P : lp (Z, X) → lp (Z, X) refer to the projection operator xk if k ≥ 0 (P x)k := 0 if k < 0, and set Q := I − P . If necessary, we will write also PX in place of P . Then, for each band-dominated operator on lp (Z, C), the operators P AQ and QAP are compact. This is obvious for band operators in which case P AQ and QAP are of ﬁnite rank, and it follows for general band-dominated operators by an obvious approximation argument. Consequently, the operators A − (P AP + Q)(P + QAQ)

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and A− (P + QAQ)(P AP + Q) are compact, which implies that a band-dominated operator on lp (Z, C) is Fredholm if and only if both operators P AP + Q and P + QAQ are Fredholm and that ind A = ind (P AP + Q) + ind (P + QAQ). In this case we call ind+ A := ind (P AP + Q) and ind− A := ind (P + QAQ) the plus- and the minus-index of A. Finally, let σop (A) = σ+ (A)∪σ− (A), the latter components collecting the limit operators of A with respect to sequences h tending to +∞ and to −∞, respectively, and note that in case X = C all band-dominated operators are rich. Theorem 6. Let X = C, and let A be a Fredholm band-dominated operator on lp (Z). Then, for arbitrary operators B+ ∈ σ+ (A) and B− ∈ σ− (A), ind+ B+ = ind+ A

and

ind− B− = ind− A

(5)

and, consequently, ind A = ind+ B+ + ind− B− .

(6)

In particular, all operators in σ+ (A) have the same plus-index, and all operators in σ− (A) have the same minus-index. It is the goal of the present section to generalize the assertion of Theorem 6 to operators acting on E = lp (Z, X) with a general Banach space X with sap $ . A ﬁrst observation is that for these which are of the form I + K with K ∈ CE operators P-Fredholmness and common Fredholmness coincide. Proposition 7. An operator in I + CE is Fredholm if and only if it is P-Fredholm. Proof. We claim that CE ∩ K(E, P) = K(E). (7) The inclusion K(E) ⊆ CE is evident, and the inclusion K(E) ⊆ K(E, P) holds since the projections Pn and Pn∗ converge strongly to the identity operators on E and E ∗ , respectively. Thus, K(E) ⊆ CE ∩ K(E, P). For the reverse inclusion, let K ∈ CE ∩ K(E, P). Since K ∈ CE , one has Pn K ∈ K(E) for every n, and since K ∈ K(E, P), one has K − Pn K → 0. Thus, K ∈ K(E), which veriﬁes (7). Since K(E) ⊆ K(E, P) by (7), every Fredholm operator in L(E, P) is PFredholm. For the reverse implication, let A := I + K with K ∈ CE be a PFredholm operator. Then there are operators B ∈ L(E, P) and L ∈ K(E, P) such that BA = I − L. Set R := I − KB. Then RA − I = A − I − KBA = K − KBA = K(I − BA) = KL. Since KL ∈ CE ∩K(E, P) is compact by (7), the operator R is a left Fredholm regularizer for A. Similarly one checks that A possesses a right Fredholm regularizer. Thus, the operator A is Fredholm. Combining Proposition 7 with Theorem 5 one gets the following.

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$ Corollary 8. Let A := I + K with K ∈ CE . Then the operator A is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded.

We will make use of the following lemma several times. $ Lemma 9. Every band-dominated operator in CE (resp. in CE ) is the norm limit $ of a sequence of band operators CE (resp. in CE ).

This can be proved in exactly the way as we derived Theorem 2.1.18 in [15] which states that every rich band-dominated operators is the norm limit of a sequence of rich band operators. As a ﬁrst consequence of the CE -version of Lemma 9 we conclude that P AQ and QAP are compact operators for each operator A ∈ I + CE . Indeed, this is obvious for A being a band operators in which case P AQ and QAP have only a ﬁnite number of non-vanishing entries, and these are compact. The case of general A follows by an obvious approximation argument. Consequently, the operators A − (P AP + Q)(P + QAQ) and A − (P + QAQ)(P AP + Q) are compact, which implies that an operator A ∈ I + CE is Fredholm if and only if both operators P AP + Q and P + QAQ are Fredholm. In this case, the integers ind+ A := ind (P AP + Q) and ind− A := ind (P + QAQ) are called the plus- and the minus-index of A. Clearly, ind A = ind+ A + ind− A.

(8)

Finally, let σop (A) = σ+ (A) ∪ σ− (A) in analogy with the case X = C. $ . Here is the announced result for the indices of Fredholm operators in I + CE $ Theorem 10. Let A ∈ I + CE be a Fredholm operator. Then, for arbitrary operators B+ ∈ σ+ (A) and B− ∈ σ− (A),

ind+ B+ = ind+ A

and

ind− B− = ind− A

(9)

and, consequently, ind A = ind+ B+ + ind− B− .

(10)

The remainder of this section is devoted to the proof of Theorem 10. We will verify this theorem by reducing its assertion step by step until we will arrive at operators on lp (Z, C) (with scalar entries) for which the result is known (Theorem 6). The ﬁrst step of the reduction procedure is based on the following observation. Proposition 11. Let F be a dense subset of the set of all Fredholm operators in $ . If the assertion of Theorem 10 holds for all operators in F , then it holds I + CE $ for all Fredholm operators in I + CE . $ , and let B ∈ σ+ (A). We will show Proof. Let A be a Fredholm operator in I + CE that (11) ind+ B = ind+ A,

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which settles the plus-assertion of (9). The minus-assertion follows similarly, and (9) implies (10) via (8). To prove (11), choose a sequence (An ) of operators in F which converges to A in the operator norm, and let h be a sequence tending to +∞ such that B = Ah . Employing Cantor’s diagonal method, we construct a subsequence g of h for which all limit operators (An )g exist. For the details of this construction, consult the proof of Proposition 1.2.6 in [15]. From Proposition 1.2.2 (e) in [15] we conclude that B − (An )g = Ag − (An )g → 0. Now one has ind+ (An )g = ind+ An

for all n ∈ N

and this implies (11) by letting n go to inﬁnity due to the continuity of the index. $ -version of Lemma 9 allows one Our choice of the set F is as follows. The CE $ by a to approximate each band-dominated operator A = I + K with K ∈ CE $ sequence of band operators An := I + Kn with Kn ∈ CE . Each band operator Kn can be written as a sum Kn(k) Vk (12) Kn = k∈Z (k)

with only ﬁnitely many non-vanishing items. The coeﬃcients Kn in (12) are operators of multiplication by sequences of compact operators on X, and these multiplication operators are rich whenever Kn is rich. From Theorem 2.1.16 in [15] we know that a multiplication operator is rich if and only if the set of its entries is relatively compact in L(X). So we conclude from the equivalence between (a) (k) and (d) in Proposition 3 that each coeﬃcient Kn in (12) can be approximated as (k) closely as desired by a sequence (Kn, N )N ∈N of multiplication operators the entries of which map im ΠN into itself and act on im (IX − ΠN ) as the zero operator. Thus, one can approximate the operator A = I + K as closely as desired by band operators An, N = I + Kn, N where the entries of Kn, N map im ΠN into itself and act as the zero operator on im (IX − ΠN ). We denote the set of all operators Kn, N of this form by CE, N . Note that the operators in CE, N are automatically rich. Further, if A = I + K is a Fredholm operator, then the operators An, N = I + Kn, N are Fredholm for all suﬃciently large n and N . Thus, we can choose F as the set of all Fredholm operators I + Kn, N with Kn, N ∈ CE, N . By Proposition 11, it remains to prove Theorem 10 for these operators. We agree upon writing XN in place of im ΠN if we want to consider im ΠN as a Banach space in its own right, not as a subspace of X. Further we introduce the mappings R : lp (Z, X) → lp (Z, XN ),

(xn ) → (ΠN xn )

where ΠN xn is considered as an element of XN , and L : lp (Z, XN ) → lp (Z, X),

(xn ) → (xn )

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where the xn on the right-hand side are considered as elements of X. Clearly, RL is the identity operator on lp (Z, XN ), whereas LR is the projection Π : lp (Z, X) → lp (Z, X),

(xn ) → (ΠN xn ),

now with the ΠN xn being considered as elements of X. We are going to show that the operators A = I + Kn, N as well as their limit operators behave well under the mapping A → RAL. Proposition 12. Let A = I + Kn, N with Kn, N ∈ CE, N . (a) If A is a Fredholm operator on lp (Z, X), then RAL is a Fredholm operator on lp (Z, XN ), and the Fredholm indices of A and RAL coincide. (b) If the limit operator of A with respect to a sequence h : N → Z exists, then the limit operator of RAL with respect to h exists, too, and (RAL)h = RAh L. Proof. Since A is Fredholm, there are operators B, T on lp (Z, X) with T compact such that BA = I + T. (13) For x ∈ ker A one gets x + T x = 0, whence x ∈ im T . Hence, dim ker A ≤ rank T for each pair (B, T ) such that (13) holds. One can choose the pair (B, T ) even in such a way that dim ker A = rank T . For write X as a direct sum ker A ⊕ X0 and let Pker A refer to the projection from X onto ker A parallel to X0 . Then A(I − Pker A ) : im (I − Pker A ) → im A is an invertible operator. Let B denote its inverse. Then BA(I −Pker A ) = I −Pker A and BA = I − Pker A + BAPker A = I − (I − BA)Pker A . Clearly, rank (I − BA)Pker A ≤ dim ker A. Thus, one can indeed assume that (13) holds with dim ker A = rank T . From (13) we get RBAL = RL + RT L = I + RT L, and since L = ΠL and A commutes with Π, we obtain RBL RAL = I + RT L.

(14)

In the same way, AB = I +T with T compact implies that RAL RBL = I +RT L with RT L compact. Hence, RAL is Fredholm, and (14) moreover shows that dim ker RAL ≤ rank RT L ≤ rank T = dim ker A. For the reverse estimate, let B, T be operators on lp (Z, XN ) with BRAL = I + T and dim ker RAL = rank T . Then LBRALR = LR + LT R, whence (LBRΠ + I − Π)A = I + LT R (take into account that AΠ = ΠA = A − (I − Π)). This identity shows that dim ker A ≤ rank LT R ≤ rank T = dim ker RAL,

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whence ﬁnally dim ker A = dim ker RAL. In the same way one gets dim ker A∗ = dim ker (RAL)∗ . Since dim ker A∗ = dim im A for each Fredholm operator A, we arrive at assertion (a). (b) Let Ah be a limit operator of A. Then, by deﬁnition, (Ah − V−h(n), X AVh(n), X )Pk, X → 0 for each k ∈ N. Thus, R(Ah − V−h(n), X AVh(n), X )Pk, X L → 0 for each k ∈ N. Since the projection Π commutes with each of the operators Pk, X , Vh(n), X and A, and since RVh(n), X L = Vh(n), XN and RPk, X L = Pk, XN , one concludes that (RAh L − V−h(n), XN RAL Vh(n), XN )Pk, XN → 0 for each k ∈ N. Similarly one obtains Pk, XN (RAh L − V−h(n), XN RAL Vh(n), XN ) → 0 for each k ∈ N. Thus, RAh L is the limit operator of RAL with respect to the sequence h.

Since the projections P and Π also commute, it is an immediate consequence of the preceding proposition and its proof that ind+ A = ind+ RAL and ind+ Ah = ind+ RAh L = ind+ (RAL)h for each limit operator Ah ∈ σ+ (A). Thus, the assertion of Theorem 6 will follow once we have proved this theorem for band-dominated operators on lp (Z, XN ) in place of lp (Z, X). Proposition 13. The assertion of Theorem 10 holds for all Fredholm band-dominated operators on lp (Z, XN ) (with fixed N ∈ N). Proof. Let d < ∞ be the dimension of XN , and let e1 , . . . , ed be a basis of XN . Then there are positive constants C1 , C2 such that C1 (x1 , . . . , xd )lp ≤ x1 e1 + . . . + xd ed XN ≤ C2 (x1 , . . . , xd )lp

(15)

for each vector (x1 , . . . , xd ) ∈ Cd . Deﬁne J : lp (Z, XN ) → lp (Z, C) by (Jx)nd+r := (xn )r ,

0≤ r ≤ d−1

where (xn )r refers to the rth coordinate of the nth entry xn ∈ XN of the sequence x. It follows from (15) that C1 Jxlp (Z, C) ≤ xlp (Z, XN ) ≤ C2 Jxlp (Z, C) , i.e., J is a topological isomorphism from lp (Z, XN ) onto lp (Z, C). The deﬁnition of J implies that if A is a Fredholm band operator on lp (Z, XN ), then JAJ −1

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is a Fredholm band operator on lp (Z, C), and conversely. Moreover, ind A = ind JAJ −1 in this case. This identity holds for the plus- and minus-indices as well, since JPXN J −1 = PC . Moreover, one has JVn, XN J −1 = Vdn, C

and JPk, XN J −1 = Pdk, C

for all n ∈ Z and k ∈ N. These equalities imply that if Ah is the limit operator of the band-dominated operator A ∈ lp (Z, XN ) with respect to the sequence h, then JAh J −1 is the limit operator of JAJ −1 with respect to the sequence dh : N → Z, m → dh(m), i.e., (JAJ −1 )dh = JAh J −1 . Summarizing, we obtain ind+ A = ind+ JAJ −1 and ind+ Ah = ind+ JAh J −1 = ind+ (JAJ −1 )dh for each Fredholm band-dominated operator A on lp (Z, XN ) and for each of its limit operators Ah ∈ σ+ (A). Since dh tends to +∞ whenever h does, one has (JAJ −1 )dh ∈ σ+ (JAJ −1 ), and from Theorem 6 we infer that ind+ JAJ −1 = ind+ (JAJ −1 )dh . Thus, ind+ A = ind+ Ah for each Fredholm band-dominated operator A on lp (Z, XN ) and for each of its limit operators Ah ∈ σ+ (A). The minuscounterpart of this assertion follows analogously. This proves the proposition and ﬁnishes the proof of Theorem 10.

4. The Fredholm index of locally compact band-dominated operators on Lp (R) This section is devoted to the proof of Theorem 1. As in the discrete case, the ˆ limit operators approach provides us with a criterion for the P-Fredholmness of ˆ an operator rather than for its common Fredholmness. Here, P = (Pˆn )n≥0 where Pˆn : Lp (R) → Lp (R) is the operator of multiplication by the characteristic function of the interval [−n, n], i.e., f (x) if x ∈ [−n, n] ˆ (Pn f )(x) = 0 else, ˆ ˆ and P-compactness and P-Fredholmness are deﬁned literally as in the discrete case. The following proposition can be proved as its discrete counterpart Proposition 7. Proposition 14. An operator A ∈ L(Lp (R)) of the form A = I + K with K ∈ Lp ˆ is Fredholm if and only if it is P-Fredholm.

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We prove Theorem 1 via a suitable discretization. Let χ0 denote the characteristic function of the interval [0, 1]. The mapping G : Lp (R) → lp (Z, Lp [0, 1]) which sends the function f ∈ Lp (R) to the sequence Gf = ((Gf )n )n∈Z

where

(Gf )n := χ0 U−n f

is a bijective isometry the inverse of which maps the sequence x = (xn )n∈Z to the function G−1 x = Un xn χ0 , n∈Z

the series converging in Lp (R). Thus, the mapping Γ : L(Lp (R)) → L(lp (Z, Lp [0, 1])),

A → GAG−1

is an isometric algebra isomorphism. It is shown in Proposition 3.1.4 in [15] that Γ(Ah ) = (Γ(A))h for each limit operator Ah of an operator A ∈ Bp , whereas Proposition 3.1.6 in [15] states that Γ maps Bp$ onto A$E with E = lp (Z, Lp [0, 1]). Further, if A ∈ Lp (R) is a locally compact operator, then the entries of the matrix representation of its $ . Finally, discretization Γ(A) are compact operators. Thus, Γ maps L$p into I + CE one evidently has ind A = ind Γ(A) for each operator A ∈ L(Lp (R)), and the Banach space Lp [0, 1] has the sap as already mentioned. Thus, the assertions of Theorem 1 follow immediately from their discrete counterparts Corollary 8 and Theorem 10.

5. Applications As an application of Theorem 1, we are going to examine the Fredholm properties of operators of the form I + K with K ∈ Kp (BU C). The latter stands for the smallest closed subalgebra of L(Lp (R)) which contains all operators of the form aCbI where a, b ∈ BU C and where C is a Fourier convolution operator with L1 -kernel k. Thus, (Cf )(x) = (k ∗ f )(x) = k(x − y)f (y) dy, x ∈ R. R

In Proposition 3.3.6 in [15] it is veriﬁed that Kp (BU C) ⊆ L$p . Hence, Theorem 1 applies to operators in Kp (BU C), and it yields the following. Theorem 15. Let A ∈ L(Lp (R)) be a convolution type operator of the form I + K with K ∈ Kp (BU C). Then (a) A is Fredholm if and only if all of its limit operators are invertible, and if the norms of their inverses are uniformly bounded.

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(b) if A is Fredholm then, for arbitrary limit operators B± ∈ σ± (A), ind A = ind+ B+ + ind− B− . One cannot say much about the limit operators of a general operator A ∈ I + Kp (BU C). It is only clear that they belong to I + Kp (BU C) again. Thus, the computation of the plus- and minus indices of the limit operators of convolution type operators will remain a serious problem in general. In what follows we will discuss some instances where this computation can be easily done (slowly oscillating coeﬃcients) or is at least manageable (slowly oscillating plus periodic coeﬃcients). Let SO stand for the set of all functions f ∈ BU C which are slowly oscillating in the sense that lim

sup |f (t) − f (t + h)| = 0.

t→±∞ h∈[0, 1]

This set forms a C ∗ -subalgebra of BU C. Let Kp (SO) stand for the smallest closed subalgebra of Kp (BU C) which contains all operators of the form aCbI where a, b ∈ SO and where C is a Fourier convolution with L1 -kernel. Further, we write P ER for the C ∗ -subalgebra of BU C which consists of all continuous functions of period 1 on R. By Kp (P ER, SO) we denote the smallest closed subalgebra of Kp (BU C) which contains all operators of the form aCbI where now a, b ∈ P ER + SO and where C is again a Fourier convolution with L1 -kernel. Similarly, Kp (P ER) refers to the smallest closed subalgebra of Kp (BU C) which contains all operators aCbI with a, b ∈ P ER and with a Fourier convolution C with L1 -kernel. Lemma 16. The limit operators of operators in Kp (SO) are operators of Fourier convolution with L1 -kernel, and all limit operators of operators in Kp (P ER, SO) belong to Kp (P ER). Proof. Operators of convolution are shift invariant with respect to arbitrary shifts, and operators of multiplications by functions in P ER are invariant with respect to integer shifts. Hence, operators of this form as well as there sums and products possess exactly one limit operator, namely the operator itself. Further, as it has been pointed out in Proposition 3.3.9 in [15], all limit operators of operators of multiplication by slowly oscillating functions are constant multiples of the identity operator, whence the assertion. Hence, the determination of the index of a Fredholm operator in I + Kp (SO) requires the computation of the plus- and the minus index of an operator of the form I + C where C is a Fourier convolution with kernel k ∈ L1 (R). Equivalently, one has to determine the common Fredholm index of operators of the form I + χ± Cχ± I. The operator I + χ+ Cχ+ I is the Wiener-Hopf operator with generating function 1+a where a is the Fourier transform of the kernel k of C. After reﬂection at the origin, the operator I + χ− Cχ− I also becomes a Wiener-Hopf operator.

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The Fredholm property of Wiener-Hopf operators of this type is well understood (see [2, 4, 8]). Since lim a(x) = lim a(x) = 0,

x→+∞

x→−∞

one can consider 1 + a as a continuous function on the one-point compactiﬁcation ˙ of the real line, which is also called the symbol of the operator. It turns out R that the Wiener-Hopf operator with symbol 1 + a is Fredholm if and only if the ˙ and that in this case its Fredholm index is function 1 + a does not vanish on R, ˙ around the origin. This the negative winding number of the closed curve 1 + a(R) solves the problem of computing the Fredholm index of an operator in I + Kp (SO) completely and in an easy way. Let us now turn over to the setting of operators in I + Kp (P ER, SO). Here we are left with the problem to determine the Fredholm index of operators of the form χ+ (I + K)χ+ I on Lp (R+ ) where K ∈ Kp (P ER). The proofs of Theorems 1 and 10 given above oﬀer a way to perform this calculation. The decisive point is that, due to the periodicity, the operator Γ(χ+ (I + K)χ+ I) ∈ L(lp (Z+ , Lp [0, 1]))

(16)

is a band-dominated Toeplitz operator the entries of which are of the form I + compact if they are located on the main diagonal, whereas they are compact when located outside the main diagonal. Recall that a Toeplitz operator on lp (Z+ , X) is an operator with matrix representation (Ai−j )i, j∈Z+ , i.e., the entries of the matrix are constant along each diagonal which is parallel to the main diagonal. If now I + K is Fredholm on Lp (R), then the Toeplitz operator (16) is Fredholm, too, and it has the same index. Employing the reduction procedure used in the proof of Theorem 10, one can further approximate the Toeplitz operator (16) by a Toeplitz operator on lp (Z+ , CN ) with band structure which is also Fredholm and has the same index as the original operator I + K. Thus, we are left with the determination of the index of a common Toeplitz operator T (g) on lp (Z+ , CN ) where each entry gij of the generating function g : T → CN ×N is a trigonometric polynomial. This operator can be identiﬁed with an operator matrix (T (gij ))N i, j=1 where each T (gij ) is a Toeplitz band operator on lp (Z+ , C) = lp (Z+ ). As it is well known (see, e.g., Theorem 6.12 in [2]), this operator is Fredholm if and only if the common Toeplitz operator (with scalar-valued polynomial generating function) T (det g) is Fredholm, and the indices of these operators coincide. Moreover, the index of T (det g) is equal to the negative winding number of the function det g with respect to the origin. For a general account on matrix functions and the Toeplitz and Wiener-Hopf operators generated by them, we refer to the monographs [3] and [10]. Convolution and Wiener-Hopf operators with almost periodic matrix-valued generating functions are thoroughly treated in the monograph [1]. For general results about relations between the Fredholmness of a block operator and its determinant one should consult Chapter 1 in [9].

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A similar approach is possible for operators in I + Kp (P ERZ , SO) where P ERZ stands for the set of all functions with integer period. After discretization and approximation as above, one ﬁnally arrives at a block Toeplitz operator in place of (16) which again can be reduced to a matrix of Toeplitz operators on lp (Z+ ). The results of Theorems 1, 10 and 15 can be completed by an observation made in [16] for the case of band-dominated operators on lp (Z, C). This observation concerns the independence of the Fredholm index on p. To make this statement precise we have to explain what is meant by a band-dominated operator which acts on diﬀerent lp -spaces (notice that the class of all band operators is independent of p whereas the algebra AE of all band-dominated operators depends on the parameter p of E = lp (Z, X) heavily). Every inﬁnite matrix (aij )i, j∈Z induces an operator A on the linear space c00 (Z, X) of all functions x : Z → X with compact support by i → (Ax)i :=

aij xj .

j∈Z

We say that A extends to a bounded linear operator on lp (Z, X) or that A acts on lp (Z, X) if Ax ∈ lp (Z, X) for each x ∈ c00 (Z, X) and if there is a constant C such that Axp ≤ Cxp for each x ∈ c00 (Z, X). If A extends to a banddominated operator on both lp (Z, X) and lr (Z, X), then we say that A is a banddominated operator on lp (Z, X) and lr (Z, X). Otherwise stated: we consider two band-dominated operators B and C acting on lp (Z, X) and lr (Z, X), respectively, as identical, and we denote them by the same letter, if their matrix representations coincide. $ be a Fredholm band-dominated operator both Proposition 17. Let A ∈ I + CE p on E = l (Z, X) and on E = lr (Z, X) with 1 < r < p < ∞. Then A is a Fredholm band-dominated operator on each space ls (Z, X) with r < s < p, and the Fredholm index inds A of A, considered as an operator on ls (Z, X), is independent of s ∈ [r, p].

The proof follows exactly the line of the proof of Theorem 10, ﬁnally reducing the assertion of the proposition to the case X = C which is treated in [16]. It should be also mentioned that Proposition 17 remains valid for band-dominated operators on Lp (ZN , X) with N a positive integer which also follows from [16]. In combination with Theorems 1 and 15 one gets the following corollary. Corollary 18. (a) Let A ∈ I + L$q be a Fredholm band-dominated operator both for q = p and for q = r with 1 < r < p < ∞. Then A is a Fredholm banddominated operator on each space Ls (R) with r < s < p, and the Fredholm index inds A of A, considered as an operator on Ls (R), is independent of s ∈ [r, p].

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(b) Let A ∈ I + Kq (BU C) be a Fredholm convolution type operator both for q = p and for q = r with 1 < r < p < ∞. Then A is a Fredholm convolution type operator on each space Ls (R) with r < s < p, and the Fredholm index inds A of A, considered as an operator on Ls (R), is independent of s ∈ [r, p].

References [1] A. B¨ ottcher, Yu. I. Karlovich, I. M. Spitkovsky, Convolution operators and Factorization of Almost Periodic Matrix Functions. Birkh¨ auser Verlag, Basel 2002. [2] A. B¨ ottcher, B. Silbermann, Analysis of Toeplitz operators. Springer-Verlag, Berlin, Heidelberg, New York 1990. [3] K. F. Clancey, I. Gohberg, Factorization of matrix functions and singular integral operators. Birkh¨ auser Verlag, Basel 1981. [4] I. Gohberg, I. Feldman, Convolution Equations and Projection Methods for Their Solution. Nauka, Moskva 1971 (Russian, Engl. transl.: Amer. Math. Soc. Transl. of Math. Monographs, Vol. 41, Providence, Rhode Island, 1974). [5] I. Gohberg, M. Krein, Systems of integral equations on the semi-axis with kernels depending on the diﬀerence of the arguments. Usp. Mat. Nauk 13(1958), 5, 3 – 72 (Russian). [6] N. Karapetiants, S. Samko, A certain class of convolution type integral equations and its applications. Izv. Akad. Nauk SSSR, Ser. Mat. 35(1971), 3, 714 – 726 (Russian). [7] N. Karapetiants, S. Samko, Equations with Involutive Operators. Birkh¨ auser Verlag, Boston, Basel, Berlin 2001. [8] M. G. Krein, Integral equations on the semi-axis with kernels depending on the difference of the arguments. Usp. Mat. Nauk 13(1958), 2, 3 – 120 (Russian). [9] N. Ya. Krupnik, Banach algebras with symbol and singular integral operators. Shtiintsa, Kishinev 1984 (Russian, English transl.: Birkh¨ auser Verlag, Basel 1987). [10] G. S. Litvinchuk, I. M. Spitkovski, Factorization of measurable matrix functions. Birkh¨ auser Verlag, Basel 1987. [11] V. S. Rabinovich, S. Roch, Fredholmness of convolution type operators. In: Operator Theory: Advances and Applications 147, 423–455, Birkh¨ auser Verlag, Basel, Boston, Berlin 2004. [12] V. S. Rabinovich, S. Roch, J. Roe, Fredholm indices of band-dominated operators. Integral Equations Oper. Theory 49(2004), 2, 221 – 238. [13] V. S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and ﬁnite section method for band-dominated operators. Integral Equations Oper. Theory 30(1998), 4, 452 – 495. [14] V. S. Rabinovich, S. Roch, B. Silbermann, Band-dominated operators with operatorvalued coeﬃcients, their Fredholm properties and ﬁnite sections. Integral Equations Oper. Theory 40(2001), 3, 342 – 381. [15] V. S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory. Birkh¨ auser Verlag, Basel, Boston, Berlin, 2004.

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[16] S. Roch, Band-dominated operators on lp -spaces: Fredholm indices and ﬁnite sections. Acta Sci. Math. (Szeged) 70(2004), 3 - 4, 783 – 797. Vladimir S. Rabinovich Instituto Politechnico National ESIME-Zacatenco, Ed.1, 2-do piso Av. IPN Mexico, D.F., 07738 Mexico e-mail: [email protected] Steﬀen Roch Technische Universit¨ at Darmstadt Fachbereich Mathematik Schlossgartenstrasse 7 D-64289 Darmstadt Germany e-mail: [email protected] Submitted: April 6, 2006

Integr. equ. oper. theory 57 (2007), 283–301 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020283-19, published online August 8, 2006 DOI 10.1007/s00020-006-1449-x

Integral Equations and Operator Theory

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators A. Rogozhin Abstract. In this paper we estimate the norm of the Moore-Penrose inverse T (a)+ of a Fredholm Toeplitz operator T (a) with a matrix-valued symbol a ∈ L∞ N×N defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality T (a)+ > a−1 ∞ holds. Moreover, we discuss the asymptotic behavior of T (tr a)+ for r ∈ Z. The results are illustrated by numerical experiments. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47A30. Keywords. Moore-Penrose inverses, Toeplitz operators, singular values.

1. Introduction Let L∞ := L∞ (T) denote the usual Lebesgue space on the unit circle T = {t ∈ ∗ C : |t| = 1}, and let L∞ N ×N be the C -algebra of N × N – matrices with entries ∞ 2 from L . Further, we denote by N the Hilbert space of all sequences x : Z+ → CN , Z+ = {i ∈ Z : i ≥ 0}, such that ∞ 1/2 2 xi CN < ∞, x2N := i=0

xi 2CN

2 (x1i , x2i , . . . , xN i )CN

2 = = |x1i |2 + |x2i |2 + . . . + |xN where i | . 2 2 The block Toeplitz operator T (a) : N → N is deﬁned by the matrix representation   a0 a−1 a−2 · · ·  a1 a0 a−1 · · · ∞ , T (a) = ai−j i,j=0 =   a2 a1 a0 · · · ... ... ... ...

where {ak }k∈Z is the sequence of the Fourier coeﬃcients of a function a ∈ L∞ N ×N . This work was supported by the Deutsche Forschungsgemeinschaft, DFG project SI 474110-1.

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It is well known (see e.g. [4]) that for all generating functions a ∈ L∞ N ×N the 2 Toeplitz operator T (a) is a bounded linear operator on N , i.e. T (a) ∈ L(2N ), and T (a)L(2N ) = a∞ . Here L(H) stands for the C ∗ − algebra of all bounded linear operators acting on a Hilbert space H. Moreover, if the Toeplitz operator T (a) ∈ L(2N ), a ∈ L∞ N ×N , is Fredholm (i.e. invertible modulo compact operators) then the generating function a is invertible in L∞ N ×N (see e.g. [4]). Further, we recall some basic facts concerning Moore-Penrose invertibility of linear operators. Let H be a Hilbert space. An operator A ∈ L(H) is called Moore-Penrose invertible if there is an operator A+ ∈ L(H) such that AA+ A = A

A+ AA+ = A+

(AA+ )∗ = AA+

(A+ A)∗ = A+ A.

If such an operator A+ exists, then it is uniquely determined. Clearly, if A is invertible then it is Moore-Penrose invertible and the operator A+ coincides with the inverse of A. It is well known that an operator is Moore-Penrose invertible if and only if its range is closed. In particular, if A is Fredholm, then it is MoorePenrose invertible. Clearly, the norm of the Moore-Penrose inverse of a Toeplitz operator T (a) is of great interest (for example, it plays a signiﬁcant role for the determination of the least square solution of the equation T (a)x = y). Unfortunately, the norm T (a)+ is available in rare cases only and it is even diﬃcult to estimate the norm of the usual inverse of T (a). Using the so-called k−splitting property of the singular values of truncated Toeplitz matrices we show in Section 2 that one can compute approximately the norm of the Moore-Penrose inverse of a Fredholm Toeplitz operators (see (3)). Note that, on the other hand, this result provides one reason more to estimate the norm of Moore-Penrose inverses (see the convergence speed estimate (2)). In Section 3 we present several simple estimates of the norm of Moore-Penrose inverses. In some special cases, e.g. for triangular Toeplitz matrices, we were able to get upper estimates of the norm T (a)+ (see Lemmas 3.2-3.4). The main result is obtained in Section 4. Let us denote by F L∞ N ×N the set of all functions a ∈ L∞ for which the Toeplitz operator T (a) is Fredholm. N ×N Theorem 1.1. The the set of all functions a ∈ F L∞ N ×N for which the strict inequality T (a)+ > a−1 ∞ holds is an open and dense subset of F L∞ N ×N . In Section 5 we discuss how the norm of the Moore-Penrose inverse of a Toeplitz operator changes when the generating function is multiplied by the diagonal matrix diag(tr , tr , . . . , tr ). Theorem 1.1 means that generically the norm of the Moore-Penrose inverse of a Fredholm Toeplitz operator is strictly greater than the norm of a−1 . However, for continuous generating functions a we have proved the following interesting result. Theorem 1.2. If a ∈ F L∞ N ×N is continuous, then lim T (tr a)+ = lim T (t−r a)+ = a−1 ∞ .

r→+∞

r→+∞

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Finally, in Section 6 we illustrate the results of the paper by numerical experiments.

2. Truncated Toeplitz matrices First, we recall that the singular values of a matrix An ∈ CnN ×nN are deﬁned as the nonnegative square roots of the spectrum points of A∗n An . We denote the (ordered) singular values of An by s1 (An ), s2 (An ), . . . snN (An ), i.e. we have 0 ≤ s1 (An ) ≤ s2 (An ) ≤ . . . ≤ snN (An ) = An . Further, given a function a ∈ L∞ N ×N and a number r ∈ {0, 1, . . .} let us denote by Tn,r (a) the following rectangular truncated Toeplitz matrices   a−n+r a0 a−1 · · · · · ·  a1 a0 · · · · · · a−n+r+1    Tn,r (a) =  .  .. .. ..   .. . . an

······

an−1

ar

If r equal zero, then we get the usual square ﬁnite sections. Note also that to evaluate the singular values of Tn,r (a) these matrices are extended to square nN × nN matrices by ﬁlling in zeros in the remaining places. A function a ∈ L∞ N ×N is called piecewise continuous if it has one-sided limits a(t ± 0) for all t ∈ T. Several years ago S. Roch and B. Silbermann proved the next result Theorem 2.1 (see [8], Theorem 3.1). If a ∈ F L∞ N ×N is piecewise continuous, then the singular values of Tn,r (a) have the k−splitting property, that is lim sk (Tn,r (a)) = 0

n→∞

and

lim inf sk+1 (Tn,r (a)) > 0 n→∞

with

k = dim ker T (a) + dim ker T (tr a ˜)T (t−r E), where E = diag(1, 1, . . . , 1) is the identity N × N matrix, and the function a ˜ is deﬁned by a ˜(t) = a(1/t).

Further, it turns out that the sequence (sk+1 (Tn (a))) of the (k+1)-st singular values of Tn,r (a) even converges as n → ∞ and (see [7], Theorem 7.2) dr := lim sk+1 (Tn,r (a)) = n→∞

1 . max (T (a)+ , [T (tr a ˜)T (t−r E)]+ )

(1)

Moreover, one can estimate the speed of this convergence by the smoothness of the function a. For instance, Theorem 7.3 of [7] shows that if a is rational then  √ −γ n   dr < a−1 −1 O e ∞  dr − sk+1 Tn,r (a) = (2)   O ln n otherwise. n

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√ We remark that in a related but diﬀerent context such O e−γ n and O (ln n/n) were ﬁrst established by A. B¨ottcher, S. Grudsky, A. Kozak, and B. Silbermann in [3], [1]. Note also that in view of Lemma 5.1 below [T (tr a ˜)T (t−r E)]+ = T (tr a ˜)+ . ˜)+ Besides, if the generating function a is continuous, then the norms T (tr a −1 converge to the norm of the function a (see Theorem 1.2 or Theorem 5.4 below). Thus, at least for continuous functions a we have lim dr = T (a)+ −1 .

r→∞

(3)

Here we use the fact that the norm T (a)+ is always greater than or equal to the norm a−1 ∞ (see Theorem 3.1 below). Thus, if we know the splitting number k, then we can compute approximately the quantity dr (and consequently the norm T (a)+ ). But the problem is that, in general, we do not know the kernel dimensions of Toeplitz operators. However, if we would be able to decide whether the singular values converge to zero or not, then we would know the splitting number k. Consequently, it would be good to have some information about the size of dr . For instance, if dr ≈ 10−1 , then we perhaps have a possibility to decide whether the singular values converge to zero or to 10−1 . But if dr ≈ 10−6 , then one has no chance. Therefore, we want that the norm T (a)+ is not large. But, on the other hand, if the norm T (a)+ is strictly greater than the norm of a−1 then we get more quick convergence sk+1 (Tn,r (a)) → dr (see estimate (2)).

3. Simple estimates We need some additional information concerning Moore-Penrose invertibility of linear operators. Again, let H be a Hilbert space and let A be a bounded linear operator acting on H. An operator A(−1) ∈ L(H) satisfying only the condition AA(−1) A = A will be called generalized inverse of A ∈ L(H). One can show that if an operator A has a generalized inverse A(−1) then it is Moore-Penrose invertible and (see [6], eq. IV.13.14, p. 182) A+ = (I − Pker A )A(−1) Pim A ,

(4)

where Pker A and Pim A are the orthogonal projectors onto the kernel and the image of A, respectively. Moreover, in case an operator A ∈ L(H) is Moore-Penrose invertible, the following equalities take place A+ A = I − Pker A

and

AA+ = Pim A .

(5)

∞ Further, we denote by T (L∞ N ×N ) the set of all Toeplitz operators with LN ×N generating functions, i.e. ∞ T (L∞ N ×N ) := T (a) : a ∈ LN ×N ,

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∗ 2 ∞ by alg T (L∞ N ×N ) the closed C -subalgebra of L(N ) generated by T (LN ×N ), i.e.   n m   , T (aij ) : aij ∈ L∞ alg T (L∞ N ×N ) := clos N ×N   i=1 j=1

QT (L∞ N ×N )

and by the quasicommutator ideal of alg T (L∞ N ×N ), i.e. the smallest closed two-sided ideal of alg T (L∞ N ×N ) containing all elements of the form T (ab) − T (a)T (b),

a, b ∈ L∞ N ×N .

It is well known that the quasicommutator ideal QT (L∞ N ×N ) contains the ideal K(2N ) of all compact operators acting on 2N (see e.g. [4], Proposition 4.5). Moreover, in [4], Corollary 4.3 it was shown that the algebra alg T (L∞ N ×N ) decomposes ∞ into the direct sum of T (L∞ ) and Q (L ) : T N ×N N ×N ∞ ∞ alg T (L∞ N ×N ) = T (LN ×N ) ⊕ QT (LN ×N ). ∞ We denote by ST the projection of alg T (L∞ N ×N ) onto T (LN ×N ) parallel to ∞ QT (LN ×N ). One can show that ST = 1 (see [4], Proposition 4.1). + −1 Theorem 3.1. If a ∈ F L∞ ∞ . N ×N , then T (a) ≥ a 2 Proof. First, we note that the quotient algebra alg T (L∞ N ×N )/K(N ) is inverse 2 2 ∞ closed in the Calkin algebra L(N )/K(N ), since alg T (LN ×N )/K(2N ) is a C ∗ subalgebra of L(2N )/K(2N ). Taking into account this fact, the equality (5), and the compactness of the orthogonal projection Pker T (a) onto the kernel of T (a), we conclude that the Moore-Penrose inverse T (a)+ belongs to the algebra alg T (L∞ N ×N ). + −1 Hence, the operator T (a) T (a)T (a ) belongs to the algebra alg T (L∞ N ×N ) and we can apply the projection ST to it. On the one hand we have

ST (T (a)+ T (a)T (a−1 )) = ST (T (a)+ [I + T (a)T (a−1 ) − T (aa−1 )]) = ST (T (a)+ ), since the operator [T (a)T (a−1 ) − T (aa−1 )] belongs to the ideal QT (L∞ N ×N ). On the other hand, using that Pker T (a) ∈ K(2N ) ⊂ QT (L∞ ), we get N ×N ST (T (a)+ T (a)T (a−1 )) = ST ([I − Pker T (a) ]T (a−1 )) = ST (T (a−1 )) = T (a−1 ). Thus, we obtain a−1 ∞ = T (a−1 ) = ST (T (a)+ ) ≤ T (a)+ . Now we recall the well-known relation (see e.g. [4], Proposition 2.14) T (a)T (b) = T (ab) − H(a)H(˜b),

(6)

L∞ N ×N

where the matrix functions a, b ∈ and where the block Hankel operator H(a) : 2N → 2N is deﬁned by the inﬁnite matrix ∞ H(a) = ai+j+1 i,j=0 . Notice that, for all a ∈ L∞ N ×N , the Hankel operator is a bounded linear operator on 2N and H(a) ≤ a∞ .

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∞ + ∞ − Further, let us denote by HN ×N and HN ×N the Hardy spaces ∞ + ∞ HN ×N := {a ∈ LN ×N : ak = 0 for k < 0}, ∞ − ∞ HN ×N = {a ∈ LN ×N : ak = 0 for k > 0}. ∞ + Obviously, if a ∈ HN a) = ×N , then the matrix T (a) is lower block triangular and H(˜ ∞ − 0. Analogously, if a ∈ HN ×N , then the matrix T (a) is upper block triangular and H(a) = 0. This observation together with the relation (6) imply that if a function ∞ + ∞ − ∞ −1 ) is a a ∈ HN ×N (HN ×N ) is invertible in LN ×N , then the Toeplitz operator T (a left (right) inverse of T (a). ∞ + ∞ − ∞ Lemma 3.2. Let a ∈ HN ×N (HN ×N ) be invertible in LN ×N . Then

T (a)+ = a−1 ∞ . Proof. Since the Toeplitz operator T (a−1 ) is a generalized inverse of T (a), we get that the Toeplitz operator T (a) is Moore-Penrose invertible and (see (4)) T (a)+ = (I − Pker T (a) )T (a−1 )Pim T (a) ≤ T (a−1) = a−1 ∞ .

Now the assertion follows immediately from the previous theorem.

Now we assume that a function a ∈ L∞ N ×N admits a factorization a− da+ such ∞ ± that the factorization factors a± and their inverses a−1 ± belong to HN ×N (here d χ1 χN is a diagonal matrix function of the form d(t) = diag(t , . . . , t ) with integers χ1 , χ2 , . . . , χN ). For example, every invertible rational matrix function or every invertible H¨ older continuous matrix function admits such a factorization (see e.g. [5]). Lemma 3.3. Let a function a ∈ L∞ N ×N admit a factorization as above. Then the Toeplitz operator T (a) is Moore-Penrose invertible and −1 T (a)+ ≤ a−1 + ∞ a− ∞ .

(7)

Proof. With the help of the relation (6) one can easily check that T (a) = T (a− )T (d)T (a+ ) and that the operator

−1 ) T (a−1 + )T (d

T (a−1 − ) is a generalized inverse of T (a). Hence

−1 −1 −1 −1 −1 T (a)+ ≤ T (a−1 )T (a−1 + )T (d − ) ≤ T (a+ )T (a− ) = a+ ∞ a− ∞ .

We can estimate T (a)+ also by the norms of the factorization factors themselves. Lemma 3.4. Let a ∈ L∞ N ×N be as in the previous lemma. Then T (a)+ ≤ a−1 2∞ a+ ∞ a− ∞ , T (a)+ ≤ a−1 ∞ a− ∞ a−1 − ∞ , T (a)+ ≤ a−1 ∞ a+ ∞ a−1 + ∞ .

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Proof. All the inequalities follow immediately from the previous lemma and the relations −1 −1 −1 a−1 a− a− d∞ = a−1 a− d∞ ≤ a−1 ∞ a− ∞ , + ∞ = a+ d −1 −1 −1 a−1 a− ∞ = da+ a−1 ∞ ≤ a−1 ∞ a+ ∞ . − ∞ = da+ a+ d

4. Proof of Theorem 1.1 Let us denote by Pn and Qn the projectors acting on the space pN and deﬁned by Pn x = Pn (x0 , x1 , . . . , xn−1 , xn , xn+1 , . . .) := (x0 , x1 , . . . , xn−1 , 0, 0, . . .), Qn x := (I − Pn )x = (0, 0, . . . , 0, xn , xn+1 , . . .). First we show that the set of all functions a ∈ F L∞ N ×N for which the strict inequality T (a)+ > a−1 ∞ is a dense subset of F L∞ N ×N . Lemma 4.1. Let a ∈ F L∞ N ×N . Then in any neighborhood of a there is a function such that b ∈ F L∞ N ×N b−1 ∞ = a−1 ∞ ,

b∞ = a∞ , but

T (b)+ > T (a)+ . Proof. Without loss of generality assume that T (a)+ = 1. We ﬁx a δ < 1/4 and an n ∈ N and introduce two diagonal matrix functions f and g by 1 g(t) := f (t) := 1 − δ + δt−n E, E. 1 − δ + δtn One can easily check that ∞ − f, f −1 ∈ HN ×N

and

∞ + g, g −1 ∈ HN ×N .

In particular, the Toeplitz operators T (f ) and T (g) are invertible and T (f )−1 = T (f −1 )

and

T (g)−1 = T (g −1 ).

By the deﬁnition of the norm, we have T (a)+ =

T (a)+ y = 1. y y∈2N ,y=0 sup

A little thought shows that T (a)+ =

T (a)+ y x = sup . y x⊥ker T (a),x=0 T (a)x y⊥ker T (a)+ ,y=0 sup

Hence, there is an xδ ∈ 2N such that xδ ⊥ ker T (a),

xδ = 1,

and

T (a)xδ ≤ 1 + δ 3 .

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Now let us consider the matrix function b = f ag. Clearly, the function b is invertible and b∞ = a∞ , b−1 ∞ = a−1 ∞ . Moreover a − b∞

1 − δ + δt−n ≤ a∞ max 1 − ≤ 4a∞ δ. t∈T 1 − δ + δtn

∞ − ∞ + Note also that since f ∈ HN ×N and g ∈ HN ×N we get

T (b) = T (f )T (a)T (g).

(8)

Further, we put yδ := (I − Pker T (b) )T (g −1 )xδ , where Pker T (b) is the orthogonal projection onto the kernel of T (b). We obtain T (b)+ 2 ≥

(I − Pker T (b) )T (g −1 )xδ 2 yδ 2 = . T (b)yδ 2 T (f )T (a)xδ 2

From (8) we conclude that zδ := T (g)Pker T (b) T (g −1 )xδ ∈ ker T (a). Moreover, we have zδ ≤ T (g)T (g −1) =

1 maxt∈T |1 − δ + δtn | ≤ . mint∈T |1 − δ + δtn | 1 − 2δ

Now, taking into account that Pn T (tn ) = 0, Qn = T (tn )T (t−n ), xδ ⊥ zδ , and xδ = 1, we get a lower estimate of yδ 2 (I − Pker T (b) )T (g −1 )xδ 2 = T (g −1 )xδ 2 − (Pker T (b) T (g −1 )xδ , T (g −1 )xδ ) − (T (g −1 )xδ , Pker T (b) T (g −1 )xδ ) + Pker T (b) T (g −1 )xδ 2 ≥ T (g −1 )xδ 2 − 2|(T (g −1 )xδ , Pker T (b) T (g −1 )xδ )| ≥ Pn T (1 − δ + δtn )xδ 2 + Qn T (1 − δ + δtn )xδ 2 − 2|(T (1 − δ + δtn )xδ , T (1 − δ + δtn )zδ )| = (1 − δ)2 Pn xδ 2 + (1 − δ)Qn xδ + δT (tn )xδ 2 − 2|((1 − δ)xδ , δT (tn )zδ )| − 2|(δT (tn )xδ , (1 − δ)zδ )| ≥ (1 − δ)2 (1 − Qn xδ 2 ) + δ 2 − 2|((1 − δ)Qn xδ , δT (tn )xδ )| − 2|((1 − δ)Qn xδ , δT (tn )zδ )| − 2|(δT (tn )xδ , (1 − δ)Qn zδ )| ≥ (1 − δ)2 (1 − Qn xδ 2 ) + δ 2 − 2δ(1 − δ)Qn xδ 2δ(1 − δ) Qn xδ − 2δ(1 − δ)Qn zδ 1 − 2δ ≥ 1 − 2δ + 2δ 2 − Qn xδ 2 − 6δQn xδ − 2δQn zδ . −

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Further, the equality T (t−n )Pn = 0 implies T (f )T (a)xδ = T (1 − δ + δt−n )T (a)xδ ≤ (1 − δ)T (a)xδ + δT (t−n )(Pn + Qn )T (a)xδ ≤ (1 − δ)(1 + δ 3 ) + δQn T (a)xδ . Hence T (f )T (a)xδ 2 ≤ (1 − δ)2 (1 + δ 3 )2 + δ 2 Qn T (a)xδ 2 + 2(1 − δ)(1 + δ 3 )δQn T (a)xδ ≤ 1 − 2δ + δ 2 + 2δ 3 + δ 6 + 4δQn T (a)xδ + δ 2 Qn T (a)xδ 2 . Obviously, for any δ > 0 we can ﬁnd an n ∈ N such that Qn xδ , Qnzδ , Qn T (a)xδ ≤ δ 2 . Thus, we arrive at the inequality 1 − 2δ + 2δ 2 − δ 4 − 6δ 3 − 2δ 3 1 − 2δ + 2δ 2 − 9δ 3 ≥ 1 − 2δ + δ 2 + 2δ 3 + δ 6 + 4δ 3 + δ 4 1 − 2δ + δ 2 + 8δ 3 2 3 δ − 17δ =1+ > 1 = T (a)+ 2 , 1 − 2δ + δ 2 + 8δ 3

T (b)+ 2 ≥

whenever δ < 1/17.

Corollary 4.2. The set of all functions a ∈ F L∞ N ×N for which the strict inequality T (a)+ > a−1 ∞ holds is a dense subset of F L∞ N ×N . Proof. This follows immediately from the previous lemma and Theorem 3.1.

Now we turn to the proof of the second part of Theorem 1.1. Note that this assertion is not so trivial as it can seem. The main problem is that the MoorePenrose inversion is not continuous (in contrast with the usual inversion). To have an example, let us consider the compact projection P1 ∈ L(2N ). Clearly, this operator is Moore-Penrose invertible and P1+ = P1 . Further, we denote by Aλ the operators (λI + P1 ) ∈ L(2N ). One can easily check that for all λ ∈ R \ {0, −1} the operators Aλ are invertible in the usual sense and A−1 λ =

1 1 I− P1 . λ λ(λ + 1)

Thus, the operators Aλ converge to P1 as λ → 0, but their Moore-Penrose inverses −1 + + + A+ λ = Aλ do not converge to P1 (even Aλ P1 ). However the following result takes place. Lemma 4.3. Let H be a Hilbert space and let A, B ∈ L(H) be two Moore-Penrose invertible operators. Then B + ≥ A+

1 − A+ A − B . 1 + A+ A − B

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Proof. Again we will use that A+ =

x . x⊥ker A,x=0 Ax sup

First, this formula implies that for every x ∈ H x − Pker A x ≤ A+ Ax, where Pker A is the orthogonal projection onto the kernel of the operator A. Moreover, for any ε > 0 there exists an xε ∈ H such that 1+ε . xε = 1, xε ⊥ ker A, and Axε ≤ A+ Further, we have Bxε ≤ Axε + A − B and (with Pker B being the orthogonal projection onto the kernel of the operator B) xε − Pker B xε ≥ xε − Pker A Pker B xε − Pker A Pker B xε − Pker B xε ≥ xε − Pker A Pker B xε − A+ APker B xε ≥ xε − A+ (A − B)Pker B xε ≥ 1 − A+ A − B. Combining all, we get B + ≥

1 − A+ A − B xε − Pker B xε ≥ 1+ε Bxε A+ + A − B

= A+

1 − A+ A − B . 1 + ε + A+ A − B

Finally, taking into account that ε is arbitrary we arrive at the assertion of the lemma. Corollary 4.4. The set of all functions a ∈ F L∞ N ×N for which the strict inequality T (a)+ > a−1 ∞ holds is an open subset of F L∞ N ×N . + −1 ∞ . We put Proof. Let a ∈ F L∞ N ×N be any function such that T (a) > a

δ :=

T (a)+ − 1. a−1 ∞

Since the set of all Fredholm operators is open in L(2N ) and since the norm of a Toeplitz operator coincides with the norm of the generating function, there is a neighborhood U of the function a which is contained in F L∞ N ×N . From the previous lemma we conclude that for all functions b ∈ U we have 1 − T (a)+ a − b∞ T (b)+ ≥ T (a)+ 1 + T (a)+ a − b∞ 1 − T (a)+ a − b∞ = (1 + δ)a−1 ∞ . 1 + T (a)+ a − b∞

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Further, we get b−1 ∞ ≤ a−1 ∞ + b−1 − a−1 ∞ = a−1 ∞ + b−1 (a − b)a−1 ∞ ≤ a−1 ∞ + b−1 ∞ a − b∞ a−1 ∞ . Now let δ1 ∈ (0, 1) be arbitrary, let b ∈ U, and let a − b∞ ≤ δ1 T (a)+ −1 . We obtain 1 − T (a)+a − b∞ T (b)+ ≥ (1 + δ)b−1 ∞ (1 − a − b∞ a−1 ∞ ) 1 + T (a)+a − b∞ 2 (1 − δ1 ) ≥ b−1 ∞ (1 + δ) ≥ b−1 ∞ (1 + δ)(1 − δ1 )3 1 + δ1 ≥ b−1 ∞ (1 + δ)(1 − δ1 ) > b−1 ∞

whenever δ1 < δ/(1 + δ).

It should be mentioned that a special case of Theorem 1.1 (namely the assertion that for scalar continuous generating functions a the strict inequality T (a)−1 > a−1 ∞ represents the generic case) was ﬁrst established in [3]. The proof given there contained a mistake which subsequently repaired by A. B¨ottcher and S. Grudsky in [2], Theorem 6.15. Our proof of Theorem 1.1 is diﬀerent from the proof of Theorem 6.15 of [2] but makes use of some ideas of that proof.

5. Relationships between T (a)+ and T (ta)+ First, we show that [T (a)T (t−r E)]+ = T (a)+ for all r ∈ Z+ (cf. the deﬁnition of the quantity dr ). Taking into account that for all r ∈ Z+ the Toeplitz operator T (t−r E) is a left inverse of the Toeplitz operator T (tr E), one can easily check the next result. Lemma 5.1. Let the Toeplitz operator T (a), a ∈ L∞ N ×N , be Moore-Penrose invert−r ible. Then, for all r ∈ Z+ , the operator T (a)T (t E) is also Moore-Penrose invertible and [T (a)T (t−r E)]+ = T (tr E)T (a)+ . Further, we analyze the particular cases when a Moore-Penrose invertible Toeplitz operator has a trivial kernel or cokernel. Lemma 5.2. Let the Toeplitz operator T (a), a ∈ L∞ N ×N , be Moore-Penrose invertible. If T (a) has a trivial kernel, then T (ta) is also Moore-Penrose invertible and T (ta)+ ≤ T (a)+ . Proof. Since ker T (a) = {0} it follows from (5) that the operator T (a)+ is a left inverse of T (a). Applying (6) we obtain that the operator T (t−1 E)T (a)+ is a generalized inverse of T (ta) T (ta)T (t−1 E)T (a)+ T (ta) = T (ta)T (t−1 E)T (a)+ T (a)T (tE) = T (ta)T (t−1 E)T (tE) = T (ta).

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Hence the operator T (ta) is Moore-Penrose invertible and T (ta)+ = (I − Pker T (ta) )T (t−1 E)T (a)+ Pim T (ta) ≤ T (a)+ .

Analogously one can prove the following lemma. Lemma 5.3. Let the Toeplitz operator T (a), a ∈ L∞ N ×N , be Moore-Penrose invert2 ible. If the image of T (a) coincides with N , then T (t−1 a) is also Moore-Penrose invertible and T (t−1 a)+ ≤ T (a)+ . From the Wiener-Hopf factorization theory it is known (see e.g. [5], Corollary r ∞ 2.3) that if a function a ∈ F L∞ N ×N , then t a ∈ F LN ×N for all r ∈ Z. Moreover (see ∞ e.g. [5], Theorem 3.1), for any function a ∈ F LN ×N there exist integer numbers m and l (l ≤ m) such that the kernel of T (tr a) is trivial for all r ≥ m, and the cokernel of T (tr a) is trivial for all r ≤ l. Hence, Lemmas 5.2 and 5.3 imply that the function ”the norm of the Moore-Penrose inverse of T (tr a)” attains its maximum in the interval [l, m] (see Figure 1), i.e. sup T (tr a)+ = max T (tr a)+ . r∈[l,m]

r∈Z

If a is a scalar function (i.e. N = 1), then one can even take m = l, that is the Toeplitz operator with the generating function tm a is invertible in the usual sense. Thus, in this case the norm of the inverse of T (tm a) is just the maximum of the norms of the Moore-Penrose inverses sup T (tr a)+ = T (tm a)−1 . r∈Z

Now we present several results for continuous matrix functions. Let CN ×N ⊂ L∞ N ×N denote the algebra of complex valued continuous matrix functions on T. First, we note that the group GCN ×N of invertible elements in CN ×N is contained ∞ in F L∞ N ×N (see e.g. [4]), that is, a continuous matrix function a belongs to F LN ×N if and only if it is invertible. Further, for every continuous matrix function a, we put Er (a) :=

inf

r p∈PN ×N

a − p∞ ,

r ∈ Z+ ,

r where PN ×N is the set of all trigonometric polynomials p on T of the form

p(t) =

r

p k tk ,

pk ∈ CN ×N .

k=−r

It is well-known (see e.g. [9], Chapter 3.13) that, for any a ∈ CN ×N and r ∈ Z+ , r there is a polynomial pr (a) ∈ PN ×N such that Er (a) = a − pr (a)∞ and Er (a) → 0 as r → ∞. The next theorem describes the asymptotic behavior of the norm T (tr a)+ as |r| → ∞. We know already that in the generic case the norm of the MoorePenrose inverse T (tr a)+ is strictly greater than the norm of a−1 , but it turns out that the norms T (tr a)+ converge to the norm a−1 ∞ .

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T (tm a)+ T (tl a)+

a−1

m

l

Figure 1. T (tr a)+ for r ∈ Z.

Theorem 5.4. Let a ∈ GCN ×N . Then, for all suﬃciently large r ∈ N, T (tr a)+ ≤ a−1 ∞ [1 − Er (a)Er (a−1 )]−1 and T (t−r a)+ ≤ a−1 ∞ [1 − Er (a)Er (a−1 )]−1 . In particular, lim T (tr a)+ = lim T (t−r a)+ = a−1 ∞ .

r→∞

r→∞

(9)

Proof. First, we make use of relations (5) and (6) to obtain (I − H(t−r a−1 )H(t−r a ˜))T (tr a)+ = T (t−r a−1 )T (tr a)T (tr a)+ = T (t−r a−1 )Pim T (tr a) . Further, for any continuous matrix function c ∈ CN ×N , we get H(t−r c) = H(t−r [c − pr (c) + pr (c)] = H(t−r [c − pr (c)] ≤ c − pr (c)∞ = Er (c). a) = Er (a) converge to zero as r → ∞ we deduce that for Since Er (a−1 ) and Er (˜ ˜)] is invertible and r large enough the operator [I − H(t−r a−1 )H(t−r a ˜)]−1 ≤ [1 − Er (a)Er (a−1 )]−1 . [I − H(t−r a−1 )H(t−r a

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Hence we obtain that for all suﬃciently large r ˜)]−1 T (t−r a−1 )Pim T (tr a) T (tr a)+ = [I − H(t−r a−1 )H(t−r a ≤ a−1 ∞ [1 − Er (a)Er (a−1 )]−1 .

The second estimate can be proved analogously.

For smooth matrix functions we can estimate the speed of the convergence α (9). Let CN ×N , α > 0, denote the algebra of all continuous functions on T which are [α] times continuously diﬀerentiable and whose [α]−th derivative satisﬁes a H¨ older condition with the exponent {α} = α − [α] (0 ≤ {α} < 1). α Corollary 5.5. Let a ∈ GCN ×N . If the function a belongs to CN ×N for some α > 0, then T (tr a)+ − a−1 ∞ = O(|r|−2α ), |r| → ∞. If the function a is rational, then there exists a δ > 0 such that even

T (tr a)+ − a−1 ∞ = O(e−δ|r| ),

|r| → ∞.

Proof. Due to Theorem 5.4 it remains only to estimate the quantities Er (a) and Er (a−1 ). α −1 In case a ∈ CN ) = O(r−α ) for r ∈ N (see e.g. [9], ×N , we have Er (a), Er (a Chapter 3.13). If a is rational, then Er (a), Er (a−1 ) = O(e−µr ) with some µ > 0, since the Fourier coeﬃcients of rational functions decay exponentially. Thus, for rational functions we have very high (exponential) speed of the convergence (9). However, the following result shows that, in the general case, one can say nothing about the relation between the norms of the Moore-Penrose inverses of the Toeplitz operators with the symbols a and ta. Lemma 5.6. For any M > 0 there exists a rational function aM ∈ F L∞ 1×1 such that + aM ∞ = a−1 M ∞ = T (aM ) = 1, but T (t−1 aM )+ ∞ > M (t−1 aM ∞ = (t−1 aM )−1 ∞ = 1). Proof. We ﬁx an n ∈ N and introduce a scalar rational function b on the unit circle n + 1 + nt−1 . n + 1 + nt One can see that |b(t)| = 1 for all t ∈ T, and b = b− b+ with b(t) :=

b− (t) = n + 1 + nt−1 Hence

and

b+ (t) = (n + 1 + nt)−1 .

b∞ = b−1 ∞ = 1.

−1 Moreover, the Toeplitz operator T (b) is invertible (T (b)−1 = T (b−1 + )T (b− )), since ∞ − b− , b−1 − ∈ H1×1

and

∞ + b+ , b−1 + ∈ H1×1 .

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Let {bk }k∈Z be the Fourier coeﬃcients of the function b. The Parseval’s equality implies 2π ∞ 1 |bk |2 = b2L2(T) = |b(eiϕ )|2 dϕ = 1. 2π 0 k=−∞

Further, we estimate the Fourier coeﬃcient of b for k = −1. We have 2π 2π (n + 1)eiϕ + n 1 1 b−1 = a(eiϕ )eiϕ dϕ = dϕ 2π 0 2π 0 n + 1 + neiϕ 2π 1 − eiϕ 1 dϕ. =1− 2π 0 n + 1 + neiϕ Putting ϕ± := π(1 ± n−1/2 ), we get for any ϕ ∈ [0, ϕ− ] ∪ [ϕ+ , 2π] |n + 1 + neiϕ |2 = (n + 1 + n cos ϕ)2 + (n sin ϕ)2 = (n + 1)2 + n2 + 2n(n + 1) cos ϕ ≥ 2n(n + 1)(1 + cos ϕ) ≥ 2n(n + 1)(1 − cos πn−1/2 ) π = 4n(n + 1) sin2 √ ≥ 4n(n + 1)n−1 ≥ 4n. 2 n Moreover, we have for all ϕ ∈ [0, 2π] |1 − eiϕ | ≤ 2

and

|n + 1 + neiϕ | ≥ 1.

Thus, it follows that 2π ϕ+ ϕ− 2π 1 1 3 1 − eiϕ dϕ dϕ 1 1 √ + √ ≤√ . dϕ ≤ 2dϕ + 2π iϕ n + 1 + ne 2π 2π n 2π n n 0 ϕ+ ϕ− 0 Hence, we deduce that |b−1 | ≥ 1 − 3n−1/2 . Now let e0 = (1, 0, 0, . . .). Since T (b)e0 = (b0 , b1 , b2 , . . .), we obtain T (b)e02 =

∞

|bk |2 ≤

k=0

∞

|bk |2 − |b−1 |2

k=−∞

6 ≤ 1 − (1 − 3n−1/2 )2 ≤ √ . n Thus, we arrive at the estimate T (b)+ = T (b)−1 ≥

n1/4 e0 ≥ √ . T (b)e0 6

∞ + Finally, we note that the function tb ∈ H1×1 . This shows that (see Lemma 3.2)

T (tb)+ = (tb)−1 ∞ = tb∞ = 1.

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6. Numerical results In this section we present three examples in which the behavior of T (tl a)+ for l ∈ Z is reﬂected. To evaluate the quantities T (tl a)+ we use the convergence of sk+1 (Tn,r (a)) to min(T (a)+ −1 , T (tr a ˜)+ −1 ) and compute the singular values sk+1 (Tn,r (a)) for large n. Example 1

First we consider the scalar function a1 (t) =

1 6 − 5t = t−2 (6 − 5t). 2 6t − 5t 6 − 5t−1

˜1 ) = |l − 2|, l ∈ Z, we Taking into account that dim ker T (tl a1 ) + dim ker T (t−l a obtain that the singular values of the ﬁnite sections Tn,0 (tl a1 ) have the k− splitting property with k = |l − 2|. Moreover since T (t−l a ˜1 ) is the transposed operator ˜1 )+ . Hence s|l−2|+1 (Tn,0 (tl a1 )) of T (tl a1 ) we have that T (tl a1 )+ = T (t−l a converge to 1/T (tl a1 )+ for l ∈ Z (see (1)). Note also that a−1 1 ∞ = 1, because for all t ∈ T |6 − 5t| |6 − 5t| = |a1 (t)| = = 1. |6 − 5t−1 | |6 − 5t| In the next ﬁgure we plotted the quantities 1/s|l−2|+1 (Tn,0 (tl a1 )) for n = 500. 2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 −5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

Figure 2. T (tl a1 )+ for −5 ≤ l ≤ 10. The ﬁgure shows that the norms T (tl a1 )+ converge very quickly to a−1 1 ∞ as l → ∞ (see Theorem 5.4 and Corollary 5.5). Further, in accordance with Lemmas 5.2 and 5.3 the sequence {T (tl a1 )+ }l∈Z has the maximum by l = 2, since

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the operator T (t2 a1 ) is invertible. Moreover we see that T (tl a1 )+ = a−1 1 ∞ for ∞ − ). l ≤ 1, what stems from Lemma 3.2 (note that ta1 ∈ H1×1 Example 2

Next we consider the 2 × 2 matrix function 10−9t t3 + t2 sin t−1 10t−9 10t−9 a2 (t) = 0 1 − 4t 1 −1 t sin t−1 t 0 10 − 9t = 10−9t−1 0 0 t 0 t−1 − 4

t3 .. 1

∞ − ∞ + Since t−2 a2 ∈ H2×2 , or equivalently t2 a ˜2 ∈ H2×2 , we obtain that ker T (t2 a ˜2 ) = {0}. Hence, the singular values of the ﬁnite sections Tn,r (tl a2 ), where r = max(0, 2 + l), have the k− splitting property with k = dim ker T (tl a2 ). Further, from ˜2 )+ = a−1 Lemma 3.2 it follows that T (tl a 2 ∞ for all l ≥ 2. Thus, by virtue of Theorem 3.1, we have that sk+1 (Tn,r (tl a2 )), where r = max(0, 2 + l), converge to 1/T (tla2 )+ for l ∈ Z (see (1)). It should be also noted that a−1 2 ∞ ≈ 1.187. In Figure 3 we plotted the quantities 1/sk+1 (Tn,r (tl a2 )) and 1/sk+1 (Tn,˜r (tl a ˜2 )) for n = 250. ˜ 4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −5

−4

−3

−2

−1

0

1

2

3

4

5

0 −5

−4

−3

−2

−1

0

1

2

3

4

5

Figure 3. T (tl a2 )+ and T (tl a ˜2 )+ for −5 ≤ l ≤ 5. To evaluate the norms T (tl a ˜2 )+ we compute the ﬁrst singular value of the l ﬁnite sections Tn,˜r (t a ˜2 ) not going to zero, where r˜ = 10 + l. Here we use that (Tn,˜r (tl a ˜2 )) converge to 1/T (tl a ˜2 )+ , since T (t10 a2 )+ ≤ T (t5 a2 )+ ≈ sk+1 ˜ a−1 2 ∞ . Again we obtained that the norms T (tl a2 )+ and T (tl a ˜2 )+ converge quick−1 ly to a2 ∞ as l → ∞ (see Theorem 5.4 and Corollary 5.5). Further, Figure 3 ˜2 )+ = a−1 shows that, in accordance with Lemma 3.2, T (tl a2 )+ = T (t−l a 2 ∞ + l + for l ≤ −2. Moreover, we see that T (ta2 ) ≥ T (t a2 ) for l ≥ 1 and

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T (t−1 a2 )+ ≥ T (tl a2 )+ for l ≤ −1, as prescribed by Lemmas 5.2 and 5.3. Note also that in this case we got the strict inequality max(T (tl a2 )+ : l ∈ Z) < max(T (tl a ˜2 )+ : l ∈ Z). Example 3

Now we consider the following matrix function

−2 t + 3 − t + (1 − i)t2 + it3 3t−3 + t−2 + 9t−1 + 3 + t2 + it3 t−1 − i − 2t2 + 2t3 3t−2 + (1 − 3i)t−1 − i + 2t3 −1 t + i t−2 + 3 t2 t−1 t 0 = .. t t+3 2 t−1 − i 0 t−1

a3 (t) =

In the next ﬁgure we plotted the quantities 1/sk+1 (Tn,0 (tl a3 )) for n = 250.

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 −5

−4

−3

−2

−1

0

1

2

3

4

5

Figure 4. max(T (tl a3 )+ , T (t−l a ˜3 )+ ) for −5 ≤ l ≤ 5.

Looking at the ﬁgure one can not see any diﬀerence between the norms of the Moore-Penrose inverses and the norm of the function a−1 3 . We would like to note that numerical experiments show that the situation described in Example 3 occurs relatively often. Moreover, it is diﬃcult to ﬁnd a function a for which one can convincingly see that the strict inequality T (a)+ > a−1 ∞ holds. That is, numerical experiments show that in the generic case the diﬀerence between the norms of the Moore-Penrose inverses and the norm of a−1 is too small (much less than the norm of a−1 ) to be perceptible visually.

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References [1] A. B¨ ottcher, S. Grudsky, Toeplitz Matrices, Asymptotical Linear Algebra and Functional Analysis, Hindustan Book Agency, New Delhi, 2000 and Birkh¨ auser Verlag, Basel, 2000. [2] A. B¨ ottcher, S. Grudsky, Spectral Properties of Banded Toeplitz Matrices, SIAM, Philadelphia, 2005. [3] A. B¨ ottcher, S. Grudsky, A. Kozak, and B. Silbermann, Convergence speed estimates for the norms of the inverses of large truncated Toeplitz matrices, Calcolo, 36 (1999), pp. 103–122. [4] A. B¨ ottcher, B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989 and Springer-Verlag, Berlin, Heidelberg, New York, 1990. [5] K. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications, 3, Birkh¨ auser Verlag, Basel, Boston, Stuttgart, 1981. [6] I. Gohberg and N. Krupnik, One-dimensional Linear Singular Integral Equations, Birkh¨ auser Verlag, Basel, Boston, Berlin, 1992. [7] A. Rogozhin, The singular value behavior of the finite sections of block Toeplitz operators, to appear in SIAM J. Matrix Anal. Appl. [8] B. Silbermann, Modified finite sections for Toeplitz operators and their singular values, SIAM J. Math. Anal. Appl., 24 (2003), pp. 678–692. [9] A. Zygmund, Trigonometric series, Vol. I, Cambridge Univ. Press, New York, 1959. A. Rogozhin Department of Mathematics Chemnitz University of Technology D-09107 Chemnitz Germany e-mail: [email protected] Submitted: November 11, 2005 Revised: July 13, 2006

Integr. equ. oper. theory 57 (2007), 303–308 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030303-6, published online December 26, 2006 DOI 10.1007/s00020-006-1463-z

Integral Equations and Operator Theory

A Remark on Support of the Principal Function for Class A Operators Muneo Ch¯o, Mariko Giga, Tadasi Huruya and Takeaki Yamazaki Dedicated to Professors Eizaburo Kamei and Masatoshi Fujii on their sixtieth birthdays

Abstract. Let T = U |T | be an invertible class A operator such that [T ∗ , T ] ∈ C1 . Then we show that supp(gT ) ⊆ σ(T ), where gT is the principal function of T . Moreover, we show that if T is pure, then supp(gT ) = σ(T ). Mathematics Subject Classification (2000). Primary 47B20; Secondary 47A10. Keywords. Principal function, support, spectrum.

1. Introduction The principal functions were introduced by Pincus [17] concerning the diagonalization of self-adjoint singular integral operators. Carey and Pincus [3] studied the principal function gT of an operator T with trace class self-commutator. Operator means a bounded linear operator on a complex Hilbert space H. We denote by supp(gT ) the (essential) support of gT in C (identiﬁed with R2 ). Thus a complex number z is in supp(gT ) if and only if every neighborhood of z intersects the set { x+ iy : gT (x, y) = 0 } in a set of positive measure. It is known that supp(gT ) of a pure hyponormal operator T with trace class self-commutator is equal to σ(T ) (the spectrum of T ) (see [16, p.243,(4)]). The relation between the principal function gT and the spectrum σ(T ) of T is not fully understood (see a comment in p.105 of [11]). An operator T belongs to class A if |T 2 | ≥ |T |2 [12]. If T is hyponormal, then it belongs to class A. In this short note, we show that if T is an invertible class A operator with trace class self-commutator, then the support of the principal function gT of T is contained in σ(T ). Let C1 denote the trace class operators on H. A function φ(r, z) is called Laurent polynomial if there exist a non-negative integer N and polyN nomials pk (r) such that φ(r, z) = k=−N pk (r)z k . For diﬀerentiable functions P, Q This research is partially supported by Grant-in-Aid Research No.17540176.

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of two variables (x, y), let J(P, Q)(x, y) = Px (x, y) · Qy (x, y) − Py (x, y) · Qx (x, y). For an operator T = X + iY = U |T |, we consider the following trace formulae: Tr([P (X, Y ), Q(X, Y )]) =

1 2πi

J(P, Q)(x, y)gT (x, y)dxdy,

(∗)

for polynomials P and Q. 1 (∗∗) Tr([φ(|T |, U ), ψ(|T |, U )]) = J(φ, ψ)(r, eiθ )eiθ gTP (eiθ , r)drdθ 2π for Laurent polynomials φ and ψ. If formula (∗) holds, the function gT is called the principal function related to the Cartesian decomposition T = X + iY. If formula (∗∗) holds, the function gTP is called the principal function related to the polar decomposition T = U |T |.

2. Result We show the following Theorem 2.1. Let T = U |T | be an invertible class A operator such that [T ∗ , T ] ∈ C1 . Then supp(gT ) ⊆ σ(T ). For the proof, we need following result. Lemma 2.2. Let T = U |T | be an invertible operator such that [T ∗ , T ] ∈ C1 and let T˜ = |T |1/2 U |T |1/2 be the Aluthge transformation of T . Then, for the polar decomposition T˜ = V |T˜ |, it holds [|T |, U ], [T˜ ∗ , T˜ ] and [|T˜ |, V ] ∈ C1 . Lemma 2.2 is a modiﬁcation of [5, Theorems 3, 4]. Proof. By [|T |2 , U ] = [T ∗ , T ]U ∈ C1 and [5, Theorem 2], we have [|T |, U ] ∈ C1 . Conversely, by invertibility of U , we have [T ∗ , T ] = |T |[|T |, U ]U ∗ + [|T |, U ]|T |U ∗ ∈ C1 . Hence we obtain that, for an invertible operator T , [T ∗ , T ] ∈ C1

if and only if [|T |, U ] ∈ C1 .

(2.1) ˜∗

Then by [5, Theorems 3, 4], we have [|T˜ |, V ] ∈ C1 , and we obtain [T , T˜ ] ∈ C1 by (2.1). Proof of Theorem 2.1. Since T is invertible, the operator |T | is invertible and U is unitary. Put S = U |T |2 . Then since [S ∗ , S] = [T ∗ , T ]|T |2 + |T ∗ |2 [T ∗ , T ], we have ˜ be the Aluthge and the 2-nd Aluthge transformations [S ∗ , S] ∈ C1 . Let S˜ and S˜ ˜ and S˜˜ = W |S| ˜˜ be the polar decompositions of S˜ of S, respectively. Let S˜ = V |S| ˜ ˜ respectively. Then by using Lemma 2.2 two times, we have and S, ˜˜ [|S|, ˜˜ W ] ∈ C . ˜ [|S|, ˜ V ], [S˜˜∗ , S], [S˜∗ , S], 1

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It follows from [7, Theorem 4] that there exist the principal functions gS˜ , gSP˜ , gS˜˜ , g P˜˜ satisfying S

gS˜ (x, y) = gSP˜ (eiθ , r) and gS˜˜ (x, y) = g P˜˜ (eiθ , r) for x + iy = reiθ . S

˜ ˜ |S| ˜ are invertible and V, W are unitary. Let S˜ = X1 + iY1 Note that operators |S|, ˜ ˜˜ respectively. Since and S˜ = X2 + iY2 be the Cartesian decompositions of S˜ and S, ˜ V ] ∈ C1 , [|S|, ˜ 1/2 , V ] ∈ C1 , so that by [5, Theorem 2], we have [|S| ˜ 1/2 ]|S| ˜ 1/2 + |S| ˜ 1/2 [|S| ˜ 1/2 , V ∗ ] ∈ C1 . 2(X1 − X2 ) = [V, |S| Similarly, we have Y1 − Y2 ∈ C1 . Hence, we have, for any polynomials P, Q of two variables, Tr([P (X1 , Y1 ), Q(X1 , Y1 )]) = Tr([P (X2 , Y2 ), Q(X2 , Y2 )]), so that,

J(P, Q)(x, y)gS˜ (x, y)dxdy =

J(P, Q)(x, y)gS˜˜ (x, y)dxdy.

Since P, Q are arbitrary, we have gS˜ = gS˜˜ .

(2.2)

By the proof of [10, Theorem 2.1], S˜ is semi-hyponormal and Aluthge’s theorem ˜˜ we have ˜ is hyponormal. Applying [3, Theorem 5.2] to S, [1] implies that S˜ ˜˜ supp(gS˜˜ ) ⊆ σ(S).

(2.3)

Let S = X0 + iY0 be the Cartesian decomposition of S. Since S = U |T |2 satisﬁes [|T |, U ] ∈ C1 , similarly we have X0 − X1 ∈ C1 and Y0 − Y1 ∈ C1 . Then by a similar argument above, gS = gS˜ ,

(2.4)

so that, we obtain, by (2.2) and (2.4), gS˜ = gS˜˜ = gS .

(2.5)

˜ ˜ = σ(S), then we see that by (2.3) and (2.5) Since σ(S) supp(gS ) ⊆ σ(S). Applying [7, Theorem 4] to T = U |T |, we have 1 J(P, Q)(r, eiθ )eiθ gTP (eiθ , r)dθdr. Tr([P (|T |, U ), Q(|T |, U )]) = 2π

(2.6)

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Hence, 2

2

Tr([P (|T | , U ), Q(|T | , U )])

= =

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1 2rJ(P, Q)(r2 , eiθ )eiθ gTP (eiθ , r)dθdr 2π 1 √ J(P, Q)(ρ, eiθ )eiθ gTP (eiθ , ρ)dθdρ. 2π

Also, it holds 1 J(P, Q)(ρ, eiθ )eiθ gSP (eiθ , ρ)dθdρ. 2π Since polynomials P and Q are arbitrary, we have √ (2.7) gTP (eiθ , ρ) = gSP (eiθ , ρ). Tr([P (|T |2 , U ), Q(|T |2 , U )]) =

By [8, Corollary 2], it holds that gT (x, y) = gTP (eiθ , r) and gS (x, y) = gSP (eiθ , r) for x + iy = reiθ . By the proof of [10, Theorem 2.2], we have (2.8) σ(S) = σ(U |T |2 ) = {r2 eiθ : reiθ ∈ σ(U |T |)}. Using (2.5), (2.6), (2.7) and (2.8), we have that supp(gT ) ⊆ σ(U |T |) = σ(T ). Next we study in the case of pure. First we show the following Lemma 2.3. Let T = U |T | be an invertible class A operator. If T is pure, then S˜ = |T |U |T | is also pure. Proof. Put S = U |T |2 . First we show that S is pure. Since T is invertible, S is also invertible. Let X be the reducing subspace for S such that S ∗ Sx = SS ∗ x for x ∈ X . Then, we have |T |x = |T ∗ |x ∈ X for x ∈ X . Since |T | is invertible, by [6, Lemma 2] it holds |T |(X ) = X . Hence, T (X ) = U |T |(|T |(X )) = S(X ) ⊂ X . Since |T |x = |T ∗ |x, we have U ∗ |T |x = |T |U ∗ x (x ∈ X ). Hence, T ∗ (X ) = |T |U ∗ (|T |(X )) = |T |2 U ∗ (X ) = S ∗ (X ) ⊂ X . Therefore, the space X is the reducing subspace for T such that T ∗ T x = T T ∗ x for x ∈ X . Since T is pure, X = {0} or X = H, so that, S is pure. Next, since the operator T is class A, by [15, Theorem 1, (i)] it holds (|T |U ∗ |T |2 U |T |)1/2 ≥ |T |2 ≥ (|T |U |T |2 U ∗ |T |)1/2 , ˜ ≥ |S| ≥ |S˜∗ |. Since S is pure and S˜ is the Aluthge transformation of so that, |S| S, the operator S˜ is also pure by [9, Lemma 2]. Corollary 2.4. Under the assumptions of Theorem 2.1, if the operator T is pure, then supp(gT ) = σ(T ). ˜ be operators in the proof of Theorem 2.1. Since S˜ is pure by Proof. Let S˜ and S˜ ˜ is pure. Since S˜˜ is a pure hyponormal operator, we Lemma 2.3, the operator S˜ ˜ ˜ By (2.5), (2.7) and (2.8), we have supp(gT ) = σ(T ). So the have supp(gS˜˜ ) = σ(S). proof is complete.

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References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integr. Equat. Oper. Th. 13(1990), 307-315. [2] C. A. Berger, Intertwined operators and the Pincus principal function, Integr. Equat. Oper. Th. 4(1981), 1-9. [3] R. W. Carey and J.D. Pincus, Mosaics, principal functions, and mean motion in von-Neumann algebras, Acta Math. 138(1977), 153-218. [4] M. Ch¯ o and T. Huruya, Trace formulae of p-hyponormal operators, Studia Math. 161 (2004), 1-18. [5] M. Ch¯ o and T. Huruya, Relations between principal functions of p-hyponormal operators, J. Math. Soc. Japan 57 (2005), 605-618. [6] M. Ch¯ o, T. Huruya and M. Itoh, Spectra of completely log-hyponormal operators, Integr. Equat. Oper. Th. 37(2000), 1-8. [7] M. Ch¯ o, T. Huruya and C. Li, Trace formulae associated with the polar decomposition of operators, Math. Proc. Royal Irish Acad. 105(2005), 57-69. [8] M. Ch¯ o, T. Huruya, A. H. Kim and C. Li, Principal functions for high powers of operators, Tokyo J. Math. 29(2006), 111-116. [9] M. Ch¯ o, T. Huruya and T. Yamazaki, Mosaic and principal functions of loghyponormal operators, Integr. Equat. Oper. Th. 48(2004), 295-304. [10] M. Ch¯ o and T. Yamazaki, An operator transformation from class A to the class of hyponormal operators and its application, Integr. Equat. Oper. Th. 53(2005), 497508. [11] K. F. Clancey, Seminormal operators, Springer Verlag Lecture Notes No. 742, BerlinHeidelberg-New York, 1979. [12] F. Furuta, M. Ito and T. Yamazaki, A subclass of paranormal operators including class of log-hyponormal and several related classes, Sci. Math., 1(1998), 389-403. [13] T. Furuta, Invitation to linear operators, Taylor & Francis Inc, London and New York, 2001. [14] J. W. Helton and R. Howe, Integral operators, commutator traces, index and homology, Proceedings of a conference on operator theory, Springer Verlag Lecture Notes No. 345, Berlin-Heidelberg-New York, 1973. r

r

r

[15] M. Ito and T. Yamazaki, Relations between two inequalities (B 2 Ap B 2 ) p+r ≥ B r p p p and Ap ≥ (A 2 B r A 2 ) p+r and their applications, Integr. Equat. Oper. Th. 44 (2002), 442–450. [16] M. Martin and M. Putinar, Lectures on hyponormal operators, Birkh¨ auser Verlag, Basel, 1989. [17] J. D. Pincus, Commutators and systems of singular integral equations, I, Acta Math. 121(1968), 219-249. [18] D. Xia, Spectral theory of hyponormal operators, Birkh¨ auser Verlag, Basel, 1983.

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Muneo Ch¯ o Department of Mathematics Kanagawa University Yokohama, 221-8686 Japan e-mail: [email protected] Mariko Giga Department of Mathematics Nippon Medical School Kawasaki, 221-0063 Japan e-mail: [email protected] Tadasi Huruya Faculty of Education and Human Sciences Niigata University Niigata, 950-2181 Japan e-mail: [email protected] Takeaki Yamazaki Department of Mathematics Kanagawa University Yokohama, 221-8686 Japan e-mail: [email protected] Submitted: September 28, 2005 Revised: July 17, 2006

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Integr. equ. oper. theory 57 (2007), 309–326 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030309-18, published online August 8, 2006 DOI 10.1007/s00020-006-1454-0

Integral Equations and Operator Theory

The σg-Drazin Inverse and the Generalized Mbekhta Decomposition A. Daji´c and J. J. Koliha Abstract. In this paper we deﬁne and study an extension of the g-Drazin for elements of a Banach algebra and for bounded linear operators based on an isolated spectral set rather than on an isolated spectral point. We investigate salient properties of the new inverse and its continuity, and illustrate its usefulness with an application to diﬀerential equations. Generalized Mbekhta subspaces are introduced and the corresponding extended Mbekhta decomposition gives a characterization of circularly isolated spectral sets. Mathematics Subject Classification (2000). 46H30, 47A10, 47A62, 34G10. Keywords. Isolated spectral set, σg-Drazin inverse, Mbekhta decomposition.

1. Isolated spectral sets The g-Drazin inverse in a unital Banach algebra A studied in detail in [8] is a useful generalization of the ordinary inverse in the case when a ∈ A is quasipolar, that is when 0 is not an accumulation point of the spectrum of a. The g-Drazin inverse ﬁnds many applications, in particular to singular diﬀerential equations in the case of operator algebras. The special form of a is a limitation on this versatile concept. In many situation a similar generalized inverse can be deﬁned even in the case when the element is not quasipolar. The ﬁrst mention of this generalized concept appeared in [9]; it was subsequently used by Tran in [16] applied to diﬀerential equations. In this paper we deﬁne and study such a generalization of the g-Drazin of an element of Banach algebra based on isolated spectral sets rather than isolated spectral points. For any a ∈ A, Sp(a), r(a) and Res(a) denote the spectrum, spectral radius and resolvent set of a relative to the algebra A, respectively. We write R(λ; a) = (λ−a)−1 for the resolvent of a. We write Sp(B; a) for the spectrum of a ∈ B with respect to a subalgebra B of A; we attach a similar meaning to Res(B; a). The group of invertibles in A will be denoted by Ainv .

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First we recall the following well known concept. Definition 1.1. A (possibly empty) subset σ of Sp(a) is called an isolated spectral set of a if σ and Sp(a) \ σ are both closed. When σ is an isolated spectral set, Sp(a) \ σ is also an isolated spectral set called the isolated spectral set complementary to σ. Isolated spectral sets were introduced and studied by Dunford and Schwartz in [3], where they are called spectral sets. We prefer the term isolated spectral sets to distinguish them from spectral sets in the sense of von Neumann. Equivalently, σ is an isolated spectral subset of a if there exist disjoint open sets ∆ ⊃ σ and Ω ⊃ Sp(a) \ σ. This is especially useful in applications of the holomorphic calculus to the element a. The element p = f (a), where f is the characteristic function of ∆ in ∆∪Ω, is called the spectral idempotent of a corresponding to σ. If σ = ∅, then p = 0; if σ = Sp(a), then p = 1. When dealing with a set σ in the complex plane, we use the symbol |σ| for the set {|λ| : λ ∈ σ}, and σ n for the set {λn : λ ∈ σ}. If p is an idempotent in A, then pAp is a closed subalgebra of A with the unit p. If ap = pa, then Sp(pAp; ap) ∪ {0} = Sp(ap),

(p = 1)

(1.1)

(see, for instance, [9, Lemma 2.3]). In this paper we will consistently use the notation p = 1 − p for the complementary idempotent to p. It is well known that Sp(a) = Sp(pAp; ap) ∪ Sp(p Ap ; ap ).

(1.2)

Indeed, if u and v are the inverses of λp − ap and λp − ap in the algebras pAp and p Ap , respectively, then u + v is the inverse of λ − a in A. Conversely, if w is the inverse of λ − a in A, then w commutes with p and p , and wp, wp are the inverses of λp − ap, λp − ap in the appropriate algebras. This proves the equation Res(a) = Res(pAp; ap) ∩ Res(p Ap ; ap ),

(1.3)

which is equivalent to (1.2), and shows that R(λ; a) = Rp (λ; ap) + Rp (λ; ap ),

(1.4)

where Rp and Rp are the resolvents in pAp and p Ap , respectively. We give a characterization of isolated spectral sets of a ∈ A, and a characterization of spectral idempotents which does not require functional calculus (though the proof does). Theorem 1.2. Let a ∈ A. A set σ ⊂ C is an isolated spectral set of a if and only if there exists an idempotent p ∈ A commuting with a such that Sp(pAp; ap) ∩ Sp(p Ap ; ap ) = ∅

and

Sp(pAp; ap) = σ.

(1.5)

In this case p is the spectral idempotent of a corresponding to σ, and p the spectral idempotent corresponding to τ = Sp(a) \ σ.

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Proof. Let σ be an isolated spectral set of a with the spectral idempotent p and the complementary isolated spectral set τ . We show that Sp(pAp; ap) ⊂ σ: Let ξ ∈ / σ. Choose disjoint open sets ∆ and Ω such that σ ⊂ ∆, ξ ∈ / ∆ and τ ⊂ Ω. Deﬁne h on ∆ ∪ Ω by setting h(λ) = (ξ − λ)−1 if λ ∈ ∆, and h(λ) = 0 if λ ∈ Ω. Let f be the characteristic function of ∆ in ∆ ∪ Ω; then p = f (a). By the holomorphic calculus, (ξp − ap)h(a) = f (a) = p. Note that h(a) ∈ pAp as h(a)p = h(a). Since everything in sight commutes, ξp − ap is invertible in pAp. Thus Sp(pAp; a) ⊂ σ. Since p is the spectral idempotent for τ , the preceding argument yields Sp(p Ap ; ap ) ⊂ τ . From (1.2) and σ ∩ τ = ∅, σ = Sp(a) ∩ σ = Sp(pAp; ap) ∩ σ = Sp(pAp; ap). Similarly, τ = Sp(p Ap ; ap ). This proves (1.5). Conversely assume that p is an idempotent satisfying (1.5). Then we have Sp(pAp; ap) = σ, and τ = Sp(p Ap ; ap ) are compact disjoint sets satisfying Sp(a) = σ ∪ τ in view of (1.2). Hence σ, τ are isolated spectral sets of a. To prove that p is the spectral idempotent of a corresponding to σ, choose disjoint open sets ∆ ⊃ σ and Ω ⊃ τ . If f is the characteristic function of ∆ in ∆ ∪ Ω, then in view of (1.4), f (a) = fp (ap) + fp (ap ) = fp (ap) = 1pAp = p, where fp and fp denote the application of f in pAp and p Ap , respectively.

Let us comment on two special cases of the preceding theorem. If p = 0, then pAp = {0}, 1 = 0, and every element in pAp is invertible with the inverse 0. Then p = 1, and equation (1.5) is fulﬁlled with σ = ∅ and Sp(p Ap ; ap ) = Sp(a). If p = 1, then (1.5) holds with σ = Sp(a) and Sp(p Ap ; ap ) = Sp({0}; 0) = ∅. The case σ = {0} is described in the following corollary. Recall that a ∈ A is quasipolar if 0 is not an accumulation point of the spectrum of a. Corollary 1.3. An element a ∈ A is quasipolar if and only if there exists an idempotent p ∈ A commuting with a such that Sp(ap) = {0}

and

ap + ξp ∈ Ainv for some ξ = 0.

(1.6)

Proof. An element a ∈ A is invertible if and only if (1.6) holds with p = 0. By Theorem 1.2, σ = {0} is an isolated spectral set of a if and only if there exists an idempotent p (= 0) commuting with a such that Sp(pAp; ap) = {0} and 0 ∈ / Sp(p Ap ; ap ). The second condition says that ap is invertible in p Ap . It is not diﬃcult to show that this happens if and only if ap + ξp is invertible in A for some ξ = 0. When we observe that ap + ξp ∈ Ainv if and only if a + ξp ∈ Ainv for some ξ = 0 (under the assumption Sp(ap) = {0}), we recover [8, Theorem 4.2]. The characterization of isolated spectral sets given in Theorem 1.2 goes outside the Banach algebra A, as it depends on the spectra of elements in the algebras

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pAp and p Ap . The following result characterizes isolated spectral sets in terms of the Banach algebra A alone (see also Lemma 2.3 and Theorem 3.2 in [9]). Theorem 1.4. Let a ∈ A. A set σ ⊂ C is an isolated spectral set of a if and only if there exists an idempotent p ∈ A commuting with a such that the following conditions are satisﬁed:  Sp(ap) / Ainv , if ap + p ∈ (i) σ = Sp(ap) \ {0} if ap + p ∈ Ainv ; (ii) there exists ξ ∈ C such that a − µ − ξp ∈ Ainv for all µ ∈ σ. Proof. First we check that the formula in condition (i) describes the spectrum of ap in the algebra pAp. This follows from (1.1) and the observation that ap ∈ (pAp)inv if and only if ap + p ∈ Ainv . Suppose ﬁrst that σ is an isolated spectral set of a with the spectral idempotent p and the complementary spectral set τ . Then p commutes with a, and (i) holds by Theorem 1.2. Let r = sup {|λ| : λ ∈ σ}, let ξ be a complex number satisfying |ξ| > 2r, and let µ ∈ σ. Choose disjoint open sets ∆ ⊃ σ and Ω ⊃ τ , and deﬁne h by h(λ) = λ − µ − ξ if λ ∈ ∆, and h(λ) = λ − µ if λ ∈ Ω. Then h(a) = a − µ − ξp. We show that h(λ) = 0 for all λ ∈ Sp(a). Let λ ∈ σ. Then |h(λ)| ≥ |ξ| − |µ + λ| > 0 as |ξ| > 2r. If λ ∈ τ , then h(λ) = λ − µ as σ ∩ τ = ∅. Hence h(a) is invertible, which proves (ii). Conversely, assume that an idempotent p commutes with a and satisﬁes conditions (i)–(ii). Condition (i) is equivalent to Sp(pAp; ap) = σ. For every µ ∈ σ, ξp + µ − a ∈ Ainv . Then µp − ap = (ξp + µ − a)p is invertible in p Ap , which shows that Sp(p Ap ; ap ) ∩ σ = ∅. By Theorem 1.2, σ is an isolated spectral point of a. If 0 ∈ σ, then condition (i) in the preceding theorem simpliﬁes to σ = Sp(ap) as in this case Sp(pAp; ap) = Sp(ap). A variant of this result for closed operators was given in [16, Theorem 2.1].

2. The σg-Drazin inverse Definition 2.1. We say that a ∈ A is g-Drazin invertible if there exists an idempotent p commuting with a such that ap is quasinilpotent in pAp (Sp(pAp; ap) = {0} or ap = 0) and ap is invertible in p Ap . The g-Drazin inverse aD of such a is the inverse of ap in p Ap : (ap )aD = p = aD (ap ). From Corollary 1.3 it follows that a is g-Drazin invertible if and only if a is quasipolar with the spectral idempotent p. A diﬀerent, but equivalent, deﬁnition was given in [8], and the properties of the g-Drazin inverse were studied there in some detail. A special case of the g-Drazin inverse is the Drazin inverse arising

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when ak p = 0 for some integer k ≥ 0. If k = 1, the Drazin inverse b of a is called the group inverse. It is characterized by the equations ab = ba,

bab = b,

aba = a.

(2.1)

In order to further generalize the g-Drazin inverse we replace {0} in the preceding deﬁnition by an isolated spectral set. An example of such a construction was given in [9, Example 3.3], and for closed linear operators in [16]. In this paper we present a detailed study of the properties of this generalized g-Drazin inverse, and give applications. Definition 2.2. Let σ be an isolated spectral set of a ∈ A such that 0 ∈ Res(a) ∪ σ, and let p be the spectral idempotent of σ. The inverse of ap in the Banach algebra p Ap is called the σg-Drazin inverse of a, written aD,σ . The spectral idempotent of a corresponding to σ will be denoted by aπ,σ . We say that a ∈ A is σg-Drazin invertible if a possesses an isolated spectral set σ such that 0 ∈ Res(a) ∪ σ. We must check that ap ∈ (p Ap )inv . For this we observe that the spectrum Sp(p Ap ; ap ) coincides with τ , the complementary isolated spectral set of σ, and that 0 ∈ / τ by the assumption 0 ∈ Res(a) ∪ σ. We observe that aD,{0} = aD if inv a∈ / A , and aD,∅ = aD = a−1 if a ∈ Ainv . Further, aD,Sp(a) = 0. We observe that Sp(pAp; ap) = σ for any σg-Drazin invertible element a. The following theorem gives an explicit formula for the σg-Drazin inverse of a.

Theorem 2.3. Let a ∈ A be σg-Drazin invertible with the spectral idempotent p. Then for any ξ ∈ / σ ∪ {0} and any η = 0, aD,σ = (a − ξp)−1 p = (ap − ηp)−1 p .

(2.2)

Proof. Let b = aD,σ . By the deﬁnition of the σg-Drazin inverse, ab = (ap )b = p , and bp = 0. Recall that Sp(ap) = Sp(pAp; ap) ∪ {0} = σ ∪ {0}. If ξ ∈ / σ ∪ {0}, then (a − ξp)(ξb − p) = ξab − ap − ξ 2 bp + ξp = ξ − ap ∈ Ainv . Thus a − ξp ∈ Ainv , and the ﬁrst equality in (2.2) follows from (a − ξp)b = p . The second equality is obtained from the ﬁrst observing that ap ∈ (p Ap )inv if and only if ap + ηp ∈ Ainv for some η = 0. If a is σg-Drazin invertible, we have 1 − aaD,σ = aπ,σ . Indeed, for p = a

π,σ

(2.3)

,

1 − a(a − ξp)−1 p = 1 − (a − ξp)−1 (a − ξp)p = 1 − p = p. The following characterization of the σg-Drazin inverse is reminiscent of the classical deﬁnition of the g-Drazin inverse. Theorem 2.4. An element a ∈ A is σg-Drazin invertible for some set σ if and only if there exists b ∈ A such that ab = ba,

bab = b,

Sp(a − aba) ∩ Sp(aba) = {0}.

(2.4)

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In this case b = aD,σ with σ = Sp(pAp; ap), where p = 1 − ab. Proof. Suppose that an element b satisfying (2.4) exists. Let p = 1−ab and p = ab. According to the ﬁrst two equations in (2.4), p and p are idempotents commuting with a and b. The third equation in (2.4) is equivalent to Sp(ap) ∩ Sp(ap ) = {0}. We observe that (ap )b = (aab)b = a(bab) = ab = p , that is, ap is invertible in p Ap with the inverse b = bp . Then 0 ∈ / Sp(p Ap ; ap ), and Sp(pAp; ap) ∪ {0} ∩ Sp(p Ap ; ap ) ∪ {0} = Sp(ap) ∩ Sp(ap ) = {0}, that is, Sp(pAp; ap) ∩ Sp(p Ap ; ap ) = ∅. Set σ = Sp(pAp; ap). By Theorem 1.2, σ is an isolated spectral set of a with the spectral idempotent p. By deﬁnition, aD,σ = b, the inverse of ap in p Ap . We note that if 0 ∈ / σ, then 0 ∈ / Sp(a) = σ ∪ Sp(p Ap ; ap ); hence 0 ∈ Res(a) ∪ σ. Conversely, if σ is an isolated spectral set of a with the spectral idempotent p such that 0 ∈ Res(0) ∪ σ, then b = aD,σ satisﬁes (2.4): Indeed, ab = p = ba (as b = bp is the inverse of ap in p Ap ). From this we get ab = ba and bab = b. Further, Sp(pAp; ap) ∩ Sp(p Ap ; ap ) = ∅, which combined with (1.1) implies Sp(a − aba) ∩ Sp(aba) = Sp(ap) ∩ Sp(ap ) = {0}

as required. From the preceding theorem and its proof we glean the following result:

Corollary 2.5. An element a ∈ A is σg-Drazin invertible for some set σ if and only if there exists an idempotent p ∈ A commuting with a such that ap ∈ (p Ap )inv

and

Sp(ap) ∩ Sp(ap ) = {0}.

In this case aD,σ = (ap + p)−1 p and σ = Sp(pAp; ap). We can give a representation of aD,σ in terms of the holomorphic calculus: If σ is an isolated spetral set of a such that 0 ∈ Res(a) ∪ σ, choose disjoint open sets ∆ ⊃ σ and Ω ⊃ τ , 0 ∈ / Ω, and deﬁne 0 if λ ∈ ∆, h(λ) = (2.5) −1 if λ ∈ Ω. λ We show that aD,σ = h(a). If f is the characteristic function of Ω in ∆ ∪ Ω, then λf (λ)h(λ) = f (λ), and ap h(a) = f (a) = p . Since h(a) ∈ p Ap and commutes with ap , it is the inverse of ap in p Ap . Thus we have the following Proposition 2.6. If a ∈ A is σg-Drazin invertible, then aD,σ = h(a), where h is deﬁned by (2.5).

(2.6)

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If the last condition in (2.4) is replaced by Sp(a − aba) = {0}, we obtain the usual g-Drazin inverse aD of a. In this case it makes sense to deﬁne the Drazin index of a, ind (a) = inf {k ∈ N ∪ {0} : ak+1 aD = ak }. It follows that ind (a) = 0 if and only if a ∈ Ainv , ind (a) is ﬁnite if and only if a − aba is nilpotent, and ind (a) = ∞ if and only if a − aba is quasinilpotent but not nilpotent. Further, 0 < ind (a) = k < ∞ if and only if 0 is a pole of order k of the resolvent R(λ; a). However, such a deﬁnition of the index of a relative to the σg-Drazin inverse with a general isolated spectral set σ is no longer possible. By (2.6), ak+1 aD,σ = g(a), where g is equal to 0 on σ and to λk on τ . If σ = {0}, g(a) = ak . Next we look at some properties of the σg-Drazin inverse. Theorem 2.7. Let a ∈ A be σg-Drazin invertible. Then: (i) If σ n ∩τ n = ∅, where τ is the complementary spectral set of σ, then (an )D,σ n n exists, and (an )D,σ = (aD,σ )n . In this case (an )π,σ = aπ,σ .

n

(ii) (aD,σ )D exists, and (aD,σ )D = a2 aD,σ . (iii) ((aD,σ )D )D = aD,σ . (iv) aD,σ (aD,σ )D = aaD,σ = 1 − aπ,σ . Proof. (i) Write b = aD,σ , p = aπ,σ . Then an commutes with p and p , (ap)n = an p, (ap )n = an p , while Sp(pAp; an p) = σ n

and Sp(p Ap ; an p ) = τ n .

Since σ n and τ n are disjoint, they are isolated spectral sets of an by Theorem 1.2. Further, 0 ∈ Res(an ) ∪ σ n . By the deﬁnition of aD,σ , b(ap ) = p . Hence bn (an p ) = (bn an )p = (ba)n p = (ba)p = b(ap ) = p , n

that is, bn is the inverse of an p in p Ap , and (an )D,σ = bn . We have also shown n that (an )π,σ = p. (ii) Write b = aD,σ , p = aπ,σ . Then b commutes with p and p , bp = 0 is quasinilpotent in pAp and bp = b is invertible in p Ap with the inverse ap . By Deﬁnition 2.1, b is g-Drazin invertible with bD = ap = a2 b. (iii) and (iv) follow from the preceding result. Property (i) of the preceding theorem can be generalized as follows: Corollary 2.8. If a ∈ A is σg-Drazin invertible and f is an analytic function deﬁned on some open neighbourhood of Sp(a) such that f (σ) ∩ f (τ ) = ∅, where τ is the complementary isolated spectral set of σ, then f (a) is ρg-Drazin invertible with ρ = f (σ).

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By Deﬁnition 2.2, the σg-Drazin inverse of a is the inverse of ap in the Banach algebra p Ap . It is interesting to see that this inverse is equal to the g-Drazin inverse of ap in A. Lemma 2.9. Let a ∈ A be σg-Drazin invertible with the spectral idempotent p. Then ap is group invertible with aD,σ = (ap )D

and

aπ,σ = (ap )π .

(2.7)

Proof. We see that (ap )p = 0 is quasinilpotent in pAp, and ap is invertible in p Ap with the inverse aD,σ . Hence aD,σ = (ap )D . Further, 1 − aπ,σ = aaD,σ = (ap )(ap )D = 1 − (ap )π . The following decomposition of a σg-Drazin invertible element generalizes the important core-quasinilpotent decomposition of a g-Drazin invertible element. Theorem 2.10. An element a ∈ A is σg-Drazin invertible if and only if a = x + y, where xy = 0 = yx, x is group invertible, and Sp(x) ∩ Sp(y) = {0}. Moreover, such a decomposition is unique. Proof. If aD,σ exists, we can set x = ap , y = ap, where p = aπ,σ . Then xy = 0 = yx. By the preceding lemma, x is g-Drazin invertible with xp = (ap )p = 0, that is, x is group invertible. Further, Sp(x) ∩ Sp(y) = {0} as in the proof of Theorem 2.4. For the converse assume that the decomposition a = x + y with the given properties exists. Let p = xπ . Then yp = y (as yp = yxxD = 0), and xp = 0 (x is group invertible). Let b = xD . Then ab = ba, bab = b, a − aba = ap = y, and aba = ap = x; hence Sp(a − aba) ∩ Sp(aba) = Sp(y) ∩ Sp(x) = {0}. By Theorem 2.4, b = aD,σ , where σ = Sp(pAp; ap). Next, we prove the uniqueness of such decomposition. Suppose that a has decompositions a = x + y and a = v + w, satisfying the conditions of the theorem. Then x = (xD )D = (aD,σ )D = a2 aD,σ = (v 2 + w2 )v D = v since v is group invertible and wv D = wv(v D )2 = 0. Consequently y = w. This completes the proof. We shall call the decomposition a = x + y from the preceding theorem the core decomposition of a, and x the core of a. Corollary 2.11. The core decomposition a = x+y of a σg-Drazin invertible element a ∈ A has the following properties: aD,σ = xD ,

aπ,σ = xπ ,

Sp(x) = τ ∪ {0},

Sp(y) = σ ∪ {0},

where τ is the complementary isolated spectral set of σ.

3. Representation of the σg-Drazin inverse In this section we consider sequences of σg-Drazin invertible elements an convergent to a σg-Drazin invertible element a.

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Theorem 3.1. Let a ∈ A be σg-Drazin invertible with aπ,σ = p. Then aD,σ = lim (ap − λ)−1 p . λ→0

(3.1)

Proof. The element ap is quasipolar, and some punctured neighbourhood 0 < |λ| < r lies in the resolvent set of ap . From the Laurent expansion for the resolvent of a quasipolar element, ∞ λn ((ap )D )n+1 , 0 < |λ| < r. (ap − λ)−1 p = n=0

Hence the limit in (3.1) is equal to (ap )D = aD,σ .

For σ = {0} we recover [7, Theorem 6.1]. Next we give an integral representation of the σg-Drazin inverse. Theorem 3.2. Let a ∈ A be σg-Drazin invertible with the spectral projection p and with σ satisfying (Re σ) \ {0} < 0. Then ∞ exp(ta)p dt. (3.2) aD,σ = − 0

Proof. Let a = x + y be the core decomposition of a. Then Sp(x) \ {0} = σ \ {0} lies in the open left half plane and x is group invertible. By [7, Theorem 6.3], ∞ exp(tx)p dt. xD = − 0

Since x = ap , we have exp(tx)p = exp(tap )p = exp(ta)p . The conclusion of the theorem follows from the equation aD,σ = xD .

For σ = {0} we recover [2, Theorem 2.2].

4. Continuity of the σg-Drazin inverse The continuity of the g-Drazin inverse was studied in detail in [10]. Our ﬁrst result proves the continuity of the σg-Drazin inverse under an additional condition on the convergence of (an ). We use the following notation: We write Sp(an ) = σ ∪ τn with the disjoint union of isolated spectral sets of an ; similarly Sp(a) = σ ∪ τ . Theorem 4.1. Let an and a be σg-Drazin invertible elements of A with the spectral idempotents pn and p, respectively. Let an → a and an pn → ap. Then the following conditions are equivalent: → aD,σ , (i) aD,σ n (ii) supn aD,σ n < ∞, (iii) supn r(aD,σ n ) < ∞,

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(iv) pn → p, → aaD,σ . (v) an aD,σ n Proof. Let xn and x be the core of an and a, respectively. Recall that xn = an − an pn and x = a − ap. Hence an → a and an pn → ap imply that xn → x. We recall that xn D = an D,σ , xn π = an π,σ = pn , xD = aD,σ and xπ = aπ,σ = p. In D D,σ = 1 − p. Since xn and x are addition, xn xn D = an aD n = 1 − pn and xx = aa Drazin invertible, the result follows from [10, Theorem 2.4]. Remark 4.2. We observe that the conclusions of the preceding theorem remain in force if we merely assume that xn → x, where xn and x is the core of an and a. In the case of a mere convergence an → a we have the following rather modest result. Proposition 4.3. Let an and a be σg-Drazin invertible elements of A, and let pn and p be the spectral idempotents of an and a, respectively. If an → a, the following conditions are equivalent: → aD,σ , (i) aD,σ n (ii) an aD,σ → aaD,σ , n (iii) pn → p. Proof. Condition (i) implies (ii) in view of the continuity of the product in A. = 1 − pn and aaD,σ = 1 − p; hence (iii) follows from (ii). Further, an aD,σ n Suppose that (iii) holds. According to Theorem 2.3, for any ξ ∈ / σ ∪ {0}, aD,σ = (an − ξpn )−1 pn → (a − ξp)−1 p = aD,σ . n This proves (i), and the proof of the theorem is complete.

Let us consider the situation when an → a with all the elements σg-Drazin → aD,σ . invertible. Clearly, condition (iii) of Theorem 4.1 is necessary for aD,σ n However, it need not be suﬃcient. Suppose that (iii) holds with s = supn r(aD,σ n )> 0. If λ ∈ τn and hn is a holomorphic function such that aD,σ = hn (an ), then n |λ−1 | = |hn (λ)| ≤ r(hn (an )) = r(aD,σ n ) ≤ s. This means that all the sets τn lie in the annulus |λ| ≥ s−1 . From the spectral mapping theorem we deduce that 1 rn := inf |τn | = if τn = ∅. r(aD,σ n ) Following [9], we say that a set σ ⊂ Sp(a) is circularly isolated about µ in Sp(a) if there is a cirle |λ − µ| = r whose interior contains σ and whose exterior contains τ = Sp(a) \ σ. Suppose that σ is circularly isolated about 0 in Sp(an ) for all n, and write r = sup |σ|. Then each τn and σ are isolated by a circle |λ| = ρn , where rn > ρn > r. However, we may have rn → s−1 , in which case there is no single

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circle |λ| = ρ isolating all the τn from σ, and in general we cannot conclude that pn = fn (an ) → f (a) = p. If σ = {0}, then such an isolation obviously exists, and condition (iii) of Theorem 4.1 is seen to be suﬃcient for pn → p; thus we recover [10, Theorem 2.4]. The analysis carried out in the preceding paragraph motivates the following result. Theorem 4.4. Let an and a be σg-Drazin invertible elements of A, and let there be a Jordan curve Γ such that σ belongs to the interior of Γ, and each τn and τ to the exterior of Γ. Let pn and p be the spectral idempotents of an and a, respectively. If an → a, then aD,σ → aD,σ . (4.1) n Proof. Let ∆ = int Γ and Ω = ext Γ. Let h be the function deﬁned on ∆ ∪ Ω by setting h(λ) = 0 if λ ∈ ∆, and h(λ) = λ−1 if λ ∈ Ω. (Recall that the hypothesis 0 ∈ Res(a)∪σ guarantees that 0 ∈ / τ , and that we can remove 0 from Ω is necessary.) Since τ ⊂ Ω, τn ⊂ Ω for all n, and σ ⊂ ∆, a single holomorphic function h can be used to deﬁne aD,σ = h(a), aD,σ = h(an ), n ∈ N. n Since an → a, we conclude that h(an ) → h(a) by [13, Theorem 3.3.7]. Hence (4.1) holds. Corollary 4.5. Let an and a be σg-Drazin invertible elements of A such that an → a and 1 max {r(aD,σ ), sup r(aD,σ . n )} sup |σ|. Then there exists a circle |λ| = ρ whose interior contains σ and whose exterior contains all the τn and τ . Then the preceding theorem applies. If σ = {0} and A = Cn×n is the Banach algebra of all n×n complex matrices, the preceding theorem leads to the following result: D Corollary 4.6. Let An and A be matrices in Cn×n . Then AD n → A if and only if there exists a constant α > 0 such that the nonzero eigenvalues of An satisfy |λ| ≥ α for all n.

We close the section with the following theorem. Theorem 4.7. Let a ∈ A be σg-Drazin invertible, let there be a Jordan curve Γ such that σ belongs to the interior of Γ, and τ into the exterior of Γ, and let an → a. Then there exists N such that for all n ≥ N each an is σn g-Drazin invertible for some σn , and n aD,σ → aD,σ . (4.2) n

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Proof. By a theorem of Kato [5, Theorem IV.3.16], there exists N such that for all n ≥ N the spectrum Sp(an ) is separated into two isolated spectral sets σn ⊂ int Γ and τn ⊂ ext Γ. Note that 0 ∈ Res(an ) ∪ σn . Hence each an with n ≥ N is σn gDrazin invertible, and (4.2) holds by an argument analogous to the one given in the proof of Theorem 4.4.

5. The σg-Drazin inverse for operators If A is the Banach algebra B(X) of all bounded linear operators on a complex Banach space X, we can characterize the σg-Drazin inverse of A ∈ B(X) in terms of the direct sum of operators. Theorem 5.1. An operator A ∈ B(X) is σg-Drazin invertible for some set σ if and only if A = A1 ⊕ A2 ,

Sp(A1 ) ∩ Sp(A2 ) = ∅, D,σ

In this case Sp(A1 ) = σ and A

=0⊕

A2 is invertible.

(5.1)

A−1 2 .

Proof. By Theorem 1.2, A is σg-Drazin invertible in B(X) if and only there exists an idempotent operator P ∈ B(X) commuting with A such that AP is invertible in the algebra P B(X)P , and the spectra of AP and AP in the algebras P B(X)P and P B(X)P are disjoint. (Here, in accordance with our notation, P = I − P .) Let X1 = R(P ) and X2 = N (P ) = R(P ). Then X is the topological direct sum X = X1 ⊕ X2 , and A is decomposed as A = A1 ⊕ A2 relative to this sum. It is known that Sp(A1 ) is the spectrum of AP in the algebra P B(X)P , and Sp(A2 ) is the spectrum of AP in the algebra P B(X)P . Hence AP is invertible in P B(X)P if and only A2 is invertible in B(X2 ). By the deﬁnition of the σg-Drazin inverse, AD,σ is the inverse of AP in the algebra P B(X)P . Observe that P = I ⊕ 0 and P = 0 ⊕ I. Let B = 0 ⊕ A−1 2 . Then B = BP , that is B ∈ P B(X)P , and −1 (AP )B = (A1 ⊕ A2 )(0 ⊕ I)(0 ⊕ A−1 2 ) = (0 ⊕ A2 )(0 ⊕ A2 ) = 0 ⊕ I = P .

This completes the proof.

Let us recall that in the preceding theorem the condition 0 ∈ σ ∪ Res(A) is assumed when A is σg-Drazin invertible, and can be deduced when (5.1) is assumed. In addition, for any ξ ∈ / σ ∪ {0} and any η = 0, AD,σ = (A − ξP )−1 P = (AP − ηP )−1 P .

(5.2)

If X is a ﬁnite dimensional, then every linear operator T on X has the σgDrazin inverse for every admissible subset σ of its spectrum as every such set is an isolated spectral set of T .

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Example 5.2. We give an example of a  2 −2 12 1 5 −1  1 3 0   1 −2 12 A=  1 1 0  2  1 1 0 2 1

1

0

321

σ-Drazin inverse of a matrix. Let 1 3  − 23 0 2 2 2 2 −2 −5   1 2 −1 −4   1 5   − 21 −1 2 2 . 1 3 1 2 −2 −2 2  1 − 21 0 − 12  2 1

−1

1

−1

The eigenvalues of A are 0, 2 and 1 with algebraic multiplicities 1, 2 and 4, respectively. Then σ = {0, 2} is an isolated spectral set of A with 0 ∈ σ, and AD,{0,2} exists. Let P = Aπ,{0,2} be the spectral projection relative to σ = {0, 2}. We have Aπ,{0,2} + Aπ,1 = Aπ,0 + Aπ,2 + Aπ,1 = I, that is, P = I − P = Aπ,1 :   1 3 −1 1 1 −1 −3 1 1 0 1 0 − 21 − 12    2 2  1 1 1 1 0 1  0 − − 2 2 2 2  π,1  1 1 5 3 P = A = 0 2 − 2 2 1 − 2 − 2  .   1 3 7 3 0 3 − 2 2 2 − 2 − 2    0 1 0 1 0 0 −1  0

2

0

1

1

−1

−2

We can calculate AD,{0,2} as the ordinary Drazin inverse of AP , that is,   3 1 − 21 − 52 0 2 − 21 2 3 7   0 −2 − 32 −2 12  2 2   3 3 1 7   0 −2  − 2 −2 2 2 2   D,{0,2} D 1 1 7 5  A = (AP ) = −1 2 − 2 2 −2 −2 . 2   1 1 1 1 0  0 −2 0 −2 2 2   −1 1 0 2 1 0 −2  0 −1 1 −1 −1 0 2

6. The generalized Mbekhta decomposition Mbekhta’s subspaces H0 (T ) and K(T ) originally deﬁned in [12] for closed linear operators in Hilbert spaces recently received new attention from several authors in [1, 4, 15]. For brevity we shall write r(T ; x) = lim supn→∞ T n x1/n , and S(T ; x) for the set of all sequences (xn ) in X such that T xn+1 = xn for all n ≥ 1 and

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T x1 = x. Mbekhta [12] deﬁned H0 (T ) = {x ∈ X : lim T n x1/n = 0},

(6.1)

n→∞

K(T ) = {x ∈ X : (∃ (xn ) ∈ S(T ; x)) (∃c > 0) with xn ≤ cn x}.

(6.2)

Following [9], for any T ∈ B(X) we deﬁne the generalized quasinilpotent part Hr (T ) and the generalized analytic core Kr (T ) of T as follows: Definition 6.1. Let T ∈ B(X) and r > 0. We deﬁne Hr (T ) = {x ∈ X : r(T ; x) < r},

(6.3)

Kr (T ) = {x ∈ X : ∃ (xn ) ∈ S(T ; x) with lim sup xn

1/n

n→∞

−1

}.

(6.4)

It is not diﬃcult to show that for any r > 0, Hr (T ) and Kr (T ) are (not necessarily closed) hyperinvariant subspaces of X, and that T −1 Hr (T ) = Hr (T ),

T Kr (T ) = Kr (T );

(6.5)

(they are invariant under any operator commuting with T ). These spaces are linked with Mbekhta’s subspaces as follows. Lemma 6.2. For any T ∈ B(X) we have H0 (T ) = Hr (T ), r>0

K(T ) =

Kr (T ).

(6.6)

r>0

Proof. The equality is easily seen for H0 (T ). Let x ∈ r>0 Kr (T ). Then x ∈ Kr (T ) for some r > 0, and there exists (xn ) ∈ S(T ; x) such that lim supn xn 1/n < r−1 . We may assume that x = 0. There exists n0 such that xn 1/n ≤ r−1 x1/n for all n ≥ n0 . Setting c = max {r−1 , max {(xk /x)1/k : k = 1, . . . , n0 }} we get xn ≤ cn x for all n. This proves that x ∈ K(T ). Conversely, if x ∈ K(T ), then there is a sequence (xn ) ∈ S(T ; x) and a positive constant c such that xn ≤ cn x for all n. This implies lim supn xn 1/n ≤ c, and x ∈ Kr (T ) for any r satisfying 0 < r < c−1 . From the lemma and the deﬁnition of Hr (T ) and Kr (T ) we see that Hr (T ) H0 (T ) and Kr (T ) K(T ) as r → 0 + .

(6.7)

For completeness we may set K0 (T ) = K(T ), H∞ (T ) = X and K∞ (T ) = {0}. We also observe that Kr (T ) ⊂ K(T ) ⊂ R(T n ) and N (T n ) ⊂ H0 (T ) ⊂ Hr (T )

(6.8)

for all n and all r ∈ [0, ∞]. The proof of Lemma 6.2 yields an alternative deﬁnition of K(T ): K(T ) = {x ∈ X : ∃ (xn ) ∈ S(T ; x) with lim sup xn 1/n < ∞}. n→∞

(6.9)

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The main result involving the generalized quasinilpotent part and the generalized analytic core of an operator T is the following theorem ﬁrst obtained in [9, Theorem 5.6]. We give a proof based on the foregoing results. Theorem 6.3. Let T ∈ B(X) and 0 < r < ∞. Then T has a spectral set σ circularly isolated by the circle |λ| = r if and only if X is the topological direct sum X = Hr (T ) ⊕ Kr (T ).

(6.10)

Proof. It will be convenient to write Dr for the open disc |λ| < r, Cr for the circle |λ| = r and Ar for the annulus |λ| > r. Suppose ﬁrst that there exists a spectral set σ of T circularly isolated by Cr . We show that R(P ) = Hr (T ), N (P ) = Kr (T ), (6.11) where P is the spectral projection corresponding to σ. Let S = T D,σ be the σg– Drazin inverse of T . Then r(T P ) < r, and r(S) < r−1 by the spectral mapping theorem. The inclusion R(P ) ⊂ Hr (T ) can be easily veriﬁed. For the converse inclusion assume that x ∈ Hr (T ). Then P = (P )n = (ST )n = S n T n . Further, lim supn P x1/n ≤ r(S)r(T ; x) < r−1 r = 1, which implies P x = 0. Hence x ∈ R(P ). To prove N (P ) ⊂ Kr (T ), for a given x ∈ N (P ) construct a S(T ; x) sequence xn = S n x, and check that lim supn xn 1/n < r−1 . For the reverse inclusion show that any x ∈ Kr (T ) satisﬁes lim supn P x1/n ≤ r(T P ) lim supn xn 1/n < 1, that is, P x = 0. Conversely assume that X is the topological direct sum X = Hr (T ) ⊕ Kr (T ). Then T = T1 ⊕ T2 relative to this sum as the subspaces Hr (T ), Kr (T ) are invariant under T . For each x ∈ Hr (T ), r(T1 ; x) = r(T ; x) < r. According to [6, Corollary 2.1], this implies r(T1 ) < r, and Sp(T1 ) ⊂ Dr . The operator T2 is bijective, and hence invertible, on Kr (T ) in view of (6.5). For any x ∈ Kr (T ) we use T n xn = x to show that r(T2−1 ; x) = lim sup T2−n x1/n = lim sup xn 1/n < r−1 n→∞

n→∞

where (xn ) ∈ S(T ; x). By [6, Corollary 2.1], r(T2−1 ) < r−1 , so Sp(T2 ) ⊂ Ar . Combining this with Sp(T1 ) ⊂ Dr , we conclude that σ is circularly isolated by Cr . It is interesting to observe that the condition that X = Hr (T ) ⊕ Kr (T ) is a topological direct sum in the preceding theorem can be relaxed: Corollary 6.4. Let T ∈ B(X) and for some r > 0 let X = Hr (T ) ⊕ Kr (T ) be an algebraic direct sum with at least one of the component spaces closed. Then T has a spectral set σ circularly isolated in Sp(T ) by the circle |λ| = r. This is proved in [9, Theorem 5.9]. Specializing σ to {0}, we have the following result [9, Corollary 5.11]:

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Corollary 6.5. An operator T ∈ B(X) is quasipolar if and only if for each suﬃciently small r > 0, X = Hr (T ) ⊕ Kr (T ) with at least one of the component spaces closed. The original result of Mbekhta in [12] states that 0 is not an accumulation point of Sp(T ) if and only if X is the topological sum X = H0 (T ) ⊕ K(T ). In [14], Schmoeger improved this result by showing that it is enough to have an algebraic direct sum with K(T ) closed to ensure that 0 is not an accumulation spectral point of T . We show that it is enough to have only H0 (T ) closed. (See also [9, Corollary 5.11].) First a useful known lemma. Lemma 6.6. Let H be a closed subspace of a Banach space X, invariant under T ∈ B(X). Deﬁne A : H → H by Ax = T x for all x ∈ H, and B : X/H → X/H by B(x + H) = T x + H. If both A and B are invertible, then so is T . Proof. We note that A, B are bounded linear operators acting on Banach spaces. Assuming that they are invertible, we show that T is bijective. If T x = 0, then B(x + H) = H, and x ∈ H by the injectivity of B. Hence Ax = 0, and x = 0. Let y ∈ X be arbitrary. By the surjectivity of B, there exists x ∈ X such that T x + H = B(x + H) = y + H. Hence y − T x ∈ H, and T (x + A−1 (y − T x)) = T x + (y − T x) = y. Theorem 6.7. An operator T ∈ B(X) is quasipolar if and only if X = H0 (T )⊕K(T ) with at least one of the component spaces closed. Proof. The topological direct decomposition X = H0 (T ) ⊕ K(T ) for a quasipolar operator T was proved in [9, 12, 14]. Let X = H0 (T ) ⊕ K(T ) be an algebraic direct sum. Schmoeger [14] proved that T is quasipolar if K(T ) is closed. Suppose that H0 (T ) is closed. If A is the restriction of T to H = H0 (T ), then A is quasinilpotent. Let B be the operator on X/H deﬁned by B(x + H) = T x + H, and let B(x + H) = T x + H = H. Then T x ∈ H, and x ∈ H since T −1 H = H. Hence B is injective. Let y ∈ X. By hypothesis, y = h + k with h ∈ H and k ∈ K(T ). Since K(T ) = T K(T ), there exists x ∈ K(T ) such that T x = k. Then y + H = T x + H = B(x + H), and B is surjective. Thus B is invertible. There exists δ > 0 such that A(λ) = λIH − A and B(λ) = λIX/H − B are both invertible whenever 0 < |λ| < δ. Applying Lemma 6.6 with T (λ) = λI − T , A(λ) and B(λ) in place of T, A, B, we conclude that T (λ) is invertible whenever 0 < |λ| < δ. This proves that 0 is not an accumulation point of Sp(T ). Several authors recently applied Mbekhta’s subspace theorem to derive certain properties of special classes of operators. Gong and Wang [4] were concerned with compact operators, Bouamama [1] studied Riesz operators, while Schmoeger extended some of the results of [1] and [4] to meromorphic operators. Several of their results can be further generalized when Corollary 6.5 and Theorem 6.7 are taken into account.

The σg-Drazin Inverse

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7. Application The concept of the σg-Drazin inverse can be applied to closed linear operators on Banach spaces. Trung Dinh Tran used the special case of the inverse to study the solutions of a certain diﬀerential equation in a Banach space. His deﬁnition is based on an analogue of our equation (5.2). Tran considered the diﬀerential equation studied by S. G. Krein [11, Chapter 3], d2 x(t) = B 2 x(t) + f (t), t ∈ [0, T ], (7.1) dt2 where B is the inﬁnitesimal generator of a strongly continuous group of bounded linear operators T (t) on a Banach space X, and f is continuously diﬀerentiable on [0, T ]. Assuming that B has an isolated spectral set σ containing 0 and circularly isolated by |λ| = r, Tran obtained the following result involving the σg-Drazin inverse of the generator B: Theorem 7.1. (Tran [16, Theorem 3.1]) The unique solution of the equation (7.1) with T = r−1 and the initial conditions d x(0) = u0 and x(t) = v0 dt 0 is given by ∞

x(t) =

B 2(j−1) P F (2) (t)

j=1

1 1 + (T (t) + T (−t))(I − P )u0 + B D,σ (T (t) − T (−t))(I − P )v0 2 2 t + B D,σ (T (t − s) − T (s − t))(I − P )f (s) ds, t ∈ [0, r−1 ], 0

provided that u0 ∈ D(B 2 ), v0 ∈ D(B), and ∞ j=1

B 2(j−1) P F (2j) (0) = P u0 ,

∞

B 2(j−1) P F (2j−1) (0) = P v0 ,

j=1

where P = B π,σ and F (k) is the kth primitive of f . In closing we comment that other results on diﬀerential equations in Banach spaces involving isolated spectral points of the inﬁnitesimal generator can be extended to the case when the generator has the σg-Drazin inverse for some isolated spectral set σ. Work on such problems is in progress.

References [1] W. Bouamama, Op´erateurs de Riesz dont le coeur analytique est ferm´e, Studia Math. 162 (2004), 15–23.

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[2] N. Castro-Gonz´ alez, J. J. Koliha and Yimin Wei, On integral representation of the Drazin inverse in Banach algebras, Proc. Edinburgh Math. Soc. 45 (2002), 327–331. [3] N. Dunford and J. T. Schwartz, Linear Operators I: General Theory, Wiley Interscience, New York, 1958 [4] W. Gong and L. Wang, Mbekhta’s subspaces and a spectral theory of compact operators, Proc. Amer. Math. Soc. 133 (2002), 587–592. [5] Tosio Kato, Perturbations of Bounded Linear Operators, 2nd ed., Springer, Berlin 1980. [6] J. J. Koliha, Power convergence and pseudoinverses of operators in Banach spaces, J. Math. Anal. Appl. 48 (1974), 446–469. [7] J. J. Koliha, Isolated spectral points, Proc. Amer. Math. Soc. 124 (1996), 3417–3424. [8] J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367–381. [9] J. J. Koliha, P. W. Poon, Spectral sets II, Rend. Circ. Mat. Palermo (2) 47 (1998), 293-310. [10] J. J. Koliha and V. Rakoˇcevi´c, Continuity of the Drazin inverse II, Studia Math. 131 (1998), 167–177. [11] S. G. Krein, Linear Diﬀerential Equations in Banach Spaces, Translations of Mathematical Monographs, Vol. 29. Amer. Math. Soc., Providence, R.I., 1971. [12] M. Mbekhta, G´en´eralisation de la d´ecomposition de Kato aux op´erateurs paranormaux et spectraux, Glasgow Math. J. 29 (1987),159–175. [13] T. W. Palmer, Banach Algebras and the General Theory of ∗-Algebras, vol. I, Encyclopaedia Math. Appl. 49, Cambridge Univ. Press, Cambridge, 1994. [14] C. Schmoeger, On isolated points of the spectrum of a bounded linear operator, Proc. Amer. Math. Soc. 117 (2003), 715–719. [15] C. Schmoeger, A note on meromorphic operators, Proc. Amer. Math. Soc. 133 (2005), 511–518. [16] T. D. Tran, Spectral sets and the Drazin inverse with applications to second order diﬀerential equations, Appl. Math. 47 (2002), 1–8. A. Daji´c and J. J. Koliha Department of Mathematics University of Melbourne VIC 3010, Australia e-mail: [email protected] [email protected] Submitted: November 12, 2005 Revised: April 6, 2006

Integr. equ. oper. theory 57 (2007), 327–337 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030327-11, published online December 26, 2006 DOI 10.1007/s00020-006-1457-x

Integral Equations and Operator Theory

Extensions of Well-Boundedness and C m-Scalarity Jos´e E. Gal´e, Pedro J. Miana and Detlef M¨ uller Dedicated to the memory of Tadeusz Pytlik

Abstract. We consider some extensions of well-boundedness and C m -scalarity by using fractional calculus, and prove some theorems accordingly. These results are applied to the usual Laplacian on Rn and sub-Laplacians on nilpotent Lie groups. Mathematics Subject Classification (2000). Primary 47B40, 26A33; Secondary 47D03, 34L40. Keywords. Well-bounded operators, C m -scalar operators, functional calculus, fractional derivation, Laplacian.

1. Well-boundedness and C m scalarity Let A be a possibly unbounded, closed, densely deﬁned operator on a Banach space X, with spectrum σ(A) contained in (a, b) ⊂ R, where −∞ ≤ a < b ≤ ∞. Let denote B(X) the usual Banach algebra of bounded operators on X. When X is a Hilbert space and A is a self-adjoint operator on X then there exists a projectionvalued measure Ω → E(Ω) from the Borel subsets Ω of (a, b) into B(X), such that (1) A= λ dE(λ). σ(A)

The research of first and second authors has been partly supported by the Project MTM200403036 of the M.C.YT.-DGI/F.E.D.E.R., Spain, and the Project E-12/25, D. G. Arag´ on, Spain. Part of the research of second author was developed in the Christian-Albrechts Universit¨ at in Kiel, while he was enjoying a HARP-postdoctoral position in the European Harmonic Analysis Network, HARP, IHP 2002-06.

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This is the so-called spectral theorem, which provides us with the L∞ functional calculus for A given by the formula (2) f (A)x := f (λ) dE(λ)x (x ∈ X) σ(A)

where f is any bounded, Borel measurable function on σ(A). Projection-valued measures are deﬁned almost verbatim for arbitrary Banach spaces, and then the operator A is said to be a scalar-type spectral operator (or scalar operator, for short) if it satisﬁes (1), for some projection-valued measure. As a consequence, A also enjoys the corresponding L∞ calculus (2). Scalar operators were introduced by Dunford. They have a rich structure, which implies strong properties of series convergence, see [8], [9]. But only few diﬀerential operators are scalar. For instance, the operator i d/dx is scalar on Lp (T) if and only if p = 2, and the same occurs to Laplacian on R [18]. In order to handle wider classes of interesting operators, more general functional calculi are needed. Integrating by parts in (2) suggests some suitable notions both concerning operating functions and projection-valued families. The following deﬁnitions are taken from [7] (for bounded operators, see [8]). A decomposition of the identity for X on (a, b) is a family {E(s)}s∈R of projections on X ∗ such that (i) E(s) = 0 for any s < a; E(s) = I for s ≥ b (ii) E(s)E(t) = E(t)E(s) = E(s) for s ≤ t. (iii) sup{E(s) : s ∈ R} < ∞. (iv) The map s → (E(s)φ)x is Lebesgue measurable for every x ∈ X, and φ ∈ X ∗ . t (v) If, for x ∈ X and φ ∈ X ∗ , the function t → 0 (E(r)φ)xdr is right diﬀerentiable at s ∈ R, then the right derivative at s is (E(s)φ)x. (vi) For any x ∈ X, the map φ → (E(·)φ)x, from X ∗ into L∞ ((a, b)) both with their weak∗ -topologies, is continuous. When X is reﬂexive, then there exists a family of projections (F (s))s∈R on X, such that E(s) = F (s)∗ (s ∈ R). Let denote AC 1 [a, b] the Banach algebra, endowed with pointwise multiplication and norm given by the integral below, of absolutely continuous functions b f on (a, b) satisfying a |f (s)|ds < ∞. An operator A on X as before is said to be well-bounded if it has an AC 1 [a, b] functional calculus, that is, there exists a bounded algebra homomorphism f → f (A), AC 1 [a, b] → B(X) such that (z + · )−1 → (z + A)−1 for every z ∈ C \ (a, b). It turns out that the existence of a decomposition of the identity E(s) is equivalent to having a well-bounded operator A (see [7]). Moreover the functional calculus is given by b f (s)(E(s)φ)x ds (x ∈ X; φ ∈ X ∗ ), φ(f (A)x) = a 1

for every f ∈ AC [a, b].

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The above calculus widens the area of application of the L∞ calculus in general Banach spaces. Turning back to former examples, we have that i d/dx is p well-bounded on Lp (T), and corresponding i d/dx is well-bounded (R), if n on L 2 and only if 1 < p < ∞ [1]. Also, the usual Laplacian ∆ := − j=1 ∂ /∂x2j on Rn (−∆ is the inﬁnitesimal generator of √ the Gaussian semigroup in L1 (Rn )) and √ the square root ∆ of the Laplacian (− ∆ is the inﬁnitesimal generator of the Poisson semigroup) are well-bounded on Lp (R) if 1 1. Thus looking for other functional calculi is in order. For instance, if A is a bounded operator with σ(A) ⊂ [0, 1], and m ≥ 1, then A is said to be C m -scalar if A admits a functional calculus with C m functions on [0, 1], see [13], [14]. We are mainly interested in operators having the spectrum in the half line [0, ∞). Using the change of variable u → (1 + u)−1 gives rise to the notion of, possibly unbounded, C m -scalar operator in this setting: Put Crm [0, ∞) := {f = h((1 + t)−1 ) : h ∈ C m [0, The space Crm [0, ∞) is a Banach algebra with respect to 1]. m the norm f := k=0 (1/k!)h(k) ∞ . A closed, densely deﬁned operator A on X with σ(A) ⊂ [0, ∞) is C m -scalar if there exists a functional calculus Crm [0, ∞) → B(X), in the sense that the above mapping is a bounded homomorphism such that (z + · )−1 → (z + A)−1 for every z ∈ C \ (−∞, 0]. We have that A is C m -scalar on [0, ∞) if and only if (1 + A)−1 is C m -scalar on [0, 1]. Deﬁnitions and the last property have been taken from [6]. Concerning well-boundedness in itself, we have the following straightforward generalisation. For m > 1, the operator A, with σ(A) ⊂ [0, ∞), is said to have an AC m functional calculus if there exists a functional calculus f → f (A), AC m → B(X). Here, AC m denotes the Banach algebra of C m−1 functions f on [0, ∞) whose m − 1 derivative is absolutely continuous and has an a. e. derivative f (m) ∞ satisfying 0 |f (m) (s)|sm−1 ds < ∞. It is easy to check that Crm [0, ∞) is continuously contained in AC m , so the AC m calculus implies C m - scalarity. On the other hand, it has been proved that −i d/dx is C 1 -scalar on H 1 (T), see [6], and, moreover, on L1 (T) and C(T) [5] (deﬁnitions are similar when the spectrum of A is not necessarily contained in [0, ∞)). Since −i d/dx on H 1 (T) (as well as −i d/dx semigroup on on H 1 (R)) coincides with the inﬁnitesimal generator of the Poisson √ H 1 (T) (respectively, on H 1 (R)), the following question is natural: Is ∆ C 1 -scalar on L1 (R) and L1 (T) ? There is a one-dimensional argument which allows us to answer the above question in the aﬃrmative: In [6] some characterizations of well-boundedness and C 1 -scalarity are given for generators of uniformly bounded analytic semigroups on z > 0. The group of translations on H 1 (T) (respectively, H 1 (R)) is the

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group of boundary values of the Poisson semigroup (P z )z>0 on H 1 (T) (respectively, H 1 (R)) which means in particular that supz>0 P z B(H 1 ) < ∞. So The√ orem 13 a) of [6] applies to show that −i d/dx ≡ ∆ on H 1 (T) is C 1 -scalar (the same proof works for −i d/dx on H 1 (R)). In the L1 (R) case the Poisson semigroup is not uniformly bounded but it satisﬁes the estimate P z 1 ≤ C log(2 +

|z| ),

z

z > 0,

and this is enough to get, along the same lines as in [6], that the operator C 1 -scalar on L1 (R) if and only if the function g(u) 1 u → φ ∗ f ≡ (φ ∗ f ) g(u), (1 + u)2 (1 + u)2 R

√ ∆ is

is of bounded variation on [0, ∞) for every φ ∈ L∞ (R) and f ∈ L1 (R). Here, g 1 − cos(ur) if u > 0 and r ∈ R; g(u)(r) = 0 is the function deﬁned as g(u)(r) := r2 if u ≤ 0, r ∈ R. Thus the problem is reduced to show that the mapping T : L∞ (R+ ) → L1 (R+ , (1 + u)−2 du) given by ∞ sin (ur) T (h)(u) = h(r) dr, h ∈ L∞ (R+ ), r 0 is bounded. This is accomplished by using some Fourier analysis which implies + 2 du) → L1 (R+ ) is bounded. (In that the transpose mapping T ∗ : L∞ (R √ , (1 + u) 1 1 a similar way, it can be proved that ∆ is C -scalar on √ C0 (R), that ∆1 is C 1 scalar on L (R) and C0 (R), and that correspondingly ∆ and ∆ are C -scalar on √ L1 (T), C(T). In all these cases the basic fact is that the semigroup generated by ∆ or ∆ is O((|z|/ z)1/2 ) on z > 0.) Well-bounded operators (and C m -scalar operators) have been extensively studied because of their rich spectral theory. In this setting, it is clear the interest of having a number of signiﬁcant examples. Thus we may ask about results like the previous ones for a Laplacian or the square root of a Laplacian in higher dimensions. An a priori diﬃculty to deal with these problems is that the above argument relies heavily on the precise description of the integral kernel g(u)(r) involved in the C 1 calculus, see [6]. As we will see, such a kernel also exists on Rn , but it cannot be handled directly as before [4]. In the next section we overcome this diﬃculty by using some versions of stronger functional calculi. In this way, it will become clear that the subject enters the classical theory of multipliers.

2. Extension of well-boundedness Let Cc∞ [0, ∞) denote the algebra of inﬁnitely diﬀerentiable functions on [0, ∞) with compact support. Fix f in Cc∞ [0, ∞). The Weyl fractional integral of f of

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Extensions of Well-Boundedness and C m -Scalarity

order ν > 0, W −ν f , is deﬁned by W −ν f (t) :=

1 Γ(ν)

∞

(t − s)ν−1 f (s)ds,

331

t > 0.

t

The Weyl fractional derivative W ν f of f , of order ν > 0, is then deﬁned by dn −(n−ν) W f (t), t > 0, dtn where n is any natural number greater than α. We have that W ν W µ f = W ν+µ f for every ν, µ ∈ R, and W n f = (−1)n f (n) if n is a non negative integer. The two following spaces are deﬁned in [10], see also [19] for the ﬁrst one. Take ν > 0. By AC ν we denote the Banach space obtained as the completion of Cc∞ [0, ∞) in the norm ∞ 1 |W ν f (t)|tν−1 dt. f (ν) := Γ(ν) 0 ν Analogously, we denote by AC2,1 the completion of Cc∞ [0, ∞) in the norm 1/2 ∞ 2y dx dy . |W ν f (x)xν |2 f (ν);2,1 := x y 0 y W ν f (t) := (−1)n

µ ν We have that AC ν ⊂ AC µ if ν > µ > 0, AC2,1 ⊂ AC2,1 for ν > µ > 1/2, µ and AC ν+1/2 ⊂ AC2,1 ⊂ AC µ if ν > µ > 0 [10, Proposition 3.7]. The space AC ν is in fact a Banach algebra with respect to pointwise multiν plication, whenever ν ≥ 1 [19], [10], and, similarly, AC2,1 is closed under pointwise multiplication, provided that ν > 1/2 [10]. These algebras are both invariant under the change of variable t → tθ , θ > 0 [10]. For an operator A with spectrum ν in [0, ∞), the notions of AC ν and AC2,1 functional calculi can be introduced in an obvious way, and these calculi are naturally linked to generators A of certain analytic semigroups [10, pp.344 and ss.]. Let (az )z>0 be an analytic C0 semigroup of bounded operators on X with generator −A. We will assume that the semigroup, or alternatively the operator A, satisﬁes property (HGα ): α |z| az ≤ Cα , ( z > 0), (HGα )

z

for some α ≥ 0, where Cα is some positive constant. Property (HGα ) implies that σ(A) ⊂ [0, ∞). Moreover, the inverse Laplace transform of z −(ν+1) az x, 1 Gν (u)x := z −(ν+1) euz az xdz, 2πi z=1 is well deﬁned for every u ∈ R, x ∈ X and ν > α, and it is such that Gν (u) ∈ B(X), Gν (u) = 0 if u ≤ 0 and Gν (u) ≤ Cν uν (u > 0) [10, Lemma 6.1]. The formula ∞ Φν (f )x := W ν+1 f (u)Gν (u)xdu (x ∈ X; f ∈ AC ν+1 ) 0

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deﬁnes a functional calculus AC ν+1 → B(X), for A, such that Φν = Φµ for all ν, µ > α [10]. We will write f (A) ≡ Φ(f ) := Φν (f ). The AC ν+1 calculus is a natural extension of well-boundedness. The kernel Gν (u) may be regarded a “ν-integrated” projection-valued measure. We do observe that Gν (u) ≡ (u − A)ν+ is the Bochner-Riesz mean of A of order ν. Usual powers √ Aθ , θ > 0, can be recovered from the calculus. If θ = 1/2 then A1/2 = A. ν+ 1

Moreover, the calculus Φ extends automatically to an AC2,1 2 calculus for A via the formula ∞ 1 1 1 Gν+ 2 (u − h) − Gν+ 2 (u) x du, f (A)x ≡ Φ(f )x := lim W ν+ 2 f (u) h h→0+ 0 ν+ 1

where x ∈ X, f ∈ AC2,1 2 , and whenever ν > α [10, Theorem 6.4]. We will say ν+ 1

ν+ 1

that A is AC2,1 2 -bounded if A has a AC2,1 2 functional calculus, and similarly with AC ν+1 . There are many Laplace or sub-Laplace operators on Lie groups, or LaplaceBeltrami operators on certain manifolds, which generate semigroups satisfying ν+ 1 property (HGα ) and so are AC2,1 2 -bounded. To illustrate what is behind the matter of well-boundedness we focus on Rn and the Heisenberg group Hn . For convenience, we ﬁrst give an abstract Lp version of extended well-boundedness. Theorem 2.1. Let M be a measure space, and let A be a positive definite operator on L2 (M ). Suppose that −A generates a holomorphic semigroup (e−zA )z>0 in B(L1 (M )) which satisfies (HGα ), for some α > 0. Let θ > 0. Then α+ 1

(1) Aθ is AC2,1 2 -bounded on Lp (M ) for all p such that 1 0 is a holomorphic semigroup in B(L2 (M )) satisfying (HG0 ). Then, by interpolation, e−zA enjoys the property (HGβ(p) ) on Lp (M ), with β(p) := 2α| p1 − 12 | if 1 ≤ p ≤ ∞. Thus, for 1 β(p) and therefore A has an AC2,1

calculus on Lp (M ) [10, α+ 1

Theorem 6.3]. For Aθ , the result follows from the invariance of AC2,1 2 under the transformation u → uθ of [0, ∞) [10, Proposition 3.9]. (2) As above, if 1 β(p) and so Gα (u) is in p B(L (M )). Then the mapping ∞ f → W α+1 f (u)Gα (u)du, AC α+1 → B(Lp (M )) 0

α+1

deﬁnes the AC calculus on Lp (M ). Finally, for Aθ , we use again the invariance α+1 under u → uθ . of AC Remark 2.2. (i) If A is as in Theorem 2.1 and p = 1 or p = ∞ then a direct application of [10, Theorem 6.3 and Proposition 3.7 (ii)] gives us that Aθ is

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333

α+ 1 +ε

AC2,1 2 -bounded and AC α+1+ε -bounded, for every ε > 0. This arbitrarily small ε makes the diﬀerence, in this setting, between L1 (or L∞ ) and Lp , 1 < p < ∞. (ii) Theorem 2.1 applies automatically to holomorphic semigroups on metric spaces M having balls of polynomial growth [3]. This includes a fairly large amount of operators A to be found as sub-Laplacians on Lie groups or Laplace-Beltrami operators on more general manifolds. √ (iii) If M = Rn , n ≥ 2, the Poisson semigroup P z := e−z ∆ satisﬁes (HGα ) with α = (n − 1)/2 whence, by Theorem 2.1 again, we obtain that all n/2 fractional powers ∆θ , θ > 0 (so Laplacian ∆ in particular), are AC2,1 -bounded and AC (n+1)/2 -bounded on Lp (Rn ) when 1 < p < ∞. Notice that for n = 1 this just means the well-boundedness of the Laplacian on Lp (R). Incidentally, the argument considered in Theorem 2.1 does not work for n = 1 (there is no suitable gap between α and β(p) in this case) and so the well-boundedness of ∆ must be established speciﬁcally, as in [17] for example. According to Remark (i), a better, lower, degree of diﬀerentiability is admitted when dealing with ∆ (with any power 1

+ε

2 -bounded on every Lp (R) if 1 ≤ p ≤ ∞. ∆θ , indeed) on R: ∆θ is AC2,1 (iv) The order of diﬀerentiability in Theorem 2.1 has validity for all p in the range 1 < p < ∞. A natural question is to ask for the best index γ(p) for which A is AC γ(p) -bounded on Lp (M ). This problem involves ﬁnding out the best degree ν , so γ(p) can be smaller than β(p) + 1 for which the kernel Gν (u) is bounded on Lp√ in particular cases: When M = Rn and A = ∆, the boundedness of such a kernel is exactly the “Bochner-Riesz conjecture” problem [4].

Let now Hn be the Heisenberg group Cn × R endowed with the group law (z, t)·(z , t ) := (z +z , t+t − 12 (z ·z )). It is well kown that Hn is a (stratiﬁed) Lie group of dimension d = 2n + 1, with homogeneous dimension Q = 2n + 2. Let us ∂ ∂ ∂ , Xj := ∂x − 12 yj ∂u , Yj := consider the left invariant vector ﬁelds given by U := ∂u j n ∂ 1 ∂ 2 2 j=1 (Xj + Yj ) deﬁnes a sub∂yj + 2 xj ∂u , (j = 1, · · · , n). Then the sum L := Laplacian on Hn which is homogeneous of degree 2 with respect to the family of dilations δr : (z, u) → (rz, r2 u), r > 0. D. M¨ uller and E. M. Stein have proved that √ −it L α α/2 e f 1 ≤ Cα (1 + |t| )(1 + L) f 1 for every f ∈ L1 (Hn ), provided that α > (d − 1)/2, √ see [16], [15]. Let (pz )z>0 be the analytic semigroup in L1 (Hn ) generated by − L. Lemma 2.3. For every α > (d − 1)/2 there exists a constant Cα > 0 such that α |z| z p 1 ≤ Cα ,

z > 0.

z Proof. By homogeneity, it is enough to get the estimate in z = 1 + it, t ∈ R. In eﬀect, p1+it ∗ f 1

= e−(1+it)

√ L

√

f 1 = (e−it

α/2 1

≤ Cα (1 + |t| )(1 + L) α

√ L 1

L 1

p ) ∗ f 1 ≤ e−it

p 1 f 1 =

Cα (1

p 1 f 1

+ |t| )f 1 , α

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for every f ∈ L1 (Hn ). Since L1 (Hn ) has a bounded approximate identity, the required estimate follows automatically. According to the lemma, Theorem 2.1 applies on Hn along the same lines as for Rn . Thus we obtain the following. Corollary 2.4. If L is the above sub-Laplacian on Hn and θ > 0 then Lθ is AC d/2 bounded and AC (d+1)/2 -bounded on Lp (Hn ) when 1 0 then Lθ is AC d/2+ε -bounded on L1 (Hn ), L∞ (Hn ). Assume that p in (1, ∞) is ﬁxed. Finding the best β for which L is an AC β bounded operator on Lp (Hn ) is again linked to the Bochner-Riesz conjecture, this time on Hn .

3. Extension of C m -scalarity Recall the algebras Crm [0, ∞) of the ﬁrst section. Note that Cr1 [0, ∞) is the Banach algebra of all functions f in C 1 ∩ L∞ such that supu≥0 |f (u)|(u + 1)2 < ∞, when endowed with the norm f (u)(u + 1)2 ∞ + f ∞ . We will introduce now ν algebras Cr,ε [0, ∞) of the same type, but of fractional order of diﬀerentiablity, which generalizes the algebras Crm [0, ∞). Let ν > 0. Weyl fractional derivatives have sense for more functions than ν those belonging to the algebras AC2,1 . Indeed, for a locally integrable function h (ν) on R, an a. e. derivative h can be deﬁned which generalizes, whenever it exists, the derivative W ν h, see [2]. In the sequel, we will use the notation h(ν) instead of ν W ν h. Let Cr,ε [0, ∞) denote the Banach space of functions h in L∞ ∩ C ∞ [0, ∞) for which the derivative h(ν) exists and hν,ε := h∞ + h(ν) (u)(u + 1)ν+ε ∞ < ∞, where ε > 0. This space is a slightly diﬀerent version of spaces of so-called “weak bounded variation”, see [12], [2]. ∞ 1 Using the formula f (µ) (x) = Γ(ν−µ) (y − x)ν−µ−1 f (ν) (y) dy (x ∈ R+ ), see x [2], we get ∞ (t − x)ν−µ−1 f ν,ε (x + 1)µ+δ |f (µ) (x)| ≤ (x + 1)µ+δ f ν,ε dt ≤ C , ν+ε (t + 1) (x + 1)ε−δ x µ ν for x > 0. Hence, Cr,ε [0, ∞) → Cr,δ [0, ∞) if ν ≥ µ and ε ≥ δ. ν Theorem 3.1. For ν > 1/2 and ε > 0, Cr,ε [0, ∞) is a Banach algebra and it is ν continuously included in AC2,1 .

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ν Proof. Take h ∈ Cr,ε [0, ∞). Then the function g on [0, ∞) deﬁned by the integral ∞ ∞ 1 g(x) := Γ(ν) x (y −x)ν−1 h(ν) (y) dy, x ≥ 0, exists. Moreover, 0 h(ν) (x)xν−1 dx <

∞ and so, by [10, p. 318], we obtain that g ∈ AC ν with W ν g = h(ν) . It follows ν ν that g = h and therefore Cr,ε [0, ∞) ⊂ AC2,1 . Moreover, 1/2 ∞ 2y x2ν dx h(ν);2,1 ≤ dy hν,ε (x + 1)2ν+2ε x 0 y ∞ yν dy ≈ hν,ε = Cε hν,ε . (y + 1)ν+ε y 0

Finally, using [10, Lemma 2.2] in the Leibniz formula proven in [10, Proposition 2.5] we obtain that ν+ε (ν) (1 + x)ν+ε |(f g)(ν) (x)| ≤ |(1 + x)ν+ε f (ν) g (x)| (x)| |g(x)| + |f (x)| |(1 + x)

+ C(1 + x)ν+ε

∞

u

x

(u − x)ν−2 |f (ν) (t)| |g (ν) (u)|du dt,

x

for x ≥ 0. Take δ such that 1/2 < δ < min{1, ν}. Then (u − x)δ−1 ≤ (t − x)δ−1 if t ≤ u, and therefore the double integral is dominated by ∞ ∞ C(x + 1)ν+ε (t − x)δ−1 |f (ν) (t)|dt (u − x)ν−δ−1 |g (ν) (u)|du x

x

≤ C (x + 1)ν−δ+ε f˜(ν−δ) (x)(x + 1)δ g˜(δ) (x) ≤ C f˜ν−δ,ε ˜ gδ,ε ≤ C f˜ν,ε ˜ gν,ε = C f ν,ε gν,ε ˜ plus a similar term where the role of u and t is exchanged. (Here, we denote by h ∞ 1 ν−1 (ν) (ν) (ν) ˜ ˜ |h (y)| dy, for x ≥ 0, so that h = |h |.) the function h(x) := Γ(ν) x (y−x) We have used the inclusion prior to the theorem. Thus we obtain that f gν,ε ≤ Cf ν,ε gν,ε as it was required. Looking at functional calculi the above theorem suggests to introduce the following deﬁnition. Definition 3.2. A closed, densely deﬁned operator A on a Banach space X is called ν Cεν -scalar if it admits a functional calculus Cr,ε [0, ∞) → B(X). Note that if A is C 1 -scalar it means that A is C11 -scalar in our notation. Corollary 3.3. If A generates a holomorphic semigroup which satisfies property (HGα ), for some α ≥ 0, then A is Cεν -scalar for every ν > α + (1/2) and ε > 0. Proof. This is a consequence of Theorem 3.1 and [10, Theorem 6.3].

Applied to the usual Laplacian ∆ on R , Corollary 3.3 gives us the following. n

Theorem 3.4. For every θ > 0, ∆θ is Cεν -scalar on L1 (Rn ), provided that ν > n/2. √ Proof. It is suﬃcient to recall that ∆ generates the Poisson semigroup and that this semigroup satisﬁes (HGα ) with α = (n − 1)/2.

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In particular, for n = 1, this result is a non trivial improvement of the fact that ∆ is C 1 -scalar (see Section 1). The analog to Theorem 3.4 holds for the usual Laplacian operator on Hn , which has been considered formerly, whenever ν > d/2 where d = 2n + 1. n/2 According to Theorem 2.1 and Theorem 3.1, ∆θ is a Cε -scalar operator on Lp (Rn ) for 1 < p < ∞. Let us remark that it is possible to improve weights using multiplier theorems: Let W BV∞,α be the set of functions in L∞ ∩ C ∞ [0, ∞) such that h(α) exists a. e., and h(α) (u)uα ∞ < ∞. Then W BV∞,α is a Banach (α) space with norm h∞ + h(α) (u) uα ∞ , see [12], [2]. Let M∞ denote the closure (α) of W BV∞,α ∩ C ∞ in W BV∞,α . Then M∞ is a Banach algebra with pointwise (α) α [0, ∞) ⊂ M∞ for all multiplication, for all α ≥ 0 [11]. (It is readily seen that Cr,ε ε > 0.) Theorems on Mikhlin-type multipliers for suitable sub-Laplacian L and Lie (α) group G give rise to functional calculi M∞ → B(Lp (G)). This of course includes n the case of radial multipliers on R through ∆. (Note that we can also consider spaces W BV2,α , see [2], which are intrinsically related to H¨ormander multiplier conditions.) In this sense, it seems that the interest of scalarity of general order mainly concerns L1 or C0 spaces rather than Lp spaces with 1 < p < ∞.

References [1] H. Benzinger, E. Berkson and T.A. Gillespie, Spectral families of projections, semigroups and diﬀerential operators, Trans Amer. Math. Soc. 275 (1983), 431–475. [2] A. Carbery, G. Gasper and W. Trebels, On localized potential spaces, Trans Amer. Math. Soc. 48 (1986), 251–261. [3] E. B. Davies, Lp spectral independence and L1 analyticity, J. London Math. Soc., 52 (1995), 177–184. [4] K. M. Davies and Y.-C.Chang, Lectures on Bochner-Riesz means, Cambridge UP, Cambridge, 1987. [5] R. deLaubenfels, The operator id/dx, on C[0, 1], is C 1 -scalar, Proc. Amer. Math. Soc. 103 (1988), 215–221. [6] R. deLaubenfels, Functional calculus for generators of uniformly bounded holomorphic semigroups, Semigroup Forum 38 (1989), 91–103. [7] R. de Laubenfels, Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform, Studia Math. 103 (1992), 143–159. [8] H. R. Dowson, Spectral theory of linear operators, Academic Press, New York, 1978. [9] N. Dunford and J.T. Schwartz, Linear operators, Parts I, III, Interscience, New York, 1971. [10] J.E. Gal´e and T. Pytlik, Functional calculus for inﬁnitesimal generators of holomorphic semigroups, Journal of Func. Anal. 150 (1997), 307–355. [11] J.E. Gal´e and P. J. Miana, Mikhlin-Type theorems for quasimultipliers, to appear in Canad. J. Math. [12] G. Gasper and W. Trebels, A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math. 65 (1979), 243–278.

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[13] S. Kantorovitz, Spectral Theory of Banach Space Operators, Lecture Notes in Math. 1012, Springer-Verlag, Berlin, 1983. [14] E. Marschall, Functional calculi for closed linear operators in Banach spaces, Manuscripta Math. 35 (1981), 277–310. [15] D. M¨ uller, Functional calculus for Lie groups and wave propagation, Doc. Math., International Congress Math., Berlin 1998. [16] D. M¨ uller, E.M. Stein, Lp estimates for the wave equation on Hn , Revista Matem´ atica Iberoamericana 15 (1999), 293–334. [17] D. J. Ralph, Semigroups of well-bounded operators and multipliers, Thesis. University of Edinburgh, 1977. [18] W. Ricker, Spectral properties of the Laplace operator in Lp (R), Osaka J. Math. 25 (1988), 399–410. [19] W. Trebels, Some Fourier multiplier criteria and spherical Bochner-Riesz kernel, Rev. Roumaine Math. Pures Appl., 20 (1975), 1173–1185. Jos´e E. Gal´e and Pedro J. Miana Departamento de Matem´ aticas Universidad de Zaragoza E-50009 Zaragoza Spain e-mail: [email protected] [email protected] Detlef M¨ uller Mathematisches Seminar Christian-Albrechts-Universit¨ at D-24098 Kiel Germany e-mail: [email protected] Submitted: March 14, 2005 Revised: October 19, 2006

Integr. equ. oper. theory 57 (2007), 339–379 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030339-41, published online December 26, 2006 DOI 10.1007/s00020-006-1458-9

Integral Equations and Operator Theory

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators, Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory Kiril P. Kirchev and Galina S. Borisova Abstract. In this paper a triangular model of a class of unbounded nonselfadjoint K r -operators A presented as a coupling of dissipative and antidissipative operators in a Hilbert space with real absolutely continuous spectra and with diﬀerent domains of A and A∗ is considered. The asymptotic behaviour of the corresponding non-dissipative processes Tt f = eitA f , generated from the semigroups Tt with generators iA, as t → ±∞ are obtained. The strong wave operators, the scattering operator for the couple (A∗ , A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model. Mathematics Subject Classification (2000). Primary 47A48; Secondary 60G12. Keywords. Unbounded operator, Brodskii-Livˇsic operator colligation, characteristic function, coupling, non-dissipative process, wave operator, scattering operator.

1. Introduction This paper is dedicated to triangular models, characteristic functions and applications in the scattering theory for semigroups with generators presented as a coupling of dissipative and anti-dissipative unbounded operators with real spectra. Partially supported by Grant MM-1403/04 of MESC and by Scientific Research Grant 27/25.02.2005 of Shumen University.

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The study of non-selfadjoint operators is based on the methods of the characteristic functions and its development began with the works of M.S. Livˇsic [22] and his associates in 1950’s [8, 7]. Later on M.S. Livˇsic and A.A. Yantsevich in their book [25] proposed an interesting idea for an investigation of non-stationary (random) processes (or more generally continuous curves — following the terminology introduced in [15, 16] and used in [12]) with the help of the theory of the non-selfadjoint operators and their characteristic operator functions. This idea was expanded and developed in the works of L.A. Sakhnovich [31], K. Kirchev and V. Zolotarev [11, 15, 16], and others. This paper is a continuation of the works of the authors [5, 12, 13], where the triangular models of couplings of bounded dissipative and anti-dissipative operators with real absolutely continuous spectra have been considered with the help of the characteristic functions of M.S. Livˇsic and the channel representations of the imaginary part of the models, the asymptotics of the corresponding non-stationary processes (continuous curves) have been obtained and these results have been applied in the scattering theory. In this paper we use the triangular model A of a class of unbounded K r operators, introduced in [14] by the authors with the help of the characteristic function of A. Kuzhel (see [21]). This model describes the class of regular couplings of dissipative and anti-dissipative K r -operators with real spectra (i.e. closed operators in a Hilbert space whose Hermitian part has deﬁciency index (r, r) (0 < r < ∞) and a nonempty resolvent set). This model and the obtained properties by the authors in [14] are suitable for an investigation of the asymptotic behaviour of the corresponding non-stationary processes (continuous curves) and for an application in the scattering theory in the case when the domains of A and A∗ are diﬀerent. These problems for the model A and A∗ with diﬀerent domains are considered in this paper. The characteristic function of an unbounded operator was ﬁrst deﬁned by M.S. Livˇsic [22] under the condition that this operator is a non-selfadjoint extension of a symmetric (densely deﬁned) operator with deﬁciency index (1, 1). In the papers [17, 18, 19] the characteristic function was deﬁned for the quasiHermitian extensions of an Hermitian operator H (which is not necessarily densely deﬁned) with ﬁnite and equal defect numbers (the K r - operators). In the same papers the general properties of characteristic functions are studied, the triangular model for the K r -operators is constructed and the spectrum of these operators is investigated. In the survey by E. Tsekanovskii and Yu. Shmulyan [34] Tsekanovskii’s definition of the characteristic function is presented in terms of the rigged Hilbert spaces. This deﬁnition coincides formally with the deﬁnition of the characteristic function in the case of bounded operators (see (1.2) and (1.4) below). These ideas were developed in the connection with the theory of conservative linear systems, realization and factorization problems of J-contractive operator-valued functions, triangular models as well as the problems of conservative systems coupling [2, 3, 33], etc.

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341

Many realizations in diﬀerent ﬁelds including system theory, scattering theory, electrical engineering involve unbounded main operators. For applications in investigations of random processes (or more generally continuous curves) and scattering theory with corresponding unbounded generators (for example, asymptotics, wave operators, a scattering operator) is suitable to apply the characteristic function and the model of A. Kuzhel. In our considerations we use the generalized deﬁnition of the characteristic function of A. Kuzhel [20] (see also [21] and (1.6) below) to the case where the operator A is closed and the condition J ∗ = J −1 is not necessarily satisﬁed. A. Kuzhel has introduced the characteristic function of the K r -operators (see [21]) with the help of the fundamental Theorem of Potapov [30] which is analogue of the Riesz-Herglotz (Riesz-Nevanlinna) formula in the case of J-non-expanding matrix functions. For a bounded linear operator A in a Hilbert space H with an imaginary part, presented as Im A = Φ∗ JΦ where Φ ∈ B(H, E) (i.e. Φ is a linear bounded operator from H to E) and J ∈ B(E) is self-adjoint and unitary, the array A Φ∗ J (1.1) X= H E deﬁnes a Brodskii-Livˇsic operator colligation and a function WX (λ) = I − 2iΦ(A − λI)−1 Φ∗ J

(1.2)

is the characteristic (transfer) function of X. In the system theory WX (λ) is interpreted as a transfer function of the conservative system of the form (A − λI)x = Φ∗ Jϕ− and ϕ+ = ϕ− − 2iΦx, where ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x is a state space vector on H, so that ϕ+ = WX (λ)ϕ− . The system is said to be the minimal if the main operator A of X is completely nonselfadjoint (i.e. there are no nontrivial invariant subspaces on which A induces self-adjoint operators) ([7, 23, 8, 24]). The characterization of Herglotz-Nevanlinna functions is to identify them as (linear fractional transformation of ) transfer function VX (λ) = i(WX (λ) + I)−1 (WX (λ) − I)J where J = J ∗ = J −1 and W (λ) is the transfer function (of some generalized linear stationary conservative dynamical system). For a non-hermitian operator T in a Hilbert space H a linear stationary conservative dynamical system X of the form A Φ∗ J X= (1.3) H+ ⊂ H ⊂ H− E (where A ∈ B(H+ , H− ), H+ ⊂ H ⊂ H− is a rigged Hilbert space, A ⊃ T ⊃ A, A∗ ⊃ T ∗ ⊃ A, A is a Hermitian operator in H, T is a non-hermitian operator in H, Φ∗ ∈ B(E, H− ), J = J ∗ = J −1 , and Im A = Φ∗ JΦ) is said to be a BrodskiiLivˇsic rigged operator colligation. The transfer function WX (λ) of X in (1.3) and

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its linear fractional transformation VX (λ) are given by WX (λ) = I − 2iΦ(A − λI)−1 Φ∗ J, −1

VX (λ) = Φ(AR − λI)

Φ

∗

(1.4) (1.5)

where AR is the real part of A (see, for example, [2]). The signiﬁcance of rigged Hilbert spaces in system theory and natural appearing of systems (1.1), (1.3) in electrical engineering and the scattering theory can be seen in [23]. The characteristic function WA (λ) of A. Kuzhel of K r -operator A is deﬁned by the equality WA (λ)L−1 WA∗ (α) = L−1 − i(λ + α)ΦTαλ Φ∗

(1.6)

where Tαλ = (A − αI)(A − λI)−1 , L∗ = L, det L = 0, the condition L∗ = L−1 is not necessarily satisﬁed, α is a ﬁxed point from the resolvent set of A and the auxiliary operator ∗ ∗ + 2Im αRα Rα Bα = iRα − iRα

(Rλ = (A − λI)−1 ) has the representation Bα = Φ∗ LΦ onto the Hilbert space H. In the case of a bounded non-selfadjoint operator A in [8, 31, 6] the application of the linear transformation VX (λ) of the transfer function WX (λ) is considered. The function VX (λ) is applied in the scattering theory for obtaining of the asymptotics of the process eitAγ f (where Aγ = γA+γA∗ ) for operators A with ﬁnite dimensional imaginary part and γ = 1/2 in [8], for dissipative operators A with trace class imaginary part and Re γ = 1/2 in [31] and for operators Aγ with Re γ = 1/2, where A is a coupling of dissipative and anti-dissipative operators with purely real absolutely continuous spectra in [6] and for constructing of the scattering theory for the couples (Aγ , A) and (A, Aγ ). For two given Brodskii-Livˇsic operator colligations Ak Φ∗k J Xk = , k = 1, 2, (1.7) Hk E with the same external part (E, J) their coupling is deﬁned A Φ∗ J X = X1 ∨ X2 = H E where H = H1 ⊕ H2 and operators A1 0 , iΦ∗1 JΦ2 A2

Φ = (Φ1 Φ2 )

(1.8)

being written in the block form with respect to the direct sum decomposition H = H1 ⊕ H2 . The system-theoretic signiﬁcance of the colligation allows to consider the coupling of colligations (connected with the multiplication theorem of the corresponding transfer functions) as a correspondence to the cascade connection of systems (see [23]).

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In this connection it is worthwhile to mention the natural consideration of the broad class of operators, presented as a coupling of dissipative and anti-dissipative operators. Really, let us consider (for example) the Brodskii-Livˇsic operator colligation A Φ∗ L X= L2 (0, l; Cn ) Cm where the operator x Af (x) = α(x)f (x) + i

f (ξ)Π(ξ)LΠ∗ (x)dξ

(1.9)

0

is the triangular model of the class of all bounded non-selfadjoint operators with a ﬁnite dimensional imaginary part and purely real spectrum, introduced by M.S. Livˇsic (where L : Cm −→ Cm , L∗ = L, det L = 0, α(x) is a bounded nondecreasing function on a ﬁnite interval [0, l] and Π(x) is a measurable n × m (1 ≤ n ≤ m) matrix function on [0, l], whose rows are linearly independent at each point of a set of a positive measure, f belongs to the Hilbert space H = L2 (0, l; Cn )). We can suppose without loss of generality that L has the representation L = J1 − J2 , where I 0 0 0 J1 = r1 , J2 = , 0 Im−r1 0 0 r1 and m − r1 are the numbers of the positive and negative eigenvalues of L correspondingly. Then the imaginary part of (1.9) takes the form 1 A − A∗ = 2i 2

l 0

1 f (ξ)Π(ξ)J1 Π (x)dξ − 2 ∗

l

f (ξ)Π(ξ)J2 Π∗ (x)dξ.

(1.10)

0

The representation (1.10) implies that A always can be considered as a sum of dissipative and anti-dissipative operators. But this sum is not always a coupling of dissipative and anti-dissipative operators. In other words the coupling of two operators implies that the Hilbert space H is presented as H1 ⊕H2 and one of these subspaces is invariant by the operator A. The coupling is connected also with the multiplication theorem of the characteristic matrix functions of the projections of A onto the subspaces H1 and H2 and connected with cascade connection of systems. The representation (1.9) is not suitable for the investigation of the asymptotic behaviour of eitA f (f ∈ H) in the general case of the considered model A. Let now A with imaginary part (1.10) be a coupling of a dissipative operator and an anti-dissipative one, i.e. A has the representation A = P1 AP1 + P2 AP2 + P1 AP2 , where P1 , P2 are orthogonal projectors in H, P1 A is dissipative onto P1 H and P2 A is anti-dissipative onto P2 H. Then from (1.7), (1.8) with J = L and E = Cm it

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follows that (P1 AP1 − P1 A∗ P1 )/i = Φ∗1 LΦ1 ≥ 0, (P2 AP2 − P2 A∗ P2 )/i = Φ∗2 LΦ2 ≤ 0, P1 AP2 = iΦ∗1 LΦ2 . Using the Generalized inertia law (see, for example, [8], I §2 Lemma 5) it follows that Φ∗1 LΦ1 , Φ∗2 LΦ2 can be presented in the form 1, ∗ Q Ir1 0 R LR∗ Ir1 0 Q∗ Φ (1.11) Φ∗1 LΦ1 = Φ 1 0 0 0 0 0 0 ∗ ∗ 0 ∗ 0 2 Φ2 LΦ2 = Φ2 Q (1.12) R LR Q∗ Φ 0 Im−r1 0 Im−r1 and consequently 0 0 ∗ ∗ Ir1 ∗ 0 2, P1 AP2 = iΦ1 LΦ2 = iΦ1 Q (1.13) R LR Q∗ Φ 0 0 0 Im−r1 where Q , Q , R , R are invertible matrices. If we present R and R in the block form R1 R2 R1 R2 R = , R = R3 R4 R3 R4 R1 R2 then direct calculations show that for the matrix R = the equalities R3 R4 (1.11), (1.12), (1.13) take the form R ∗ J1 Q∗ Φ 1, ∗ Q J1 RL Φ∗1 LΦ1 = Φ 1 ∗ ∗ ∗ ∗ Φ2 LΦ2 = Φ2 Q J2 RLR J2 Q Φ2 , R ∗ J2 Q∗ Φ 2, ∗1 Q J1 RL iΦ∗1 LΦ2 = iΦ ∗ = RL R ∗ = J1 S L and J1 ≥ 0, J2 ≤ 0. S J2 This implies that in the case of a coupling it is more suitableto consider the Ir1 0 matrix from the form L instead of the matrix in the form L = . 0 Im−r1 Now without loss of generality we can assume that L has the form Ir1 S∗ ∗ L = J1 − J2 + S + S = S −Im−r1 where r1 , m − r1 are correspondingly the number of the positive and negative eigenvalues of L. In the case of unbounded operators analogous reasons lead to the consideration of the model of a coupling of dissipative and anti-dissipative operators. This paper is dedicated to the model A, describing the class of K r -operators presented as a coupling of dissipative and anti-dissipative operators with real absolutely continuous spectra and with diﬀerent domains of A and A∗ . This model generates semigroups of operators {Tt }t≤0 and {Tt }t≥0 from the class (C0 ) with

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generators iA. The explicit obtaining of the asymptotics of the corresponding nondissipative processes Tt f as t → ±∞ allows to construct the scattering theory (as in the bounded case of the model A in [12]) for the couple (A∗ , A): in other words to obtain the wave operators W± (A∗ , A), the scattering operator and the similarity of A and the operator Q of multiplication by the independent variable. All results are obtained explicitly using the multiplicative integrals and matrix generalization of the classical gamma-function. In part 2 the preliminary results, concerning the triangular model A of regular couplings of dissipative and anti-dissipative K r -operators with real spectra, are reminded. These results are obtained by the authors in [14]. The considered model A is describing using the model of a coupling of A. Kuzhel and characteristic matrix function of A. Kuzhel. The explicit representation of resolvent of A, the characteristic function of A and the conditions under which DA = DA∗ are presented. In part 3 we introduce appropriately families of operators {Tt }t≥0 , {Tt }t≤0 with the generator iA with the help of the resolvent of A and the integral in the sense of a principal value. The proved properties of these families imply that {Tt }t≥0 and {Tt }t≤0 are semigroups from the class (C0 ) (only the boundedness of Tt is proved in part 4). In part 4 the asymptotics of the corresponding non-dissipative processes Tt f (f ∈ L2 (R, Cn )) as t → ±∞ are obtained. In the course of proving of these asymptotics we have used the properties of the multiplicative integrals and the suitable inequalities, concerning the multiplicative integrals describing the characteristic function of the considered model model A. These inequalities are presented by the authors in [13] and they have been obtained with the help of the existence and the form of the limit values of the multiplicative integrals from the form (4.1). The explicit expression of the asymptotics of Tt f in terms of the multiplicative integrals and matrix generalization of gamma-function allows to prove the uniformly boundedness of the families {Tt }t≥0 , {Tt }t≤0 . This ﬁnishes the description of these families as semigroups of operators from the class (C0 ) and show that {Tt }t∈R is a group of operators with generator iA which determine the exponential function eitA = Tt . In part 5 the explicit representation and the existence as a strong limits of the wave operators ∗

W± (A∗ , A) = s − lim eitA e−itA t→±∞

are obtained with the help of the asymptotics. We obtain also the similarity of the simple part of the considered model A and the operator Q of multiplying by the independent variable, we obtain the explicit form of B −1 and B where A = BQB −1, and the scattering operator for the couple (A∗ , A). In part 6 we consider an example of diﬀerential operator presented as a coupling of dissipative and anti-dissipative Schr¨odinger operators (K 1 -operators) with diﬀerent domains of the operator and its adjoint. This example shows that

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the considered class of unbounded K r -operators in this paper is too essential for the diﬀerential operators. Finally it has to mention that the obtained asymptotics of the processes, generated from the considered model A can be applied for the constructing of the ∗ scattering theory for the couples (A1/2 , A) and (A, A1/2 ), where A1/2 = A+A (i.e. 2 the real part of A) for example in the case of DA = DA∗ , using the ideas in [6] ∗+ (λ) = W (λ)W −1 (−i) − iΦ(A for γ = 1/2. On the one hand if we denote W −1 ∗ iI) Φ L, where W (λ) is the characteristic function of A (deﬁned by (1.6) for = Φ(A0 − iI) (where Φ is from the representation of Bi = Φ∗ LΦ) α = i), Φ 1/2 − λI)−1 Φ ∗ from (1.5) takes the form V (λ) = then the function V (λ) = 12 Φ(A −1 −1 (λ)) (I − W (λ))L . This implies that V (λ) has the form of the linear −2i(I + W (λ). The connection between W (λ), V (λ) and W (λ) fractional transformation of W allows to construct the scattering theory for the couples (A1/2 , A) and (A, A1/2 ) as in [6]. Other applications in the scattering theory for non-selfadjoint operators can be seen in the work of S.N. Naboko [28], where the functional models are used, in the work of M.Sh. Birman and D.R. Yafaev [4], and others. It turns out that the considered class of operators in this paper (a coupling of dissipative and anti-dissipative operators with real spectra) possesses properties and asymptotics close to the case of the dissipative operators.

2. A model of a coupling — preliminary results In this paper we consider a triangular model A of a class of unbounded operators using the characteristic function of A. Kuzhel ([21]). This model is introduced by the authors in [14] and it describes the class of regular couplings of dissipative and anti-dissipative K r -operators with real spectra. (i.e. closed operators in a Hilbert space whose Hermitian part has deﬁciency index (r, r) (0 < r < ∞) and a nonempty resolvent set). Let the Hilbert space H be H = H1 ⊕ H2 and A1 and A2 be operators acting in the spaces H1 and H2 respectively. In H we consider an invertible operator G = I + KP2 (K ∈ B(H2 , H1 )) with G−1 = I − KP2 , where P2 is the orthogonal projector in H onto H2 . The operator A deﬁned by the equality A = A1 P1 G−1 + A2 P2 + αKP2

(DA = G(D1 ⊕ D2 )),

(2.1)

where α is a ﬁxed point from the resolvent set ρ(A) of A, is called a coupling of operators A1 and A2 (see, for example, [21]). The coupling A is called a regular coupling if the auxiliary operator ∗ ∗ Bα = iRα − iRα + 2Im αRα Rα

(Rλ = (A − λI)−1 ) has the representation Bα = Φ∗1 LΦ1 P1 + Φ∗2 LΦ2 P2 + iT1α ΨP2 − iΨ∗ T1α P1 + 2Im αΨ∗ ΨP2 ,

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where L is m × m (m ≥ r) matrix with L∗ = L, det L = 0 (L−1 = L∗ is not necessarily satisﬁed) and Ψ is deﬁned by the equality ∗ A1 (α)Φ2 . Φ∗1 LW Ψ = −iT1α

A1 (λ) in the last equality is deﬁned by the relation The matrix function W ∗ (α) = L−1 + i(α − λ)Φ1 T1αλ Φ∗ A1 (λ)L−1 W W A1 1 (T1α = (A − αI)R1α , T1αλ = (A1 − αI)R1λ , R1λ = (A1 − λI)−1 ). In this case Bα = Φ∗ LΦ where

A1 (α)Φ2 P2 Φ = Φ1 P1 + W

(2.2) (see, for example, [21]). Let now α(x) be a nondecreasing unbounded real function, deﬁned in (a, b) (−∞ ≤ a < b ≤ +∞). Let Π(x) be a measurable n × m (1 ≤ n ≤ m, r ≤ m) matrix function whose rows are linearly independent on each point of a set with a positive measure and satisfying the conditions b

b tr B(x)dx < +∞,

a

||Π(x)||2 dx < +∞,

(2.3)

a

where B(x) = Π∗ (x)Π(x). Let L : Cm −→ Cm , L∗ = L, det L = 0. According to the remarks in part 1 without loss of generality we can suppose that L has the representation L = J1 − J2 + S + S ∗ , ∗

where J1 , J2 , S, S : C −→ C , Ir1 0 0 J1 = , J2 = 0 0 0 m

m

0 Im−r1

(2.4)

, S=

0 0 S 0

,

(2.5)

Ik is the identity matrix in Ck (k = r1 , m − r1 ), S is a (m − r1 ) × r1 matrix, r1 is the number of the positive eigenvalues and m − r1 is the number of the negative eigenvalues of the matrix L. Let the matrix function B(x) satisfy also the conditions B(x)J1 = J1 B(x) and α(x)B(x)J2 be an integrable matrix function on (a, b) (−∞ ≤ a < b ≤ +∞). Let us consider a Hilbert space L2 (a, b; Cn ), whose elements are vector functions f (x) from the form f (x) = (f1 (x), f2 (x), . . . , fn (x)),

(fk ∈ L2 (a, b)).

The scalar product in L2 (a, b; Cn ) is deﬁned by the formula b (f, g) = a

f (x)g ∗ (x)dx.

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In this paper we will denote by || || the norm of a matrix function in Cn and by || ||L2 — the norm in L2 (a, b; Cn ). Let Q(x) be a measurable matrix function on (a, b) satisfying the condition Π(x)Q(x) = I

(2.6)

for almost all x ∈ (a, b). Then the operators P1 and P2 , deﬁned by the equalities P1 f (x) = f (x)Π(x)J1 Q(x), 2

P2 f (x) = f (x)Π(x)J2 Q(x)

onto L (a, b; C ), are orthogonal projectors in L2 (a, b; Cn ). We consider the model A, introduced in [14], by the equality n

Af (x) = AGg(x) = α(x)g(x) x

x

→ +i g(ξ)(α(ξ) + i)Π(ξ)J1 eiα(v)B1 (v)dv J1 Π∗ (x)(α(x) − i)dξ a

ξ

x

→

x

−i g(ξ)(α(ξ) + i)Π(ξ)J2 e−iα(v)B2 (v)dv J2 Π∗ (x)(α(x) − i)dξ a

(2.7)

ξ b

b

x

→

→ + g(ξ)(α(ξ) + i)Π(ξ)J2 e−iα(v)B2 (v)dv dξS eB1 (v)dv J1 Π∗ (x), a

a

ξ 2

where G is an invertible operator in L (R; C ) and n

G = I + P1 KP2 ,

(2.8)

KP2 g(x) b

x

b

→

→ = −i g(ξ)(α(ξ) + i)Π(ξ)J2 e−iα(v)B2 (v)dv dξS eB1 (v)dv J1 Π∗ (x), a

(2.9)

a

ξ

B1 (x) = B(x)J1 , B2 (x) = B(x)J2 , for each f (x) = Gg(x) ∈ L2 (R; Cn ) such that Af ∈ L2 (R; Cn ). Using the form of the projectors P1 , P2 the model A, deﬁned by (2.7), takes the the form Af (x) = AGg(x) = P1 AP1 g(x) + P2 AP2 g(x) + iKP2 g(x),

(2.10)

where K is deﬁned by (2.9) and P1 AP1 g(x) = α(x)g(x)Π(x)J1 Q(x) x

x

→ iα(v)B (v)dv 1 J1 Π∗ (x)(α(x) − i)dξ, +i g(ξ)(α(ξ) + i)Π(ξ)J1 e a

ξ

P2 AP2 g(x) = α(x)g(x)Π(x)J2 Q(x) x

x

→ −i g(ξ)(α(ξ) + i)Π(ξ)J2 e−iα(v)B2 (v)dv J2 Π∗ (x)(α(x) − i)dξ, a

ξ

P1 AP2 g(x) =

b

b

x

→

→ g(ξ)(α(ξ) + i)Π(ξ)J2 e−iα(v)B2 (v)dv dξS eB1 (v)dv J1 Π∗ (x),

a

P2 AP1 g(x) = 0,

ξ

a

(2.11)

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and DA = G(DA1 ⊕ DA2 ) ⊂ L2 (R; Cn ). The representation (2.11) and straightforward calculations show that A1 = P1 A is a dissipative operator onto P1 G−1 DA and A2 = P2 A is an anti-dissipative operator onto P2 G−1 DA = P2 DA = DA2 . In other words Im (A1 P1 G−1 f (x), P1 G−1 f (x)) ≥ 0, (f ∈ DA ), Im (A2 P2 G−1 f (x), P2 G−1 f (x)) ≤ 0, (f ∈ DA ). The representation (2.10) implies that A is a regular coupling of a dissipative operator and an anti-dissipative one, i.e. A = A1 P1 G−1 + A2 P2 + iKP2 and A = A1 ∨ A2 . The model A, deﬁned by (2.7), is a closed densely deﬁned operator as a coupling of a dissipative operator and an anti-dissipative one with real spectra. The consideration of the case α = i in the model (2.1) is non-essentially diﬀerent from the case of an arbitrary regular point α of A. We present this case to avoid complications of writing. It has to mention that in the case when the function α(x) in (2.7) (α : (a, b) −→ R) satisﬁes the conditions 1) α(x) is strictly increasing unbounded continuous function; 2) the inverse function σ(u) of α(x) is absolutely continuous on R; 3) σ (u) is bounded on R; the model A, deﬁned by (2.12), can be written in the form Af(u) = AG g (u) = u g(u) u u

→ ivB(v)J 1 dv ∗ (u)(u − i)dη +i g(η)(η + i)Π(η)J1 e J1 Π −∞

−i +

u −∞

+∞

−∞

η

u

→ −ivB(v)J 2 dv ∗ (u)(u − i)dη g(η)(η + i)Π(η)J J2 Π 2 e η +∞

g(η)(η + i)Π(η)J 2

→ η

e

u

−iv B(v)J 2 dv

dηS

→

−∞

∗ (u) eB(v)J1 dv J1 Π

(where g ∈ L2 (R; Cn , σ(u)), ||Π(u)|| ∈ L2 (R; σ(u)) when the function g(σ(u)) ∈ 2 n 2 L (R; C ), ||Π(σ(u))|| ∈ L (R)).

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In order to avoid complications of writing we can consider without loss of generality the model Af (x) = AGg(x) x

x

→ ivB (v)dv = xg(x) + i g(w)(w + i)Π(w)J1 e 1 J1 Π∗ (x)(x − i)dw −∞

−i +

x −∞

+∞

−∞

w

x

→ g(w)(w + i)Π(w)J2 e−ivB2 (v)dv J2 Π∗ (x)(x − i)dw

(2.12)

w +∞

g(w)(w + i)Π(w)J2

→

x

e

−ivB2 (v)dv

dwS

→

−∞

w

eB1 (v)dv J1 Π∗ (x)

(i.e. α(x) = x, a = −∞, b = +∞) with a domain DA = G(DA1 ⊕ DA2 ) (where G is deﬁned by (2.8)) which is dense in L2 (R; Cn ). It has to mention also that analogously it can be considered the general case of a increasing unbounded function α(x) and the case when α(x) maps (a, b) onto (−∞, b] or [a, +∞). We suppose also that the model (2.12) is simple (i.e. there is no non-trivial reducing subspaces where A generates a self-adjoint operator). If A is not simple then all considerations in this paper are valid for the simple part As of A, where As = A|DA ∩HA , HA is the closure of the linear span of Rik Φ∗ ep , k = 0, 1, . . ., m p = 1, 2, . . . , m ({ep }m 1 is an orthonormal basis in C ), Φ is deﬁned by (2.2) (see, for example, [21]). In [14] (see Theorem 1.1) we have obtained that for each λ : Im λ = 0 the operator A − λI is invertible and the explicit form of the resolvent. In the considered case α(x) = x when λ = i the resolvent (A − λI)−1 takes the form (x) (A − λI)−1 f (x) = fx−λ x

x ξ+i

→ −i 1+λv B1 (v)dv ξ−i v−λ −i dξJ1 Π∗ (x) ξ−λ ξ−λ f (ξ)Π(ξ)J1 e −∞

+i

x −∞

i − λ−i x

×

→

−∞

ξ

x

→ i 1+λv B2 (v)dv ξ−i ξ+i v−λ dξJ2 Π∗ (x) ξ−λ ξ−λ f (ξ)Π(ξ)J2 e ξ

+∞

−∞

(2.13)

+∞

ξ+i ξ−λ f (ξ)Π(ξ)J2

→

1+λv

ei v−λ B2 (v)dv dξS

ξ

1+λv

x−i e−i v−λ B1 (v)dv J1 Π∗ (x) x−λ

for each λ : Im λ = 0, for each f ∈ L2 (R; Cn ) and the resolvent is a bounded operator in L2 (R; Cn ). The auxiliary self-adjoint operator Bi = iRi − iRi∗ + 2Ri∗ Ri −1

2

(2.14)

(Ri = (A−iI) ) maps the space L (R; C ) in the defect space Ni of the Hermitian part A0 of the operator A (dim Ni = r ≤ m). Then using the form of the resolvent n

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Ri , obtained after calculations, we obtain the next representation for Bi Bi f = Φ∗ LΦf

∀f ∈ L2 (R; Cn ),

(2.15)

where Φ = Φ1 + ω ∗ Φ2 , Φk f =

+∞

−∞

(2.16)

+∞

f (ξ) ξ+i ξ−i Π(ξ)Jk

→

eB(v)Jk LJk dv dξ,

∀f ∈ L2 (R; Cn )

(2.17)

ξ

+∞

∗

Φ h=h

← x

m eB(v)Jk LJk Jk Π∗ (x) x+i x−i ∀h ∈ C ,

(2.18)

∗ (i), where W A1 (λ) is deﬁned by the operator equality ω=W A1 A1 (λ)L−1 W ∗ (−i) = L−1 + i(i − λ)Φ1 T T T ∗ Φ∗ , W A1 1i 1iλ 1i 1

(2.19)

and T1i = (A1 − iI)(A1 + iI)−1 . Direct calculations show that the matrix +∞

A1 (λ) = I − L + L W

→

−∞

1+λv

e−i v−λ B1 (v)dv J1

(2.20)

is a solution of the equation (2.19) and consequently +∞

∗ (i) = I − L + ω=W A1

←

eB1 (v)dv L.

(2.21)

−∞

The representation (2.15) of the auxiliary self-adjoint operator Bi and (2.16) imply that the model A, deﬁned by (2.12), is a regular coupling of the dissipative operator A1 and the anti-dissipative operator A2 . The model A is a closed densely deﬁned operator as a coupling of dissipative and anti-dissipative operators with real spectra. The regular coupling A = A1 ∨ A2 , deﬁned by (2.12), has the characteristic matrix function +∞

WA (λ) = (I + L − L

→

−∞

+∞ 1+λv

ei v−λ B2 (v)dv )(I − L + L

→

1+λv

e−i v−λ B1 (v)dv ).

−∞

(see Theorem 2.1 [14]) when Φ1 L2 (R; Cn ) is the external space of A. If the matrix √ function S from the representation (2.4) of L satisﬁes the condition ||S|| < ( 5 − 1)/4, then the domains DA and DA∗ of the model A and A∗ coincide if and only if +∞

−∞

(see Theorem 3.1 [14]).

tr (x2 + 1)B(x)dx < +∞.

(2.22)

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Under the condition (2.22) (i.e. DA = DA∗ ) the considered model A, deﬁned with the help of the characteristic function of A. Kuzhel, takes the form

x ∗ g(ξ)Π(ξ)J Ag(x) = xg(x) + i 1 L1 Π (x)dξ −∞ (2.23) +∞

x

∗ ∗ +i g(ξ)Π(ξ)J2 LJ2 Π (x)dξ + i g(ξ)Π(ξ)J2 LJ1 Π (x)dξ, −∞

−∞

where ξ

Π(ξ) = (ξ + i)Π(ξ)(

←

−∞

ξ

e−ivB1 (v)dv J1 +

← −∞

eivB2 (v)dv J2 ),

+∞

S = J2

←

−∞

= J1 − J2 + S + S∗ . eivB2 (v)dv SJ1 , L

The representation (2.23) implies that the model A, deﬁned by (2.12) as a coupling of a dissipative operator and an anti-dissipative one with real spectra, using the characteristic function of A. Kuzhel in the case of DA = DA∗ coincides with the model of a coupling of a dissipative operator and an anti-dissipative one with real spectra, using the characteristic function of M.S. Livˇsic. The asymptotic behaviour of the corresponding processes (continuous curves) is similar to the case of bounded non-selfadjoint operators [5, 12] and is presented by the authors in [13], using the characteristic function of M.S. Livˇsic. In this paper we will consider the case when DA = DA∗ and we will show that the model A, deﬁned by (2.12) generates non-dissipative processes (or nondissipative continuous curves) whose asymptotics can be obtained with the help of the properties of the multiplicative integrals from the form x

x

→ −i 1+λv B1 (v)dv e v−λ ,

→ i 1+λv B2 (v)dv e v−λ .

ξ

ξ

(2.24)

It has to mention that the case of dissipative K r -operators A with diﬀerent domains of A and A∗ is considered by the authors in [13].

3. The families of operators {Tt }t≥0 and {Tt }t≤0 In this part we will introduce families of operators {Tt }t≥0 and {Tt }t≤0 which posses properties as a semigroup of operators with generator iA, where A is deﬁned by (2.12) and the domains of A and A∗ are diﬀerent, i.e. DA = DA∗ . Let Π(x), B(x), Q(x) be stated as above in part 2 and satisfy the conditions: 1)

+∞

−∞

||Π(x)||2 dx < +∞,

+∞

−∞

tr B(x)dx < +∞;

2) xB(x) is an integrable matrix function on R. 3) B(x)J1 = J1 B(x).

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Let the model A be deﬁned by the equality (2.12) onto DA = G(DA1 ⊕ DA2 ), where G is deﬁned by (2.8). Let√us consider the case when tr (x2 + 1)B(x) is not integrable on R and ||S|| < ( 5 − 1)/4. This implies that DA = DA∗ . Let us denote D1 = {f ∈ L2 (R; Cn ) : xf (x) ∈ L2 (R; Cn )}. Let the next additional condition for the matrix function B(x) = Π∗ (x)Π(x) hold: 4) ||(x2 + 1)B(x)|| ≤ M for all x ∈ R (M > 0 is a constant). Before continuing with the consideration of the main problem of this part it has to mention that the resolvent (2.13) of the model A can be presented in the form −i

+∞

−∞

f (x) x−λ −

(A − λI)−1 f (x) = w+i w−λ f (w)Π(w)Y

x−i (w, λ, x)dwΠ∗ (x) x−λ ,

(3.1)

where Y (w, λ, x)w w

← i 1+λv B1 (v)dv

← −i 1+λv B2 (v)dv J1 − J2 e v−λ J2 χ(−∞,x] (w) = J1 e v−λ −∞ −∞ +∞ x x

→ i 1+λv B2 (v)dv

→ −i 1+λv B1 (v)dv

→ i 1+λv B2 (v)dv 1 + λ−i J2 e v−λ SJ1  J1 e v−λ + J2 e v−λ . −∞

w

−∞

From the properties of the multiplicative integrals from the form (2.24) in the right hand side of the last equality and from the conditions that B(x) satisﬁed it follows that ||Y (w, ξ ± iδ, x)|| ≤ M (3.2) > 0 is a constant). (∀w, x ∈ R, ∀ξ ∈ R, ∀δ > 0 and M Let us consider the next families of operators, deﬁned by the equality 1 Tt f (x) = − 2πi 1 + 2πi

+∞

−∞

e

+∞

−∞

it(ξ+iδ)

eit(ξ−iδ) (A − (ξ − iδ)I)−1 f (x)dξ (3.3) −1

(A − (ξ + iδ)I)

f (x)dξ

in the sense of a principal value where f = (A − λ0 I)−1 g for all g ∈ D1 , λ0 is an arbitrary ﬁxed number with Im λ0 > 0, δ is an arbitrary number with 0 < δ < Im λ0 when t > 0 and Im λ0 < 0, 0 < δ < −Im λ0 when t < 0. The next two theorems describe the properties of the families {Tt }t<0 and {Tt }t>0 .

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Theorem 3.1. The operators Tt , deﬁned by (3.3), satisfy the conditions: 1) the integral in (3.3) exists in the sense of a principal value for each f = (A−λ0 I)−1 g, where g ∈ D1 (Im λ0 > δ > 0 when t > 0 and −Im λ0 > δ > 0 when t < 0); 2) Tt f (x) ∈ L2 (R; Cn ) for all f = (A − λ0 I)−1 g, g ∈ D1 ; 3) Tt f (x) does not depend on the choice of the suﬃciently small number δ > 0 for f = (A − λ0 I)−1 g, g ∈ D1 . Proof. Let us consider at ﬁrst the case when t > 0. Let λ0 be a ﬁxed number with Im λ0 > 0 and δ is an arbitrary number such that 0 < δ < Im λ0 . Let g ∈ D1 and f = (A − λ0 I)−1 g. Then after calculations using the Resolvent equation Tt f (x), deﬁned by (3.3), takes the form: 1 Tt f (x) = − 2πi 1 + 2πi 1 − 2πi

+∞

−∞ +∞

−∞

+∞

−∞

eit(ξ−iδ) ξ−iδ−λ0 (A

eit(ξ−iδ) ξ−iδ−λ0 dξf (x)

+

1 2πi

− (ξ − iδ)I)−1 g(x)dξ +∞

−∞

eit(ξ+iδ) ξ+iδ−λ0 (A

− (ξ + iδ)I)−1 g(x)dξ

(3.4)

eit(ξ+iδ) ξ+iδ−λ0 dξf (x).

For the calculation of the last three integrals (in the sense of a principal value) on the right hand side of (3.4) we use the Residue theorem for suitable domains in C. The ﬁrst integral on the right hand side of (3.4) exists, which follows from the representation (3.1) and the inequality (3.2). This implies that (3.4) takes the form +∞ eit(ξ−iδ) 1 (A − (ξ − iδ)I)−1 g(x)dξ + eitλ0 f (x), (3.5) Tt f (x) = − 2πi ξ − iδ − λ0 −∞

i.e. Tt f (x) is deﬁned for all f = (A − λ0 I)−1 g, g ∈ D1 in the sense of a principal value. The condition 2) follows from the representation (3.5), (3.1) and (3.2). Finally, using that when t > 0 Tt f (x), deﬁned by (2.23), takes the form 1 Tt f (x) = − 2πi

+∞ eit(ξ−iδ) (A − (ξ − iδ)I)−1 f (x)dξ −∞

and applying the Residue theorem for the function eitz (A−zI)−1 and for a suitable domain in the lower half plane and (3.1), (3.2) we obtain +∞ +∞ it(ξ−iτ ) −1 e (A − (ξ − iτ )I) f (x)dξ = eit(ξ−iδ) (A − (ξ − iδ)I)−1 f (x)dξ −∞

−∞

for arbitrary τ, δ : 0 < δ < τ < Im λ0 , f = (A − λ0 I)−1 g, g ∈ D1 . The last equality gives the independence of the deﬁnition of Tt f by (3.3) on the choice of

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δ > 0 (0 < δ < Im λ0 ). Analogously it can be considered the case when t < 0 (then λ0 : Im λ0 < 0) and the proof is complete. Theorem 3.1 implies that the operators {Tt }t<0 , {Tt }t>0 , deﬁned by (3.3), are well deﬁned onto the sets 0 I)−1 g ∀g ∈ D1 }, Im λ 0 < 0, D0− = {f ∈ DA : f = (A − λ + −1 D0 = {f ∈ DA : f = (A − λ0 I) g ∀g ∈ D1 }, Im λ0 > 0. Next we will show that the families {Tt }t<0 , {Tt }t>0 , deﬁned by (3.3), possess the properties of the semigroups of operators from the class (C0 ) with a diﬀerentiability and a generator iA. Theorem 3.2. The families of operators {Tt }t>0 and {Tt }t<0 , deﬁned by (3.3), satisfy the conditions (3.6) Ts Tt f = Ts+t f + − ∀f ∈ D0 , ∀t, s > 0 and ∀f ∈ D0 , ∀t, s < 0 and lim

t→0,t>0

Tt f = f, ∀f ∈ D0+ ,

lim Tt f = f, ∀f ∈ D0− .

t→0,t<0

(3.7)

The proof of this theorem is analogous to the proof of an analogous theorem in the case of the dissipative K r -operators with real spectra (see Theorem 6.2, [13]). In the course of proving of Theorem 3.2 we have used the representation (3.1) and the inequality (3.2). The relations (3.7) allow to deﬁne the operator Tt in the case when t = 0 by the equality (3.8) T0 f (x) = f (x) ∀f ∈ D0+ , ∀f ∈ D0− . The next theorem solves the question of a diﬀerentiability of the families {Tt }t≥0 , {Tt }t≤0 , deﬁned by (3.3) and (3.8), and determines the generators of the considered families. Theorem 3.3. The families of operators {Tt }t≥0 , {Tt }t≤0 , deﬁned by (3.3) and (3.8), satisfy the conditions:

+ ∀f ∈ D 0

d Tt f (x) = iATt f (x) dt − when t < 0, when t > 0 and ∀f ∈ D 0 Tt f − f +, = iAf ∀f ∈ D 0 t Tt f − f −, = iAf ∀f ∈ D lim 0 t→0,t<0 t lim

t→0,t>0

(3.9)

(3.10) (3.11)

where + = {f ∈ L2 (R; Cn ) : f = (A − λ0 I)−1 (A − µ0 I)−1 h, h ∈ D1 } D 0

(3.12)

(µ0 = λ0 , Im λ0 > 0, Im µ0 > 0) 0 I)−1 (A − µ − = {f ∈ L2 (R; Cn ) : f = (A − λ D 0 I)−1 h, h ∈ D1 } 0

(3.13)

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0 , Im λ 0 < 0, Im µ ( µ0 = λ 0 < 0). The proof of Theorem 3.3 can be obtained using the ideas of the proof of Theorem 6.3 in [13] concerning the semigroup generated by a dissipative K r operator and in the course of proving we have used the representation (3.1) of the resolvent of the model A, the Residue theorem and the Lebesgue convergence theorem. The presented properties by Theorem 3.2 and Theorem 3.3 imply that the families of operators {Tt }t≥0 and {Tt }t≤0 possess the properties of the semigroups from the class (C0 ) except the boundedness of Tt and the uniformly boundedness of Tt which will be proved in the part 4.

4. Asymptotics of Tt f as t → ±∞ The proved properties of the families {Tt }t≥0 and {Tt }t≤0 , given by Theorem 3.2 and Theorem 3.3, allow us to consider non-dissipative processes (or non-dissipative + when t ≥ 0 and for each f ∈ D − when t ≤ 0 with curves) Tt f for each f ∈ D 0 0 generator iA and Tt are deﬁned by (3.3). Let B(x), Q(x), L be stated as above in part 2 and satisfy the conditions 1), 2), 3), 4) (page 352). Let the model A be deﬁned by (2.12). Next we will introduce some appropriate denotations as in the case of dissipative K r -operators ([13]) and preliminary properties, concerning the multiplicative integrals, describing the characteristic function of the model A from the form (2.12). These properties are presented by the authors in [13]. For the obtaining we have used the existence and the form of the limit values of the multiplicative integrals for almost all x ∈ [a, b] (τ > 0): b

→ 1+(x±iτ )v s − lim e−i v−(x±iτ ) T (v)dv τ →0 a

= s − lim

x−ε

→

ε→0 a

e

(4.1)

b

2 −i 1+xv v−x T (v)dv ±π(1+x )T (x)

e

→

e

−i 1+xv v−x T (v)dv

,

x+ε

where −∞ ≤ a < b ≤ +∞ (for a nonnegative or non-positive integrable matrix function T (x)). The limits (4.1) are presented by the authors in [13] and in the course of proving of (4.1) we have used the ideas of L.A. Sakhnovich ([31]) for the obtaining of the limit values b

s − lim

→ −i 1 T (v)dv e v−(x±iτ ) .

τ →0 a

Let us denote the next operators using the multiplicative integrals and the limit values from the form (4.1): T(x) = (1 + x2 )T (x),

(4.2)

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u

→ −i 1+(x±iδ)v T (v)dv e v−(x±iδ) ,

Fw (x ± iδ, u) =

(4.3)

w u

→ −i 1+(x±iδ)v T (v)dv e v−(x±iδ)

Fw± (x, u) = s − lim

(4.4)

δ→0 w

for all w, u, x ∈ R such that −∞ ≤ w < u ≤ +∞ and x−δ

Fw± (x, u)

= s − lim

→

δ→0 w

e

u

→

−i 1+vx v−x T (v)dv ±π T (x)

e

1+vx

e−i v−x T (v)dv ,

(4.5)

x+δ

Pw (x) = Fw+ (x, u) − Fw− (x, u), (4.6)  −1 x−δ x−δ

→ −i 1+vx T (v)dv ±πT (x)

→ −i 1+vx T (v)dv ±1   , (4.7) v−x Rw (x) = s − lim e e e v−x s − lim δ→0 w

δ→0 w

x−δ

U2w (x) = s − lim

→

δ→0 w

e

−i 1+vx v−x T (v)dv

e

iT (x)

x−δ

w

1+vx v−x dv

e−iT (x)x(x−δ−w) ,

x−w

iT (x) ln u−x ± P2w (x, u) = R±1 , w (x)U2w (x)e

U3 (x, u) = lim e

−iT (x)x(u−x−δ)

δ→0

iT (x)

e

u x+δ

1+vx v−x dv

(4.8) (4.9)

u

→

1+vx

e−i v−x T (v)dv ,

(4.10)

x+δ

± iT (x) ln(u−x) −iT (u) ln(u−x) Q± e , w (x, v) = P2w (x, u)e x−δ

V−∞ (x) = lim

→

δ→0 −∞

1+vx

(4.11)

e−i v−x T (v)dv eiT (x) ln δ

(4.12)

for all w, u, x such that −∞ ≤ w < x < u ≤ +∞. Let the matrix function T (x) satisfy the conditions: 1) ||T (x)|| ≤ C, ||xT (x)|| ≤ C ∀x ∈ R; 2) T (x) ∈ Cα1 (R), xT (x) ∈ Cα2 (R) (0 < α1 ≤ 1, 0 < α2 ≤ 1) (i.e. ||T (x1 ) − T (x2 )|| ≤ C|x1 − x2 |α1 , ||x1 T (x1 ) − x2 T (x2 )|| ≤ C|x1 − x2 |α2 ∀x1 , x2 ∈ R). If α = min{α1 , α2 } then the next inequalities, presented in [13], hold (for > 0): some constant C ||eiT (x)(1+x

2

) ln(x−ξ)

− eiT (ξ)(1+ξ

2

) ln(x−ξ)

+ |x|)|x − ξ|α || ≤ C(1

for x, ξ : 0 < x − ξ < 1, ||eiT (ξ)(1+ξ

2

) ln(ξ−w)

− eiT (x)(1+x

2

) ln(x−w)

+ |x|) || ≤ C(1

x−ξ ξ−w

(4.13)

α (4.14)

for w < ξ < x, 0 < x − w < 1, α ||Fw± (ξ; x) − Q± w (ξ; x)|| ≤ C(1 + |x|)(x − ξ)

(4.15)

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for w < ξ < x, 0 < x − w < 1 and ∀α : 0 < α < α, α x−ξ ||U2w (x) − U2w (ξ)|| ≤ C(1 + |x|) ξ−w for w < ξ < x, 0 < x − w < 1 and α = α/(1 + α). ±1 ||R±1 w (ξ) − Rw (x)|| ≤ C(1 + |x|)

for w < ξ < x, 0 < x − w < 1 and α = α/(1 + α). ||Q± w (x)

−

Q± w (ξ)||

+ |x|) ≤ C(1

for w < ξ < x, 0 < x − w < 1 and α = α/(1 + α). + |x|) ||U3 (x, u) − U3 (ξ, u)|| ≤ C(1

x−ξ ξ−w

x−ξ ξ−w

(4.16)

α (4.17)

α

x−ξ u−x

(4.18) α

for ξ < x < u, 0 < u − ξ < 1 and α = α/(1 + α). α α x−ξ x−ξ ± ± ||Fw (ξ, u) − Fw (x, u)|| ≤ C(1 + |x|) + , ξ−w u−x for w < ξ < x < u, 0 < u − w < 1, α = α/(1 + α) α α x−ξ x−ξ ± ± ||Uw (x, u) − Uw (ξ, u)|| ≤ C(1 + |x|) + , ξ−w u−x

(4.19)

(4.20)

(4.21)

for w < ξ < x < u, 0 < u − w < 1, α = α/(1 + α), where Uw (x, u) is deﬁned by the equality ± Uw± (x, u) = R∓1 w (x)Fw (x, u), x → 1+vξ ξ−x −i v−ξ T (v)dv −iT (ξ)(1+ξ 2 ) ln ξ−w + |x|)(ξ − x)α − U2w (ξ)e (4.22) ≤ C(1 e w for w < x < ξ < x + β, β < 1 and for each α : 0 < α ≤ α ≤ 1. For our further applications we shall denote the matrix functions deﬁned by (4.3), (4.4) (or (4.5)), (4.6), (4.7), (4.8), (4.9), (4.10), (4.11), (4.12) with Fw (x ± ± ± iδ, u), Fw± (x, u), Pw (x, u), R±1 w (x), U2w (x), P2w (x, u), U3 (x, u), Qw (x, u), V−∞ (x) respectively for the nonnegative matrix function T (x) = B(x)J1 on R and with w (x, u), R ±1 (x), U2w (x), P ± (x, u), U3 (x, u), Q ± (x, u), Fw (x ± iδ, u), Fw± (x, u), P w w 2w −∞ (x) respectively for the non-positive matrix function T (x) = −B(x)J2 on R. V Let us denote also k (x) = (1 + x2 )B(x)Jk = (1 + x2 )Bk (x), k = 1, 2. (4.23) B Let the matrix functions B(x) and Q(x) satisfy the conditions: (i) ||B(x)|| ≤ C, ||xB(x)|| ≤ C, ||x2 B(x)|| ≤ C ∀x ∈ R;

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(ii) B(x) ∈ Cα1 (R), xB(x) ∈ Cα2 (R), (0 < α1 ≤ 1, 0 < α2 ≤ 1), ||B(x)|| ∈ L(R), ||xB(x)|| ∈ L(R); (iii) Q∗ (x) is a smooth matrix function on R, ||Q∗ (x)|| ≤ C|x|p1 , ||Q∗ (x)|| ≤ C|x|p2 (for some p1 , p2 > 0). Let α = min{α1 , α2 }. Let us denote the set of all fast decreasing functions f (x) = (f1 (x), . . . , fn (x)) on R by S(R, Cn ) (i.e. fk (x) is a fast decreasing function, k = 1, 2, . . . , n). For each f ∈ S(R, Cn ) and the resolvent (A − λI)−1 f from (2.13) after direct calculations (using the properties of the multiplicative integrals) takes the form −1

(A − λI)

+∞ x−i , f= Zf (w, x, λ)dwΠ∗ (x) x−λ

(4.24)

−∞

where Zf (w, x, λ)

x

x

→ 1+λv

→ 1+λv = (f1 (w)J1 e−i v−λ B1 (v)dv J1 + f2 (w)J2 ei v−λ B2 (v)dv J2 )χ(−∞;x] (w) w

+∞

−f2 (w)J2

→

e

w

x

i 1+λv v−λ B2 (v)dv

w

S

→

e

−i 1+λv v−λ B1 (v)dv

−∞

(4.25)

J1 ,

1 ix ∗ f (x)Π(x) + f (x)Q (x) , f1 (x) = − x−i x−i 1 ix ∗ f (x)Π(x) + f (x)Q (x) . f2 (x) = x−i x−i

(4.26)

(4.27)

Now the existence of the limits (4.1), the inequalities x → 1+λv ei v−λ B(v)Jk LJk dv ≤ M, k = 1, 2 −∞ and the Lebesgue convergence theorem show that there exist the limits Yf± (x, ξ) = lim Yf (x, ξ ± iδ) δ→0

(4.28)

where we have denoted +∞ Zf (w, x, λ)dw. Yf (x, λ) = −∞

The next lemma presents a suitable representation of Tt f which allows to obtain the asymptotics of the processes Tt f as t → +∞ and t → −∞ when f belongs to a suitable subset of L2 (R; Cn ).

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Lemma 4.1. The operator Tt , deﬁned by (3.3), possesses the next representation Tt f (x) = Tt (A − λ0 I)−1 (A − µ0 I)−1 h(x)

eitξ 1 x−i − 1 ∗ = − 2πi ξ−λ0 ξ−µ0 x−ξ Yh (x, ξ)dξΠ (x) R\∆ 1 ε→0 2πi

− lim

∆ −eitµ0 λ0 −µ0

itλ0

+e

− x−i 1 eitξ ∗ (x−ξ)1−ε ξ−λ0 ξ−µ0 Yh (x, ξ)dξΠ (x)

(4.29)

g(x)(A − µ0 I)−1 h(x) + eitλ0 f (x)

when t > 0, where f (x) = (A − λ0 I)−1 (A − µ0 I)−1 h(x), h ∈ S(R, Cn ), ∆ = [x − β; x + β], β is an arbitrary ﬁxed number: 0 < β < 1, λ0 = µ0 , Im λ0 > 0, Im µ0 > 0 and h1 (x), h2 (x) are deﬁned as in (4.26) and (4.28) and I)−1 (A − µ Tt f (x) = Tt (A − λ 0 I)−1 h(x)

eitξ 1 0 x−i − 1 ∗ = 2πi ξ− µ0 x−ξ Yh (x, ξ)dξΠ (x) R\∆

ξ−λ0

− x−i 1 1 eitξ ∗ + lim 2πi 1−ε 0 ξ− (x−ξ) µ0 Yh (x, ξ)dξΠ (x) ξ−λ ε→0 ∆ itλ 0 0 −eitµ (A − µ 0 I)−1 h(x) + eitλ0 f (x) − e λ − 0 µ0

(4.30)

0 I)−1 (A − µ when t < 0, where f (x) = (A − λ 0 I)−1 h(x), h ∈ S(R, Cn ), ∆ = 0 = µ 0 < 0, [x − β; x + β], β is an arbitrary ﬁxed number: 0 < β < 1, λ 0 , Im λ Im µ 0 < 0. Proof. We will consider in detail the case when t > 0. (The other case can be considered analogously.) Straightforward calculations show that the operator Tt , deﬁned by (3.3), for f (x) = (A − λ0 I)−1 (A − µ0 I)−1 h(x) (h ∈ S(R, Cn )) takes the form +∞ 1 eit(ξ−iδ) (A − (ξ − iδ)I)−1 f (x)dξ (4.31) Tt f (x) = − 2πi −∞

where we have used the resolvent equation and Residue theorem. Next we continue the relations (4.31) and obtain that 1 Tt f (x) = − 2πi 1 − 2πi

=

∆ 1 − 2πi

−1 −1 eit(ξ−iδ) ((A−(ξ−iδ)I) h(x)−(A−µ0 I) h(x)) dξ ξ−iδ−λ0 ξ−iδ−µ0

R\∆ eit(ξ−iδ) ξ−iδ−λ0 (A −

(ξ − iδ)I)−1 g(x)dξ + eitλ0 f (x)

1 eit(ξ−iδ) ξ−iδ−λ0 ξ−iδ−µ0 (A

− (ξ − iδ)I)−1 h(x)dξ

R\∆ it(ξ−iδ) +∞

eit(ξ−iδ) 1 e 1 1 1 dξ g(x) − 2πi + 2πi ξ−iδ−λ0 ξ−iδ−µ0 ξ−iδ−λ0 ξ−iδ−µ0 dξg(x) −∞ ∆

it(ξ−iδ)

eit(ξ−iδ) 1 1 itλ0 − 2πi e (A − (ξ − iδ)I)−1 f (x)dξ + 2πi f (x) ξ−iδ−λ0 dξf (x) + e ∆ ∆

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for all δ > 0 suﬃciently small (0 < δ < Im λ0 , 0 < δ < Im µ0 ) where g(x) = (A − µ0 I)−1 h(x). Consequently from the last relations it follows that

eit(ξ−iδ) 1 1 −1 Tt f (x) = 2πi lim h(x)dξ ξ−iδ−λ0 ξ−iδ−µ0 (A − (ξ − iδ)I) itλ0 itµ0 + e λ0 −e −µ0

1 + 2πi lim

δ→0 R\∆

1 1 eit(ξ−iδ) lim ξ−iδ−λ dξg(x) 2πi δ→0 0 ξ−iδ−µ0 ∆ 1 eit(ξ−iδ) −1 h(x) ξ−iδ−λ0 ξ−iδ−µ0 dξ(A − µ0 I)

g(x) + eitλ0 f (x) −

δ→0 ∆

eit(ξ−iδ) 1 1 − 2πi lim ξ−iδ−λ (A 0 ξ−iδ−µ0 δ→0 ∆

(4.32)

− (ξ − iδ)I)−1 h(x)dξ.

But for the last integral on the right hand side of (4.32) after straightforward calculations we have

it(ξ−iδ) 1 lim e (A − (ξ − iδ)I)−1 h(x)dξ δ→0 ∆ ξ−iδ−λ0 ξ−iδ−µ0

− x−i 1 eitξ ∗ = lim (x−ξ) 1−ε ξ−λ ξ−µ Yh (x, ξ)dξΠ (x) 0 0 ε→0 ∆

using the representation (4.24) of the resolvent, Lebesgue convergence theorem and Reside theorem for suitable domains. Now from the last relations and from (4.32) it follows that the representation (4.29) is proved. The representation (4.30) of the operators Tt when t < 0 can be obtained analogously. This proves the lemma. The representation (4.29) and (4.30) of the operators from the considered families {Tt }t≥0 and {Tt }t≤0 give the possibility to ﬁnd the asymptotics of the processes Tt f when f (x) = (A − λ0 I)−1 (A − µ0 I)−1 h(x), h ∈ S(R, Cn ) (t > 0) 0 I)−1 (A − µ and f (x) = (A − λ 0 I)−1 h(x) h ∈ S(R, Cn ) (t > 0). Before continuing with the asymptotics of the non-dissipative processes Tt f generated by unbounded K r - operators with a triangular model A, deﬁned by (2.12), it has to mention that in the course of proving of the asymptotics we use the analogue in Cm of the classical gamma-function +∞ Γ(εI − iT (u)) = e−x e((ε−1)I−iT (u)) ln x dx

(ε > 0)

(4.33)

0

(where T (u) is a matrix function) and its properties, introduced and considered by the authors in [12]. Next we will denote by T1 (t) ∼ T2 (t) as t → ±∞ that ||T1 (t)−T2 (t)||L2 −→ 0 for matrix functions T1 (t) and T2 (t). Theorem 4.2. Let for the model A, deﬁned by (2.12), the conditions (i), (ii), (iii) (page 358) hold. Then the process Tt f for each f = (A − λ0 I)−1 (A − µ0 I)−1 h, h ∈ S(R, Cn ) has the next asymptotics ||Tt f (x) − eitx S+ f (x)||L2 → 0

(4.34)

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0 I)−1 (A − µ as t → +∞ and Tt f for each f = (A − λ 0 I)−1 h, h ∈ S(R, Cn ), has the asymptotics ||Tt f (x) − eitx S− f (x)||L2 → 0

(4.35)

as t → −∞, where S± are deﬁned by the equalities: S± f (x) =

+

x −∞

w

h1 (w)

−∞

1+vx

π

ei v−x B1 (v)dv dwJ1 V−∞ (x)tiB1 (x) e− 2 B1 (x) .

1 (x))J1 Π∗ (x) x−i 1 .Γ−1 (I + iB x−λ0 x−µ0 w

x −∞

h2 (w)

←

+∞

−∞

1+vx −∞ (x)t−iB 2 (x) e π2 B 2 (x) . e−i v−x B2 (v)dv dwJ2 V

−∞ −1

.Γ −

←

(4.36)

2 (x))J2 Π∗ (x) x−i 1 (I − iB x−λ0 x−µ0

π h2 (w)Fw− (x; +∞)dwSJ1 V−∞ (x)tiB1 (x) e− 2 B1 (x) .

1 (x))J1 Π∗ (x) x−i 1 . .Γ−1 (I + iB x−λ0 x−µ0

Proof. We consider at ﬁrst the case when t > 0. Let λ0 , µ0 ∈ C, λ0 = µ0 and Im λ0 > 0, Im µ0 > 0. Let h ∈ S(R, Cn ) and f = (A − λ0 I)−1 (A − µ0 I)−1 h. Then the representation (4.29) of Tt f imply that the asymptotic behaviour as t → +∞ depends only on the behaviour of 1 eitξ x−i 1 lim · · Y − (x, ξ)dξΠ∗ (x) (4.37) ε→0 2πi (x − ξ)1−ε ξ − λ0 ξ − µ0 h ∆

where ∆ = [x − β, x + β], β is a ﬁxed number with 0 < β < 1. The other addends in (4.29) tend to 0 as t → +∞ which follows directly (using the Lebesgue lemma for Fourier transform and Im λ0 > 0, Im µ0 > 0). Next we use the methods and ideas as in the bounded case ([12, 6]) and the unbounded case ([13]) together with the presented inequalities (4.13),(4.14),. . . ,(4.22). We consider separately the projections P1 Tt P1 f , P2 Tt P2 f , P1 Tt P2 f of Tt f . The projection P1 Tt P1 f is a dissipative process which follows from the deﬁnition of the model A as a coupling of dissipative and anti-dissipative operators. The asymptotics as t → +∞ of 1 P1 Tt P1 f = − 2πi

+∞ eit(ξ−iδ P1 (A − (ξ − iδ)I)−1 P1 f (x)dξ

(4.38)

−∞

is obtained in detail in [13] for weaker conditions for B(x). In this case we have considered the dissipative K r - operators with real spectra and have obtained that ||P1 Tt P1 f (x) − eitx ×V−∞ (x)t

x

w

h1 (w)

−∞ 1 (x) 1 (x) − π B iB 2

e

−1

Γ

←

−∞

1+vx

ei v−x B1 (v)dv J1 dw

1 (x))J1 Π∗ (x) x−i · (I + iB x−λ0

(4.39) 1 2 x−µ0 ||L

→0

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as t → +∞, V−∞ (x) is deﬁned by (4.12) and h1 (x) is deﬁned by (4.26) (see [13], proof of Theorem 1.7, p. 156). Following the ideas of these proofs analogously we obtain the asymptotics of the anti-dissipative curve P2 Tt P2 f (x) when t → +∞: ||P2 Tt P2 f (x) − eitx

x −∞

w

h2 (w)

←

−∞

1+vx

e−i v−x B2 (v)dv J2 dw·

2 (x))J2 Π∗ (x) x−i · −∞ (x)t−iB 2 (x) e π2 B 2 (x) Γ−1 (I − iB ·V x−λ0

1 2 x−µ0 ||L

(4.40) → 0.

−∞ (x) is deﬁned by (4.12) and where V h2 (x) is deﬁned by (4.27). Using these methods and the asymptotic behaviour of the third addend in the right hand side of (4.29) for the projection P1 Tt P2 f (x) we obtain the next asymptotics P1 Tt P2 f (x) ∼ −eitx ×V−∞ (x)t

+∞

h2 (w)Fw− (x; +∞)dwS

−∞ 1 (x) 1 (x) − π B iB 2

e

1 (x))J1 Π∗ (x) x−i · Γ−1 (I + iB x−λ0

(4.41) 1 x−µ0

as t → +∞. Finally from the asymptotics (4.39), (4.40) and (4.41) it follows that Tt f (x) ∼e

itx

+∞

w

(( h1 (w)

−∞ w

←

+ h2 (w)

←

−∞

1+vx

ei v−x B1 (v)dv J1

−i 1+vx v−x B2 (v)dv

e J2 )χ(−∞,x] (w) −∞ x−i 1 · x−µ − h2 (w)Fw− (x, +∞)S)dw x−λ 0 0 π 1 (x))J1 ×(J1 V−∞ (x)e− 2 B1 (x) Γ−1 (I + iB −∞ (x)e π2 B 2 (x) Γ−1 (I − iB 2 (x))J2 )Π∗ (x) +J2 V ×Π(x)(J1 tiB1 (x) J1 + J2 t−iB2 (x) J2 )Q(x).

(4.42)

Analogously it can be obtained the asymptotics of Tt f as t → −∞ (when f ∈ D0− ) Tt f (x) w +∞

← i 1+vx B1 (v)dv itx ((h1 (w) e v−x J1 ∼e −∞ w

←

+ h2 (w)

−∞

−i 1+vx v−x B2 (v)dv

e J2 )χ(−∞,x] (w) −∞ x−i 1 − h2 (w)Fw+ (x, +∞)S)dw x− 0 · x− µ0 λ π 1 (x))J1 ×(J1 V−∞ (x)e 2 B1 (x) Γ−1 (I + iB −∞ (x)e− π2 B 2 (x) Γ−1 (I − iB 2 (x))J2 )Π∗ (x) +J2 V 1 (x) 2 (x) iB −iB J1 + J2 |t| J2 )Q(x). ×Π(x)(J1 |t| This proves the theorem.

(4.43)

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Let us introduce also the next denotations x) = Π(x)(J1 |t|iB 1 (x) J1 + J2 |t|−iB 2 (x) J2 )Q(x), Z(t, ∓π 2 B1 (x)

1 (x))J1 T± p = p(J1 V−∞ (x)e Γ−1 (I + iB π B ± (x) −1 −∞ (x)e 2 2 Γ (I − iB 2 (x))J2 )Π∗ (x)(x − i) ∀p ∈ Cm , +J2 V x

S11 f (x) =

w

−∞

−∞

x

S22 f (x) =

−∞

± S12 f (x) = −

x

−∞

+

+∞

1+vx

ei v−x B1 (v)dv dwJ1

1 1 x − λ0 x − µ0

w

←

h2 (w)

−∞

= −(

←

h1 (w)

+∞

1+vx

e−i v−x B2 (v)dv dwJ2

1 1 x − λ0 x − µ0

(4.44) (4.45)

(4.46)

(4.47)

1 1 h2 (w)Fw∓ (x, +∞)dwS x−λ 0 x−µ0

−∞ w

←

h2 (w)

−∞

1+vx

∓ e−i v−x B2 (v)dv dwF−∞ (x, +∞)

(4.48)

+∞

h2 (w)

x

→

w

1+vx

1 1 ei v−x B2 (v)dv dw)S x−λ , 0 x−µ0

± S± f (x) = S11 f (x) + S22 f (x) + S12 f (x)

(4.49)

S± f (x) = T± S± f (x)

(4.50)

x)T± S± f (x) = Z(t, x)S± f (x). S± f (x) = Z(t,

(4.51)

Then the asymptotics (4.34) and (4.35) can be written in the form ||Tt f (x) − eitx S± f (x)||L2 → 0 as t → ±∞ for all f ∈

± , D 0

(4.52)

where

+ = {f ∈ L2 (R; Cn ) : f = (A − λ0 I)−1 (A − µ0 I)−1 h, h ∈ S(R, Cn )}, D 0

(4.53)

0 I)−1 (A − µ − = {f ∈ L2 (R; Cn ) : f = (A − λ D 0 I)−1 h, h ∈ S(R, Cn )}. (4.54) 0 The next theorem proves the boundness of the operators S± and S± , describing the asymptotics (4.52). Theorem 4.3. If ||e−2πBk (x) ||L2 < 1 (k = 1, 2) then the operators S± and S± , deﬁned by (4.49) and (4.51) correspondingly, are bounded operators in L2 (R; Cn ).

Proof. From the dissipativness of P1 AP1 it follows that the function ||P1 Tt P1 f ||2L2 is decreasing on R and from anti-dissipativness of P2 AP2 it follows that the functions ||P2 Tt P2 f ||2L2 is increasing on R. Consequently ||P1 Tt P1 f ||L2 ≤ ||P1 f ||L2

+ , ∀t ≥ 0, ∀f ∈ D 0

(4.55)

||P2 Tt P2 f ||L2 ≤ ||P2 f ||L2

− . ∀t ≤ 0, ∀f ∈ D 0

(4.56)

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From the obtained asymptotics (4.52), the introduced denotations (4.46), (4.47), (4.48) (4.49), (4.50), (4.51) and the properties of the gamma-function ([12], Lemma6, Lemma7) we have (see [13], proof of Theorem 6.15) ||P1 Tt P1 f ||L2 − √1 ||S11 P1 f (x)J1 V−∞ (x) I − e−2πB 1 (x) ||L2 → 0 (4.57) 2π as t → +∞, where

1 x)T+ S11 f (x)||L2 √ ||S11 f (x)J1 V−∞ (x) I − e−2πB 1 (x) ||L2 = ||eitx Z(t, 2π + . From (4.55) and (4.57) it follows that for all f ∈ D 0 1 √ ||S11 f (x)J1 V−∞ (x) I − e−2πB 1 (x) ||L2 ≤ ||P1 f ||L2 (4.58) 2π which together with the invertibility of I − e−2πB 1 (x) gives that + ||S11 f ||L2 ≤ M ||f ||L2 ∀f ∈ D (4.59) 0

(where M > 0 is a suitable constant). − we have Analogously for P2 Tt P2 f when t < 0 and f ∈ D 0 ||P2 Tt P2 f ||L2 − √1 ||S22 f (x)J2 V −∞ (x) I − e−2πB 2 (x) ||L2 → 0 2π t → −∞ and consequently

1 −∞ (x) I − e−2πB 2 (x) ||L2 ≤ ||P2 f ||L2 . √ ||S22 f (x)J2 V (4.60) 2π Now the inequality (4.60) and the invertibility of I − e−2πB 2 (x) imply that ||S22 f ||L2 ≤ M ||f ||L2

− . ∀f ∈ D 0

(4.61)

− are dense in L2 (R; Cn ) which together with (4.59) and (4.61) + and D But D 0 0 gives the boundness of the operators S11 and S22 onto L2 (R; Cn ). ± On the other hand the operators S12 , deﬁned by (4.48), can be presented in the form ± S12 f (x) = −(

w

x

−∞

+

+∞

x

+∞

h2 (w)

→

e

h2 (w)

←

−∞

1+vx

∓ e−i v−x B2 (v)dv dwF−∞ (x, +∞)

i 1+vx v−x B2 (v)dv

w

(4.62)

1 1 dw)S x−λ 0 x−µ0

∓ = −((S22 f (x))F−∞ (x, +∞)S + (S22 f (x))S)

± correspondingly where for all f ∈ D 0 S22 f (x) =

+∞ +∞

→ 1+vx h2 (w) ei v−x B2 (v)dv dw x

w

1 1 . x − λ0 x − µ0

(4.63)

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But S22 is a bounded operator, describing the asymptotics of the dissipative pro∗ cess P2 eitA P2 f which can be obtained analogously as in the bounded case (see [12], Theorem 6). The boundness of S22 and S22 and the representation (4.62) ± are bounded operators onto L2 (R; Cn ). The proof is complete. imply that S12 The next theorem ﬁnishes the description of the families {Tt }t≥0 and {Tt }t≤0 , deﬁned by (3.3) as semigroups of operators from the class (C0 ) — the next theorem proves the uniformly boundness of the operators Tt . Theorem 4.4. The operators Tt , deﬁned by (3.3), are uniformly bounded onto the space L2 (R; Cn ). Proof. Let f = (A − iI)−1 (A − λ0 I)−1 h = (A − iI)−1 g, where h ∈ S(R, Cn ). Then from the diﬀerentiability of the operators Tt and (3.9) it follows that d Tt f (x) = iATt f (x) dt and hence after calculations from (2.14), (2.15) we have d (Tt f, Tt f ) = −(Bi Tt g, Tt g) = −(Φ∗ LΦTt g, Tt g) (4.64) dt for t > 0 where Φ is deﬁned by (2.16). After straightforward calculations for ΦTt g we obtain the representation λ0 I)−1 h(x) ΦTt g(x) = ΦTt (A − +∞

ξ eitξ 1 w (ξ, +∞)ω ∗ )dw ( h1 (w)J1 Pw (ξ, +∞) − h2 (w)J2 P = 2π (ξ−i)(ξ−λ0 ) −∞

−∞

+∞

+∞

→ B (v)dv w (ξ, +∞)S(F + (ξ, +∞)J1 − h2 (w)J2 (P − e 1 J1 ) −∞ −∞ −∞ + Fw (ξ, +∞)SP−∞ (ξ, +∞)J1 )dw +∞

itξ 1 e Gf (ξ)dξ, = 2π −∞

(4.65) where Gf (ξ) can be written in the form −∞ ω ∗ Gf (ξ)dξ = S11 f (ξ)P−∞ − S22 f (ξ)P +∞

+ −S22 f (ξ)(F−∞ (ξ, +∞) −

→

−∞

eB1 (v)dv )J1 − S22 f (ξ)SP−∞ (ξ, +∞)J1 ,

(4.66)

where S11 , S22 , S22 are deﬁned by (4.46), (4.47), (4.63) for f = (A − iI)−1 (A − λ0 I)−1 h and ω is deﬁned by (2.21). Hence −1

ΦTt g(x) = ΦTt (A − λ0 I)

1 h(x) = 2π

+∞ eitξ Gf (ξ)dξ, −∞

(t > 0),

(4.67)

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where Gf (ξ) has the form (4.66), Gf (ξ) ∈ L2 (R; Cm ) and ||Gf (ξ)||L2 ≤ M ||f ||L2 which follows from the boundness of S11 , S22 , S22 (proved in Theorem 4.3). Analogously it can be obtained that for t < 0 and f (x) = (A − iI)−1 (A − −1 0 I) k = (A − iI)−1 g where k ∈ S(R, Cn ), Im λ 0 < 0 the next representation λ hold d (Tt f, Tt f ) = −(LΦTt g , ΦTt g). (4.68) dt Direct calculations show that +∞ 1 −1 f (ξ)dξ, (t < 0), 0 I) k(x) = eitξ G (4.69) ΦTt g(x) = ΦTt (A − λ 2π −∞

f (ξ) ∈ L2 (R; Cm ) and ||G f (ξ)||L2 ≤ M ||f ||L2 which follows from the where G 2 boundness of S11 , S22 , S22 onto L (R; Cn ). 0 I)−1 k. Now from (4.64) and (4.68) we have Let now f = (A − iI)−1 (A − λ ||Tt f ||2L2

=

||f ||2L2

t −

(LΦTτ (A − λ0 I)−1 h, Φ(A − λ0 I)−1 h)dτ, t > 0,

(4.70)

0 I)−1 k, Φ(A − λ 0 I)−1 k)dτ, t < 0. (LΦTτ (A − λ

(4.71)

0

||Tt f ||2L2

=

||f ||2L2

0 + t

From (4.70) it follows that

t ||Tt f ||2L2 ≤ ||f ||2L2 + | (LΦTτ (A − λ0 I)−1 h, Φ(A − λ0 I)−1 h)dτ | ≤

||f ||2L2

+ ||L||

= ||f ||2L2 + ||L||

+∞

0 +∞

0

0

|ΦTτ (A − λ0 I)−1 h|2 dτ |Gf (τ )|2 dτ ≤ ||f ||2L2 + ||L||

= ||f ||2L2 + ||L||.||Gf (ξ)||2L2 ≤ ||f ||2L2 +

+∞

||Gf (ξ)||2 dξ −∞ ||L||.||Gf ||2L2 .||f ||2L2

and hence ||Tt f ||L2 ≤ (1 + ||L||.||Gf ||2L2 )1/2 ||f ||L2 (t > 0) −1

−1

(4.72)

for all f = (A − iI) (A − λ0 I) h, h ∈ S(R; C ). Analogously we obtain that n

f ||2 2 )1/2 ||f ||L2 (t < 0) ||Tt f ||L2 ≤ (1 + ||L||.||G L

(4.73)

0 I)−1 h, h ∈ S(R; Cn ). The inequalities (4.72) and (4.73) for all f = (A−iI)−1 (A− λ imply that {Tt }t∈R is uniformly bounded family of operators and together with the proved properties of Tt (Theorem 3.2, Theorem 3.3) imply that the semigroups {Tt }t≥0 , {Tt }t≤0 belong to the class (C0 ). The theorem is proved.

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Theorem 4.4 together with the proved properties of the families {Tt }t≤0 and {Tt }t≥0 gives also that {Tt }t∈R is a group. The boundness of operators Tt and S± in a dense set of L2 (R; Cn ) allow to extend Tt and S± by continuity onto L2 (R; Cn ). Hence the next relation holds ||Tt f (x) − eitx S± f (x)||L2 → 0 as t → ±∞ ∀f ∈ L2 (R; Cn ).

(4.74)

Now we can deﬁne an exponential function for t ∈ R by the equality eitA = Tt and consider eitA f = Tt f for all f ∈ L2 (R; Cn ) (i.e. non-dissipative processes following the terminology of M.S. Livˇsic [25] or non-dissipative curves following the terminology of K. Kirchev and V. Zolotarev [15, 16]).

5. Scattering theory for the couple (A∗ , A) As in the case of a bounded model of a coupling of a dissipative operator and an anti-dissipative one (see [12]) for the considered model A, deﬁned by (2.12) with the help of the obtained asymptotics (4.74) we can obtain the wave operators and the scattering operator for the couple (A∗ , A) explicitly and solve the important question of a similarity of A and the operator of multiplying by the independent variable in a suitable subspace of L2 (R; Cn ) in an explicit form. In this connection it is quite interesting to study the problem of reduction of non-selfadjoint operators with continuous spectra to the simplest form. For a broad class of operators (dissipative bounded operators with absolutely continuous spectra) this problem was solved by L.A. Sakhnovich ([31]) – the similarity of the simple part As of the model, describing this class of bounded dissipative operators with real spectra and the operator Q of multiplication by an independent variable is established by L.A. Sakhnovich: As = BQB −1 and B is determined in explicit form. An analogous problem has been studied by the authors in [12] for the model, describing the class of all bounded operators, presented as a coupling of a dissipative and an anti-dissipative operators with real absolutely continuous spectra and the similarity of the simple part of this model and the operator Q of multiplication by an independent variable is established in an explicit form. A similar problem was studied for various classes of operators by many authors and by diﬀerent methods. In particular this problem was investigated by Sz.-Nagy and Foias [32], Gohberg and Krein [9], Naboko [27], Malamud [26] and others. As a rule these authors use only the corresponding characteristic function or the resolvent. The transformation B is not determined but its existence is established. In [29] (§30, 5) for the linear diﬀerential operator L: Lu = −u + P (x)u

u ∈ L2 (0, +∞)

(where P (x) is a complex valued function, deﬁned on [0, +∞)) without spectral specialities the representation U Lc U −1 = Λ is established, where Lc is the operator

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L onto a suitable subspace Hc and Λ is the operator of multiplication by the independent variable in the space L2 (0, +∞). In this part we obtain the similarity of the simple part of the model A, deﬁned by (2.12), and the operator Q of multiplication by the independent variable on L2 (R; Cn ) and obtain the explicit form of the operator, describing this similarity. For the obtaining of these results we essentially use the obtained asymptotics of the processes Tt f . Let the modelA be deﬁned by (2.12) and Π(x), B(x), Q(x) be stated as above in part 2 and let they satisfy the conditions (i), (ii), (iii) (page 358). The ﬁrst theorem presents the existence and the explicit form of bounded inverse operators of S± , deﬁned by (4.50), describing the asymptotics of eitA f = Tt f . Theorem 5.1. The operators S± , deﬁned by (4.50), have bounded inverse operators, deﬁned in the range R(S± ) of S± . Proof. Let us consider the operators x 1 1 G11 g(x) = lim g(τ )P−∞ (τ, x) dτ J1 Π∗ (x)(x − i), ε→0 2πi (x − τ )1−ε

(5.1)

−∞

1 G22 g(x) = lim ε→0 2πi

x

g(τ )P−∞ (τ, x)

−∞

1 dτ J2 Π∗ (x)(x − i), (x − τ )1−ε

(5.2)

± G± 12 = −G11 S± G22 ,

G± = G11 + G22 + G± (5.3) 12 ± 2 n , where S11 is deﬁned deﬁned for g ∈ L (R; C ). Let g(τ ) = S11 f (τ ) for f ∈ D 0 by (4.46). Then from (5.1) and (4.46) we have (for f ∈ D0+ ) G11 (S11 f (x)) w

x

τ

← i 1+vτ B1 (v)dv 1 dw = lim 2πi h1 (w) e v−τ ε→0

−∞

−∞

−∞

1 × (x−τ1)1−ε P−∞ (τ, x) τ −λ · 0

= =

x 1 lim 2πi ε→0 −∞

h1 (w)

x

w

1 τ −λ0

1 ∗ τ −µ0 dτ J1 Π (x)(x

·

1 τ −µ0

·

Pw (τ,x) (x−τ )1−ε dτ

− i)

(5.4)

dwJ1 Π∗ (x)(x − i).

For the function 1 1 1 1 · Fw (z, x) · · (5.5) z − λ0 z − µ0 x − z z − λ and the domain with a contour ΓR = LR ∪ lδ described positively according to the domain, where LR = {z = Reiϕ ; 0 ≤ ϕ ≤ 2π}, lδ = l1 ∪ l2 ∪ l3 ∪ l4 , G(z) =

l1 l2 l3 l4

= {z = {z = {z = {z

= ξ − iδ : w − a ≤ ξ ≤ x + b}, = w − a + iν : −δ ≤ ν ≤ δ}, = ξ + iδ : w − a ≤ ξ ≤ x + b}, = x + b + iν : −δ ≤ ν ≤ δ},

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(a, b > 0, R > 0 is suﬃciently large, δ > 0 is suﬃciently small, λ ∈ C is a ﬁxed number with suﬃciently large |λ| and Residue theorem it follows that G(z)dz = 2πiγ (5.6) ΓR

where we have denoted γ = Res (G(z), λ0 ) + Res (G(z), µ0 ) + Res (G(z), λ). From the properties of the multiplicative integrals it follows that G(z)dz → 0 as R → +∞, (5.7) LR

G(z)dz → 0,

G(z)dz → 0 as δ → 0

l2

(5.8)

l4

according to the norm || ||. Now letting R → +∞, δ → 0 in the equality (5.6) and using (5.7), (5.8) we obtain lim ( G(z)dz − G(z)dz) = 2πiγ. (5.9) δ→0

l3

l1

But using again the Residue theorem it is easy to see that x

x

1 1 1 1 ∗ lim h1 (w) τ −λ0 · τ −µ0 · (x−τ )1−ε Pw (τ, x)dτ dwJ1 Π (x)(x − i) ε→0 2πi −∞ w x+b

x

1 1 1 1 = lim 1 ( · · · Fw (ξ + iδ, x) h1 (w) δ→0 2πi −∞

−

1 ξ−iδ−λ0

ξ+iδ−λ0

w−a

·

1 ξ−iδ−µ0

·

1 x−ξ+iδ

·

ξ+iδ−µ0

1 ξ−iδ−λ Fw (ξ

x−ξ−iδ

ξ+iδ−λ

− iδ, x))dξ dwJ1 Π∗ (x)(x − i).

Now from (5.9), the last relations and calculation of the right hand side of (5.6) it follows that x

x

1 1 1 1 ∗ h1 (w) lim τ −λ0 · τ −µ0 · (x−τ )1−ε Pw (τ, x)dτ dwJ1 Π (x)(x − i) ε→0 2πi −∞ w x

x

→ −i 1+vλ0 B1 (v)dv 1 = · 1 · 1 e v−λ0 h1 (w) λ0 −µ0

−∞

x−λ0

λ0 −λ

x

−

1 µ0 −λ0

+

1 λ−λ0

·

·

1 x−µ0

1 λ−µ0

· ·

1 µ0 −λ

1 x−λ

→

e

−i

w

1+vµ0 v−µ0

B1 (v)dv

w x

→ −i 1+vλ B1 (v)dv v−λ

e

w

dwJ1 Π∗ (x)(x − i).

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Laurent expansion of both sides of the last equality in a neighbourhood of ∞ and comparing of the coeﬃcients of λ−1 show that x

x

1 Pw (τ,x) 1 1 ∗ lim h1 (w) τ −λ0 · τ −µ0 · (x−τ )1−ε dτ dwJ1 Π (x)(x − i) ε→0 2πi −∞ w x

x

→ −i 1+vλ0 B1 (v)dv 1 1 = e v−λ0 h1 (w) (5.10) λ0 −µ0

x

−

x−λ0

−∞

1 x−µ0

→

e

1+vµ −i v−µ 0 0

w

B1 (v)dv

dwJ1 Π∗ (x)(x − i)

w

But the right hand side of (5.10) takes the form 1 λ0 −µ0 ((P1 AP1

− λ0 I)−1 P1 h(x) − (P1 AP1 − µ0 I)−1 P1 h(x)) = P1 (A − λ0 I) (A − µ0 I)−1 P1 h(x) = P1 f (x) −1

because of the form of the resolvent (P1 AP1 − νI)−1 P1 h(x) (for ν : Im ν = 0) (see (4.24), (4.25)) and the Resolvent equation. Consequently (5.10) implies that x

x

1 1 1 1 · · P (τ, x)dτ dw h1 (w) lim 2πi 1−ε w τ −λ0 τ −µ0 (x−τ ) ε→0 (5.11) −∞ w ×J1 Π∗ (x)(x − i) = P1 f (x). Hence from (5.2) and (5.11) it follows that G11 (S11 f (x)) = P1 f (x)

+ . ∀f ∈ D 0

(5.12)

Similarly we obtain also that G11 (S11 f (x)) = P1 f (x)

− . f ∈D 0

On the other side for P1 Tt P1 f (x) we obtain +∞

eitξ 1

x ∗ (x)(x−i)dξ 1 P1 Tt P1 f (x) = 2πi lim h1 (w)J1 Pw (ξ, x)dw J1 Π(x−ξ) 1−ε ε→0 −∞ ξ−λ0 ξ−µ0 −∞

x eitξ 1

ξ ∗ = 1 lim dξ h1 (w)J1 Pw (ξ, x)dw J1 Π (x)(x−i) 1−ε =

2πi ε→0 ξ−λ0 ξ−µ0 −∞ −∞ x

Π∗ (x)(x−i) 1 lim eitξ (S11 f (ξ)P−∞ (ξ, x) J1(x−ξ) dξ 1−ε 2πi ε→0 −∞

(x−ξ)

= G11 (eitx S11 ),

i.e. P1 Tt P1 f (x) = G11 (eitx S11 f (x)).

(5.13)

From the dissipativness of the operator P1 AP1 and the properties of Tt it follows that the function ||P1 Tt P1 f (x)||2L2 = ||G11 (eitx S11 f (x))||2L2 is decreasing and analogously as in the proof of Theorem 4.3 (see (4.57), (4.58) but for t < 0) we

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have ||G11 (eitx S11 f (x))||2L2 π 2 ≤ ||(S11 f (x))J1 V−∞ (x)e 2 B1 (x) Γ−1 (I + iB1 (x))J1 (x − i)||L2 1 = 2π ||(S11 f (x))J1 V−∞ (x)eπB1 (x) I − e−2πB1 (x) ||2L2 C ≤ 2π ||(S11 f (x))||2L2

(5.14)

(C > 0 is a suitable constant). The inequalities (5.14) imply that G11 is a bounded + which together with the operator onto the subspace of all S11 f where f ∈ D 0 − boundness of S11 onto the dense set D0 gives the boundness of G11 onto L2 (R; Cn ). Analogously it can be obtained that G22 (S22 g) = P2 f and G22 is a bounded operator Now from the deﬁnition of

(5.15)

− and consequently onto L2 (R; Cn ). onto S22 D 0 ± , S± , G11 , G22 , G± S11 , S22 , S12 12 , G± we have that

Skk f = (Skk f )Jk , Gkk g = Pk Gkk Jk g, ± ± ± S12 f = (S12 f )J1 , G± g 12 = P1 (−G11 S12 G22 J2 g) (k = 1, 2) and hence (from (5.12) and (5.15)) ± f G± S± f = G(J1 S11 f + J2 S22 f + J1 S12 ± ± = P1 G11 J1 S11 f + P1 G11 J1 S12 f + P2 G22 J2 S22 f − P1 G11 J1 S12 G22 J2 S22 f ± ± = P1 G11 S11 f + P2 G22 S22 f + P1 G11 J1 S12 f − P1 G11 J1 S12 f = P1 f + P2 f = f,

i.e.

G± S± f = f

∀f ∈ L2 (R; Cn ) and G± is a bounded operator onto S± L2 (R; Cn ). But S± f = T± S± f and for the operator T± , deﬁned by π 1 (x))J1 T± h = h(J1 V−∞ e∓ 2 B1 (x) Γ−1 (I + iB π ± 2 B2 (x) −1 +J2 V−∞ e Γ (I − iB2 (x))J2 )Π∗ (x)(x − i)

there exists the bounded operator (using the properties of the gamma-function)

∗ 1 (x))e± 2 B1 (x) V−∞ (x)J1 T±−1 h = h(x + i)Π(x)(J1 i lim Γ(εI + iB π

+J2 (−i) lim Γ(εI − ε→0

ε→0 ∗ (x)J2 ). 2 (x))e± π2 B 1 (x) V iB −∞

Hence S± has bounded inverse operator and −1 S± g = (S± )−1 T±−1 g = G± T±−1 g

and the proof is complete.

(5.16)

The next two theorems give the form and the existence of the wave operators for the couple (A∗ , A).

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Theorem 5.2. Let for the model A, deﬁned by (2.12) the conditions (i), (ii), (iii) (page 358) hold. Then there exist the limits ∗ ∗ lim (eitA e−itA f, g) = (S∓ S∓ f, g)

t→±∞

(5.17)

for all f, g ∈ L2 (R; Cn ) and the operator A satisﬁes the equality ∗ ∗ S± A = S± S± QS±

(5.18)

± , where Qf (x) = xf (x) and S± are deﬁned by (4.50). onto the spaces D 0 Proof. The equality (5.17) follows from the obtained asymptotics (4.74) and the boundness and the form (4.51) of S± . Let us consider the function of t ∈ R: (eitx S± f (x), eisx S± f (x)) (f ∈ L2 (R; Cn ), s ∈ R) then d itx (e S± f (x), eisx S± f (x)) = (ixeitx S± f (x), eisx S± f (x)) dt for all t, s ∈ R. On the other side

(5.19)

d itx (e S± f (x), eisx S± f (x)) = (ieitx S± Af (x), eisx S± f (x)) (5.20) dt ± and (5.20) is obtained analogously as in the bounded case (see for each f ∈ D 0 [12], Theorem 7). The equalities (5.19) and (5.20) imply that (xeitx S± f (x), eisx S± f (x)) = (eitx S± Af (x), eisx S± f (x))

(5.21)

and for t = s = 0 from (5.21) we have (xS± f (x), S± f (x)) = (S± Af (x), S± f (x)) and

∗ ∗ S± Af (x), f (x)) QS± f (x), f (x)) = (S± (S±

± . ∀f ∈ D 0

(5.22)

Now ∗ ∗ S± Af (x), g(x)) (S± QS± f (x), gf (x)) = (S±

and hence

∗ ∗ S± A)f = 0 QS± − S± (S±

± , ∀g ∈ L2 (R; Cn ) ∀f ∈ D 0 ± ∀f ∈ D 0

and the proof is complete.

Theorem 5.3. Let for the model A, deﬁned by (2.12), the conditions (i), (ii), (iii) (page 358) hold. Then there exist the strong limits ∗

s − lim eitA e−itA t→±∞

onto L2 (R; Cn ).

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Proof. Let us denote

IEOT

∗

∗ T−t . W (t) = eitA e−itA = T−t

Let f = (A − iI)−1 (A − λ0 I)−1 h = (A − iI)−1 g, (h ∈ S(R, Cn ), Im λ0 > 0). Then after calculations it follows that d ∗ T Tt f = −(A∗ + iI)Tt∗ Bi Tt g = −(A∗ + iI)Tt∗ Φ∗ LΦTt g. dt t Now from the representation m Bi f = (f, Φ∗ eα )(Leα , eβ )Φ∗ eβ , α,β=1

where {eα }, α = 1, . . . , m is an orhonormal basis in Cm (see, for example, [21]), we obtain for t1 , t2 > 0 that

t2 ||Tt∗2 Tt2 f (x) − Tt∗1 Tt1 f (x)||2L2 = || (A∗ + iI)Tτ∗ Φ∗ LΦTτ g(x)dτ ||2L2 t1

m

t2 = || (A∗ + iI)Tτ∗ ( (Tτ g(x), Φ∗ (x)eα )(Leα , eβ )Φ∗ eβ )dτ ||2L2 α,β=1

t2

t2 |(Leα , eβ )|2 |(ΦTτ g(x), eα )|2 dτ ||(A∗ α,β=1 t1 t1

t1

≤M

m

+ iI)Tτ∗ Φ∗ eβ ||2L2 dτ

where M > 0 is a suitable constant. But from (4.65) and similarly for t1 , t2 < 0, 0 I)−1 k, Im λ 0 < 0 and (4.69) if follows that ||ΦTt g(x)||2 2 ∈ L2 (R) as g = (A − λ L 0 I)−1 k = (A − λ0 I)−1 h for all k ∈ S(R, Cn ). a function of t for all g = (A − λ On the other hand ||ΦTτ (A − iI)−1 ||L2 ∈ L2 (R) as a function of τ and from the inequality ||(A∗ + iI)Tτ∗ Φ∗ eα ||L2 ≤ ||ΦTτ (A − iI)−1 ||L2 we have that ||(A∗ + iI)Tτ∗ Φ∗ eα ||L2 ∈ L2 (R). Consequently, ||Tt∗2 Tt2 f (x) − Tt∗1 Tt1 f (x)||2L2 → 0 as t1 , t2 → ±∞ onto the dense subset of L2 (R; Cn ). Then from the uniformly ∗ bounded set of operators W (t) = T−t T−t (t ∈ R) onto the dense subset it follows (see, for example, Lemma III, 3.5, [10]) that there exist the strong limits s − lim W (t) onto L2 (R; Cn ). t→±∞

The proof is complete.

From Theorem (5.2) and Theorem 5.3 it follows that there exist the wave ∗ S∓ . On the other operators W± (A∗ , A) as strong limits and W± (A∗ , A) = S∓ hand the existence of the bounded inverse operator of S± allows to determine the scattering operator by the equality ∗ −1 ∗ S+ ) S− S− W−−1 (A∗ , A)W+ (A∗ , A) = (S+

onto L2 (R; Cn ), where S± are deﬁned by (4.50).

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The equality (5.18) and the invertibility of S± give the similarity of A and the operator Q of multiplication by the independent variable and −1 Q S± A = S±

± and S−1 is deﬁned by (5.16). onto D ± 0

6. Example In this part we present an example of non-selfadjoint Schr¨ odinger operator that are a coupling of a dissipative operator and an anti-dissipative one with diﬀerent domains of the operator and its adjoint. We consider non-selfadjoint Schr¨ odinger operators on a half-line [0, ∞). Let in L2 (0, ∞) the operator l(y) be deﬁne as l(y) = −y + p(x)y, where p(x) is an integrable real valued function which realizes the case of a limit point. The symmetric operator Ay = −y + p(x)y y(0) = y (0) = 0 has deﬁciency indices (1, 1). Now we consider the regular extensions Ah1 and Ah2 of the operator A, deﬁned by Ah1 y = −y + p(x)y (6.1) y (0) − h1 y(0) = 0, Ah1 y = −y + p(x)y (6.2) y (0) − h2 y(0) = 0 with domains DAh1 and DAh2 , where Im h1 > 0, Im h2 < 0 and DAhk (k = 1, 2) is the set of all functions f ∈ L2 (0, ∞) with an absolutely continuous f , l(f ) ∈ L2 (0, ∞) and f satisﬁes the condition f (0) − hk f (0) = 0. This implies that Im (Ah1 f, f ) ≥ 0

∀f ∈ DAh1 ,

Im (Ah2 f, f ) ≤ 0

∀f ∈ DAh2 ,

i.e. Ah1 is a dissipative operator onto DAh1 and Ah2 is an anti-dissipative operator onto DAh2 . Let θ(x, λ) and ϕ(x, λ) be the solutions of the diﬀerential equation l(y) = λy

(6.3)

which satisfy the conditions θ(0, λ) = cos β, θx (0, λ) = sin β,

ϕ(0, λ) = sin β, ϕx (0, λ) = − cos β

(for example when β = 0) and let m(λ) be the corresponding Weyl function. Then the function g(x, λ) = θ(x, λ) + m(λ)ϕ(x, λ) (6.4)

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belongs to the space L2 (0, ∞) and it is a solution of the equation (6.3). Then using the Wronskian W {f, f } (where the Wronskian of two functions is deﬁned to be W {f, g} = f g − f g) is it easy to see that ∞ Im m(λ) = −Im λ |g(x, λ)|2 dx, 0

i.e. Im m(λ) < 0 when Im λ > 0. Let α and α be regular points of Ah1 and Ah2 and Im α > 0. Let us deﬁne the operators Φ1 , Φ2 : L2 (0, ∞) −→ C by the equalities √ ∞ 2Im h1 −Im m(α) √ , Φ1 f (x) = f (x)g(x, α)dx. ||g(x, α)||L2 |h1 + m(α)| Im α 0

∞ Φ2 f (x) = 0

√ −2Im h2 −Im m(α) √ . f (x)g(x, α)dx. ||g(x, α)||L2 |h2 + m(α)| Im α

√ 2Im h1 −Im m(α) √ = kg(x, α) , ||g(x, α)||L2 |h1 + m(α)| Im α √ −2Im h2 −Im m(α) ∗ √ . Φ1 (k) = kg(x, α) ||g(x, α)||L2 |h2 + m(α)| Im α for k ∈ C. Now direct calculations show that

(6.5)

Hence

Φ∗1 (k)

(6.6)

∗ ∗ B1α f = (iR1α − iR1α + 2Im αR1α R1α )f = (f, Φ∗1 (1))Φ∗1 (1) = Φ∗1 Φ1 f, ∗ ∗ B2α f = (iR2α − iR2α + 2Im αR2α R2α )f = −(f, Φ∗2 (1))Φ∗2 (1) = −Φ∗2 Φ2 f where Rkα = (Ahk − α)−1 , k = 1, 2. Let the matrix L have the form 1 s L= , (s ∈ C). s −1

We consider the model Ah (h = (h1 , h2 )), deﬁned by the next relations Ah f (x) = Ah Gg(x) = Ah G(g1 (x), g2 (x)) d2 − dx2 + p(x) αK g1 (x) = 2 g2 (x) 0 − d 2 + p(x)

(6.7)

dx

where Kg2 f (x) = −is

m(α) + h1 (1 + 2iIm α(A∗h1 − α)−1 )Φ∗1 Φ2 (Ah2 − α)g2 (x), (6.8) m(α) + h1

G = I + P1 KP2 , G−1 = I − P1 KP2 , P1 (f1 (x), f2 (x)) = (f1 (x), 0), P2 (f2 (x), f2 (x)) = (0, f2 (x)) are orthogonal projectors in L2 (0, ∞; C2 ) and Ah has a domain DAh = G(DAh1 ⊕ DAh2 ).

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It can be considered the operators from the form (6.1) and (6.2) with diﬀerent potentials p1 (x) and p2 (x) correspondingly which does not change the essence of the considered example Ah . By the usual variation of parameters formula which express a solution of an inhomogeneous second order ODE in terms of two linearly independent solutions θ and ϕ of the corresponding homogeneous equation, the resolvent (A∗h2 − α)−1 has the form

∞ 1 (A∗h2 − α)−1 u(x) = (ϕ(x, α) − m(α)+h g(x, α)) g(ξ, α)u(ξ)dξ

x

+g(x, α) (ϕ(ξ, α) − 0

2

1 g(ξ, α))u(ξ)dξ m(α)+h2

x

(u ∈ L2 (0, ∞)).

The operator Ah , deﬁned by (6.7) is a regular coupling of the dissipative operator Ah1 and of the anti-dissipative operator Ah2 with diﬀerent domains DAh of Ah and DA∗h of its adjoint A∗h . The absolutely continuous part of the operator Ah is unitary equivalent to the model A considered in this paper and this diﬀerential operator possesses the properties obtained above. The absolutely continuous part of Ah is similar to the operator of multiplying by the independent variable (after a suitable change of the variables), there exist the wave operators as strong limits W± (A∗h , Ah ) = ∗ s − lim eitAh e−itAh and the scattering operator W−−1 (A∗h , Ah )W+ (A∗h , Ah ). t→±∞

The presented example shows that a broad class of diﬀerential operators are operators from the considered class of operators A of a coupling of dissipative and anti-dissipative K r -operators with diﬀerent domains of A and its adjoint A∗ . It has to mention also that the characteristic function of the operator (6.1) is obtained in two diﬀerent ways — in terms of the characteristic function of A. Kuzhel in [21] and in terms of the rigged Hilbert spaces in [1]. Finally, the obtained results in this paper concern the class of K r -operators presented as a coupling of dissipative and anti-dissipative operators with real absolutely continuous spectra. In the general case of the model A with an arbitrary real spectrum the similar results can be obtained. Because the singular part of the spectrum probably does not change the asymptotics of the corresponding nondissipative processes. The considerations of these questions are forthcoming. Acknowledgements We would like to express our thanks to the referee for his useful comments and advice.

References [1] S.V. Belyi, S. Hassi, H.S.V. de Snoo, E.R. Tsekanovskii, On the realization of inverse Stieltjes functions, Proceedings of MTNS-2002, University of Notre Dame (2002). [2] S.V. Belyi, E.R. Tsekanovskii, Realization theorems for operator-valued R-functions, Oper. Theory Adv. Appl., 98 (1997), 55-91.

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[3] S. Belyi, E. Tsekanovskii, Multiplication theorems for J-contractive operator-valued functions, Fields Institute Communications, vol. 25 (2000), 187-210. [4] M.Sh. Birman, D.R. Yafaev, A general scheme in the stationary scattering theory, Wave propagation, scattering theory, 2, vol. 157 (1993), 87-113. [5] G.S. Borisova, A new form of the triangular model of M.S.Livˇsic for a class of nondissipative operators, Comptes rendus de l’Acad`emie bulgare des Sciences, Tome 53, No 10 (2000), 9-12. [6] G.S. Borisova, The operators Aγ = γA + γA∗ for a class of nondissipative operators A with a limit of the corresponding correlation function, Serdica Math. J., 29 (2003), 109-140. [7] M.S. Brodskij, Triangular and Jordan representations of linear operators, Transl. Math. Monographs 32, Amer. Math. Soc., Providence, 1971. [8] M.S. Brodskii, M.S. Livsic, Spectral analysis of nonselfadjoint operators and intermediate systems, Uspechi Mat. Nauk 13 (1958), 3-85, Engl. Transl. Amer. Math. Soc. Transl. (2), 13 (1960), 265-346. [9] I.C. Gohberg, M.G. Krein, On a description of contraction operators similar to unitary ones, Funk. Anal. i Prilozh. 1 (1967), 38-60. [10] T. Kato. Perturbation theory for linear operators, Springer-Verlag, 1966. [11] K.P. Kirchev, On a certain class of non-stationary random processes, Theor. Funktsij Funk. Anal. i Prilozh., 14 (1971), 150-167 (Russian). [12] K.P. Kirchev, G.S. Borisova, Nondissipative curves in Hilbert spaces having a limit of the corresponding correlation function, Integral Equations Operator Theory, vol. 40 (2001), 309-341. [13] K.P. Kirchev, G.S. Borisova, Triangular models and asymptotics of continuous curves with bounded and unbounded semigroup generators, Serdica Math. J., 31, (2005), 95-174. [14] K. Kirchev, G. Borisova, A triangular model of regular couplings of dissipative and antidissipative operators, Comptes rendus de l’Acad`emie bulgare des Sciences, 58, No 5, (2005), 481-486. [15] K. Kirchev, V. Zolotarev, Nonstationary curves in Hilbert spaces and their correlation functions I, Integral Equations Operator Theory, vol. 19 (1994), 270-289. [16] K. Kirchev, V. Zolotarev, Nonstationary curves in Hilbert spaces and their correlation functions II, Integral Equations Operator Theory, vol. 19 (1994), 447-457. [17] A. Kuzhel, Reduction of unbounded nonselfadjoint operators to the triangular form, Dokl. Akad. Nauk SSSR 119 (1958), No. 5, 868-871 (Russian). [18] A. Kuzhel, Spectral analysis of unbounded nonself-adjoint operators. Dokl. Akad. Nauk SSSR, 125, 1 (1959), 35-37 (Russian). [19] A. Kuzhel, Multiplication theorem for characteristic matrix function of nonunitary operators, Nauch. Dokl. Vish. Shk. (1959), No. 3, 33-41 (Russian). [20] A. Kuzhel, Regular extensions of Hermitian operators, Dokl. Akad. Nauk SSSR 251 (1980), No.1, 30-33 (Russian). [21] A. Kuzhel, Characteristic functions and models of nonself-adjoint operators, Kluwer Academic Publishers Group, Dordrecht, 1996.

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[22] M.S. Livˇsic, On a certain class of linear operators in Hilbert space, Mat. Sb., 19 (61) (1946), 236-260 (Russian). [23] M.S. Livˇsic, Operators, oscillations, waves (Open Systems), Transl. Math. Monographs 34, 1972. [24] M.S. Livˇsic, N. Kravitsky, A. Markus, V. Vinnikov, Theory of commuting nonselfadjoint operators, Kluwer Academic Publishers Group, Dordrecht, 1995. [25] M.S. Livˇsic, A.A. Yantsevich, Operator colligations in Hilbert spaces, Engl. Transl., J.Wiley, N.Y., 1979. [26] M.M. Malamud, A criterion for a closed operator to be similar to a selfadjoint operator, Ukrain. Math. Zh. 37 (1985), No.1, 49-56 (Russian). [27] S.N. Naboko, Conditions of similarity of operators to unitary or selfadjoint ones, Funk. Anal. i Prilozh. 18 (1984), No. 1, 16-27 (Russian). [28] S.N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, Wave propagation, scattering theory, 2, vol. 157 (1993), 127-151. [29] M.A. Naimark, Linear diﬀerential opearators, Moskow, 1969 (Russian). [30] V.P. Potapov, The multiplicative structure of J-contractive matrix function, Trudy Mosk. Mat. Obshch., 4 (1955), 125-236 (Russian). [31] L.A. Sakhnovich, Dissipative operators with an absolutely continuous spectrum, Works Mosk. Mat. Soc. vol. 19, 1968, 211-270 (Russian). [32] B. Sz.-Nagy, C. Foias, Harmonic analysis of operators in Hilbert space, NorthHolland, 1970. [33] E. Tsekanovskii, Triangular models of unbounded accretive operators and regular factorization of their characteristic operator-valued functions, Dokl. Akad. Nauk SSSR, vol. 297, 3 (1987), 552-556. Engl. Transl., Soviet Math. Dokl., 36, 3 (1988), 512-515. [34] E.R. Tsekanovskii, Yu.L. Shmulyan, The theory of biextensions of operators in rgged Hilbert spaces. Unbounded operator couplings and characteristic functions, Uspekhi Mat. Nauk 32 (1977), No5, 69-124 (Russian). Kiril P. Kirchev Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G.Bonchev Str., Bl. 8 1113 Soﬁa Bulgaria e-mail: [email protected] Galina S. Borisova Faculty of Mathematics and Informatics Shumen University 9712 Shumen Bulgaria e-mail: [email protected] Submitted: July 20, 2005 Revised: December 6, 2006

Integr. equ. oper. theory 57 (2007), 381–396 c 2006 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030381-16, published online December 26, 2006 DOI 10.1007/s00020-006-1461-1

Integral Equations and Operator Theory

Boundedness of Commutators on Hardy-Type Spaces Yan Lin and Shanzhen Lu Abstract. In this paper, the boundedness of the commutator generated by strongly singular Calder´ on-Zygmund operator and a Lipschitz function is discussed on the classical Hardy spaces and the Herz-type Hardy spaces. The authors also get the boundedness of the strongly singular Calder´ on-Zygmund operator itself and the commutator generated by strongly singular Calder´ onZygmund operator and a BMO function on the Herz-type Hardy spaces. Mathematics Subject Classification (2000). Primary 42B20; Secondary 42B30. Keywords. Strongly singular Calder´ on-Zygmund operator, commutator, Lipschitz space, Hardy space, Herz-type Hardy space, BMO.

1. Introduction The introduction of the strongly singular Calder´on-Zygmund operator is motivated a by a class of multiplier operators whose symbol is given by e(i|ξ| ) /|ξ|β away from the origin, 0 < a < 1, β > 0. Fefferman and Stein [5] have enlarged the multiplier operators onto a class of convolution operators. Coifman [4] has also considered a related class of operators, for n = 1. The strongly singular non-convolution operator was introduced by Alvarez and Milman [2], whose properties are similar to those of Calder´on-Zygmund operator, but the kernel is more singular near the diagonal than those of the standard case. 0

Definition 1.1. Let T : S → S be a bounded linear operator. T is called a strongly singular Calder´ on-Zygmund operator if the following conditions are satisfied. (1) T can be extended into a continuous operator from L2 into itself. This paper is supported by the National Natural Science Foundation of China(10571014) and the Doctoral Programme Foundation of Institution of Higher Education of China (20040027001). The second author is the corresponding author.

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(2) There exists a continuous function K(x, y) away from the diagonal {(x, y) : x = y}, such that |K(x, y) − K(x, z)| + |K(y, x) − K(z, x)| ≤ C

|y − z|δ δ

|x − z|n+ α

,

if 2|y − z|α ≤ |x − z| for some 0 < δ ≤ 1, 0 < α < 1, Z and satisfies hT f, gi = K(x, y)f (y)g(x)dydx, for f, g ∈ S with disjoint supports. (3) For some n(1−α)/2 ≤ β < n/2, both T and its conjugate operator T ∗ can be extended into continuous operators from Lq to L2 , where 1/q = 1/2 + β/n. In 1986, Alvarez and Milman [2, 3] proved that the strongly singular Calder´onZygmund operator T is of (L∞ , BMO) type and weak (L1 , L1 ) type, then T is bounded on Lp (Rn ), 1 < p < ∞, by interpolation theory. The estimate of sharp maximal function (T f )] can be obtained from [3], then the weighted norm inequality was established. By the well known result of Alvarez-Bagby-Kurtz-P´erez [1], the commutator [b, T ] generated by strongly singular Calder´on-Zygmund operator T and a BMO function b is bounded on Lp (Rn ), 1 < p < ∞. There are some other links between the boundedness of the commutator [b, T ] and the smoothness of the function b. Recently, the authors in [6] have obtained the boundedness of the commutator [b, T ] generated by strongly singular Calder´onZygmund operator T and a Lipschitz function b from Lp (Rn ) to Lq (Rn ). To be precise, if b ∈ Λ˙ β0 (Rn ), 0 < β0 < 1, then [b, T ] is bounded from Lp (Rn ) to Lq (Rn ), < p < βn0 , 1q = p1 − βn0 , and if 0 < β0 < min 1, n(2β−n(1−α)) , where n(1−α)+2β 2β 2β+n(1−α) then [b, T ] is bounded from Lp (Rn ) to Lq (Rn ), where 1 < p < βn0 , 1q = p1 − βn0 . In this paper, we will focus on the case 0 < p ≤ 1. Precisely, we are interested in the boundedness of [b, T ] on Hardy type spaces, which include the classical Hardy spaces and the Herz-type Hardy spaces. Let us first recall some necessary definitions and notations. Definition 1.2. Let b be a function on Rn and 0 0; 1 1 2−p ; (2) kak 0 , r)| R L2 ≤ |B(x (3) RRn a(x)xγ dx = 0, 0 ≤ |γ| ≤ [n(1/p − 1)]; (4) Rn a(x)b(x)dx = 0. Here and in what follows, for t ∈ R, [t] is the largest integer no more than t.

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Definition 1.3. For 0 < p ≤ 1, define Hbp (Rn ) ∞ ∞ X X 0 p = f :f = λj aj in S , where aj is a (p, 2) b-atom, and |λj | < ∞ . j=−∞

j=−∞

Moreover, kf k

Hbp

∼ inf

X ∞

1/p |λj | . p

j=−∞

Here the infimum is taken over all decompositions of f as above. It is easy to see that Hbp (Rn ) ⊂ H p (Rn ) for 0 < p ≤ 1. Definition 1.4. Let Bk = {x ∈ Rn : |x| < 2k }, Ek = Bk \ Bk−1 and χk = χEk be the characteristic function of the set Ek for k ∈ Z. Let α ∈ R and 0 < p, q ≤ ∞. The inhomogeneous Herz space Kqα,p (Rn ) is defined by Kqα,p (Rn ) = {f : f ∈ Lqloc (Rn ) and kf kKqα,p < ∞}, where kf kKqα,p =

kf χB0 kpq +

∞ X

2kαp kf χk kpq

1/p

k=1

with usual modification made when p = ∞. Definition 1.5. Let α ∈ R and 0 < p, q < ∞. The inhomogeneous Herz-type Hardy space HKqα,p (Rn ) is defined by HKqα,p (Rn ) = {f ∈ S 0 (Rn ) : G(f ) ∈ Kqα,p (Rn )}, and kf kHKqα,p = kG(f )kKqα,p . Here, G(f ) is the grand maximal function of f . Obviously, HKp0,p (Rn ) = H p (Rn ) for 0 < p < ∞. Thus, the Herz-type Hardy spaces extend the classical Hardy spaces. If 0 < q < ∞, it can be proved that HKqα,p (Rn ) = Kqα,p (Rn ) when −n/q < α < n(1 − 1/q), but HKqα,p (Rn ) $ Kqα,p (Rn ) when α ≥ n(1 − 1/q); see [7, 8]. The Herz-type Hardy spaces have central atomic decomposition characterizations, which are convenient in applications. Definition 1.6. Let α ∈ R and 1 < q < ∞. A function a on Rn is called a central (α, q) atom if (1) supp a ⊂ B(0, r) for some r > 0; (2) RkakLq ≤ |B(0, r)|−α/n ; (3) Rn a(x)xγ dx = 0 for 0 ≤ |γ| ≤ [α − n(1 − 1/q)].

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Definition 1.7. Let b be a function on Rn . Let α ∈ R and 1 < q < ∞. A function a on Rn is called a central (α, q) b-atom if (1) a R is a central (α, q) atom. (2) Rn a(x)b(x)dx = 0. Definition 1.8. Let 0 < p < ∞, 1 < q < ∞ and α ≥ n(1 − 1/q). Define ∞ ∞ X X α,p n 0 p Hb Kq (R ) = f : f = λj aj in S and |λj | < ∞ , j=0

j=0

where each aj is a central (α, q) b-atom supported on Bj = B(0, 2j ). Moreover, X 1/p ∞ p kf kHb Kqα,p ∼ inf |λj | . j=0

Here the infimum is taken over all decompositions of f as above. Let us now formulate our main results. For the classical Hardy spaces, we get Theorem 1.9. Let T be a strongly singular Calder´ on-Zygmund operator, n ≥ 2 and α, β, δ be as in Definition 1.1. Denote 1/p0 = 1/2+{β(δ/α+n/2) n(δ/α−δ+β)}. If b ∈ Λ˙ β0 (Rn ), 0 < β0 < 1, then [b, T ] maps H p (Rn ) continuously into weak Lq (Rn ), where 1/q = 1/p − β0 /n, max{p0 , n/(n + β0 )} ≤ p ≤ 1. Theorem 1.10. Let T be a strongly singular Calder´ on-Zygmund operator, n ≥ 2, α, β, δ be as in Definition 1.1 and n(1 − α)/2 < β < n/2. Denote 1/p0 = 1/2 + {β(δ/α + n/2) n(δ/α − δ + β)}. If b ∈ Λ˙ β0 (Rn ), 0 < β0 < 1, then [b, T ] is bounded from Hbp (Rn ) to Lq (Rn ), where 1/q = 1/p − β0 /n, max{p0 , n/(n + β0 )} ≤ p ≤ 1. For the Herz-type Hardy spaces, we get Theorem 1.11. Let T be a strongly singular Calder´ on-Zygmund operator, n ≥ 2 and α, β, δ be as in Definition 1.1. Suppose 0 < p < ∞, (n(1−α)+2β)/2β < q1 < n/β0 , 1/q2 = 1/q1 − β0 /n and n(1 − 1/q1 ) ≤ α0 < n(1 − 1/q1 ) + δ. If b ∈ Λ˙ β0 (Rn ), 0 < β0 < 1, then [b, T ] maps Hb Kqα10 ,p (Rn ) continuously into Kqα20 ,p (Rn ). Theorem 1.12. Let T be a strongly singular Calder´ on-Zygmund operator, α, β, δ be as in Definition 1.1 and n(1−α)/2 < β < n/2. Suppose 0 < p < ∞, 1 < q1 < n/β0 , n ˙ 1/q2 = 1/q1 −n β0 /n and n(1 − o 1/q1 ) ≤ α0 < n(1 − 1/q1 ) + δ. If b ∈ Λβ0 (R ),

, then [b, T ] maps Hb Kqα10 ,p (Rn ) continuously into 0 < β0 < min 1, n(2β−n(1−α)) 2β+n(1−α) Kqα20 ,p (Rn ).

Theorem 1.13. Let T be a strongly singular Calder´ on-Zygmund operator and α, β, δ be as in Definition 1.1. Suppose 0 < p < ∞, 1 < q < ∞ and n(1 − 1/q) ≤ α0 < n(1 − 1/q) + δ. Then T maps HKqα0 ,p (Rn ) continuously into Kqα0 ,p (Rn ).

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Theorem 1.14. Let T be a strongly singular Calder´ on-Zygmund operator and α, β, δ be as in Definition 1.1. Suppose 0 < p < ∞, 1 < q < ∞ and n(1 − 1/q) ≤ α0 < n(1 − 1/q) + δ. If b ∈ BMO(Rn ), then [b, T ] maps Hb Kqα0 ,p (Rn ) continuously into Kqα0 ,p (Rn ).

2. Main lemma Lemma 2.1 ([8]). Let 0 < p < ∞, 1 < q < ∞ and α ≥ n(1 − 1/q). A P distribution f ∞ on Rn belongs to HKqα,p (Rn ) if and only if it can be written as f = k=0 λk ak in distributional sense with each ak a central (α, q) atom supported on Bk = B(0, 2k ) P∞ and k=0 |λk |p < ∞. Moreover, X 1/p ∞ p kf kHKqα,p ∼ inf |λk | . k=0

Here the infimum is taken over all decompositions of f as above. From Definition 1.8 and Lemma 2.1, it is easy to get that Hb Kqα,p (Rn ) ⊂ HKqα,p (Rn ).

3. Proofs of theorems P∞ Proof of Theorem 1.9. For f ∈ H p (Rn ), we can write f = j=−∞ λj aj with each P∞ aj a (p, 2) atom supported on Bj = B(xj , rj ) and j=−∞ |λj |p < ∞. Then X ∞ ∞ X [b, T ]f (x) = λj [b(x) − b(xj )]T aj (x) + T λj [b(xj ) − b]aj (x). j=−∞

j=−∞

For any λ > 0, |{x ∈ Rn : |[b, T ]f (x)| > λ}|1/q X 1/q ∞ λj [b(x) − b(xj )]T aj (x) > λ/2 ≤ x ∈ Rn : j=−∞

+ x ∈ Rn

∞ 1/q X λj [b(xj ) − b]aj (x) > λ/2 : T j=−∞

:= I + II. Let us estimate II first. Since 1 ≤ q < 2, by the weak (Lq , Lq ) boundedness of T and H¨ older’s inequality, we can get that ∞ X −1 II ≤ Cλ |λj |k[b(xj ) − b]aj kq j=−∞

≤ CkbkΛ˙ β λ−1

∞ X

0

j=−∞

Z |λj | Bj

|x − xj |β0 q |aj (x)|q

1/q

386

Lin and Lu ∞ X

≤ CkbkΛ˙ β λ−1

|λj |rjβ0 kaj k2 |Bj |1/q−1/2

0

≤ CkbkΛ˙ β λ−1

IEOT

j=−∞ ∞ X

0

|λj |

j=−∞

≤ CkbkΛ˙ β λ−1

X ∞

0

|λj |p

1/p .

j=−∞

Let us now estimate I. q 1/q Z X ∞ −1 λj [b(x) − b(xj )]T aj (x) dx I ≤ Cλ Rn j=−∞

≤ Cλ

∞ X

−1

Z |λj |

1/q |[b(x) − b(xj )]T aj (x)| dx . q

Rn

j=−∞

For every rj , there are two cases. Case 1: rj > 1. 1/q |[b(x) − b(xj )]T aj (x)|q dx

Z Rn

1/q |[b(x) − b(xj )]T aj (x)| dx

Z

q

≤ 2Bj

Z +

q

1/q

|[b(x) − b(xj )]T aj (x)| dx

(2Bj )c

:= I1 + I2 . By H¨ older’s inequality and the boundedness of T on L2 , we obtain that Z I1 ≤ kbkΛ˙ β

β0 q

|x − xj |

0

1/q |T aj (x)| dx q

2Bj

≤ CkbkΛ˙ β rjβ0 0

Z

2

|T aj (x)| dx

1/2

|Bj |1/q−1/2

2Bj

≤ CkbkΛ˙ β rjβ0 kaj k2 |Bj |1/q−1/2 0

≤ CkbkΛ˙ β . 0

When x ∈ (2Bj )c and y ∈ Bj , 2|y − xj |α < 2rjα < 2rj ≤ |x − xj | since rj > 1. By the cancellation condition of aj , the Minkowski inequality and (2) of Definition 1.1, we can give the estimate of I2 .

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Boundedness of Commutators on Hardy-Type Spaces

Z I2 = (2Bj )c

Z |b(x) − b(xj )|q

Bj

387

q 1/q K(x, y)aj (y)dy dx

q 1/q Z [K(x, y) − K(x, xj )]aj (y)dy dx = |b(x) − b(xj )|q Bj (2Bj )c Z 1/q Z ≤ kbkΛ˙ β |aj (y)| |x − xj |β0 q |K(x, y) − K(x, xj )|q dx dy Z

0

(2Bj )c

Bj

Z |aj (y)|

Z ≤ CkbkΛ˙ β

0

(2Bj )c

Bj

≤ CkbkΛ˙ β rjδ

Z

1 p

Z |aj (y)|dy

0

Notice that

|x − xj |β0 q

|x−xj |≥2rj

Bj 1 p0

≤

1/q |y − xj |δq dx dy |x − xj |nq+δq/α 1/q (. 3.1) |x − xj |−(n−β0 )q−δq/α dx

<

n+δ n

<

αn+δ αn ,

thus

δ 1 < 0. −(n − β0 )q − δq/α + n = q n( − 1) − p α Then n(1/p−1)−δ/α

I2 ≤ CkbkΛ˙ β rjδ kaj k2 |Bj |1/2 rj 0

δ+n(1/p−1)−δ/α−n/p+n

≤ CkbkΛ˙ β rj 0

δ(1−1/α)

= CkbkΛ˙ β rj 0

≤ CkbkΛ˙ β .

(3.2)

0

Case 2: 0 < rj ≤ 1. ej = B(xj , rσ ), where σ = Let B j

δ−n(1/p−1) δ/α−n(1/p−1) .

It is easy to see that 0 < σ ≤ α.

Thus 1/q |[b(x) − b(xj )]T aj (x)|q dx

Z Rn

1/q |[b(x) − b(xj )]T aj (x)|q dx

Z ≤ ej 2B

Z +

1/q |[b(x) − b(xj )]T aj (x)|q dx

ej )c (2B

:= I3 + I4 . since p ≥ p0 . By H¨older’s inequality and the Note that σ ≥ n(1/p−1/2)−β n(1/p−1/2) q0 2 (L , L ) boundedness of T in (3) of Definition 1.1, where 1/q0 = 1/2 + β/n, we can get that

388

Lin and Lu

Z I3 ≤ kbkΛ˙ β

0

IEOT

|x − xj |β0 q |T aj (x)|q dx

1/q

ej 2B

≤ CkbkΛ˙ β rjσβ0

Z

0

1/2 ej |1/q−1/2 |T aj (x)|2 dx |B

ej 2B

ej |1/q−1/2 ≤ CkbkΛ˙ β rjσβ0 kaj kq0 |B 0 σβ0 +σn(1/q−1/2)

≤ CkbkΛ˙ β rj 0

kaj k2 |Bj |1/q0 −1/2

σn(1/p−1/2)+n(1/2+β/n−1/p)

≤ CkbkΛ˙ β rj 0

n(1/p−1/2)−β+n(1/2+β/n−1/p)

≤ CkbkΛ˙ β rj 0

= CkbkΛ˙ β .

(3.3)

0

ej )c and y ∈ Bj , 2|y − xj |α < 2rα ≤ 2rσ ≤ |x − xj | since When x ∈ (2B j j 0 < rj ≤ 1 and 0 < σ ≤ α. As in the estimate of (3.1), we can establish the estimate of I4 . 1/q Z Z |x − xj |−(n−β0 )q−δq/α dx I4 ≤ CkbkΛ˙ β rjδ |aj (y)|dy 0

|x−xj |≥2rjσ

Bj

σ(n(1/p−1)−δ/α) ≤ CkbkΛ˙ β rjδ kaj k2 |Bj |1/2 rj 0 σ(n(1/p−1)−δ/α)+δ+n(1−1/p)

≤ CkbkΛ˙ β rj 0

= CkbkΛ˙ β .

(3.4)

0

From the estimates in the two cases above, it follows that Z 1/q q |[b(x) − b(xj )]T aj (x)| dx ≤ CkbkΛ˙ β . 0

Rn

Thus I ≤ CkbkΛ˙ β λ−1

∞ X

0

|λj | ≤ CkbkΛ˙ β λ−1

X ∞

0

j=−∞

|λj |p

1/p .

j=−∞

Combining the estimates of I and II, |{x ∈ Rn : |[b, T ]f (x)| > λ}|1/q ≤ CkbkΛ˙ β λ−1

X ∞

0

|λj |p

1/p .

j=−∞

By taking infimum over all the decompositions of f , we get |{x ∈ Rn : |[b, T ]f (x)| > λ}|1/q ≤ CkbkΛ˙ β λ−1 kf kH p . 0

This completes the proof of Theorem 1.9.

Proof of Theorem 1.10. Let a be a (p, 2) b-atom supported on B = B(x0 , r). By a standard argument, it is enough to show that there is a constant C such that k[b, T ]akq ≤ CkbkΛ˙ β , where C is independent of a. 0

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Boundedness of Commutators on Hardy-Type Spaces

389

Case 1: r > 1. 1/q |[b, T ]a(x)| dx

1/q Z + |[b, T ]a(x)| dx

Z

q

q

k[b, T ]akq ≤

(2B)c

2B

:= I + II. We can find a p1 such that p ≤ 1 < n(1−α)+2β < p1 < 2 since n(1 − α)/2 < 2β β < n/2. Denote 1/q1 = 1/p1 − β0 /n. By H¨older’s inequality and the (Lp1 , Lq1 ) boundedness of [b, T ]([6]), the estimate of I can be established. Z 1/q1 q1 I≤C |[b, T ]a(x)| dx |B|1/q−1/q1 2B Z 1/p1 p1 ≤ CkbkΛ˙ β |a(x)| dx |B|1/q−1/q1 0

B

≤ CkbkΛ˙ β kak2 |B|1/p1 −1/2 |B|1/q−1/q1 0 ≤ CkbkΛ˙ β |B|1/2−1/p |B|1/p1 −1/2 |B|1/q−1/q1 0 = CkbkΛ˙ β . 0

c

When x ∈ (2B) and y ∈ B, 2|y − x0 |α < 2rα < 2r ≤ |x − x0 | since r > 1. By the cancellation condition of b-atom, the Minkowski inequality and (2) of Definition 1.1, we can give the estimate of II. q 1/q Z Z II = K(x, y)[b(x) − b(y)]a(y)dy dx (2B)c

B

q 1/q Z [K(x, y) − K(x, x0 )][b(x) − b(y)]a(y)dy dx = (2B)c B Z 1/q Z q q ≤ |a(y)| |K(x, y) − K(x, x0 )| |b(x) − b(y)| dx dy Z

(2B)c

B

Z |a(y)|

Z ≤ CkbkΛ˙ β

0

≤ CkbkΛ˙ β r 0

|x − y|

(2B)c

B δ

β0 q

Z

1/q |y − x0 |δq dx dy |x − x0 |nq+δq/α

Z

−(n−β0 )q−δq/α

|a(y)|dy

|x − x0 |

1/q dx .

|x−x0 |≥2r

B

The rest of the estimate of II is the same as in (3.2). Thus II ≤ CkbkΛ˙ β . 0

Case 2: 0 < r ≤ 1. e = B(x0 , rσ ), where σ = Let B Write

δ−n(1/p−1) δ/α−n(1/p−1) .

It is easy to see that 0 < σ ≤ α.

[b, T ]a(x) = [b(x) − b(x0 )]T a(x) − T ([b − b(x0 )]a)(x),

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then 1/q |[b(x) − b(x0 )]T a(x)|q dx

Z k[b, T ]akq ≤ Rn

1/q |T ([b − b(x0 )]a)(x)| dx

Z

q

+ Rn

:= III + IV. Let us estimate III first. Z 1/q q III ≤ |[b(x) − b(x0 )]T a(x)| dx e 2B

Z

|[b(x) − b(x0 )]T a(x)|q dx

+

1/q

e c (2B)

:= III1 + III2 . As in the estimates of (3.3) and (3.4) in Theorem 1.9, we can get III ≤ CkbkΛ˙ β . 0

Let us now estimate IV . Z IV ≤

1/q |T ([b − b(x0 )]a)(x)|q dx

e 2B

Z +

1/q |T ([b − b(x0 )]a)(x)| dx q

e c (2B)

:= IV1 + IV2 . n(1/p−β0 /n−1/2)−β since p ≥ p0 . By H¨older’s Note that σ ≥ n(1/p−1/2)−β n(1/p−1/2) > n(1/p−β0 /n−1/2) 2 q0 inequality and the (L , L ) boundedness of T in (3) of Definition 1.1, where 1/q0 = 1/2 + β/n, we can get that Z 1/2 e 1/q−1/2 IV1 ≤ C |T ([b − b(x0 )]a)(x)|2 dx |B| e 2B

Z ≤C

1/q0 e 1/q−1/2 |b(x) − b(x0 )| |a(x)| dx |B| q0

q0

B

e 1/q−1/2 ≤ CkbkΛ˙ β rβ0 kak2 |B|1/q0 −1/2 |B| 0

≤ CkbkΛ˙ β rnσ(1/p−β0 /n−1/2)+n(1/2−1/p)+β0 +β 0

≤ CkbkΛ˙ β rn(1/p−β0 /n−1/2)−β+n(1/2−1/p)+β0 +β 0

= CkbkΛ˙ β . 0

e c and y ∈ B, 2|y−x0 |α < 2rα ≤ 2rσ ≤ |x−x0 | since 0 < r ≤ 1 When x ∈ (2B) and 0 < σ ≤ α.

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391

By the cancellation condition of b-atom, the Minkowski inequality and (2) of Definition 1.1, we can give the estimate of IV2 . q 1/q Z Z IV2 = K(x, y)[b(y) − b(x0 )]a(y)dy dx e c (2B)

B

q 1/q Z = [K(x, y) − K(x, x0 )][b(y) − b(x0 )]a(y)dy dx c e (2B) B Z 1/q Z q q ≤ |a(y)| |K(x, y) − K(x, x0 )| |b(y) − b(x0 )| dx dy Z

e c (2B)

B

Z |a(y)|

Z ≤ CkbkΛ˙ β

0

δ+β0

Z

−nq −

1 p

Z |a(y)|dy

0

Notice that

|y − x0 |

e c (2B)

B

≤ CkbkΛ˙ β r

β0 q

|x−x0 |≥2r σ

B

≤

1 p0

<

1/q |y − x0 |δq dy dx |x − x0 |nq+δq/α 1/q −nq−δq/α . |x − x0 | dx

n+δ n

<

αn+δ αn ,

thus

1 β0 δ 1 δ δq + n = q n( − − 1) − < q n( − 1) − < 0. α p n α p α

Since n/(n + β0 ) ≤ p ≤ 1 and 0 < σ ≤ α, we get IV2 ≤ CkbkΛ˙ β rδ+β0 kak2 |B|1/2 rσ(n(1/p−β0 /n−1)−δ/α) 0

≤ CkbkΛ˙ β rδ+β0 +n(1−1/p)+σ(n(1/p−β0 /n−1)−δ/α) 0

= CkbkΛ˙ β rn(1−σ)(1+β0 /n−1/p)+δ(1−σ/α) 0

≤ CkbkΛ˙ β . 0

Hence k[b, T ]akq ≤ CkbkΛ˙ β . 0

This completes the proof of Theorem 1.10.

P∞

Proof of Theorem 1.11. For f ∈ Hb Kqα10 ,p (Rn ), we can write f = k=0 λk ak with P∞ each ak a (α0 , q1 ) b-atom supported on Bk = B(0, 2k ) and k=0 |λk |p < ∞. Then k[b, T ]f kpK α0 ,p q2

= k[b, T ]f χB0 kpq2 +

∞ X

2jα0 p k[b, T ]f χj kpq2

j=1

≤

X ∞

+

k=0 ∞ X j=2

p |λk |k([b, T ]ak )χB0 kq2

+ 2α0 p

X ∞ k=0

2jα0 p

X ∞ k=0

p |λk |k([b, T ]ak )χj kq2

p |λk |k([b, T ]ak )χ1 kq2

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Lin and Lu

≤C

X ∞

p |λk |k[b, T ]ak kq2

k=0 ∞ X

+C

+C

∞ X

2jα0 p

IEOT

X j−2

j=2

2jα0 p

X ∞

j=2

p |λk |k([b, T ]ak )χj kq2

k=0

p |λk |k([b, T ]ak )χj kq2

k=j−1

:= I1 + I2 + I3 .

(3.5) q1

q2

The estimate of I1 and I3 can be established by the (L , L ) boundedness of [b, T ]. X p ∞ p I1 ≤ CkbkΛ˙ |λk |kak kq1 β0

k=0

X ∞

≤ CkbkpΛ˙

β0

≤ CkbkpΛ˙

β0

|λk |2−α0 k

p

k=0

∞ X    |λk |p 2−α0 pk , if 0 1. |λk |   k=0

k=0

≤ CkbkpΛ˙

∞ X

β0

|λk |p .

(3.6)

k=0

And ∞ X

I3 ≤ CkbkpΛ˙

β0

≤ CkbkpΛ˙ β

0

≤ CkbkpΛ˙

β0

2jα0 p

j=2 ∞ X

β0

2

jα0 p

j=2

X ∞

β0

|λk |2

−α0 k

p

k=j−1 ∞ X jα0 p

∞ X   2 |λk |p 2−α0 pk , if 0 1. |λk | 2   k=j−1

k=j−1

 ∞ k+1  X X    |λk |p 2(j−k)α0 p , if 0 1.

j=2

(3.7)

Vol. 57 (2007)

Boundedness of Commutators on Hardy-Type Spaces

393

Let us now estimate I2 . When j ≥ 2 and 0 ≤ k ≤ j − 2, 2|y|α < 2 · 2kα ≤ 2 · 2(j−2)α ≤ 2 · 2j−2 = 2j−1 ≤ |x|, for any x ∈ Ej and y ∈ Bk . By the cancellation condition of b-atom and (2) of Definition 1.1, we get Z [K(x, y) − K(x, 0)][b(x) − b(y)]ak (y)dy |[b, T ]ak (x)|χj (x) = Bk

Z ≤ CkbkΛ˙ β

0

Bk

|y|δ |x − y|β0 |ak (y)|dy |x|n+δ/α

≤ CkbkΛ˙ β |x|β0 −n−δ/α 2kδ kak kq1 |Bk |1−1/q1 0

≤ CkbkΛ˙ β |x|β0 −n−δ/α 2k(δ−α0 +n(1−1/q1 )) . 0

Therefore, k(δ−α0 +n(1−1/q1 ))

k([b, T ]ak )χj kq2 ≤ CkbkΛ˙ β 2 0

Z

(β0 −n−δ/α)q2

|x|

1/q2 dx

Ej

≤ CkbkΛ˙ β 2k(δ−α0 +n(1−1/q1 )) 2j(β0 −n−δ/α) |Bj |1/q2 0

= CkbkΛ˙ β 2−jα0 2(j−k)(α0 −n(1−1/q1 )−δ) 2j(δ−δ/α) 0

≤ CkbkΛ˙ β 2−jα0 2(j−k)(α0 −n(1−1/q1 )−δ) 0

:= CkbkΛ˙ β 2−jα0 W (j, k). 0

Notice that α0 < n(1 − 1/q1 ) + δ, then p j−2 ∞ X X |λk |W (j, k) I2 ≤ CkbkpΛ˙ β0

≤ CkbkpΛ˙

β0

≤ CkbkpΛ˙

β0

≤ CkbkpΛ˙

β0

j=2

k=0

 j−2 ∞ X  X    |λk |p W (j, k)p , if 0 1.   j=2 k=0 k=0 ∞ ∞ X X  p   |λ | W (j, k)p , if 0 1.

(3.8)

394

Lin and Lu

IEOT

In conclusion, k[b, T ]f kKqα0 ,p ≤ CkbkΛ˙ β 2

X ∞

0

p

1/p

|λk |

.

k=0

of f .

And the desired estimate follows from taking infimum over all decompositions

Proof of Theorem 1.12. When 0 < β0 < min{1, n(2β−n(1−α)) 2β+n(1−α) }, [b, T ] is bounded from Lq1 (Rn ) to Lq2 (Rn ), 1 < q1 < the same as in Theorem 1.11.

n 1 β0 , q2

=

1 q1

−

β0 n ([6]).

Thus the proof here is P∞ Proof of Theorem 1.13. By Lemma 2.1, we can write f = k=0 λk ak with each ak P ∞ a central (α0 , q) atom supported on Bk = B(0, 2k ) and k=0 |λk |p < ∞. By using the (Lq , Lq ) boundedness of T instead of the (Lq1 , Lq2 ) boundedness of [b, T ] in the proof of Theorem 1.11, the rest of the proof here is similar to that in Theorem 1.11. P ∞ Proof of Theorem 1.14. For f ∈ Hb Kqα0 ,p (Rn ), we can write f = k=0 λk ak with P ∞ each ak a (α0 , q) b-atom supported on Bk = B(0, 2k ) and k=0 |λk |p < ∞. As the same as in (3.5), we have k[b, T ]f kpK α0 ,p q X X p p j−2 ∞ ∞ X |λk |k[b, T ]ak kq + C 2jα0 p |λk |k([b, T ]ak )χj kq ≤C +C

k=0 ∞ X

j=2

2jα0 p

X ∞

j=2

k=0

p |λk |k([b, T ]ak )χj kq

k=j−1

:= I1 + I2 + I3 . By using the boundedness of [b, T ] on Lq (Rn ), it is easy to see that the estimate of I1 and I3 are similar to (3.6) and (3.7). Thus I1 ≤ CkbkpBMO I3 ≤ CkbkpBMO

∞ X k=0 ∞ X

|λk |p , |λk |p .

k=0

Let us now estimate I2 . When j ≥ 2 and 0 ≤ k ≤ j − 2, 2|y|α < 2 · 2kα ≤ 2 · 2(j−2)α ≤ 2 · 2j−2 = 2j−1 ≤ |x|, for any x ∈ Ej and y ∈ Bk . By the cancellation condition of b-atom and (2) of Definition 1.1, we get

Vol. 57 (2007)

Boundedness of Commutators on Hardy-Type Spaces

Z |[b, T ]ak (x)|χj (x) =

Bk

[K(x, y) − K(x, 0)][b(x) − b(y)]ak (y)dy

|y|δ |b(x) − b(y)||ak (y)|dy n+δ/α Bk |x| Z ≤ C2kδ |x|−n−δ/α |b(x) − b(y)||ak (y)|dy. Z

≤C

Bk

Therefore, we obtain the following estimate by the Minkowski inequality. k([b, T ]ak )χj kq Z Z ≤ C2kδ |x|(−n−δ/α)q Ej

≤ C2kδ 2j(−n−δ/α)

q |b(x) − b(y)||ak (y)|dy

Bk

1/q dy |b(x) − b(y)|q dx

Z |ak (y)|

Z Bk

≤ C2kδ 2j(−n−δ/α)

1/q dx

Bj

Z |ak (y)|

Z Bk

1/q |b(x) − bBj |q dx

Bj

1/q

1/q

+|bBj − bBk ||Bj |

dy

+ |bBk − b(y)||Bj | Z kδ j(−n−δ/α) 1/q ≤ C2 2 |Bj | kbkBMO + (j − k)kbkBMO |ak (y)|dy Bk Z + |ak (y)||bBk − b(y)|dy Bk kδ j(−n(1−1/q)−δ/α) ≤ C2 2 (j − k + 1)kbkBMO kak kq |Bk |1−1/q + kak kq Z ·

q0

1/q0

|bBk − b(y)| dy

Bk

≤ CkbkBMO 2kδ 2j(−n(1−1/q)−δ/α) 2−kα0 2kn(1−1/q) (j − k + 1) = CkbkBMO 2−jα0 2(j−k)(α0 −n(1−1/q)−δ) 2j(δ−δ/α) (j − k + 1) ≤ CkbkBMO 2−jα0 2(j−k)(α0 −n(1−1/q)−δ) (j − k + 1) := CkbkBMO 2−jα0 W 0 (j, k). As in the estimate of (3.8), we can get I2 ≤ CkbkpBMO

∞ X

|λk |p .

k=0

Combining the estimates of I1 , I2 and I3 , X 1/p ∞ p α ,p k[b, T ]f kKq 0 ≤ CkbkBMO |λk | . k=0

395

396

Lin and Lu

IEOT

By taking infimum over all the decompositions of f , we get k[b, T ]f kKqα0 ,p ≤ CkbkBMO kf kHb Kqα0 ,p . This completes the proof of Theorem 1.14.

Acknowledgment The authors thank the referee for many valuable comments.

References [1] J. Alvarez, R. J. Bagby, D. S. Kurtz and C. P´erez, Weighted estimates for commutators of linear operators. Studia Math. 104 (1993), 195-209. [2] J. Alvarez and M. Milman, H p continuity properties of Calder´ on-Zygmund-type operators. J. Math. Anal. Appl. 118 (1986), 63-79. [3] J. Alvarez and M. Milman, Vector valued inequalities for strongly singular Calder´ onZygmund operators. Rev. Mat. Iberoamericana 2 (1986), 405-426. [4] R. Coifman, A real variable characterization of H p . Studia Math. 51 (1974), 269-274. [5] C. Fefferman and E. M. Stein, H p spaces of several variables. Acta Math. 129 (1972), 137-193. [6] Y. Lin and S. Z. Lu, Toeplitz operators related to strongly singular Calder´ on-Zygmund operators. To appear in Science in China (Ser. A). [7] X. W. Li and D. C. Yang, Bounedeness of some sublinear operators on Herz spaces. Illinois J. Math. 40 (1996), 484-501. [8] S. Z. Lu and D. C. Yang, The weighted Herz-type Hardy space and its applications. Science in China (Ser. A) 38 (1995), 662-673. Yan Lin and Shanzhen Lu School of Mathematical Sciences Beijing Normal University Beijing, 100875 P.R. China e-mail: [email protected] [email protected] Submitted: August 31, 2005 Revised: September 15, 2006

Integr. equ. oper. theory 57 (2007), 397–412 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030397-16, published online December 26, 2006 DOI 10.1007/s00020-006-1462-0

Integral Equations and Operator Theory

The Bargmann Transform and Windowed Fourier Localization Min-Lin Lo Abstract. We consider the relationship between Gabor-Daubechies windowed Fourier localization operators Lw ϕ and Berezin-Toeplitz operators Tϕ , using the Bargmann isometry β. For “window” w a ﬁnite linear combination of Hermite functions, and a very general class of functions ϕ, we prove an equivalence of the form −1 = C ∗ Mϕ C = T(I+D)ϕ βLw ϕβ by obtaining the exact formulas for the operator C and the linear diﬀerential operator D. Mathematics Subject Classification (2000). 47B35, 42C40, 81R30. Keywords. Berezin-Toeplitz operator, Bargmann isometry, windowed Fourier localization.

1. Introduction We begin with two Hilbert spaces. The ﬁrst is the usual space L2 (Rn , dv) of Lebesgue square-integrable complex functions on real Euclidean space. The second is the space H2 (Cn , dµ) of entire functions on complex n-space which are squareintegrable with respect to the measure dµ(z) = (2π)−n exp{−|z|2 /2}dv(z) (dv is Lebesgue volume measure). Consider the “natural” orthonormal basis for L2 (Rn , dv) consisting of Hermite functions ([18], p.51; [21], pp. 105, 106) n

1

1

2

wk = π − 4 (2|k| k!)− 2 Hk (x)e− 2 x , where k ranges over all n-tuples of non-negative integers, k! = k1 !k2 ! · · · kn ! z with |z|2 = z · z |k| = k1 + · · · + kn , az = a1 z1 + a2 z2 + · · · + an zn , z 2 = zz, a · z = a¯ This analysis is based on some papers by L. A. Coburn [8, 9, 10].

398

Lo

IEOT

for z¯j the complex conjugate of zj and 2

2

Hk (x) = (−1)|k| ex Dk e−x , ∂ k1 ) · · · ( ∂x∂ n )kn . There is a corresponding “natural” orthonormal for Dk = ( ∂x 1 2 n basis for H (C , dµ) consisting of monomials

ek = (2|k| k!)−1/2 z k , where z k = z1k1 z2k2 · · · znkn . In what follows, we will also write ∂ k = ∂1k1 · · · ∂nkn = ( ∂z∂ 1 )k1 · · · ( ∂z∂n )kn as well as ∂¯k = ∂¯1k1 · · · ∂¯nkn = ( ∂∂z¯1 )k1 · · · ( ∂∂z¯n )kn . For f in H2 (Cn , dµ), [3] began a systematic study of densely deﬁned operators of the form: (Tϕ f )(z) = ez·a/2 ϕ(a)f (a)dµ(a), Cn

e(·)·z/2 ϕ(·) is assumed to be in L2 (Cn , dµ) for all z in Cn . The (possibly unbounded) operator Tϕ is called the Berezin-Toeplitz operator with symbol ϕ. Note that the Berezin-Toeplitz operator Tϕ maps its domain into H2 (Cn , dµ). The orthogonal projection operator from L2 (Cn , dµ) onto the closed subspace H2 (Cn , dµ) is given by a·z/2 = ez·a/2 f (a)dµ(a). (P f )(z) = f, e Cn

Restriction to f in H2 (Cn , dµ) shows that H2 (Cn , dµ) is a “reproducing kernel Hilbert space” with reproducing kernel ez·a/2 . Thus, for Mϕ the operator of “multiplication by ϕ” on L2 (Cn , dµ), we see that Tϕ = P Mϕ P . Moreover, Tz¯i = 2∂i and Tzj = Mzj , at least on ﬁnite linear combinations of the {ek }. There is an interesting connection between Berezin-Toeplitz operators on H2 (Cn , dµ) and Gabor-Daubechies windowed Fourier localization operators on L2 (Rn , dv). To be precise, the Gabor-Daubechies localization operator Lw ϕ with “symbol” (or “weight function”) ϕ and “window” w, is densely deﬁned on L2 (Rn , dv) by −n f, g = (2π) ϕ(a)βf, Wa βwWa βw, βgdv(a), Lw ϕ Cn

where β is the Bargmann isometry ([1]; [18], pp. 40, 47), β : L2 (Rn , dv) → H2 (Cn , dµ), which is surjective, with βwk = ek and (Wa f )(z) = ka (z)f (z − a) = ez·a/2−|a|

2

/4

f (z − a).

Typically, the “window” w is chosen to be a unit vector in L2 (Rn , dv). For 2 n 1 0 −1 w0 = π − 4 e− 2 |x| , βw0 = 1 and it is not diﬃcult to verify that βLw = Tϕ . ϕ β Thus, the Bargmann isometry allows the amalgamation of the substantial work 0 already done in analyzing Tϕ [2, 6, 3, 4, 5, 7, 8, 11, 20, 22] and Lw ϕ [12, 13, 14, 15, 16, 17].

Vol. 57 (2007)

Windowed Fourier Localization

399

In a paper by L. A. Coburn [10], he checked for ϕ either an arbitrary polynomial in z and z¯ , or a function in Ba (Cn ) (the algebra of Fourier-Stieltjes transforms of compactly supported, complex, bounded regular Borel measures on Cn ), that w

βLϕ (1,0,··· ,0) β −1 w βLϕ (2,0,··· ,0) β −1

= T(I+2∂¯1 ∂1 )ϕ , = T(I+4∂¯1 ∂1 +2(∂¯1 ∂1 )2 )ϕ .

His most general result was: Let p(z) be an arbitrary polynomial in H2 (Cn , dµ) and let ϕ be a function such that e(·)·a/2 ϕ(·) is in L2 (Cn , dµ) for all a in Cn . Then, for windoww = k ak wk with k ranging over a ﬁnite set of non-negative multi-indices and k |ak |2 = 1, −1 βLw p(a) = C ∗ Mϕ Cp(a), ϕβ

for a precisely determined operator C = C(w). It follows that there is a linear diﬀerential operator D = D(w), with coeﬃcients which are polynomials in z and z¯ and with no derivative- free term, such that −1 βLw p(z) = C ∗ Mϕ Cp(z) = T(I+D)ϕ p(z), ϕβ

for ϕ an arbitrary polynomial in z and z¯. Coburn conjectured that D(w) was actually a constant coeﬃcient linear diﬀerential operator and that the stated result held for all ϕ in Ba (Cn ) as well. In the next section of this article, we begin with proofs of the result outlined above for w= Hermite function wk , and ϕ in the Schwartz space S. The third section deals with more general windows w which are ﬁnite linear combinations of Hermite functions for the same type of “symbol” ϕ as in section two. In the ﬁnal section, we extend the above result to ϕ in a more general class of functions E(Cn ) which includes the space Ba (Cn ), as well as all polynomials ϕ.

2. Computations for Hermite Windows For p(z) an arbitrary polynomial in H2 (Cn , dµ), and β the Bargmann isometry, our ﬁrst result is Theorem 2.1. For ϕ such that e(·)·a/2 ϕ(·) is in L2 (Cn , dµ) for all a in Cn , we have 1 k −1 p(z) = |k| Ck∗ Mϕ Ck p(z) βLw ϕ β 2 k! with k k Ck = (−1)|i| (I − P )Mz¯i P Mz¯k−i P, i i0 k where “ i0 ” means “sum over all i with 0 i1 ≤ k1 , 0 i2 ≤ k2 , · · · , 0 in ≤ kn ” and ki = ki11 ki22 · · · kinn . Formally (up to domain considerations), k k 1 1 k ∗ C M C = (−1)|i+j| Tzk−i Tzi ϕ¯zj Tz¯k−j . ϕ k k |k| |k| j 2 k! 2 k! i,j=0 i

400

Lo

Proof. Note that βwk = ek = p(z), ez·a/2 (z − a)k

√ 1 zk . 2|k| k!

IEOT

We observe that

k

k ea·z/2 ¯i p(z)dµ(z) (−1)|i| z¯k−i a i Cn i=0 k

k ¯i P Mz¯k−i p (a) (−1)|i| a i i=0 k

k (−1)|i| (I − P )Mz¯i P Mz¯k−i P p (a) i i0 Ck p (a).

=

=

= =

Now, for p(z) and q(z) arbitrary polynomials in z, w −1 βLϕ k β p, q 1 = ϕ(a)p(z), ez·a/2 (z − a)k ez·a/2 (z − a)k , q(z)dµ(a) |k| Cn 2 k! 1 = ϕ(a) Ck p (a) Ck q (a)dµ(a) |k| Cn 2 k! 1 = |k| Mϕ Ck p, Ck q 2 k! and ϕ(a) Ck p (a) is easily seen to be in L2 (Cn , dµ) since Ck p (a) is a polynomial in a, a ¯. Thus, k −1 βLw p(z) = ϕ β

1 C ∗ Mϕ Ck p(z). 2|k| k! k

We check directly that

=

Ck∗ Mϕ Ck p(z) k

k (−1)|i| P Mzk−i P Mzi (I − P ) Mϕ i i0

k

k (−1)|j| (I − P )Mz¯j P Mz¯k−j P p(z) j j0

=

k k

k k (−1)|i| Tzk−i P Mzi − (−1)|i| Tzk−i Tzi P Mϕ i i i0

k

j0

i0

k k k (−1)|j| Mz¯j Tz¯k−j − (−1)|j| Tz¯j Tz¯k−j p(z) j j j0

Vol. 57 (2007)

=

Windowed Fourier Localization

401

k

k (−1)|i| Tzk−i P Mzi + Tzk P Mϕ i i0

k

k (−1)|j| Mz¯j Tz¯k−j + Tz¯k p(z) j j0

=

=

k k

k k |i| (−1) Tzk−i P Mzi Mϕ (−1)|j| Mz¯j Tz¯k−j p(z) i j i=0 j=0 k

k k (−1)|i+j| Tzk−i Tzi ϕ¯zj Tz¯k−j p(z). i j i,j=0

The Schwartz space S consists of those C ∞ functions which, together with all their derivatives, vanish at inﬁnity faster than any power of |z|. More precisely, for any nonnegative integer N and any multi-index α, we deﬁne f (N,α) = sup (1 + |z|)N |∂ α f (z)|; z∈Cn

and then

S = { f ∈ C ∞ : f (N,α) < ∞ for all N, α }.

Lemma 2.2. For ϕ in S,

ϕ(z) = Cn

χa (z)ϕ(a)dv(a), ˇ

where χa (z) = exp{iIm(z · a)}. Proof. For ϕ in S, ϕˇ is in S since the inverse Fourier transform is an isomorphism of S onto itself. By the Fourier Inversion Theorem, ϕ(z) = Cn χa (z)ϕ(a)dv(a). ˇ Using Theorem 2.1, Lemma 2.2, and the identity Wa = e|a| we show

2

/4

Tχa (·) from [3],

Theorem 2.3. For ϕ in S, and p(z) an arbitrary polynomial in H2 (Cn , dµ), we have k −1 βLw p(z) = T{ k ϕ β

m=0

k 1 ¯ k−m ϕ} p(z). 2|k−m| (m (∂∂) ) (k−m)!

Proof. By direct calculation, 1 C ∗ Mχa (·) Ck p(z) |k| 2 k! k k

−|a|2 k k 1 4 = e (−1)|i+j| Tzk−i Tz¯j Wa Tzi Tz¯k−j p(z) |k| i j 2 k! i,j=0 =

k 2|k| −|a|2 k k 4 e (−1)|i+j| z k−i ∂ j [Ka (z)(z − a)i ∂ k−j p(z − a)] |k| i j 2 k! i,j=0

402

Lo

(to simplify the notation, we use c to denote

=

=

c

c

l=0

k k k (−1)|k+i| z k−i (−1)|j−k| i j j=0

j j−l ∂ [Ka (z)(z − a)i ]∂ k−j+l p(z − a) l

(Reindexing by m = k − j + l) k

k c (−1)|k+i| z k−i i i=0 m k k k − r (−1)|r| k−r m−r m=0 r=0  k =( m )

=

=

=

=

=

l=0

k

i=0

∗

1 −|a|2 /4 ) k! e

j k

k k j j−l (−1)|i+j| z k−i ∂ [Ka (z)(z − a)i ]∂ k−j+l p(z − a) i j l i,j=0

j

=

IEOT

m

r=0 (−1)

  1,

|r| m r

( )=

 0,

∂ m p(z − a)∂ k−m [Ka (z)(z − a)i ]

if m = 0; otherwise.

k

k (−1)|k+i| z k−i p(z − a)∂ k [Ka (z)(z − a)i ] i i=0 k k

k k |k+i| k−i z p(z − a) (−1) ∂ k−m Ka (z) ∂ m (z − a)i c i m m=0 i=0 i! (z−a)i−m ( a¯2 )k−m Ka (z) (i−m)! k k−m k a ¯ cKa (z)p(z − a) m 2 m=0 k k i! (−1)|k+i| z k−i (z − a)i−m i (i − m)! i=m k! (k−m ) i−m (k−m)! k k−m −|a|2 k a ¯ 1 (−1)|m+k| e 4 Ka (z)p(z − a) m 2 (k − m)! m=0

c

[z − (z − a)]k−m k−m |k−m| =( −a (−2) ) 2 k

k 1 m=0

m (k − m)!

k−m a ¯ k−m −a 2

2

2 2|k−m| e−|a| /4 Wa p(z),

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Windowed Fourier Localization

403

so that 1 C ∗ Mχa (·) Ck p(z) 2|k| k! k k−m k

a k 1 ¯ k−m −a = 2|k−m| Tχa (·) p(z). m (k − m)! 2 2 m=0 Integrating both sides of the operator equation above with respect to dσ(a), for dσ of the form dσ = ϕdv ˇ for ϕ in S (see Lemma 2.2)gives 1 C ∗ Mϕ Ck p(z) = T{ k 2|k−m| ( k ) 1 (∂∂) ¯ k−m ϕ} p(z) m=0 m (k−m)! 2|k| k! k for

ϕ(z) = Cn

χa (z)dσ(a).

Operator integrals are understood in the weak sense and Fubini’s Theorem is required. Note that k 1 |k−m| k ¯ k−m ϕ(z) (∂∂) 2 m (k − m)! m=0 k−m k

a k 1 ¯ k−m −a 2|k−m| χa (z)dσ(a) 2 Cn m=0 m (k − m)! 2

=

by diﬀerentiation “under the integral”. By Theorem 2.1, we now have k −1 p(z) = βLw ϕ β

1 C ∗ Mϕ Ck p(z) = T{ k 2|k−m| ( k ) 1 (∂∂) ¯ k−m ϕ} p(z). m=0 m (k−m)! 2|k| k! k

3. Computations for more general windows For w a ﬁnite linear combination of Hermite functions wk , we can replicate the analysis of Theorem 2.1 and Theorem 2.3. Theorem 3.1. Let p(z) be an arbitrary polynomial in H2 (Cn , dµ) and let ϕ be a function such that e(·)·a/2 ϕ(·) is in L2 (Cn , dµ) for all a in Cn . Then, for window Ak wk w= k

with k ranging over a ﬁnite set of non-negative multi-indices, we have A¯ A −1 √ k Ck∗ Mϕ √ h Ch p(z) βLw p(z) = ϕβ 2|k| k! 2|h| h! k h with (Ck p)(a) = p(z), ez·a/2 (z − a)k ,

404

Lo

IEOT

and h ranging over the same ﬁnite set of non-negative multi-indices. Formally, k h k h ∗ Ck Mϕ Ch p(z) = (−1)|i+j| Tzk−i Tzi ϕ¯zj Tz¯h−j p(z). i j i=0 j=0 Proof. Note that βw =

k

√ Ak z k . 2|k| k!

p(z), ez·a/2 (βw)(z − a)

We observe that Ak √ (z − a)k = p(z), ez·a/2 |k| 2 k! k A¯k √ = p(z), ez·a/2 (z − a)k |k| 2 k! k A¯ √ k Ck p (a). = 2|k| k! k

Hence, for p(z), q(z) arbitrary polynomials in z, −1 p, q βLw ϕβ = ϕ(a)p(z), ez·a/2 (βw)(z − a)ez·a/2 (βw)(z − a), q(z)dµ(a) Cn

A¯ A¯ h k √ √ Ch p (a) Ck q (a)dµ(a) ϕ(a) 2|h| h! 2|k| k! Cn h k A¯k A¯h √ √ = Mϕ Ch p, Ck q 2|h| h! 2|k| k! h k

=

and ϕ(a) (Ch p) (a) is easily seen to be in L2 (Cn , dµ), since (Ch p) (a) is a polynomial in a,¯ a. Thus, A¯ A k h −1 √ √ Ck∗ Mϕ Ch p(z). p(z) = βLw ϕβ 2|k| k! 2|h| h! k h We check directly that (similar to the proof of Theorem 2.1)

=

=

Ck∗ Mϕ Ch p(z)   k

h k h (−1)|i| Tzk−i P Mzi Mϕ  (−1)|j| Mz¯j Tz¯h−j  p(z) i j i=0 j=0 k h k h (−1)|i+j| Tzk−i Tzi ϕ¯zj Tz¯h−j p(z). i j i=0 j=0

Theorem 3.2. For ϕ in S, p(z) an arbitrary polynomial in H 2 (Cn , dµ), and w = k Ak wk with k ranging over a ﬁnite set of non-negative multi-indices, we have −1 βLw p(z) = T{ ϕβ

h,k

√

¯ Ak A √h (−2)|h+k| 2|k| k! 2|h| h!

min(h, k) j=0

¯k−j ∂ h−j ϕ} 2|−j| (kj)(h j )j!∂

p(z),

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with h ranging over the same ﬁnite set of non-negative multi-indices as k. Proof. From Theorem 3.1, and using the identity Wa = e|a|

=

=

=

2

/4

Tχa (·) , we ﬁnd

Ck∗ Mχa (·) Ch p(z) k h k h (−1)|i+j| Tzk−i Tz¯j Tχa (·) Tzi Tz¯h−j p(z) i j i=0 j=0 |h| −|a|2 /4

2 e

k h ! " k h (−1)|i+j| z k−i ∂ j Ka (z)(z − a)i ∂ h−j p(z − a) i j i=0 j=0

(to simplify notation, we use c to denote 2|h| e−|a| k h k h |i+h| k−i z c (−1) (−1)|j−h| i j i=0 j=0

2

/4

in this proof)

j " j j−l ! ∂ Ka (z)(z − a)i ∂ h−(j−l) p(z − a) l l=0

=

(using step (∗) in Theorem 2.3) k k c (−1)|i+h| z k−i i i=0 t h h h−r |r| (−1) h−r t−r t=0 r=0 

=

=

  1,

if t = 0; otherwise. k h k h h−j |i+h| k−i (−1) ∂ z Ka (z)∂ j (z − a)i p(z − a) c i j i=0 j=0 =(h t)

=

! " ∂ h−t Ka (z)(z − a)i ∂ t p(z − a)

t

r=0 (−1)

|r| t r

( )=

 0,

k k c (−1)|i+h| z k−i i i=0   i h−j h a ¯ i!  (z − a)i−j  p(z − a) Ka (z) j 2 (i − j)! j=0

k i! z k−i (z − a)i−j j i 2 (i − j)! j=0 i=j k−j k k i! = j!) (using the identity i−j j i (i − j)! cKa (z)p(z − a)

k k h−j h a ¯

(−1)|h+i|

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k j!(z − (z − a))k−j j j 2 j=0   h−j k−j min(h, k) k h −a a ¯  e−|a|2 /4 Wa p(z) = (−2)|h+k| 2|−j| j! j j 2 2 j=0 = cKa (z)p(z − a)

k h−j h a ¯

IEOT

(−1)|j+h|

Thus, we have Ck∗ Mχa (·) Ch p(z)  = (−2)|h+k|

 h−j k−j k h a ¯ −a  Tχa (·) p(z). j! 2|−j| j j 2 2

min(h, k)

j=0

Integrating both sides of the operator equation above with respect to dσ(a), for dσ of the form dσ = ϕdv ˇ for ϕ ∈ S (see Lemma 2.2) gives Ck∗ Mϕ Ch p(z) = T (−2)|h+k| min(h, k) 2|−j| j=0

for

(kj)(hj)j!∂¯k−j ∂ h−j ϕ

p(z)

ϕ(z) = Cn

χa (z)dσ(a).

Operator integrals are understood in the weak sense and Fubini’s Theorem is required. Note that min(h, k) |h+k|

(−2) =



2

|−j|

j=0

(−2)|h+k|

k h j!∂¯k−j ∂ h−j ϕ j j

min(h, k)

Cn

j=0

 h−j k−j k h −a a ¯  χa (z)dσ(a) 2|−j| j! j j 2 2

by diﬀerentiation “under the integral”. By Theorem 3.1, it follows that −1 βLw p(z) = ϕβ

h,k

Ak A¯h √ √ Ck∗ Mϕ Ch p(z) 2|k| k! 2|h| h!

= T{

h,k

¯ Ak A √h (−2)|h+k| 2|k| k! 2|h| h!

√

min(h, k) j=0

¯k−j ∂ h−j ϕ} p(z). 2|−j| (kj)(h j )j!∂

By applying the same proof above to ϕ(z) = Cn χa (z)dσ(a), where σ is a compactly supported, regular, bounded complex-valued Borel measure, we can prove that the result of Theorem 3.2 holds for ϕ in Ba (Cn ) as well.

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4. Computations for more general symbols In this section, we use “cut-oﬀ” functions to prove that Coburn’s conjectured result holds for ϕ in E(Cn ), where E(Cn ) = { ϕ ∈ C ∞ (Cn ) : for any multi-index k, there exist constants M = M (k) and α = α(k) such that |(Dk ϕ)(z)| ≤ M eα|z| }. First, we construct appropriate “cut-oﬀ” functions. Take a C ∞ - function η n on C which has the following properties: (i) 0 ≤ η ≤ 1 on Cn ; (ii) η(z) = 1 when |z| ≤ 1; (iii) η(z) = 0 when |z| ≥ 2. Clearly, η is in Cc∞ (Cn ). z ), then Now, for each m ∈ N, let ηm (z) = η( m (i) 0 ≤ ηm ≤ 1 on Cn ; (ii) ηm (z) = 1 when |z| ≤ m; (iii) ηm (z) = 0 when |z| ≥ 2m.

Note that ηm is still in Cc∞ (Cn ). If ϕ is in E(Cn ), then ηm ϕ is in Cc∞ (Cn ) ⊂ S. By Theorem 3.2, we now have n 2 n Lemma 4.1. For ϕ in E(C ), p(z) an arbitrary polynomial in H (C , dµ), and w = k Ak wk with k ranging over a ﬁnite set of non-negative multi-indices, we have −1 p(z) βLw ηm ϕ β

= T{

h,k

¯ Ak A √h (−2)|h+k| 2|k| k! 2|h| h!

√

min(h, k) j=0

¯k−j ∂ h−j (ηm ϕ)} p(z), 2|−j| (kj)(h j )j!∂

with h ranging over the same ﬁnite set of non-negative multi-indices as k. In addition to Lemma 4.1, we also need the following lemmas to prove Coburn’s conjectured result for ϕ in E(Cn ). Lemma 4.2. For p, q polynomials in H2 (Cn , dµ), there exist positive constants K, such that 2 |Wa p, q| ≤ Ke−|a| . Proof.

Wa p, q =

ez·a/2−|a|

2

/4

p(z − a)q(z)dµ(z)

ez·a/2−|a|

2

/4

p(z − a)q(z)(2π)−n e−|z|

Cn

= Cn

2

/2

dv(z).

By completing the square, we can rewrite |z|2 /2 − z · a/2 + |a|2 /4 with its real part equal to (|z|2 + |z − a|2 )/4. For p, q polynomials in H2 (Cn , dµ), we can ﬁnd positive constants M1 and M2 such that |p(z)| ≤ M1 e|z|

2

/8

,

408

Lo

IEOT

and |q(z)| ≤ M2 e|z| Thus, we have |Wa p, q| ≤ ≤ ≤ =

−n

2

/8

.

2

2

|p(z − a)||q(z)|e−|z| /4−|z−a| /4 dv(z) Cn 2 2 (2π)−n M1 M2 e−(|z| +|z−a| )/8 dv(z) n C 2 2 −n (2π) M1 M2 e−(|z| /2+|a| /4)/8 dv(z) Cn 2 −n −|a|2 /32 (2π) M1 M2 e e−|z| /16 dv(z). (2π)

Cn

Lemma 4.3. For ϕ in E(Cn ), and w, f, g ﬁnite linear combinations of Hermite functions, w Lηm ϕ f, g → Lw as m → ∞. ϕ f, g Hence, for p, q polynomials in H 2 (Cn , dµ), w −1 p, q as m → ∞. βLηm ϕ β −1 p, q → βLw ϕβ Proof. w Lηm ϕ f, g = (2π)−n

Cn

ηm (a)ϕ(a)βf, Wa βwWa βw, βgdv(a).

Let hm (a) = ηm (a)ϕ(a)βf, Wa βwWa βw, βg, observe that hm (a) → h(a) = ϕ(a)βf, Wa βwWa βw, βg pointwise, as m → ∞. And |hm (a)| ≤ |ϕ(a)|ce−|a| ≤ M eα|a|−|a| 2

2

2

for some c, > 0 by Lemma 4.2 for some M, α > 0 since ϕ is in E(Cn ).

Since M eα|a|−|a| is in L1 (Cn , dv), we can apply the dominated convergence theorem to the sequence {hm } to obtain −n lim Lw f, g = lim (2π) hm (a)dv(a) ηm ϕ m→∞ m→∞ Cn = (2π)−n h(a)dv(a) n w C = Lϕ f, g .

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Windowed Fourier Localization

For p(z), q(z) polynomials in H 2 (Cn , dµ), −1 p, q = lim βLw ηm ϕ β m→∞

= =

409

lim Lw β −1 p, β ∗ q η mϕ m→∞ w −1 Lϕ β p, β ∗ q w −1 βLϕ β p, q .

Lemma 4.4. For ϕ in E(Cn ), p(z) an arbitrary polynomial in H 2 (Cn , dµ), we have T(I+D)ηm ϕ p(z) → T(I+D)ϕ p(z)

in L2 (Cn , dµ) as m → ∞,

where the diﬀerential operator (I + D) = {

h,k

min(h, k) k h Ak A¯h √ √ 2|−j| j!∂¯k−j ∂ h−j }, (−2)|h+k| j j 2|k| k! 2|h| h! j=0

with h, k ranging over a ﬁnite set of non-negative multi-indices. Hence, for p(z), q(z) polynomials in H 2 (Cn , dµ), T(I+D)ηm ϕ p, q → T(I+D)ϕ p, q as m → ∞. Proof. It suﬃces to show that T∂¯α ∂ β ηm ϕ p(z) → T∂¯α ∂ β ϕ p(z) in L2 (Cn , dµ) as m → ∞, for p(z) an arbitrary polynomial in H 2 (Cn , dµ).

= =

=

∂ β (ηm (z)ϕ(z)) β β β−i ηm (z)∂ i ϕ(z) ∂ i i=0 z β β ∂ i ϕ(z) ηm (z)∂ ϕ(z) + ∂ β−i η i m 0≤i<β z β 1 |β−i| β ηm (z)∂ ϕ(z) + ∂ i ϕ(z). (∂ β−i η) i m m 0≤i<β

∂¯α ∂ β (ηm (z)ϕ(z))

z $ # β 1 |β−i| α β ¯ = ∂ ηm (z)∂ ϕ(z) + ∂¯α (∂ β−i η) ∂ i ϕ(z) i m m 0≤i<β z α 1 |α−j| α β ¯ = ηm (z)∂ ∂ ϕ(z) + (∂¯α−j η) ∂¯j ∂ β ϕ(z) + j m m 0≤j<α |β−i| # z $ β 1 ∂ i ϕ(z) ∂¯α (∂ β−i η) i m m 0≤i<β

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Lo

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z α 1 |α−j| (∂¯α−j η) ∂¯j ∂ β ϕ(z) + j m m 0≤j<α |β−i| α 1 |α−k| β z k i 1 ∂¯ ∂ ϕ(z). ∂¯α−k ∂ β−i η i k m m m

= ηm (z)∂¯α ∂ β ϕ(z) +

0≤i<β

0≤k≤α

For each ﬁxed pair of multi-indices α and β, ∂¯α ∂ β η (z) is bounded for all 1 |n| α β z γ δ ∂¯ ∂ η m ∂¯ ∂ ϕ(z)p(z) → 0 in L2 (Cn , dµ) z ∈ Cn . Thus, for |n| = 0, m 2 n for any polynomial p(z) in H (C , dµ), as m → ∞. We then have ∂¯α ∂ β (ηm (z)ϕ(z)) p(z) − ηm (z) ∂¯α ∂ β ϕ (z)p(z)2 → 0 as m → ∞. Since ηm (z) → 1 pointwise and ∂¯α ∂ β ϕ (z)p(z) is in L2 (Cn , dµ) for p(z) a polynomial in H 2 (Cn , dµ) and ϕ(z) in E(Cn ), ηm (z) ∂¯α ∂ β ϕ (z)p(z) − ∂¯α ∂ β ϕ (z)p(z)2 → 0 as m → ∞, by the dominated convergence theorem. Thus, limm→∞ ∂¯α ∂ β (ηm (z)ϕ(z)) p(z) − ∂¯α ∂ β ϕ(z) p(z)2 = ≤

lim ∂¯α ∂ β (ηm (z)ϕ(z)) p(z) − ηm (z) ∂¯α ∂ β ϕ(z) p(z) m→∞ +ηm (z) ∂¯α ∂ β ϕ(z) p(z) − ∂¯α ∂ β ϕ(z) p(z)2 lim ∂¯α ∂ β (ηm (z)ϕ(z)) p(z) − ηm (z) ∂¯α ∂ β ϕ(z) p(z)2 m→∞ + lim ηm (z) ∂¯α ∂ β ϕ(z) p(z) − ∂¯α ∂ β ϕ(z) p(z)2 m→∞

= So,

0.

∂¯α ∂ β (ηm (z)ϕ(z)) p(z) → ∂¯α ∂ β ϕ(z) p(z) in L2 (Cn , dµ) as m → ∞.

Since the projection operator is continuous, lim T∂¯α ∂ β (ηm (z)ϕ(z)) p(z) =

m→∞

lim P M∂¯α ∂ β (ηm (z)ϕ(z)) p(z)

m→∞

lim P (∂¯α ∂ β (ηm (z)ϕ(z)) p(z)) = P ( ∂ ∂ ϕ(z) p(z)) = P M∂¯α ∂ β ϕ(z) p(z) =

m→∞ α β ¯

= T∂¯α ∂ β ϕ(z) p(z), where the convergence is in L2 (Cn , dµ).

By Lemma 4.1, Lemma 4.3, and Lemma 4.4, we have now extended the result for ϕ in E(Cn ).

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n 2 n Theorem 4.5. For ϕ in E(C ), p(z) an arbitrary polynomial in H (C , dµ), and w = k Ak wk with k ranging over a ﬁnite set of non-negative multi-indices, we have −1 p(z) = T{ βLw ϕβ

h,k

√

¯ Ak A √h (−2)|h+k| 2|k| k! 2|h| h!

min(h, k) j=0

¯k−j ∂ h−j ϕ} p(z), 2|−j| (kj)(h j )j!∂

with h ranging over the same ﬁnite set of non-negative multi-indices as k. Corollary 4.6. For ϕ an arbitrary polynomial in z and z¯, p(z) an arbitrary poly nomial in H 2 (Cn , dµ), and w = k Ak wk with k ranging over a ﬁnite set of non-negative multi-indices, we have −1 p(z) = T{ βLw ϕβ

h,k

√

¯ Ak A √h (−2)|h+k| 2|k| k! 2|h| h!

min(h, k) j=0

¯k−j ∂ h−j ϕ} p(z), 2|−j| (kj)(h j )j!∂

with h ranging over the same ﬁnite set of non-negative multi-indices as k. Proof. This follows directly from Theorem 4.5 since ϕ is in E(Cn ).

Acknowledgment The author would like to thank Dr. Lewis A. Coburn and Dr. Jingbo Xia for their valuable advice.

References [1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Communications on Pure and Applied Mathematics, 14 (1961), 187-214. [2] F. A. Berezin, Covariant and contravariant symbols of operators, Math. USSR Izv., 6 (1972), 1117-1151. [3] C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, Journal of Functional Analysis, 68 (1986), 273-299. [4] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Transactions AMS, 301 (1987), 813-829. [5] C. A. Berger and L. A. Coburn, Heat Flow and Berezin-Toeplitz Estimates, American Journal of Mathematics,116 (1994), 563-590. [6] D. Borthwick, Microlocal techniques for semiclassical problems in geometric quantization. Perspectives on Quantization (editors: L. A. Coburn and M. A. Rieﬀel) Contemporary Math. 214, AMS, Providence, R.I. (1998), 23-37. [7] L. A. Coburn, The measure algebra of the Heisenberg group, Journal of Functional Analysis, 161 (1999), 509-525. [8] L. A. Coburn, On the Berezin-Toeplitz calculus, Proceedings of the AMS, 129 (2001), 3331-3338. [9] L. A. Coburn, The Bargmann isometry and Gabor-Daubechies wavelet localization operators, Systems, approximation, singular integrals and related topics (editors: A. Borichev and N. Nikolski) Operator Theory: Advances and Applications 129, Birkhauser, Basel (2001), 169-178. [10] L. A. Coburn, Symbol calculus for Gabor-Daubechies Windowed Fourier localization operators, preprint.

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[11] L. A. Coburn and J. Xia, Toeplitz algebras and Rieﬀel deformations, Communications in Mathematical Physics, 168 (1995), 23-38. [12] I. Daubechies, Time frequency localization operators: A geometric phase space approach, IEEE Transactions on Information Theory, 34 (1988),605-612. [13] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Transactions on Information Theory, 36 (1990), 961-1005. [14] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series 6, SIAM, Philadelphia, 1992. [15] F. De Mari, H. G. Feichtinger and K. Nowak, Uniform eigenvalue estimates for timefrequency localization operators, Journal of the London Mathematical Society, 65 (2002), 720-732. [16] J. Du and M. W. Wong, Gaussian functions and Daubechies operators, Integral Equations and Operator Theory, 38 (2000), 1-8. [17] H. G. Feichtinger and K. Nowak, A Szego type theorem for Gabor-Toeplitz localization operators, The Michigan Mathematical Journal, 49 (2001), 13-21. [18] G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Princeton Univ. Press, Princeton, 1989. [19] G. B. Folland, Real analysis, Wiley, New York, 1999. [20] V. Guillemin, Toeplitz operators in n-dimensions, Integral Equations and Operator Theory, 7 (1984), 145-205. [21] G. Szeg¨ o, Orthogonal polynomials, AMS Colloquium Publications, 23, AMS, Providence, RI., 1975. [22] J. Xia and D. Zheng, Standard deviation and Schatten class Hankel operators on the Segal-Bargmann space, Indiana University Mathematics Journal, 53 (2004), 13811399. Min-Lin Lo Department of Mathematics California State University, San Bernardino 5500 University Parkway San Bernardino, CA 92407 USA e-mail: [email protected] Submitted: October 3, 2005 Revised: October 19, 2006

Integr. equ. oper. theory 57 (2007), 413–423 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030413-11, published online December 26, 2006 DOI 10.1007/s00020-006-1459-8

Integral Equations and Operator Theory

Hypercyclic Conjugate Operators Henrik Petersson Abstract. We prove that for any weighted backward shift B = Bw on an infinite dimensional separable Hilbert space H whose weight sequence w = (wn ) satisfies supn |w1 w2 ...wn | = ∞, the conjugate operator CB : S → BSB ∗ is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there n (S) = B n SB ∗n }n≥0 is dense in S(H). exists an S ∈ S(H) such that {CB We generalize the result to more general conjugate maps S → T ST ∗ , and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N (H) of compact and nuclear operators respectively. Mathematics Subject Classification (2000). 47A16, 47B49, 47A58, 47L10, 47B10. Keywords. Hypercyclic, hypercyclicity criterion, conjugate operator, operator algebra, backward shift.

1. Introduction and Notation Definition 1.1. A continuous linear operator T , acting on a topological vector space (TVS) E, is said to be hypercyclic if there exists a vector x ∈ E, called hypercyclic for T , such that the orbit {T n x}n≥0 is dense. A necessary condition for the existence of a hypercyclic operator T is that E is separable and inﬁnite dimensional. Recall that once an operator is hypercyclic on a real or complex locally convex space, it has a dense invariant subspace of, except for zero, hypercyclic vectors, see [4]. For a general overview of the theory of hypercyclicity, and its history, we refer to Grosse-Erdmann’s survey articles [10, 11]. In all what follows, X denotes an inﬁnite dimensional separable real or complex Banach space, and when X is a Hilbert space we use the symbol H. Hypercyclic phenomena in the operator algebra B(X) of bounded linear operators have The author was supported by the The Royal Swedish Academy of Sciences.

414

Petersson

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been studied in [6, 7, 8]. In fact, Chan initiated the study in [7] by investigating hypercyclicity in B(H) of left multiplication operators LT : S → T S, T ∈ B(H), and pursued his work in [8] to the Banach space setting together with Taylor. Due to the fact that B(X) is in general not separable for the operator norm topology, they worked instead with the coarser strong operator topology (SOT), i.e., the topology of pointwise convergence. It follows that B(X) is always separable for the SOT [8, Corollary 3]. (Given T ∈ B(X), T shall always refer to the operator norm, supx≤1 T x, and the operator norm topology is the topology generated by · . The compact-open topology on B(X), i.e. the topology of uniform convergence on compact sets, is abbreviated COT.) A natural question to ask is what can be said about hypercyclicity of, say, left multiplication operators acting on diﬀerent subspaces and subalgebras of B(X), such as the ideals K(X) and N (X) of compact and nuclear operators respectively. In this way we may also experiment with alternative topologies. This motivated Bonet et. al in [6] where they improved earlier results of Chan and Taylor and obtained the following: Proposition 1.2 ([6]). Assume X has the approximation property and a separable dual X . If T ∈ B(X) satisfies the Hypercyclicity Criterion, then LT is hypercyclic on 1. B(X) provided with the COT ; 2. K(X) provided with the operator norm topology; 3. N (X) provided with the nuclear norm topology. In fact, (3 ) holds without assuming the approximation property. Let us recall the well-known Hypercyclicity Criterion (any operator that satisﬁes the Criterion is of course hypercyclic [3, 5]): Definition 1.3. An operator T ∈ B(X) is said to satisfy the Hypercyclicity Criterion (HC) with respect to an increasing sequence (nk ) ⊆ N, if there exist dense subsets Y, Z ⊆ X such that: 1. T nk z → 0 (k → ∞) for all z ∈ Z; 2. For every y ∈ Y there exists a nullsequence (xk ) in X such that T nk xk → y. T is said to satisfy the HC if it satisﬁes the HC with respect to some sequence (nk ). The nuclear norm topology (see below) is ﬁner than the operator norm topology, which in turn is ﬁner than the COT and SOTCOT, and so Proposition 1.2 canonically gives that LT is hypercyclic on N (X) for the SOT, the COT as well as for the operator norm topology and similarly, that LT is SOT and COThypercyclic on K(X). (In general, a hypercyclic operator T : (E, τ ) → (E, τ ) that is continuous for some coarser topology τ τ , is also τ -hypercyclic.) The proof of Proposition 1.2 in [6] is based on a tensor product technique which heavily rests on the work [13]. We should also mention that right multiplication operators RT (RT (S) ≡ ST ) are also studied in [6]. (Some of the obtained results for B(X),

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such as (1) in Proposition 1.2, are also extended to the algebra of continuous linear operators on certain non-normable Fr´echet spaces.) In this note we shall consider hypercyclicity of another type of construction, i.e. diﬀerent from left and right multiplication operators. Our idea originates from the following observation. An important subclass of B(H) is the (norm-closed) real subspace S(H) of self-adjoint operators. However, S(H) is not an ideal and accordingly, right and left multiplication operators do not in general map S(H) invariantly. This motivates us to instead work with ”conjugate operators” CT : Definition 1.4. Let T, S ∈ B(H), then CT (S) is the operator given by CT (S) ≡ T ST ∗ where T ∗ denotes the Hilbert adjoint of T . Indeed, we note that for any T ∈ B(H), CT deﬁnes a continuous operator on S(H) provided with topologies such as: the norm topology, the SOT and, the topology in between, the COT. Of course, any CT also deﬁnes an operator on spaces like B(H), K(H) and N (H), and in particular we shall complement Proposition 1.2 by proving a similar statement for conjugate operators CT . We formulate an important consequence of our main results. A weighted backward shift Bw is an operator in B(H) deﬁned by Bw en ≡ wn en−1 if n ≥ 1 and Bw e0 ≡ 0, where w = (wn ) is a bounded (weight) sequence ∗ is the of nonzero scalars and (en ) a given orthonormal basis of H. The adjoint Bw forward shift given by en → wn+1 en+1 , n ≥ 0. In particular we shall denote by B the unweighted backward shift, i.e. B ≡ Bw where w = (wn = 1), and F ≡ B ∗ . We shall see that the following holds: Theorem 1.5. For any weight w = (wn ) satisfying the following: sup |w1 w2 ...wn | = ∞, n

(1)

CBw is COT-hypercyclic on S(H). That is, there exists an S ∈ S(H) such that n ∗n SBw }n≥0 is COT-dense in S(H). {Bw B (or rather its weight (1, 1, ...)) does of course not satisfy the hypothesis of the theorem. However, if |λ| > 1, λB is a weighted backward shift whose weight n n = αn CB if |λ|2 = α, we (λ, λ, ...) satisﬁes (1) and hence, by noting that CλB obtain: Corollary 1.6. For any real α with α > 1 there exists an S ∈ S(H) such that {αn B n SF n }n≥0 is COT-dense in S(H). Before we start our investigation we comment and clarify some few things. The space of nuclear operators, N (H), is formed by all operators T that admit a representation T = λi ·, fi ei where (λi ) ∈ 1 and (ei ) and (fi ) are bounded sequences in H and H respectively. The nuclear norm · N on N (H) is deﬁned by T N ≡ inf |λi |fi ei , where inﬁmum is taken over all possible representations λi ·, fi ei of T . (N (X) is deﬁned analogously for a general X.) In fact, for any p ≥ 1, Np (H) is the set – and ideal – of all T ∈ B(H) such ∞ that n=0 αn (T )p < ∞, where αn (T ) ≡ inf{T − K : Rank K ≤ n}, equipped

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p 1/p with the norm T p ≡ ( ∞ . The spaces Np (H) – the Schatten-von 0 αn (T ) ) Neumann classes – are Banach spaces. It follows now that N (H) = N1 (H) (even topologically) [15, Theorem 8.3.3]. Further, when p = 2 we obtain another important class, namely the Hilbert-Schmidt class [15, Theorem 8.3.4]. We recall that N2 (H), i.e. the Hilbert-Schmidt class, is a Hilbert space, and refer to [15, Section 2.5] and [14, Appendix 1] for details. Any element of Np (H) (p ≥ 1) is compact [15, p. 14 and Proposition 8.2.6] and so, for any p ≥ 1 we have the embeddings: N (H) = N1 (H) → Np (H) → K(H) → B(H) which also are continuous with dense ranges if we provide K(H) with the operator norm topology and B(H) with the COT. Indeed, the density is a consequence of the fact that the space A(H) of ﬁnite rank operators in B(H) is dense in any of these spaces, see [15, Proposition 8.2.5] and [12, Theorem 1.e.4]. The paper is organized as follows. In Section 3 we study hypercyclicity of conjugate operators CT acting on B(H), K(H) and Np (H) (1 ≤ p < ∞). We apply here the tensor product technique developed in [6], and obtain an analogue of Proposition 1.2 for conjugate maps. Unfortunately, due to the fact that N (H) is not naturally embedded into S(H), this technique does not work for S(H). However, with another approach, we can in Section 2 prove that Proposition 1.2 indeed also extends to conjugate maps acting on S(H) provided with the COT. Thus we shall prove the following: Theorem 1.7. If T ∈ B(H) satisfies the Hypercyclicity Criterion, then CT is hypercyclic on 1. S(H) provided with the COT ; 2. B(H) provided with the COT ; 3. K(H) provided with the operator norm topology; 4. Np (H) (p ≥ 1). In the last section, Section 4, we extend our study and some of our previous results. In particular, when 0 < p < 1, Np (H), which is deﬁned in the same way as when p ≥ 1, is too ”small” in order to apply the tensor product technique. But with ideas from Section 2, we can extend the results of Section 3 for Np (H) where p ≥ 1 to the case p < 1.

2. Hypercyclicity in S(H) In this section we prove (1) of Theorem 1.7, in the proof we use some ideas from [1]: Theorem 2.1. If T ∈ B(H) satisfies the HC, then CT is COT-hypercyclic on S(H). Proof. Let As (H) denote the set of all ﬁnite rank operators in S(H), i.e., As (H) ≡ S(H)∩A(H). (One can show that As (H) in fact consists of all elements of the form an (·, xn )xn , where (an ) is a ﬁnite sequence in R and xi ∈ H. Here (·, ·) denotes

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the inner-product in H.) By A¯s (H) we denote the norm-closure of As (H) in B(H). Since S(H) is norm-closed in B(H), A¯s (H) forms a subspace of S(H). Further, As (H) is separable in norm and so A¯s (H) is a separable Banach space, where we tacitly assume that A¯s (H) is provided with the operator norm topology. We note that CT maps As (H) invariantly, and thus, CT is a well-deﬁned continuous operator on A¯s (H). We prove that CT : A¯s (H) → A¯s (H) satisﬁes the HC, and is thus hypercyclic. This will also prove our statement, because A¯s (H) is continuously and, as we shall see, densely embedded into S(H) provided with the COT, and so S forms a hypercyclic vector for CT : S(H) → S(H) for any hypercyclic vector S ∈ A¯s (H) ⊆ S(H) for CT : A¯s (H) → A¯s (H). Sublemma: As (H) is dense in S(H). Proof of Sublemma. Let (en ) be an orthonormal basis of H, and let Pn denote the n orthogonal projection onto span{e0 , ..., en }, i.e., Pn ≡ 0 (·, ei )ei . We have that Pn ≤ 1 and lim Pn = I (≡ identity on H) with respect to the COT [12, p. 30]. Let S ∈ S(H) and consider the operators CPn (S) = Pn SPn ∈ As (H), n ≥ 0. (Note that Pn ∈ S(H) so Pn = Pn∗ .) We claim that CPn (S) → S as n → ∞ with respect to the COT. Indeed, for any compact set K ⊆ H, SK is compact and so for any f ∈ K: Pn SPn f − Sf ≤ Pn S(Pn − I)f + (Pn − I)Sf ≤ ε for all n ≥ n(ε). This completes the proof of the Sublemma. Thus, we are left to prove that CT : A¯s (H) → A¯s (H) satisﬁes the HC. Let Z, Y ⊆ H, (nk ) ⊆ N, be elements for which the HC, i.e. (1) and (2) of Deﬁnition 1.3, holds for T . Let now Z˜ denote the subset of As (H) ⊆ A¯s (H) formed by all elements F of the form zi ∈ Z. F = an (·, zn )zn , ˜ Y˜ form dense subsets The space Y˜ is deﬁned analogously, and we have that Z, an (·, T nk zn )T nk zn → of As (H), and thus of A¯s (H). We have that CTnk (F ) = ˜ because the sum is ﬁnite and limk→∞ T nk zn = 0 for all n. 0 for any F ∈ Z, Next, let F = an (·, yn )yn ∈ Y˜ (yi ∈ Y ) be arbitrary. For each n we can, by (n) (n) the assumption on T , ﬁnd a nullsequence (xk ) in H such that T nk xk → xn (n) (n) (k → ∞). It follows now that with Fk ≡ an (·, xk )xk ∈ A¯s (H), Fk → 0 and nk nk (n) nk (n) CT (Fk ) = an (·, T xk )T xk → F as k → ∞. Hence CT : A¯s (H) → A¯s (H) satisﬁes the HC (with respect to (nk )). Theorem 1.5 follows now by the fact that Bw satisﬁes the HC if and only if the weight w = (wn ) satisﬁes (1). This follows from [19, Theorem 2.8] and [5, Remark 3.2]. We conclude this section with another example where Theorem 2.1 applies, and refer to [9, Sections 3 and 4] for further details:

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Example 1. Let A2 (Ω) denote the Bergman space of holomorphic L2 (Ω) = L2 (Ω, m) functions on the bounded open set Ω ⊆ C. (m denotes here the Lebesgue measure on R2 ∼ C.) Then A2 (Ω) is closed in L2 (Ω) and is thus a (separable) Hilbert space. Let ϕ be a bounded nonconstant holomorphic function on Ω and let Mϕ denote the corresponding multiplier Mϕ : f → f ϕ. It follows that Mϕ is a bounded oneto-one operator on A2 (Ω). Now if ϕ(Ω) meets the unit circle |z| = 1, the adjoint Mϕ∗ is hypercyclic on A2 (Ω), in fact, Mϕ∗ satisﬁes the HC with respect to the full sequence (nk = k) [9, Theorem 4.5] (see also [2, Corollary 2.7]). Hence Theorem 2.1 gives: If ϕ meets the unit circle, CMϕ∗ is hypercyclic on S(A2 (Ω)) provided with the COT. The conclusion extends, by [9, Theorem 4.5], to adjoints of arbitrary nonconstant multipliers (see [9, p. 250] for the deﬁnition) acting on a (separable) Hilbert space H – e.g. the Hardy space of the unit disc – of holomorphic functions.

3. Hypercyclicity in B(H), K(H) and Np (H) We prove here the remaining part of Theorem 1.7, i.e: Theorem 3.1. Assume T ∈ B(H) satisfies the HC, then CT is hypercyclic on: 1. B(H) provided with the COT ; 2. K(H) provided with the operator norm topology; 3. Np (H) (p ≥ 1). ˆ π H denote the completion Proof. First we prove (3) for N1 (H) = N (H). Let H ⊗ of H ⊗ H with respect to the projective tensor product topology π (see e.g. ˆ π H admits a representation λi fi ⊗ ei [16, Chapter 7.2]). Any element of H ⊗ where (λi ) ∈ 1 and (fi ) and (ei ) are null sequences in H and H respectively [16, Corollary 7.2.1]. Accordingly, the map i : H ⊗H → N (H) deﬁned by f ⊗e → ·, f e ˆ π H → N (H). Next, any pair of maps extends to a continuous surjection ˆi : H ⊗ ˆ on H ⊗ ˆ ˆ π H. To be speciﬁc, R⊗S R ∈ B(H ), S ∈ B(H) deﬁnes an operator R⊗S is the unique continuous linear extension of the operator R ⊗ S on H ⊗ H deﬁned by f ⊗ e → Rf ⊗ Se. Consider now the operator T . T induces an operator T∗ ∈ B(H ) by T∗ λf ≡ λT f where λf ≡ (·, f ) ∈ H and (·, ·) denotes the inner-product in H. It is clear that if T satisﬁes the HC with respect to (nk ), then so does T∗ . By [13, Corollary ˆ is hypercyclic on H ⊗ ˆ π H. But we note that ˆi ◦ T∗ ⊗T ˆ = CT ◦ ˆi and 1.10] T∗ ⊗T thus, see [6, Lemma 1.4], CT is hypercyclic on N (H). To prove the statements for B(H) and K(H) and the remaining cases of (3), we only have to apply what we just have proved, [6, Lemma 1.4] and the fact (see the Introduction) that the embeddings N (H) → Np (H), N (H) → K(H) and N (H) → B(H) are continuous and have dense ranges. In particular, if B(H) and K(H) are provided with the topologies (1) and (2), respectively, in Theorem 3.1, we have:

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Corollary 3.2. For any weighted backward shift Bw whose weight w = (wn ) satisfies (1), CBw is hypercyclic on any of the spaces B(H), K(H) and Np (H) (1 ≤ p < ∞). From the discussion in Example 1, we also conclude that CMϕ∗ , where ϕ is a nonconstant multiplier whose image ϕ(Ω) meets the unit circle, is hypercyclic on any of the spaces B(H), K(H) and Np (H), where H = A2 (Ω) or some other suitable Hilbert space of holomorphic functions (on Ω).

4. Extensions A conjugate map CT is of course nothing else than the composition LT RT ∗ = RT ∗ LT of a left and right multiplication operator. Now, given any pair of operators S, T ∈ B(X), we may consider hypercyclicity of the ”generalized” conjugate map CS,T : Definition 4.1. If S, T ∈ B(X), then CS,T ≡ LS RT (= RT LS ). Let us note that CT,I = LT and CI,T = RT , and when X is a Hilbert space, n CT,T ∗ = CT . Note also that CS,T (R) = S n RT n . By working with generalized conjugate maps, we may also study hypercyclicity on suitable subspaces of B(X) where X may not be a Hilbert space. In particular, the spaces Np (H) have analogies when we replace H by a Banach space X, and we may also allow p to be less than 1. Indeed, given 0 < p < ∞, Np (X) is the space of all T ∈ B(X) such that n≥0 αn (T )p < ∞, equipped with the quasi-norm T p ≡ ( αn (T )p )1/p . Here, and as before, αn (T ) denote the approximation numbers deﬁned by αn (T ) ≡ inf{T − K : Rank K ≤ n}. Recall that a quasi-norm · is a norm except that the triangle inequality is replaced by the following weaker condition: There is a constant C > 0 such that x + y ≤ C(x + y) holds for all x, y in the space under consideration. Any quasi-norm generates a separated TVS structure (not necessarily locally convex), and a TVS is called a quasi-Banach space if it is complete and the topology is generated by a quasi-norm. Let us also mention that any quasi-Banach space is metrizable, this, well-known fact, follows for example by [17, Theorem 1.24]. To indicate the importance of the spaces Np (X) we recall that they characterize nuclear spaces [15, Theorem 8.6.1]. We saw that when X is a Hilbert space H and p ≥ 1, ·p is in fact a norm and Np (H) a Banach space, however, when p < 1 or when X is not a Hilbert space, this is in general not the case. However, Np (X) is always complete and thus a quasi-Banach space [15, pp. 125-126]. Another important diﬀerence between the Banach and Hilbert space setting is that, even if p ≥ 1, N (X) is not contained in Np (X) – in fact N1 (X) N (X) [15, p. 138] – for a general X. (Recall that N (H) = N1 (H) ⊆ Np (H) for any p > 1.) Now, any Np (X) forms an ideal in B(X) and so any generalized conjugate operator CS,T maps Np (X) into Np (X), and this continuously by [15, Proposition 8.2.8].

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Definition 4.2. A pair (T1 , T2 ) of continuous linear operators acting on the Banach spaces X1 and X2 , respectively, is said to satisfy the Mutual Hypercyclicity Criterion (MHC) if the following holds: There exist dense subsets Zi , Yi ⊆ Xi (i = 1, 2) and an increasing sequence (nk ) ⊆ N such that 1. T1nk z1 T2nk z2 → 0 for all (z1 , z2 ) ∈ Z1 × Z2 ; (i) 2. For any (y1 , y2 ) ∈ Y1 × Y2 , there exists sequences (yk ) ⊆ Xi such that (1) (2) (i) yk yk → 0 and Tink yk → yi (i = 1, 2) as k → ∞. In particular we have that (T, I) (and thus (I, T )) satisﬁes the MHC whenever T ∈ B(X) satisﬁes the HC and I is the identity operator on some Banach space. The MHC is related to the Tensor Hypercyclicity Criterion (THC) introduced in [13]. Indeed, we note that if T1 satisﬁes the HC and T2 the THC (or vice versa) with respect to some common sequence, then (T1 , T2 ) satisﬁes the MHC. Note also that if T ∈ B(H) satisﬁes the HC, then (T, T ∗ ) satisﬁes the MHC, so the following generalizes (3) of Theorem 3.1: Theorem 4.3. Let S, T ∈ B(X), where X is separable, and assume that (S, T ) satisfies the MHC. Then CS,T is hypercyclic on Np (X) (0 < p < ∞). Proof. First we recall that any continuous linear operator on a separable complete metrizable TVS E, that satisﬁes the HC (replace X by E in Deﬁnition 1.3), is hypercyclic [5]. In order to apply this we must prove that Np (X) is separable (recall that any quasi-Banach space, and thus Np (X), is complete and metrizable). By our hypothesis, we may ﬁnd countable dense subsets Y˜ and Z˜ of X and X , respectively. We claim now that the (countable) set D formed by all ﬁnite sums ˜ is dense in Np (X). Indeed, the set A(X) of

·, zi yi , where yi ∈ Y˜ and zi ∈ Z, ﬁnite rank operators is dense in Np (X) [15, Proposition 8.2.5], thus it suﬃces to prove that D is dense in A(X). This follows from the simple fact that if zk → z ∈ X and yk → y ∈ X, then ·, zk yk → ·, zy in Np (X). Indeed

·, zk yk − ·, zy = ·, zk − zyk + ·, z(yk − y), and our claim follows now by estimates like in (2) below. Hence it suﬃces to prove that CS,T satisﬁes the HC. Since (S, T ) satisﬁes the MHC, we may ﬁnd (nk ) ⊆ N, dense subsets Yi ⊆ X, Zi ⊆ X (i = 1, 2) such that S nk yT nk z → 0 for any (y, z) ∈ Y1 × Z1 and such that for any (y, z) ∈ Y2 × Z2 , (S nk y (k) , T nk z (k) ) → (y, z) for some sequences (y (k) ) ⊆ X, (k) (k) (k) (z ) ⊆ X with y z → 0. We have now that the operators of the form

·, zn yn , where the sum is ﬁnite and (yi , zi ) ∈ Y1 × Z1 , form a dense set U nk in A(X), and hence in Np (X). We claim that CS,T → 0 pointwise on U , and to nk prove this it suﬃces to prove that CS,T (u) → 0 for any u = ·, zy ∈ U . But nk nk CS,T (u) = ·, T nk zS nk y is a rank 1 operator and hence αr (CS,T (u)) = 0 for all r ≥ 1, and for r = 0 we have nk nk αr (CS,T (u)) ≤ CS,T (u) ≤ T nk zS nk y → 0,

as k → ∞. This shows that

nk CS,T (u)

→ 0.

(2)

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Next, let V denote the dense set in Np (X) formed by the operators

·, zn yn , where the sum is ﬁnite and (yi , zi ) ∈ Y2 × Z2 . Given v =

·, zn yn ∈ V , (n) (n) we may for each n ﬁnd sequences (yk ) ⊆ X, (zk ) ⊆ X such that, when (n) (n) (n) (n) k → ∞, yk zk → 0 but S nk yk → yn and T nk zk → zn . Deﬁne now (n) (n) vk ≡ n ·, zk yk . With arguments as above we have that (vk ) is a nullsequence (n) (n) in Np (X). Also, from the ﬁrst paragraph of the proof, ·, zk yk → ·, zn yn for all n, and hence vk → v in Np (X). Thus CS,T satisﬁes the HC. In particular we conclude that LT = CT,I is hypercyclic on Np (X) (0 < p < ∞), when T ∈ B(X) satisﬁes the HC and X is separable. This complements Proposition 1.2. Note that Theorem 4.3 does not imply that CS,T is hypercyclic on N (X), because we recall that N (X) = N1 (X) in general. However, the proof above for Np (X) applies to N (X), and if we assume X has the approximation property, A(X), and thus N (X), is dense in K(X) and B(X). We assume here, and everywhere below, that K(X) and B(X) are provided with the topologies (1) and (2), respectively, of Proposition 1.2. We have thus: Theorem 4.4. Assume that X has a separable dual and that (S, T ) satisfies the MHC, where S, T ∈ B(X). Then CS,T is hypercyclic on N (X). Further, if X also has the approximation property, then any hypercyclic vector R ∈ N (X) (⊆ K(X) ⊆ B(X)) for CS,T : N (X) → N (X), forms a hypercyclic vector for CS,T acting on K(X) as well as on B(X). As an application of Theorems 4.3 and 4.4 we may consider generalized conjugate operators built up by backward and forward shifts. Recall that a basis (en ) of a Banach space X is called shrinking if the corresponding biorthogonal functionals (en ) forms a basis of X (thus X is then separable). A basis (en ) is said to be symmetric whenever (eπ(n) ) forms an equivalent basis for any permutation π of N. See also [12, Deﬁnition 3.a.1] and Page 8 in the same book. The canonical unit bases of c0 and p (1 < p < ∞) are both shrinking and symmetric, other examples of spaces with such bases are Lorentz and Orlicz sequence spaces [12, Chapter 4]. (The canonical unit basis of 1 is symmetric, but not shrinking, because 1 ∞ is not separable.) Assume now X has a symmetric basis (en ) = (en )n≥0 , we may assume it is normalized in the sense that en = 1 for all n. Then backward and forward shifts (relative to (en )) are well-deﬁned bounded operators on X. Recall that given a weight sequence w = (wn ) (i.e. a bounded sequence of nonzero scalars), ∞ the back∞ ward shift Bw is the bounded operatordeﬁned by 0 xn en → 1 wn xn en−1 , ∞ and the forward shift Fw is deﬁned by ∞ 0 xn en → 0 wn+1 xn en+1 . Theorems 4.3 and 4.4 give: Theorem 4.5. Assume X has a shrinking symmetric (normalized ) basis (en ). Let w and µ be weight sequences such that lim |w1 w2 ...wn+i ||µ1 µ2 ...µn+j | = ∞

n→∞

(3)

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for all i, j ≥ 0. Then CBw ,Fµ is hypercyclic on any of the spaces Np (X) (0 < p < ∞), N (X), K(X) and B(X), where Bw and Fµ are backward and forward shifts, respectively, relative to the basis (en ). Proof. The adjoint of Fµ is the backward shift Bµ (relative to (en )). We prove that (Bw , Bµ ) satisﬁes the MHC. Let Y and Z denote the span of {en }n≥0 and of the corresponding dual basis elements {en }n≥0 , respectively. Then Y ⊆ X and n Z ⊆ X are dense. We have also that Bw y = Bµn z = 0 for all n suﬃciently large, if (y, z) ∈ Y × Z. Next we note that Fw−1 , where w−1 ≡ (wn−1 ), forms a right inverse to Bw on Y (note that Fw−1 is indeed well-deﬁned on Y ). Similarly, Fµ−1 is a right inverse to Bµ on Z. Hence, we only have to prove that Fwn−1 yFµn−1 z → 0 (n → ∞) for all (y, z) ∈ Y ×Z. In fact, it is enough to prove that Fwn−1 ei Fµn−1 ej → 0 for any i, j ≥ 0. But it is easily seen that 1 w1 w2 ...wi Fwn−1 e0 = en , Fwn−1 ei = en+i , i ≥ 1, w1 w2 ...wn w1 w2 ...wn+i and similarly for Fµn−1 ej . Since (en ) is a normalized basis, (en ) is a bounded sequence in X [12, p. 7]. Hence (3) gives that Fwn−1 ei Fµn−1 ej goes to zero and we are done. With w = (λ, λ, ...), where |λ| > 1, and µ = (1, 1, ...), and with B ≡ B(1,1,...) and F ≡ F(1,1,...) , we obtain: Corollary 4.6. Assume that X has a shrinking symmetric basis (en ) and that |λ| > 1. Then there exists a T ∈ Np (X) such that {λn B n T F n }n≥0 is dense in Np (X) (0 < p < ∞). We can also find T ∈ N (X) such that {λn B n T F n }n≥0 is dense in any of the spaces N (X), K(X) and B(X). The ideas and results of this section extend presumably to the ideals in B(X) of p-absolutely summing operators, p > 0, which are closely related to the Schatten-von Neumann classed Np (X), see e.g. [12, Chapter 2.b], and presumably also to other operator ideals (see [18] for a beautiful elementary exposition). Acknowledgement. I would like to thank the referee for some useful remarks, especially for some arguments improving a weaker version of Theorem 2.1. The proof of Theorem 2.1 is essentially due to him/her.

References [1] R. Aron, J. B`es, F. Le´ on, A. Peris, Operators with common hypercyclic subspaces, J. Operator Theory 54 (2005), 251–260. [2] T. Berm´ udez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria, Bull. Austral. Math. Soc. 70 (2004), 45-54. [3] L. Bernal-Gonz´ alez and K.-G. Grosse-Erdmann, The Hypercyclicity Criterion for sequences of operators, Studia Math. 157 (1) (2003), 17-32.

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[4] J. B`es, Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc. 127 (1999), 1801-1804. [5] J. B`es and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), 94-112. [6] J. Bonet, F. Mart´ınez-Gim´enez and A. Peris, Universal and chaotic multipliers on spaces of operators, J. Math. Anal. Appl. 297 (2004), 599-611. [7] K. C. Chan, Hypercyclicity of the operator algebra for a separable Hilbert space, J. Operator Theory 42 (1999), 231-244. [8] K. C. Chan and R. D. Taylor, Hypercyclic subspaces of a Banach space, 41 (2001), 381-388. [9] G. Godefroy and J. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. [10] K. -G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (3) (1999), 345-381. [11] K. -G. Grosse-Erdmann, Recent developments in hypercyclicity, Rev. R. Acad. Cienc. Ser. A Mat. RAC-SAM 97 (2003), 273-286. [12] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag (Heidelberg), 1977. [13] F. Mart´ınez-Gim´enez and A. Peris, Universality and chaos for tensor products of operators, J. Approx. Theory 124 (2003), 7-24. [14] V. Peller, Hankel operators and their applications, Springer Monographs in Math. Springer-Verlag (New-York), 2003. [15] A. Pietsch, Nuclear locally convex spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66, Springer-Verlag, 1972. [16] A. J. Robertson and W. J. Robertson, Topological vector spaces, Cambridge University Press, 1964. [17] W. Rudin, Functional Analysis, McGraw-Hill, Inc. (New York), 1988. [18] R. Ryan, Introduction to Tensor Products of Banach Spaces, Springer-Verlag, 2002. [19] H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1004. Henrik Petersson Chalmers/G¨ oteborg University School of Mathematical Sciences SE-412 96 G¨ oteborg Sweden e-mail: [email protected] Submitted: August 4, 2005 Revised: November 20, 2006

Integr. equ. oper. theory 57 (2007), 425–449 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/030425-25, published online December 26, 2006 DOI 10.1007/s00020-006-1460-2

Integral Equations and Operator Theory

Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains Roberto C. Raimondo Abstract. In this paper we study the problem of the membership of Hφ in the Hilbert-Schmidt class, when φ ∈ L∞ (Ω) and Ω is a planar domain. We ﬁnd a necessary and suﬃcient condition. We apply this result to the problem of joint membership of Hϕ and Hϕ in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and suﬃcient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case Hϕ with ϕ holomorphic. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B38. Keywords. Hankel operators, Bergman Spaces.

1. Introduction Let Ω be a bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of ﬁnitely many simple closed smooth analytic curves γj (j=1,2,..,n) where γj are positively oriented with respect to Ω and γj ∩ γi = ∅ if i = j. We also assume that γ1 is the boundary of the unbounded component of C\Ω. Let Ω1 be the bounded component of C\γ1 , and Ωj (j=2,. . . ,n) the unbounded component of C\γj , respectively, so that Ω = ∩nj=1 Ωj . For dν = π1 dxdy we consider the usual L2 −space L2 (Ω) = L2 (Ω, dν). The Bergman Space L2a (Ω, dν), consisting of all holomorphic functions which are L2 −integrable, is a closed subspace of L2 (Ω, dν) with the inner product given by f (z)g(z)dν(z) f, g = Ω

for f, g ∈ L2 (Ω, dν). The Bergman Projection is the orthogonal projection P : L2 (Ω, dν) −→ L2a (Ω, dν), This work was completed with the support of our TEX-pert.

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it is well-known that for any f ∈ L2 (Ω, dν) we have f (z)K Ω (z, w)dν(z), P f (w) = Ω Ω

where K is the Bergman Reproducing kernel of Ω. For ϕ ∈ L∞ (Ω, dν) the Hankel operator Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) is deﬁned by Hϕ = (1 − P )Mϕ where Mϕ is the standard multiplication operator. A simple calculation shows that Hϕ f (z) = (ϕ(z) − ϕ(w))f (w)K Ω (w, z)dν(w) Ω

The problem of ﬁnding necessary and suﬃcient conditions on the function ϕ ∈ L∞ (Ω, dν) for the Hankel operators Hϕ to be Hilbert-Schmidt has been studied by many authors in many diﬀerent settings. We study this problem for planar domains and we give a necessary and suﬃcient condition. We apply this result to give a necessary and suﬃcient condition on the function ϕ ∈ L∞ (Ω, dν) for the Hankel operators Hϕ and Hϕ to be jointly Hilbert-Schmidt. We prove this result using the notion of Berezin Transform and a result proved by K. Zhu. Moreover, using the fact that if ϕ is holomorphic the operator Hϕ is trivial, we are able to recover a result that was proved by Arazy and Fisher (see [4]). The paper is organized as follows. In section 2 we transfer our problem from an arbitrary planar domain to a canonical one i.e. a domain whose boundary consists of circles, and we show that it is enough to solve the problem for this special type of planar domains. In Section 3 we collect results about the Bergman kernel for a planar domain and the structure of L2a (Ω, dν). In Section 4 we collect results about Topelitz operators and we prove our main results and ﬁnally we conclude with the study of Hankel operators with antiholomorphic symbols.

2. Preliminaries Let Ω be the bounded multiply-connected domain given at the beginning of Section 1 i.e. Ω = ∩nj=1 Ωj , where Ω1 is the bounded component of C\γ1 , and Ωj (j=2,. . . ,n) is the unbounded component of C\γj . We use the symbol ∆ to indicate the punctured disk {z ∈ C|0 < |z| < 1}. Let Γ be any one of the domains Ω, ∆, Ωj (j=2,. . . ,n). We call K Γ (z, w) the reproducing kernel of Γ and we use the symbol k Γ (z, w) to 1 indicate the normalized reproducing kernel i.e. k Γ (z, w) = K Γ (z, w)/K Γ (w, w) 2 . 2 ˜ the Berezin transform of A, by For any A ∈ B(La (Γ, dν)) we deﬁne A, Γ Γ Γ Γ (z)dν(z), ˜ A(w) = Akw , kw = Akw (z)kw Γ

Γ kw (·) ∞

Γ

Γ

− 12

where = K (·, w)K (w, w) . If ϕ ∈ L (Γ), then we indicate with the symbol ϕ˜ the Berezin Transform of the

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associated Toeplitz operator Tϕ and we have Γ ϕ(w) ˜ = ϕ(z)|kw (z)|2 dν(z). Γ

We remind the reader that it is well-known that A˜ ∈ Cb∞ (Γ) and we have that ˜ ∞ ≤ ||A||B(L2 (Ω)) . ||A|| Before we state the main results of this paper we need to give a deﬁnition. Definition 2.1. Let Ω = ∩ni=1 Ωi be a bounded multiply-connected domain. We say that the set of n functions P = {p1 , . . . , pn } is a ∂−partition for Ω if: 1. For every j = 1, . . . , n pj : Ω → [0, 1] is a Lipschitz, C ∞ −function. 2. For every j = 1, . . . , n there exists an open set Wj and an j > 0 such that the Uj = {ζ ∈ Ω : rj < |ζ − aj | < rj + j } is contained in Wj and pj (ζ) = 1 if ζ ∈ Uj , and pj (ζ) = 0 if ζ ∈ Wk and j = k; 3. For any ζ ∈ Ω the following n

pi (ζ) = 1

i=1

holds. We will discuss more in detail this deﬁnition at pp. 15 , for the moment we need only to point out that near each connected component of the boundary there is only one function which is diﬀerent from zero (note that this implies that the function must be equal to 1). With this deﬁnition in mind we can state the main theorem: Theorem 2.2. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the following are equivalent. 1. The operator Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) is Hilbert-Schmidt. 2. For any j = 1, . . . , n the operators Hϕj : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt. 3. For any j = 1, . . . , n the operators Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) are Hilbert-Schmidt. and

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Theorem 2.3. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the following are equivalent. 1. The operators Hϕ , Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt. 2. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt. 3. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) are Hilbert-Schmidt. and ﬁnally we recover the following (a diﬀerent proof of this fact has been established, in a more general setting, by Beatrous and Li, see [8]). Theorem 2.4. Let ϕ ∈ L∞ (Ω), and Ω be a bounded multiply-connected domain in the complex plane C, whose boundary ∂Ω consists of finitely many simple closed smooth analytic curves γj . Then the operators Hϕ , Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt if and only if 2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

holds. We remind the reader (see [21] or [16], Chapter 15) that any bounded multiply-connected domain whose boundary consists of ﬁnitely many simple closed smooth analytic curves, i.e. a regular domain, is conformally equivalent to a canonical bounded multiply connected domain whose boundary consists of ﬁnitely many circles. Moreover, it is possible to prove (see [3] and [4]) the following. Theorem 2.5. Let Ω be a regular domain and let ψ be a conformal mapping from Ω onto D then: 1. The Bergman kernels of Ω and ψ(Ω) = D are related via K D (ψ(z), ψ(w))ψ (z)ψ (w) = K Ω (z, w); 2. The operator Vψ f = ψ · f ◦ ψ is an isometry from L2 (D) onto L2 (Ω); 3. Vψ P D = P Ω Vψ ; 4. If ϕ : D → W is conformally onto, then Vϕ Vψ = Vϕ◦ψ . In particular Vψ−1 = Vψ−1 . Now we observe that if A ∈ B(L2a (Ω)) and we deﬁne AD ∈ B(L2a (D)) as Vψ−1 AVψ , where ψ is a conformal mapping from Ω onto D, then we it is possible to prove the following.

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˜ Proposition 2.6. A(z) = A˜D (ψ(z)). Since every conformal map is an open map, we can conclude that it is enough to prove the theorem when the domain is a canonical bounded multiply connected domain whose boundary consists of ﬁnitely many circles.

3. The Structure of L2a (Ω) and some Estimates about the Bergman Kernel From now on we will assume that Ω = ∩nj=1 Ωj where Ω1 = {z ∈ C : |z| < 1} and Ωj = {z ∈ C : |z − aj | > rj } for j=2,. . . ,n. Here aj ∈ Ω1 and 0 < rj < 1 with |aj − ak | > rj + rk if j = k and 1 − |aj | > rj . We will indicate with the symbol ∆ the punctured disk Ω1 \{0}. With the symbols K Ωj (z, w), K Ω (z, w), K ∆ (z, w) we denote the Bergman kernel on Ωj , Ω, and ∆ respectively. If we deﬁne Lpa (Ω1 ) = {f ∈ Lp (Ω)| f is holomorphic} and Lpa (∆) = {f ∈ Lp (∆)| f is holomorphic}, then we have the following. Theorem 3.1. There exists an isomorphism I : L2 (∆) −→ L2 (Ω1 ) such that I(L2a (∆)) = L2a (Ω1 ). Moreover for any p ≥ 2 we have that Lpa (∆) = Lpa (Ω1 ) and, for any (z, w) ∈ ∆2 , the Bergman kernels K ∆ and K Ω1 satisfy the following equation K ∆ (z, w) = K Ω1 (z, w). Proof. Suppose that f ∈ L2a (∆), this means that f is holomorphic on ∆, then we can write down the Laurent expansion of f about 0 and we have

f (z) =

∞ n=−∞

an z n .

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n m This implies that |f (z)|2 = ∞ n,m=−∞ an am z z , therefore we have ∞ |f (z)|2 dν(z) = an am z n z m dν(z) ∆

∆ n,m=−∞

2π

∞

1

= 0

=

=

0 n,m=−∞

∞

an am

n,m=−∞ ∞

2π

2

n=−∞



=

2π 

∞

|an |2

n=−1

2π

ei(n−m)θ dθ

0

|an |

an am rn+m+1 ei(n−m)θ drdθ 1

rn+m+1 dr

0 1

r2n+1 dr

0

r2n+2 2n + 2

1 0

+ |a−1 |2

0

1

 1  dr . r

The last equation, together with the fact that f is square-integrable, implies that an = 0 if n ≤ −1. Then we can conclude that f has an holomorphic extension on Ω1 . We deﬁne I : L2 (∆) −→ L2 (Ω1 ) in this way: if g ∈ L2 (∆), then (Ig)z = g(z) if z = 0 and g(z)dν(z). (Ig)0 = ∆

Then Ig ∈ L2 (Ω1 ) and ||Ig||Ω1 = ||g||∆ . If f ∈ L2a (∆), we have just shownd that If ∈ L2a (Ω1 ). Clearly I is injective and surjective, in fact if G ∈ L2 (Ω1 ), then g = G|∆ is an element of L2 (∆) and I(g) = G. Then I is an isomorphism of L2 (∆) onto L2 (Ω1 ) and I(L2a (∆)) = L2a (Ω1 ). Moreover, observing that p > 2 implies ||f ||∆,2 ≤ ||f ||∆,p for any f ∈ Lp (∆), we conclude that H p (∆) = H p (Ω1 ). Finally, it is easy to verify that for any f, g ∈ L2a (∆) we have f, g∆ = If, IgΩ1 and this fact implies, by the deﬁnition of the Bergman Reproducing kernel, that K ∆ (z, w) = K Ω1 (z, w) for any (z, w) ∈ ∆2 .

The last Theorem has an important application. In fact we have, Remark 3.2. If ∆a,r = {z ∈ C : 0 < |z − a| < r} and Oa,r = {z ∈ C : |z − a| > r} then r2 ∀(z, w) ∈ Oa,r × Oa,r . K Oa,r (z, w) = (r2 − (z − a) · (w − a))2

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The previous Theorem and the well-known fact that the reproducing kernel of the unit disk is given by (1 − z w) ¯ −2 imply that K ∆0,1 (z, w) =

1 (1 − z · w)2

∀(z, w) ∈ ∆0,1 × ∆0,1 .

Then we can conclude, by comformal mapping, that the reproducing kernel of ∆a,r is r2 K ∆a,r (z, w) = ∀(z, w) ∈ ∆a,r × ∆a,r . (r2 − (z − a) · (w − a))2 Now we deﬁne ϕ : ∆a,r → Oa,r by ϕ(z) = (z − a)−1 r2 + a, and we use the well-known fact that the Bergman kernels of ∆a,r and ψ(∆a,r ) = Oa,r are related via K Oa,r (ϕ(z), ϕ(w))ϕ (z)ϕ (w) = K ∆a,r (z, w); to obtain that K Oa,r (z, w) =

r2 (r2 − (z − a) · (w − a))2

∀(z, w) ∈ Oa,r × Oa,r .

Since Ω1 = O0,1 and, for j = 2, . . . , n, Oaj ,rj = Ωj , then the Remark implies that K Ω1 (z, w) =

1 (1 − z · w)2

and, K Ωj (z, w) =

rj2 (rj2 − (z − aj ) · (w − aj ))2

if j=2,. . . ,n. If we deﬁne E Ω = K(z, w) −

n

K Ωj (z, w)

j=1

we can prove Lemma 3.3. 1. E Ω is conjugate symmetric about z and w. For each w ∈ Ω, E Ω (·, w) is conjugate analytic on Ω and E Ω ∈ C ∞ (Ω × Ω). 2. There are neighborhoods Uj of ∂Ωj (j=1,. . . ,n) and a constant C > 0 such that Uj ∩ Uk is empty if j = k and Ω K (z, w) − K Ωj (z, w) < C for z ∈ Ω and w ∈ Uj . 3. E Ω ∈ L∞ (Ω × Ω).

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Proof. Since the Bergman kernels K Ω and K Ωj have these properties (see [26]), by the deﬁnition of E Ω we get 1. 2. The proof is given in [3] and [4]. 3. Using the fact that 1 K Ω1 (z, w) = (1 − z · w)2 and K Ωj (z, w) =

rj2 (rj2 − (z − aj ) · (w − aj ))2

for j=2,. . . ,n and 1. and 2. we get 3.

We observe that we can choose Rj > rj for j=2,. . . ,n and R1 < 1 such that Gj = {z : rj < |z − aj | < Rj } (j=2,. . . ,n) and G1 = {z : R1 < |z| < 1} , then we have Gj ⊂ Uj , where Uj is the same as in Lemma 1. We also have Lemma 3.4. There are constants D > 0 and M > 0 such that: 1. For any (z, w) ∈ Gi × Ω ∪ Ω × Gi we have |K Ω (z, w)| < D|K Ωj (z, w)| and |K Ωj (z, w)| < |K Ω (z, w)| + M. 2. For any z ∈ Ω we have K Ωj (z, z) < K Ω (z, z). Proof. By the explicit formula of the Bergman kernels K Ωi , there are constant Ci and Mi such that Ω K i (z, w) ≥ Ci for (z, w) ∈ (Gi × Ω) ∪ (Ω × Gi ) and Ω K i (z, w) ≤ Mi if (z, w) ∈ Gi × Gi for i=1,2,. . . ,n. From the last Lemma it follows that Ω K (z, w) ≤ K Ωi (z, w) + C ≤ (1 + C/Ci ) K Ωi (z, w) and

Ω K i (z, w)

Ω K j (z, w) ≤ K Ω (z, w) + E Ω (z, w) + j=i

< K Ω (z, w) + ||E Ω ||∞ + Mj i=j

whenever (z, w) ∈ (Gi × Ω) ∪ (Ω × Gi ). If we call C the biggest number among n {1 + C/Cj } and we let M = ||E Ω ||∞ + j=1 Mj , then we get the ﬁrst claimed estimate. The proof of (2) can be found in [27] and [12].

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Lemma 3.5. For f ∈ L2a (Ω), we can write it uniquely as f (z) =

n

(Pj f )(z) + (P0 f )(z)

j=1

with Pj f ∈ L2a (Ωj ), P0 f ∈ L2a (Ω) ∩ C ∞ (Ω), Pk (Pj f ) = 0 if j = k and moreover, there exists a constant M1 such that, for j=0,1,. . . ,n, we have ||Pj f ||Ω ≤ ||Pj f ||Ωj ≤ M1 ||f ||Ω . In particular, if f ∈ L2a (Ωi ), then Pi f = f and ||f ||Ωi ≤ M1 ||f ||Ω for i = 1, . . . , n. Proof. Let f be any function analytic on Ω. For any z ∈ Ω, let γi (i=1,. . . ,n) be the circles which center at ai (a1 = 0) and lie in Gi respectively so that z is exterior to γi (i=2,. . . ,n) and interior to γ1 . Using Cauchy’s Formula we can write n 1 f (ζ) · dζ. f (z) = 2πi ζ −z γj j=1 Let 1 fj (z) = · 2πi

γj

f (ζ) dζ. ζ−z

By Cauchy’s Formula, the value fj (z) does not depend on the choice of γj if n 1 ≤ j ≤ n and f (z) = j fj (z). Of course, each fj is well deﬁned for all z ∈ Ωj and analytic in Ωj . In addition, if j = 1, we have that fj (z) → 0 as |z| → ∞. Writing the Laurent expansion at aj of fj we have f1 (z) =

∞

α1,k z k

k=0

and, for j = 1, fj (z) =

−∞

αj,k (z − aj )k

k=−1

and these series converge to fj uniformly and absolutely on any compact subset of Ωj respectively. We remark that the coeﬃcients are given by the following formula f (ζ) 1 dζ αj,k = 2πi γj (ζ − aj )k+1 where k ≥ 0 if j = 1 and k ≤ −1 if j = 1 and γj ⊂ Gj , 1 ≤ j ≤ n. Moreover, if f is holomorphic in some Ωj and f (z) → 0 as |z| → ∞ when i = 1, then αjk = 0 for

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all j = i by Cauchy’s Theorem and, therefore, fj = 0. Now we deﬁne P1 f = f1 and −∞

Pj f (z) =

αjk (z − aj )k

k=−2

for j=2,3,. . . ,n and P0 f (z) =

n

α−1,j (z − aj )−1

j=2

n

then f (z) = i=0 Pi f (z) for all z ∈ Ω and Pk (Pj f ) = 0 if 0 = k = j = 0 as we have proved above. We claim that f ∈ L2a (Ω) implies that Pi f ∈ L2a (Ωj ) for j=1,2,. . . ,n, respectively. Indeed, since each annulus Gj is contained in Ω, f ∈ L2a (Ω) implies that f is an element of L2a (Gi ) for all i=1,2,. . . ,n. For any ﬁxed i, note that Pj f (0 = j = i) and P0 f − αj,−1 · (z − aj )−1 are analytic ¯ i ∪ (C/Ωi ) and lim|z|→∞ Pj f (z) = 0 for j = 1. Expanding them as Laurent on G series, it follows that: +∞ β 1. If i = 1, then Pj f = k=1 zjk k for j = 1; 2. If i = 1, then +∞ βjk (z − ai )k Pj f (z) = k=0

for 0 = j = i and P0 f (z) =

+∞

β0k (z − ai )k +

k=0

αi,−1 . z − ai

It is obvious that, in any case, these series converge uniformly and absolutely on Gi . Observing that each Gi is an annulus at ai we have, by direct computation, that f, f Gi ≥ Pi f, Pi f Gi + |αi,−1 |2 (lnRi − lnri ) if i = 1 and f, f G1 ≥ P1 f, P1 f G1 . Therefore, for any i=1,. . . ,n, there exists a constant M such that ||Pi f ||Gi ≤ ||f ||Gi ≤ ||f ||Ω

(∗)

and |αi,−1 | ≤ M · ||f ||Ω

(∗∗).

From the deﬁnition of Pj f we derive ||P1 f ||2G1 =

+∞ |α1k |2 (1 − R2k+2 ) 1

0

k+1

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Hilbert-Schmidt Operators on the Bergman Space

and ||Pi f ||2Gi =

−∞ k=−2

435

|α|2ik (ri2k+2 − Ri2k+2 ) k+1

for i=2,. . . ,n. The convergence of these series is guaranteed by the condition (∗) and (∗∗). Since R1 < 1 and ri < Ri , it follows that Pi f ∈ L2a (Ωi ) and ||P1 f ||2Ω1 = and ||Pi f ||2Ωi =

+∞ |α1k |2 0

k+1

−∞ |α1k |2 ri2k+2 k+1

k=−2

for i=2,. . . ,n. Comparing the expression of ||Pi f ||Ωi with the expression of ||Pi f ||Gi , it follows that ||Pi f ||Ωi < M · ||Pi f ||Gi for some constant M for i=1,. . . ,n. Hence, ||Pi f ||Ωi < M · ||Pi f ||Ω . Moreover, if we deﬁne M = M ax{||(z − ai )−1 ||Ω }, from the inequalities ||Pi f ||Gi ≤ ||f ||Gi ≤ ||f ||Ω and |αi,−1 | ≤ M · ||f ||Ω and the deﬁnition of P0 , it follows that ||P0 f ||Ω ≤ n · M · M · ||f ||Ω . If f ∈ L2a (Ωi ) for some i ∈ {1, 2, . . . , n}, note that lim f (z) = 0 as |z| → ∞ for i = 1, then f (z) = Pi f (z) + αi,−1 (z − ai )−1 if i = 1 and P1 f = f if i = 1. For i = 1, since f ∈ L2a (Ωi ) ⊂ L2a (Ω) implies that Pi f ∈ L2a (Ωi ), then αi,−1 · (z − ai )−1 ∈ L2a (Ωi ). We must have αi,−1 = 0 and, consequently, P0 f = 0. Hence, in any case, f ∈ L2a (Ωi ) implies f = Pi f and Pj f = 0 if i = j and this remark completes our proof. Lemma 3.6. If {fn } is a bounded sequence in L2a (Ω) and fn → 0 weakly in L2a (Ω), then Pj fn → 0 weakly on L2a (Ωj ) for j = 1, . . . , n and P0 fn → 0 uniformly on Ω. Proof. By the previous Lemma we know that the linear transformations {Pj } are bounded operators. Then fn → 0 weakly in L2a (Ω) implies that Pj fn → 0 weakly on L2a (Ωj ) for j = 1, . . . , n. For the same reason P0 fn → 0 weakly in L2a (Ω) and then P0 fn (ζ) → 0 for any ζ ∈ Ω. Since n αi,−1 (m) P0 fm = , (ζ − ai ) i=2 by the estimates given in the last lemma, we have that |αi,−1 (m)| < M ||fm ||Ω . The boundness of {||fm ||Ω } implies that the family of continuous functions {P0 fm } is ¯ then, by Arzela-Ascoli’s Theorem, uniformly bounded and equicontinuous on Ω we have that P0 fm → 0 uniformly on Ω.

4. Multiply-Connected Domains In this section we investigate, with the help of the results established in the previous section, necessary and suﬃcient conditions on the function ϕ ∈ L∞ (Ω, dν) for the Hankel operators Hϕ to be Hilbert-Schmidt. We apply these conditions to

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the problem of the joint membership of Hϕ and Hϕ in the Hilbert-Schmidt class. First we collect some well-known facts and deﬁnitions about operators. Recall that a linear operator A : H1 → H2 is in the p−th Schatten-von Neumann class Sp (H1 , H2 ) if there exists an orthonormal basis {ej } of H1 and an orthonormal basis {fj } of H2 such that ∞ Aek , fk

H2

p < ∞.

k=1

A positive operator A on an Hilbert space H is in the trace class if ∞ Aej , ej H < ∞ tr(A) = j=1

for any orthonormal basis {ej } of H. It is possible to show that tr(A) is independent from the choice of the basis and that an operator A on an Hilbert space H is p/2 p−th Schatten-von Neumann class Sp if and only if (A∗ A) is in the trace class. Moreover it is possible to prove the following. Theorem 4.1. Let 1 ≤ p < ∞ and H1 and H2 be two Hilbert spaces. An operator A : H1 → H2 belongs to the p-th Schatten-von Neumann class Sp (H1 , H2 ) if and only if, no ∞ matter how we choose the orthonormal basis {en }∞ n=1 ⊂ H1 and {fn }n=1 ⊂ H2 , we have ∞ Aen , fn H2 n=1 ∈ p . Before we state and prove the main result of this Section we remind to the reader that given Ω = ∩ni=1 Ωi a bounded canonical multiply-connected domain. We say that the set of n functions P = {p1 , . . . , pn } is a ∂−partition for Ω if: 1. For every j = 1, . . . , n pj : Ω → [0, 1] is a Lipschitz, C ∞ −function; 2. For every j = 1, . . . , n there exists an open set Wj and an j > 0 such that the Uj = {ζ ∈ Ω : rj < |ζ − aj | < rj + j } is contained in Wj and pj (ζ) = 1 if ζ∈ Uj , and pj (ζ) = 0 if ζ ∈ Wk and j = k; 3. For any ζ ∈ Ω the following n i=1 pi (ζ) = 1holds. These conditions imply that for any j the function pj is equal to 1 when we are close to ∂Ωj , and pj becomes zero when we are close to ∂Ωk if j = k. We observe that for any j it is possible to extend pj to a function p˜j : Ωj → [0, 1] in such a way that pj (ζ) = p˜j (ζ) ζ ∈Ω and p˜j (ζ) = 0 ζ ∈ Ω. Clearly p˜j ∈ C(Ωj , [0, 1]). We will write pj instead of p˜j when it is clear within what context we are working. Finally, we assume that the pj s are nice smooth functions since this makes things easier and the proofs are more natural. We do not need to work with a more complicated partition since we are only using this

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partition as a tool to localize near the boundary. The reader should observe that it is always possible to construct a ∂−partition. In fact, by construction, we have Ω = ∩nj=1 Ωj where Ω1 = {ζ ∈ C : |ζ| < 1} and, for j=2,3,. . . ,n, Ωj = {ζ ∈ C : |ζ − aj | > rj }. Since aj ∈ Ω1 , 0 < rj < 1, and |aj − ai | > rj + ri if i = j and 1 − |aj | > rj , it possible to ﬁnd a δ > 0 such that 1 − |aj | > rj + δ and |aj − ai | > rj + ri + δ if i = j. If we choose such that 0 < < δ and we deﬁne U1 V1

= =

{ζ ∈ C : 1 − δ < |ζ| < 1} {ζ ∈ C : 1 − < |ζ| < 1}

and, for j=2,3,. . . ,n, Uj

=

{ζ ∈ C : rj + δ > |ζ − aj | > rj }

Vj

=

{ζ ∈ C : rj + > |ζ − aj | > rj },

and U0 = Ω − ∪nj=1 Vj then {Ui }i=0,1,...,n is an open cover of Ω. By a well-known theorem (see [22] Chap 1) we can construct a smooth partition of unity subordinate to this cover, clearly such a partition is a ∂−partition. Before we state the next Theorem we remind the reader that n K Ω (ζ, z) = E Ω (ζ, z) + KΩ (ζ, z), =1 Ω

∞

where E ∈ L (Ω × Ω) and, ∀ = 1, . . . , n, we have KΩ (ζ, z) = K Ω (ζ, z)

∀ζ, z ∈ Ω × Ω,

where K Ω is the reproducing kernel of Ω . If we use the symbol K0Ω to indicate E Ω we can write n KΩ (ζ, z). K Ω (ζ, z) = =0

We also remind to the reader that if I : L2a (Ω) −→ L2a (Ω) is the identity operator, then n I= P , =0

L2a (Ω)

L2a (Ω)

where P : −→ is a bounded operator ∀ = 0, 1, . . . , n with P f ∈ ¯ and Pk P = 0 if k = (see Lemma 3). L2a (Ω ) if = 1, . . . , n and P0 f ∈ C ∞ (Ω) In order to make our notation a little simpler, when we use a kernel operator we will denote it by the name of its kernel function. For example, the Bergman

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projection will be denoted by the symbol K Ω . Finally, we deﬁne the operators Q : L2 (Ω) −→ L2 (Ω), for = 1, 2, . . . , n, in this way f (ζ)|KΩ (ζ, z)|dv(ζ). Q f (z) = Ω

It is important to notice that ∀ = 1, . . . , n Q is a bounded operator. In fact Lemma 4.2. The operators Q : L2 (Ω) −→ L2 (Ω) ( = 1, . . . , n) are bounded. Proof. To prove our claim we observe that there is a constant M such that, for = 1, . . . , n, 1 1 |KΩ (w, z)|KΩ (w, w) 4 dv(w) ≤ M KΩ (z, z) 4 ∀z ∈ Ω Ω

and

Ω

1

1

|KΩ (w, z)|KΩ (z, z) 4 dv(z) ≤ M KΩ (w, w) 4

∀w ∈ Ω.

It is enough to show the ﬁrst inequality for = 1. Let λ = (w − z)/(1 − z¯w), then 1 1 Ω Ω 4 |K (w, z)|K (w, w) dv(w) ≤ 1 2 ¯w| (1 − |w|2 ) 2 Ω ∆ |1 − z 1 1 = 1 1 dλ. 2 2 (1 − |z| ) ∆ |1 − z¯λ|(1 − |λ|2 ) 2 By a result of Axler (see [6]), the integral in the last equation is bounded by a constant M. Therefore, we have 1 1 |KΩ (w, z)|KΩ (w, w) 4 dv(w) ≤ M KΩ (z, z) 4 . Ω

Using the symmetry, an application of Schur’s test (see [23]) completes our proof. We prove the following Theorem 4.3. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the following are equivalent. 1. Hϕ is Hilbert-Schmidt; 2. For any j = 1, . . . , n the operators Hϕj : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt. j by Proof. For ϕ ∈ L2 (Ω) we deﬁne the operators Tm j Ω g(z) = (ϕj (z) − ϕj (w)) · (P g)(w) · Km (z, w)dv(w) Tm Ω

for all g ∈

L2a (Ω)

( m, = 0, 1, . . . , n ) and we show that Hϕj Pm ∈ S2 if m = j.

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To do so we observe that we can write the following n Hϕj P g(z) = Ω (ϕj (z) − ϕj (w)) · (P g)(w) · i=0 KiΩ (z, w)dv(w) =

n

439

..

j i=0 Ti g(z)

for all g ∈ L2a (Ω) ( m, = 1, . . . , n ). j Claim I. The operator Tt0 is Hilbert-Schmidt for any t = 0, 1, . . . , n.

Proof. We observe that the image, by construction, satisﬁes the following inclusion P0 L2 (Ω) ⊂ C ∞ (Ω) therefore the operator Tt0 has its range inside L∞ (Ω). Using the Closed Graph Theorem is possible to show that the operator P0 : L2 (Ω) → L∞ (Ω) is continuous. Finally we observe that a Theorem of Grothendieck (see [17]) implies that the inclusion L∞ (Ω) → L2 (Ω) is 2-summing, but since L2 (Ω) is an Hilbert space the set of 2-summing is the same as the set of Hilbert-Schmidt operators j and this implies that for any t = 0, 1, . . . , n Tt0 is Hilbert-Schmidt. Therefore the following n

j Tt0

t=0

is Hilbert-Schmidt and this complete the proof of the claim. Claim II. The operator Tij is Hilbert-Schmidt if i = j = 0 and i = 1, . . . , n. Proof. We observe that Tij g(z) =

Ω

(ϕj (z) − ϕj (w)) · (P g)(w) · KiΩ (z, w)dv(w).

To start we give the following def

Bji (z, w) = (ϕj (z) − ϕj (w)) · KiΩ (z, w). The Fubini Theorem and the properties of the ∂-partition implies that 2 |Bji (z, w)| dv(w)dv(z) < ∞ Ω

Ω

in fact, if we use the set Gj after Lemma 1 and we deﬁne as j = {s ∈ Ω |0 < |ϕj (s)| < 1 } ; G

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we have 2 |Bji (z, w)| Ω

=

Ω

Ω Ω

2 |Bji (z, w)| dv(w) dv(z) Ω 2 |ϕj (z) − ϕj (w)| Ki (z, w) dv(w) dv(z) 2

< Ω

Ω\Gj

Ω

Ω\Gj

2 Ω |ϕj (z)| Ki (z, w) dv(w) dv(z) 2

= Ω

2 Ω |ϕj (z) − ϕj (w)| Ki (z, w) dv(w) dv(z) 2

+

Ω\Gj

+

2 Ω |ϕj (z) − 1| Ki (z, w) dv(w) dv(z) 2

Ω

j Gj\G

Ω

j G

+

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2 |ϕj (z)) − ϕj (w)|2 KiΩ (z, w) dv(w) dv(z)

|(ϕj (z)|2 KiΩ (z, z) dv(z) Ω Ω 2 Ki (z, w)2 dv(w) dv(z) + I + |ϕj (z)) − 1|

<

Ω

Gj

since i = j both integrals are ﬁnite and clearly the last term 2 I = |ϕj (z)) − ϕj (w)|2 KiΩ (z, w) dv(w) dv(z) Ω

j G

Ω

j G

< = <

j G

Ω 2 4 Ki (z, w) dv(w) dv(z)

Ω 4 Ki (z, z) dv(w)

∞.

We now observe that XGj (w)ϕj (z) · (P g)(w) · KjΩ (z, w)dv(w) = 0.

(*)

Ω

In fact, we have XGj (w)ϕj (z) · (P g)(w) · KjΩ (z, w)dv(w) Ω = ϕj (z) XGj (w)(P g)(w) · KjΩ (z, w)dv(w) Ω

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and by expanding (P g)(w) and KjΩ (z, w) as series and keeping in mind that Gj is an annulus we can conclude that (*) is zero. Hence we have j Tj g(z) = - Ω ϕj (w) · (P g)(w) · KjΩ (z, w)dv(w)

+

Ω

XGcj (w)ϕj (z) · (P g)(w) · KjΩ (z, w)dv(w)

and a simple calculation shows that XGcj (w)ϕj (z) · (P g)(w) · KjΩ (z, w)dv(w) Ω

is Hilbert-Schmidt. Therefore we need to study Aj, g(z) = − ϕj (w) · (P g)(w) · KjΩ (z, w)dv(w). Ω

To do so we consider an orthonormal basis {fn } in L2a (Ω) and we prove that ∞

2

|Aj, fn , hn | < ∞

n=1

for any orthonormal basis {hn } in L2 (Ω). wk

We know that fn → 0 since it is a basis, therefore the continuity of P implies that P fk → 0 weakly on L2a (Ωl ) and {||P fk ||Ω } is bounded by Lemma 4. Since it is a sequence of holomorphic functions we know that {P fk } is uniformly bounded on any compact subset of Ω . Therefore the sequence {P fk } is a normal family of functions. Since P fk (ζ) → 0 for any ζ ∈ Ω , then P fk converges uniformly on any compact subset of Ω and consequently on F=supp(pj ). Now we observe that |Aj, fk (ζ)| ≤ M ax|P fk (ζ)| · |Q (|XF ϕj |)(ζ)|, ζ∈F

then, by using the fact that Q is bounded, we have ||Aj, fk ||Ω ≤ M ax|P fk (ζ)| · M · ||ϕj ||Ω,2 ζ∈F

and now we claim that

∞ k=1

M ax|P fk (ζ)|2 < ∞. ζ∈F

Actually we can prove that if {gn } is orthonormal basis in the unit disk Ω and an ∞ we deﬁne Mn = Sup |gn (z)| then n=1 Mn2 < ∞. In fact, ﬁx r0 ∈ (0, 1) with |z|≤r

r ∈ (0, r0 ). For |z| ≤ r we have

1 2π gn (r0 eiθ )r dθ |gn (z)| = 2π 0 r0 − e−iθ z

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and 1 2π

2π 0

gn (r0 eiθ )r dθ r0 −e−iθ z

≤

1 2π

2π 0

IEOT

2

|gn (r0 eiθ )| dθ

(r0 − r)−1 ×

≤

1 2π 2

Hence Mn2 and

1 1 = Sup |gn (z)| ≤ × 2 (r0 − r) 2π |z|≤r

∞ n=1

2

Mn2

=

(r0 − r)−2 ×

1 2π

0

1 2π

2π 0

r2 dθ |r0 −e−iθ z|2 2

|gn (r0 eiθ )| dθ.

gn (r0 eiθ )2 dθ

2π

0

2π ∞ gn (r0 eiθ )2 dθ 0

n=1

1 2π

(r0 − r)−2 ×

≤

2π 0

1

2

(1−r02 )

dθ

(r0 − r)−2 (1 − r02 )−2 .

≤

Finally, we observe that, using Cauchy-Schwartz Inequality and the fact that any orthonormal basis {hn } is a orthonormal basis, we have ∞ ∞ 2 2 ≤ n=1 |Aj, fn , hn | n=1 Aj, fn 2 ≤

2 · M 2 · ||ϕj ||2Ω,2 n=1 M ax|P fk (ζ)|

∞

ζ∈F

∞

≤ Therefore

n=K

∞

Mn2 · M 2 · ||ϕj ||2Ω,2 .

2

|Aj, fn , hn | < ∞

n=1

and we are done. We have proved that Hϕ

= =

n j=1

n j=1

Hϕ j

n =0

P

j Tjj +K

where K is a Hilbert-Schmidt operator and since Pt2 = Pt , Pt Ps = 0 if s = t this completes the proof of the Theorem. The last Theorem implies that Theorem 4.4. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the following are equivalent. 1. Hϕ and Hϕ are Hilbert-Schmidt;

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2. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt. We need also a proof of the following fact Theorem 4.5. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the following are equivalent. 1. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt. 2. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) are Hilbert-Schmidt. Proof. We have proved that Hϕ

= =

n j=1

n j=1

Hϕ j

n =0

P

j Tjj +K

where K is a Hilbert-Schmidt operator and Hϕ is Hilbert-Schmidt if and only if T is Hilbert-Schmidt and since, by deﬁnition, T g(z) = (ϕ (z) − ϕ (w)) · (P g)(w) · KΩ (z, w)dv(w) Ω

and, always by deﬁnition, KΩ (z, w) = K Ω (z, w). Since Pt2 = Pt , Pt Ps = 0 if s = t we conclude Hϕj : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) is Hilbert-Schmidt if and only if Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) is Hilbert-Schmidt. If we apply the same argument to ϕj we have the thesis.

Now we are ready to prove, with the help of Zhu’s result (see [38]), the following Theorem 4.6. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the following are equivalent. 1. For any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) are in S2 ;

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2 2 |ϕ | (z) − | ϕ (z)| K Ωj (z, z)dv(z) < ∞ j j

2. It holds

Ω

for any j = 1, . . . , n; 3. The inequality 2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

holds. Proof. 1 ⇔ 2 The equivalence of 1 and 2 has been proven by Zhu (see [37] ). 2 ⇔ 3 Given a function φ ∈ L∞ (Ω) we deﬁne the following function def 2 2 MO(φ, z) = |φ| (z) − φ(z) and we claim that there exists a constant C1,j such that 2 MO(ϕ, z)2 ≤ C1,j · MO(ϕj , z)2

and there exists a constant C2,j such that MO(ϕj , z)2 ≤ C2 · MO(ϕ, z)2 + (|z − aj | − rj ) . We will prove our result only in the case j = 1 since the other cases work in the same way . To start we prove that MO(ϕ, z) ≤ C12 · MO(ϕ1 , z), in fact 2 2 2 |ϕ(u) − ϕ(w)| kzΩ (u) kzΩ (w) dv(u)dv(w) 2MO(ϕ, z)2 = Ω

Ω

and, if we denote with the the symbol ϕ 1 the function ϕ j : Ω1 → C so deﬁned   ϕ(w) if w ∈ Ω ϕ 1 (w) =  0 if w ∈ /Ω we know that

Ω

Ω

is less than or equal C2 Ω1

2 2 2 |ϕ(u) − ϕ(w)| kzΩ (u) kzΩ (w) dv(u)dv(w)

to 2 2 2

ϕ1 (u) − ϕ1 (w) kzΩ1 (u) kzΩ1 (w) dv(u)dv(w)

Ω1

since Ω ⊂ Ω1 and the deﬁnition of ϕ 1 . Finally we observe that the deﬁnition of ϕ1 Ω and the fact kz j (u) is bounded when z is far away from ∂Ω1 , the boundary of Ω1 , there exists a constant C1 such that 2 2 2

C2 ϕ1 (u) − ϕj (w) kzΩ1 (u) kzΩ1 (w) dv(u)dv(w) Ω1

Ω1

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Hilbert-Schmidt Operators on the Bergman Space

Ω1

Ω1

445

2 2 2 |ϕj (u) − ϕj (w)| kzΩ1 (u) kzΩ1 (w) dv(u)dv(w)

and this is equal to

2C12 MO(ϕ1 , z)2 .

We now prove that

MO(ϕ1 , z)2 ≤ C · MO(ϕ, z)2 + (|z − aj | − rj ) .

To start let us note that 2 2MO(ϕ1 , z) = Ω1

Ω1

= Ω

Ω

2 2 2 |ϕ1 (u) − ϕ1 (w)| kzΩ1 (u) kzΩ1 (w) dv(u)dv(w)

2 2 2 |ϕ(u) − ϕ(w)| kzΩ1 (u) kzΩ1 (w) dv(u)dv(w) +

2·

(Ω1 \Ω)×Ω

ϕ(w) · kzΩ1 (u) · kzΩ1 (w)2 dv(u)dv(w).

To prove the necessity we know, see Lemma 1 and Lemma 2, that if z ∈ G1 we have ! Ω 2 Ω 1 1 1 k (u) ≤ C · k (u) + z z K Ω1 (z, z) since if for z ∈ Ω1 \Ω there is a constant M such that Ω kz 1 (u) ≤ M · (1 − |z|) and this implies that

2 2 2 2 2MO(ϕ1 , z) ≤ C · MO(ϕ, z) + (1 − |z|) |ϕ| (z) + (1 − |z|) . 2

Therefore it follows that 2 |ϕ| (z) ≤ MO(ϕ, z)2 + M ·

1 (1 − |z|)

and this implies that

2MO(ϕ1 , z)2 ≤ C · MO(ϕ, z)2 + (1 − |z|) .

It is easy to show that the same holds any j = 2, . . . , n. 2 2 Finally, we observe that Ω |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) can be written as n j=1

Ωj

Clearly

2 2 Ω |ϕ| (z) − |ϕ(z)| K (z, z) +

Ω\∪n j=1 Gj.

Ω\∪n j=1 Gj.

2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z).

2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞

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therefore we have

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2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

if and only if 2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞.

n Gj

j=1

Therefore, the proved inequalities, imply that 2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Gj

if and only if

2 2 |ϕ ϕj (z)| K Ωj (z, z)dv(z) < ∞ j | (z) − |

Gj

and we are done.

Theorem 4.7. Let ϕ ∈ L∞ (Ω), define ϕj = ϕ · pj where j = 1, . . . , n and P is a ∂−partition for Ω. Then the operators Hϕ , Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) are Hilbert-Schmidt if and only if 2 2 |ϕ| (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

holds. Proof. We know, by Theorem 7, that the operators Hϕ and Hϕ are HilbertSchmidt if and only if for any j = 1, . . . , n the operators Hϕj , Hϕj : L2a (Ωj , dν) → L2 (Ωj , dν) L2a (Ωj , dν) are Hilbert-Schmidt and we know, by Theorem 8, that this is true if and only if 2 |ϕ|2 (z) − |ϕ(z)| K Ω (z, z)dv(z) < ∞ Ω

therefore we are done.

Finally we concentrate our attention to the study of the operator Hϕ when ϕ ∈ Hol (Ω)={h : Ω → C |h is halomorphic } and we assume that sup |ϕ(w)| < ∞. w∈Ω

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This situation has been already studied by Arazy, Fisher and Peetre. It is worth to notice that our result implies the following Theorem 4.8. Let ϕ ∈ Hol (Ω) ∩ L∞ (Ω) then the following are equivalent. 1. The operator Hϕ : L2a (Ω, dν) → L2 (Ω, dν) L2a (Ω, dν) is in S2 ; 2. The function ϕ satisfies the following: 2 2 |ϕj | (z) − | ϕj (z)| K Ωj (z, z)dv(z) < ∞. Ω

Proof. It is is enough to observe that Hϕ = 0 ∞

if ϕ ∈ Hol (Ω) ∩ L (Ω) and to apply the main Theorem of this paper.

Acknowledgement. I wish to thank Professor Donald Sarason who helped me with the last estimate in Theorem 4-6.

References [1] M. B. Abrahamse, Toeplitz Operators in Multiply Connected Regions , Am. J. of Math. 96 (1972), 261-297. [2] L. Ahlfors, Complex Analysis , McGraw-Hill, 1966. [3] J. Arazy, Membership of Hankel Operators on Planar Domains in Unitary Ideals, Analysis at Urbana, vol.1, London Math. Soc. Lecture Notes Ser. 137, Cambridge University Press, 1989, 1-40 [4] J. Arazy, S. Fisher and J. Peetre, Hankel Operators on Planar Domains, Constr. Approx. 6 (1990), 113-138 [5] S. Axler, Bergman Spaces and their Operators, Surveys of Some Recent Results in Operator Theory, vol. 1, edited by J. B. Conway and B. Morrel, Pitman Research Notes in Mathematics, 1988, 1-50. [6] S. Axler, The Bergman Space, the Bloch Space, and Commutators of Moltiplication Operators, Duke Math. J. 53 (1986), 315-332. [7] S. Axler and D. Zheng, Compact Operators via the Berezin Transform, (Preprint). [8] F. Beatrous and S. Y. Li , Trace Ideal Criteria for Operators of Hankel Type, Illinois J. Math. 39 (1995), 723-754. [9] F. A. Berezin, Covariant and Controvariant Symbols of Operators, Math. USSR Izv. 6 (1972), 1117-1151. [10] V. Bergman, On a Hilbert Space of Analytic Functions and on an Associate Integral Transform, Comm. Pure Appl. Math. 14 (1961), 187-214. [11] V. Bergman, Remarks on a Hilbert Space of Analytic Functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 199-204.

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[12] V. Bergman, The Kernel Function and the Conformal Mapping, AMS Math. Surveys 5,1950. [13] C. A. Berger and L. A. Coburn, Toeplitz Operators and Quantum Mechanics , J. Funct. Anal. 68 (1986), 273-299. [14] C. A. Berger and L. A. Coburn, Toeplitz Operators on the Segal-Bergman Space, Trans. Amer. Math. Soc. 301 (1987), 813-829. [15] J. B. Conway, A Course in Functional Analysis, Springer-Verlag, 1990. [16] J. B. Conway, Functions of One Complex Variable II, Springer-Verlag, 1991. [17] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge University Press (1995) [18] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972. [19] F. Forelli and W. Rudin, Projections on Spaces of Holomorphic Functions in Balls, Indiana Math. J. 24 (1974), 593-602. [20] S. Fisher, Function Theory on Planar Domains, John Wiley & Sons, 1983. [21] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Trans. of Math. Monographs 26 Providence, R.I., 1969 [22] V. Guillemin and A. Pollack, Diﬀerential Topology, Prentice-Hall, 1974. [23] P. R. Halmos and V. S. Sunder, Bounded Integral Operators on L2 Spaces, SpringerVerlag, 1978. [24] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, SpringerVerlag (GTM 199), 2000 [25] H. Li, Hankel Operators on the Bergman Space of Multiply-Connected Domains, J. Oper. Theory 28 (1992), 321-335 [26] N. Kerzman, The Bergman Kernel Function, Diﬀerentiability at the Boundary, Math. Ann. 195 (1972), 149-158. [27] S. G. Krantz, Function Theory of Several Complex Variables, Wiley, 1982. [28] S. G. Krantz, Characterization of Smooth Domains in C by Their Biholomorphic Self-Maps, Am. Math. Monthly 90 (1983), 555-557. [29] E. Nordgren and P. Rosenthal, Boundary Values of Berezin Symbols, Operator Theory: Advances and Applications 73 (1994), Birkhauser, 362-368. [30] R. C. Raimondo, Compact Operators on the Bergman Kernel Space of MultiplyConnected Domains, Proc. Am. Math. Soc. 129 (2001) 3, 739-747 [31] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, 1980. [32] M. Reed and B. Simon, Methods of Modern Mathematical Phycics vol. I, Academic Press, 1975 [33] B. Russo, On the Hausdorﬀ-Young Theorem for Integral Operators, Paciﬁc J. of Math. 68 (1977), 241-252. [34] K. Stroethoﬀ, Compact Hankel Operators on the Begman Spaces of the Unit Ball and the Polydisk in Cn , J. Oper. Theory 23 (1990), 153-170. [35] K. Stroethoﬀ, Hankel and Toeplitz Operators on the Fock Space, Michigan Math. J. 39 (1992), 3-16. [36] U. Venugopalkrishna, Fredholm Operators Associated with Strongly Pseudoconvex Domains in Cn , J. Funct. Analysis 9 (1972), 349-373.

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[37] K. Zhu, VMO, ESV, and Toeplitz Operators on the Bergman Space, Trans. Am. Math. Soc. 302 (1987), 617-646. [38] K. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball, Amer. J. Math. 113 (1991), 147-167. Roberto C. Raimondo University of Melbourne 3010 Parkville, Victoria Australia e-mail: [email protected] Submitted: August 9, 2005 Revised: August 15, 2006

Integr. equ. oper. theory 57 (2007), 451–471 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040451-21, published online December 26, 2006 DOI 10.1007/s00020-006-1465-x

Integral Equations and Operator Theory

Diﬀerential Geometry for Nuclear Positive Operators Cristian Conde To my parents

Abstract. Let H be a Hilbert space, dim H = ∞. The set ∆1 = {1 + a : a in the trace class, 1 + a positive and invertible} is a diﬀerentiable manifold of operators, and a homogeneous space under the action of the invertible operators g which are themselves nuclear perturbations of the identity (one of the called classical Banach-Lie groups): lg (1 + a) = g(1 + a)g ∗. In this paper we introduce a Finsler metric in ∆1 , which is invariant under the action. We investigate the metric space thus induced. For instance, we prove that it is complete non-positively curved (in the sense of Busemann). Other geometric properties are derived. Mathematics Subject Classiﬁcation (2000). Primary 58B20; Secondary 22E65, 53C30, 53C45. Keywords. Nuclear positive operators, Finsler metric, reductive homogeneous space, non-positive curvature.

1. Introduction The purpose of this paper is to introduce a Finsler structure and expose several results about the geometrical structure of the set ∆1 , deﬁned by ∆1 = {1 + a ∈ L1 : 1 + a > 0}, where L1 denotes the trace class perturbations of multiples of the identity. This study relates to previous work on diﬀerential geometry of positive operators (or positive deﬁnite matrices). Mainly a series of papers [6], [7] and [8] by Corach, Porta and Recht, where the geometry of the set of positive invertible of a C∗ This work was completed with the support of CONICET.

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algebra was studied. Also this study is related to classical work on the geometry of positive matrices ([18]). Basically, there are two reasons why we have selected the set ∆1 . The ﬁrst is that the trace class operators (which are not invertible) usually appear in Physics ([21]) and other sciences. The second is the duality that exists between the space of positive functional in B(H) and the positive nuclear operators. Let Gl(H) the general linear group of all invertible bounded operators on a separable and inﬁnite dimensional Hilbert space H and Gl(H, B1 (H)) the subgroup of invertible trace class perturbations of the identity, i.e. Gl(H, B1 (H)) = {1 + a ∈ Gl(H) : a ∈ B1 (H)} = {g ∈ Gl(H) : g − 1 ∈ B1 (H)}. The subgroup Gl(H, B1 (H)) is a Banach-Lie group locally diﬀeomorphic to B1 (H). The classical reference for this subject and notation is [12]. Consider the homogeneous space Gl(H, B1 (H))/U1 , where U1 is the subgroup of Gl(H, B1 (H)) of unitary operators (this space can be identiﬁed with ∆1 ). Then there exists a Finsler metric on the tangent bundle of ∆1 which is given by the 1-norm on (T ∆1 )1 . We show that the Finsler metric induces the following metric on ∆1 1 1 d(a, b) = log(a− 2 ba− 2 )1 a, b ∈ ∆1 . The main object of this paper is to determine the properties of the metric space (∆1 , d). The material is organized as follows. Section 2 contains a survey of the topological structure and diﬀerential geometry of ∆1 , with a description of its structure as a reductive homogeneous space. In Section 3 we investigate the minimality properties of the geodesics. In Section 4 we prove that ∆1 shares some properties with Riemannian manifolds of nonpositive sectional curvature (though we can not deﬁne the sectional curvature in this space). For instance, the metric increasing property (MIP) of the exponential map (Theorem 4.3). Finally, in Section 5 we prove that ∆1 has non-positive curvature (in Busemann’s sense [14]) with respect to the Finsler metric (or geodesic distance).

2. Some aspects of the geometry of ∆1 2.1. Topological and diﬀerentiable structure of ∆1 Let B(H) denote the algebra of bounded operators acting on a complex and separable Hilbert space H. Throughout, B1 (H) stands for the bilateral ideal of trace class operators of B(H), that is the subset of compact operators with singular values in l1 . Recall that B1 (H) is a Banach algebra without unit, with the norm 1 a1 = tr |a| = tr(a∗ a) 2 = |a| ei , ei , i∈N

where {ei }i∈N is any given orthonormal basis of H.

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We consider a certain subset of Fredholm operators, namely L1 = {λ + a ∈ B(H) : λ ∈ C,

a ∈ B1 (H)},

the complex linear subalgebra consisting of the trace class perturbations of multiples of the identity. There is a natural (not quadratic) norm for this subspace λ + X(1) = |λ| + X1 . The selfadjoint part of L1 is L1R = {λ + a ∈ L1 : (λ + a)∗ = λ + a}, Remark 2.1. 1. (L1 , .(1) ) is the unitazion of (B1 (H), .1 ). 2. Note that the multiples of identity λ1 and the operators a ∈ B1 (H) are linearly independent. Therefore λ + a ∈ L1R if and only if λ ∈ R , a∗ = a. Formally, L1 = C ⊕ B1 (H)

L1R = R ⊕ B1 (H)h ,

where B1 (H)h denotes the set of selfadjoint trace class operators. 3. One has the usual estimates (a) λ + a ≤ λ + a(1) , (b) (λ + a)(µ + b)(1) ≤ λ + a(1) µ + b(1) . for all λ + a, µ + b ∈ L1 . In particular, (L1 , +, .) is a Banach algebra. Inside L1R , we consider ∆ = {λ + a ∈ L1 : λ + a > 0}, and ∆1 = {1 + a ∈ L1 : 1 + a > 0}. Apparently ∆ is an open subset of L1R , and therefore a diﬀerentiable (analytic) submanifold. The next step is to prove that ∆1 is a submanifold of ∆. For this purpose, we consider θ : ∆ → R , θ(λ + a) = λ. Lemma 2.1. θ is a submersion. Proof. It is suﬃcient to show that dθλ+a is surjective and ker(dθλ+a ) is complemented ([15], Theorem 2.2). Since L1R and R are Banach spaces and θ is a continuous linear map we get that dθλ+a = θ Apparently, dθλ+a is surjective and ker(dθλ+a ) has codimension 1 and hence is complemented.

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It follows that ∆1 is a submanifold, since ∆1 = θ−1 ({1}). These facts imply that, for 1 + a ∈ ∆1 , (T ∆1 )1+a identiﬁes with B1 (H)h . There is a natural action of Gl(H, B1 (H)) over ∆1 , deﬁned by l : Gl(H, B1 (H)) × ∆1 −→ ∆1 , lg (1 + a) = g(1 + a)g ∗ . This action is clearly diﬀerentiable and transitive, since if 1 + a, 1 + b ∈ ∆1 then lr (1 + a) = (1 + b), 1 2

− 12

for r = (1 + b) (1 + a)

∈ Gl(H, B1 (H)). For 1 + a ∈ ∆1 , let

I1+a = {g ∈ Gl(H, B1 (H)) : g(1 + a)g ∗ = 1 + a}, the isotropy group of 1 + a. In particular, for 1 ∈ ∆1 I1 = {g ∈ Gl(H, B1 (H)) : gg ∗ = 1} = U (H) ∩ Gl(H, B1 (H)) = U1 , where U (H) denotes the unitary operators on H. 2.2. Reductive structure Let us recall the deﬁnition of homogeneous reductive space Deﬁnition 2.1. A homogeneous space G/F is reductive (RHS) if there exists a vector space decomposition g = f ⊕ m of the Lie algebra g of G, such that m is invariant under the action of F In order to give an RHS structure to ∆1 (or equivalently to Gl(H, B1 (H))/U1 ) under the action of Gl(H, B1 (H)) we must ﬁnd a decomposition (T Gl(H, B1 (H)))1 = (T U1 )1 ⊕ m. g

f

Recall that g = (T Gl(H, B1 (H)))1 and f = (T U1 )1 can be identiﬁed with B1 (H) and iB1 (H)h , respectively. Then, we have B1 (H) = iB1 (H)h ⊕ m. The most natural choice is m = B1 (H)h . Note that B1 (H)h is U1 -invariant: lg (B1 (H)h ) = {gXg ∗ : X ∈ B1 (H)h } = B1 (H)h . From the above remarks, we get Proposition 2.1. ∆1 has an RHS structure under the action of Gl(H, B1 (H)). Now, in order to construct a covariant derivative in ∆1 , we use its reductive structure. We introduce the transport equation whose solutions give the horizontal lifts to Gl(H, B1 (H)) of curves on ∆1 following the lines of [16]. Deﬁnition 2.2. The diﬀerential equation 1 −1 Γ˙ = γγ ˙ Γ, 2 is called the transport equation for γ, and the solution Γ(t) with initial condition Γ(0) = 1 ∈ Gl(H, B1 (H)) is called the horizontal lift of γ(t).

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The transport equation induces a covariant derivative of a tangent ﬁeld X along γ, namely d 1 DX = Γ(t) ((T lΓ(t)−1 )γ(t) X(t))Γ(t)∗ = X˙ − (Xγ −1 γ˙ + γγ ˙ −1 X). dt dt 2 From now on, we denote with a, b, .. etc. the elements of ∆1 . The curvature tensor for this connection is 1 R(X, Y )Z = − a[[a−1 X, a−1 Y ], a−1 Z], 4 for X, Y, Z ∈ (T ∆1 )a . The corresponding exponential at a ∈ ∆1 is 1

−1 2

expa : (T ∆1 )a → ∆1 , expa X = a 2 ea

1

Xa− 2

1

a2 .

Notice that expa is a diﬀeomorphism and its inverse map is 1

1

1

1

loga : ∆1 → (T ∆1 )a , loga b = a 2 log(a− 2 ba− 2 )a 2 . The covariant derivative can now be used to deﬁne a parallel vector ﬁeld and geodesics on ∆1 as solutions to ordinary diﬀerential equations. A curve γ is a geodesic if γ˙ is parallel, i.e. ˙ γ¨ = γγ ˙ −1 γ.

(2.1)

The basics properties of the geodesics can be summarized in the following statement. Proposition 2.2. Let a ∈ ∆1 , X ∈ (T ∆1 )a and γ a geodesic. Then 1. The curve gγg ∗ is also a geodesic for all g ∈ Gl(H, B1 (H)), 2. The unique geodesic γ such that γ(0) = a and γ(0) ˙ = X, is −1 2

1

γ(t) = a 2 eta

1

Xa− 2

1

t ∈ R,

a2

3. Let b ∈ ∆1 . There is one and only one geodesic γa,b such that γa,b (0) = a and γa,b (1) = b, namely 1

1

1

1

γa,b (t) = a 2 (a− 2 ba− 2 )t a 2 t ∈ R.

Proof. The proof is straightforward. Throughout this paper, we use the following notation 1

1

1

1

a t b = a 2 (a− 2 ba− 2 )t a 2 = expa (t exp−1 a (b)), which is called the t-power mean between a and b (see [19]), and the relative operator entropy 1

1

1

1

S(a/b) = a 2 log(a− 2 ba− 2 )a 2 , deﬁned in [11]. Lemma 2 in [10] shows that for a, b ∈ ∆1 and t ∈ R a t b = b 1−t a.

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2.3. Finsler Structure We deﬁne the length of a tangent vector for X ∈ (T ∆1 )a by 1 1 Xa = a− 2 Xa− 2 . 1

where .1 denotes the norm of B1 (H). Proposition 2.3. The metric in T ∆1 is invariant for the action of the group of invertible elements, i.e. for each a ∈ ∆1 , g ∈ Gl(H, B1 (H)) and X ∈ B1 (H)h , we have Xa = gXg ∗gag∗ Proof. Let a ∈ ∆1 , g ∈ Gl(H, B1 (H)) and X ∈ B1 (H), observe that 1

1

1

1

gXg ∗ = ga 2 a− 2 Xa− 2 a 2 g ∗ . 1

Denote by z = ga 2 then 1

1

1

1

−1

1

(gag ∗ )− 2 = (ga 2 a 2 g ∗ )− 2 = (zz ∗ )− 2 = |z ∗ |

.

therefore 1

−1

1

(gag ∗ )− 2 gXg ∗(gag ∗ )− 2 = |z ∗ |

1

−1

1

za− 2 Xa− 2 z ∗ |z ∗ |

.

∗

From the polar decomposition applied to z ∈ GL(H), z = |z | ρz with ρz unitary, we have 1 1 1 1 (gag ∗ )− 2 gXg ∗ (gag ∗ )− 2 = ρz a− 2 Xa− 2 ρ∗z . Now, since |srs∗ | = s |r| s∗ for all unitary s, we get 1 1 1 1 gXg ∗ gag∗ = tr ρz a− 2 Xa− 2 ρ∗z = tr(ρz a− 2 Xa− 2 ρ∗z ) 1 1 1 1 = tr a− 2 Xa− 2 = a− 2 Xa− 2 = Xa . 1

3. Minimality of geodesics In this section we investigate the minimality properties of the geodesics; the expression “minimal” is understood in terms of the length (or more generally p-energy functional). We prove that the unique geodesic joining two points is the minimum of the p-energy functional for p ≥ 1. For a piecewise diﬀerentiable curve α : [0, 1] → ∆1 we now compute the length of the curve α by 1 α(t) ˙ l(α) = α(t) dt. 0

Note that given a, b in ∆1 , if γa,b : [0, 1] −→ ∆1 is the unique geodesic joining them, then 1

1

l(γa,b ) = log(a− 2 ba− 2 )1 .

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Deﬁnition 3.1. Let a, b ∈ ∆1 . We denote Ωa,b = {α : [0, 1] → ∆1 : α is a C 1 curve, α(0) = a and α(1) = b}. The geodesic distance between a and b (in the Finsler metric) is deﬁned by d(a, b) = inf {l(α) : α ∈ Ωa,b }. If K ⊆ ∆1 , let d(a, K) = inf {d(a, k) : k ∈ K}. The next step consists in showing that geodesics are short curves, i.e. if δ is another curve joining a to b then l(γa,b ) ≤ l(δ). and hence 1

1

d(a, b) = log(a− 2 ba− 2 )1 . The proof of this fact requires some preliminaries. We begin with the following inequalities (see [13]): Let a, b, c be Hilbert space operators with a, b ≥ 0. For any unitarily invariant norm |||.||| we have 1 1 1/2 1/2 (3.1) |||a cb ||| ≤ ||| at cb1−t dt||| ≤ |||ac + cb|||. 2 0 Theorem 3.1. For all X, Y ∈ B1 (H)h X

X

Y 1 ≤ e− 2 d expX (Y )e− 2 1 , where d expX denote the derivate, at a point X, of the exponential map. This inequality was proved by R. Bhatia for matrices ([5]). Proof. Our proof uses two ingredients. The ﬁrst is the well-known formula 1 Claim 3.1. d expX (Y ) = 0 etX Y e(1−t)X dt. We provide here a simple proof of this equality. Since d tx (1−t)y (e e ) = etx (x − y)e(1−t)y , dt we have

e −e = x

and hence

y

X+hY

lim e

h→0

h

0

−eX

1

etx (x − y)e(1−t)y dt, =

1 0

etX Y e(1−t)X dt.

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X

Let X, Y ∈ B1 (H)h . Write Y = e 2 (e− 2 Y e− 2 )e 2 , and then using the inequalities (3.1) to get from this 1 1 X tX − X −X (1−t)X −X 2 2 2 Y 1 ≤ e (e Ye )e dt1 = e etX Y e(1−t)X dte− 2 1 0

=

0

X

X

e− 2 d expX (Y )e− 2 1 .

This proves the theorem. We are now ready to prove the main result in this section.

Theorem 3.2. Let a, b ∈ ∆1 , the geodesic γa,b is the shortest curve joining them. So 1 1 d(a, b) = log(a− 2 ba− 2 )1 . Proof. Since the group Gl1 (H, B1 (H)) acts isometrically and transitively on ∆1 , is suﬃces to prove the theorem for a = 1. Then γ1,b = bt = et log b and l(γ1,b ) = log b1 . Let γ ∈ Ω1,b ; so write γ(t) = eα(t) we get 1

1

−2 ˙ 1 γ(t)− 2 γ(t)γ(t)

Finally,

e−

≥

α(t) ˙ 1.

0

˙ eα(t) e−

1

γ(t) ˙ γ(t) dt =

l(γ) =

≥

α(t) 2

=

0

1

0

1

α(t) 2

− 12

γ(t)

1 = e−

α(t) 2

− 12

γ(t)γ(t) ˙

− d expα(t) (α(t))e ˙

1 dt ≥

0

α(t) 2

1

1

α(t) ˙ 1 dt

1 α(t)dt ˙ 1 = α(t)|0 1 = α(1) − α(0)1 = log b1 .

Remark 3.1. 1. The geometrical result described above can be translated to the language of the relative entropy 1 1 d(a, b) = a− 2 S(a/b)a− 2 = S(a/b)a . 1

2. For each a ∈ ∆1 and α > 0 the exponential map expa : (T ∆1 )a → ∆1 maps the ball {X ∈ (T ∆1 )a : Xa ≤ α} onto the ball {x ∈ ∆1 : d(a, x) ≤ α}, since 1

−1 2

d(a, expa (X)) = d(a, a 2 ea

1

Xa− 2

1

a 2 ) = Xa .

Corollary 3.1. If X, Y ∈ B1 (H)h commute we have X − Y 1 = d(eX , eY ). In particular on each line RX ⊆ B1 (H)h the exponential map preserves distances.

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Deﬁnition 3.2. For every p ∈ R − {0} we deﬁne the p-energy functional 1 p Ep : Ωa,b → R+ , Ep (α) := (α(t) ˙ α(t) ) dt. 0

Remark 3.2.

1. For p = 1 we obtain the length functional 1 l(α) := α(t) ˙ α(t) dt, 0

and for p = 2 we obtain the energy functional 1 2 (α(t) ˙ E(α) := α(t) ) dt, 0

2. For any curve α such that α(t) ˙ α(t) is constant we have p

Ep (α) = (l(α))p = (E(α)) 2 . In Theorem 3.2 we have proved that the geodesic between a and b minimizes the length functional. This fact is valid also for the p-energy functional (associated with Ωa,b ) for p ∈ (1, ∞). Proposition 3.1. Let a, b ∈ ∆1 and p ∈ [1, ∞). Then the p-energy functional 1 p Ep : Ωa,b → R+ Ep (α) := (α(t) ˙ α(t) ) dt, 0

takes on its minimum global dp (a, b) precisely on γa,b . olders inequality Proof. Now, let α ∈ Ωa,b and p ∈ (1, ∞) then by H¨ 1 1 p p (l(α))p = ( α(t) ˙ (α(t) ˙ α(t) dt) ≤ α(t) ) dt = Ep (α). 0

0

On the other hand, (l(γa,b ))p = Ep (γa,b ). This implies Ep (γa,b ) = (l(γa,b ))p ≤ (l(α))p ≤ Ep (α).

Proposition 3.2. Given a, b ∈ ∆1 , g ∈ Gl(H, B1 (H)) we get 1. d(a, b) = d(a−1 , b−1 ). 2. For all t ∈ R d(a, a t b) = |t| d(a, b). 3. Invariance under the action by Gl(H, B1 (H)) d(a, b) = d(gag ∗ , gbg ∗ ). 1

1

Proof. 1. It is easy to see that S(a/b) = −a 2 log(b−1 /a−1 )a 2 , as a consequence from log(1/t) = − log(t). Then d(a, b) =

1

1

1

1

log(a− 2 ba− 2 )1 = a− 2 S(a/b)a− 2 1 1

1

1

1

1

1

=

− a− 2 a 2 log(a 2 b−1 a 2 )a− 2 a 2 1

=

− log(a 2 b−1 a 2 )1 = d(a−1 , b−1 ).

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2. It is obvious that S(a, a t b) = tS(a/b), then 1 1 1 1 d(a, a t b) = a− 2 S(a/a t b)a− 2 = |t| a− 2 S(a/b)a− 2 = |t| d(a, b). 1

1

3. Note that if γa,b is the geodesic joining a with b, then g γa,b ˙ (t)g ∗ gγa,b (t)g∗ = γa,b ˙ (t)γa,b (t) .

Deﬁnition 3.3. For a, b ∈ ∆1 , we call the midpoint of a and b, and we denote by m(a, b) (following the notation used in [14]) to m(a, b) := a 12 b. By the Proposition 3.2 and the last deﬁnition we have that: 1. m(a, b) = a 12 b = b 12 a = m(b, a). 2. d(a, m(a, b)) = 12 d(a, b) = 12 d(b, a) = d(b, m(b, a)).

4. Convexity of the geodesic distance The purpose of this section is to show that the norm of the Jacobi ﬁeld along to a geodesic γ is a convex function. Deﬁnition 4.1. A vector ﬁeld J along to a geodesic γ (i.e. J(t) ∈ (T ∆1 )γ(t) for all t) is a Jacobi field if D2 J + R(J, V )V = 0. dt2 where V (t) = γ(t) ˙ and R(X, Y )Z the curvature tensor.

(4.1)

Theorem 4.1. If J(t) is a Jacobi field along the geodesic γ(t), then J(t)γ(t) is a convex map of t ∈ R. ∈ R. The method of the following proof is based on a similar argument used in [8]. Proof. Notice that by the invariance of the connection and the metric under the action of Gl(H, B1 (H)) we may suppose that γ(t) = etX is a geodesic starting at γ(0) = 1, where X ∈ B1 (H)h . tX tX Then for the ﬁeld K(t) = e− 2 J(t)e− 2 the diﬀerential equation (4.1) changes to ¨ = KX 2 + X 2 K − 2XKX. 4K (4.2) Since the group Gl(H, B1 (H)) acts by isometries, we have 1

1

J(t)γ(t) = γ(t)− 2 J(t)γ(t)− 2 1 = K(t)1 , thus the proof reduces to show that for any solution K(t) of (4.2), the map t → K(t)1 is convex for t ∈ R. Fix u < v ∈ R and let t ∈ [u, v]. We shall prove that v−t t−u K(t)1 ≤ K(u)1 + K(v)1 . v−u v−u

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λi ., ei ei be the spectral decomposition of X ∈ B1 (H)h where {ei :

i∈N

i ∈ N} is an orthonormal basis of H. Consider the matrix valued map k(t) = (kij (t))i,j∈N , where kij (t) = K(t)ei , ej for all t ∈ R. The diﬀerential equation (4.2) is equivalent to the equations 2 kij (t), k¨ij (t) = δij

. where δij = λi −λj 2 A simple veriﬁcation shows that all solutions of f¨(t) = c2 f (t) satisfy f (t) = φ(u, v, c; t)f (u) + ψ(u, v, c; t)f (v), where

φ(u, v, c; t) =

ψ(u, v, c; t) =

Sinh c(v−t) Sinh c(v−u) (v−t) (v−u) , Sinh c(t−u) Sinh c(v−u) (t−u) (v−u) ,

if c = 0; if c=0. if c = 0; if c=0.

Then each kij (t) satisﬁes kij (t) = φij (t)kij (u) + ψij (t)kij (v), where φij (t) = φ(u, v, δij ; t) and ψij (t) = ψ(u, v, δij ; t). In matrix form k(t) = Φ(t) ◦ k(u) + Ψ(t) ◦ k(v), where Φ(t) = {φij (t)}, Ψ(t) = {ψij (t)} and ◦ denotes the Schur product of matrices, i.e. {aij } ◦ {bij } = {aij bij }. Thus we have that k(t)1 ≤ Φ(t) ◦ k(u)1 + Ψ(t) ◦ k(v)1 .

(4.3)

We make the following claim: Claim 4.1. Let Ψ(t), Φ(t) and k(t) as above, then 1. Φ(t) ◦ k(u)1 ≤ 2. Ψ(t) ◦ k(v)1 ≤

v−t v−u k(u)1 , t−u v−u k(v)1 .

Proof. We only prove the ﬁrst inequality, the second is analogous. Deﬁne for each n ∈ N and A = {aij }i,j∈N aij if 1 ≤ i, j ≤ n; An = 0 otherwise. Note that if n → ∞, .1

Φ(t) ◦ k(u)n −→ Φ(t) ◦ k(u),

(4.4)

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since Φ(t) ◦ k(u)n − Φ(t) ◦ k(u)1

≤ max|φii (t)| i>n

|kii (u)|

(4.5)

i>n

(v − t) |K(u)ei , ei | (v − u) i>n ≤ |K(u)ei , ei | → 0.

=

i>n

Next we use a theorem by Ando, Horn and Johnson ([3]), according to which if A and P are n × n matrices, with P positive semideﬁnite, then A ◦ P 1 ≤ ( max pii ) A1 . 1≤i≤n

Thus Φ(t) ◦ k(u)n 1 = Φ(t)n ◦ k(u)n 1 ≤ ( max φii (t))k(u)n 1 . 1≤i≤n

(4.6)

We conclude from (4.4) and (4.6) that Φ(t) ◦ k(u)1 ≤

v−t k(u)1 . v−u

Consequently we get k(t)1 ≤

v−t t−u k(u)1 + k(v)1 . v−u v−u

Remark 4.1. For each n ∈ N both matrices, Φ(t)n and Ψ(t)n , are positive deﬁnite. This follows from Bochner’s Theorem applied to Φ(u, v, c; t)n and Ψ(u, v, c; t)n considered as functions of c. In both cases the matrix is of the form {F (λi − λj )}n where F (c) is the Fourier transform of a positive function (see [9], formula 1.9.14, page 31). A consequence of this result follows: Theorem 4.2. Let γ(t), ρ(t) be geodesics in ∆1 , then t → d(γ(t), ρ(t)) is a convex map in R. Proof. Suppose the γ(t) and σ(t) are deﬁned in [u, v]. We consider h(s, t) deﬁned as follows: 1. the map s → h(s, u), 0 ≤ s ≤ 1 is the geodesic joining γ(u) with ρ(u); 2. the map s → h(s, v), 0 ≤ s ≤ 1 is the geodesic joining γ(v) with ρ(v); 3. for each s, the function t → h(s, t), u ≤ s ≤ v is the geodesic joining h(s, u) with h(s, v). Let J(s, t) = ∂h(s,t) ∂s . Hence, for each ﬁxed s, t → J(s, t) is Jacobi ﬁeld along the geodesic t → h(s, t). Finally, we deﬁne 1 J(s, t)h(s,t) ds. f (t) = 0

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From Theorem 4.1, t → J(s, t)h(s,t) is a convex function for each s. Hence, 1 t → f (t) ia also convex for t ∈ [u, v]. But f (u) = 0 J(s, u)h(s,u) ds is the length of s → h(s, u) and therefore f (u) = d(γ(u), ρ(u)). Similarly, f (v) = d(γ(v), ρ(v)). 1 Now, for u ≤ t ≤ v f (t) = 0 J(s, t)h(s,t) ds is the length of the curve s → h(s, t) which joins γ(t) with ρ(t) and then we get d(γ(t), ρ(t)) ≤ f (t). Convexity of d(γ(t), σ(t)) follows and the Theorem is proved. Remark 4.2. A particular consequence of the above theorem is that there are no closed nonconstant geodesics in ∆1 . Indeed if α : [0, 1] → ∆1 is a nonconstant geodesic such that α(0) = α(1) = a, then for all t ∈ (0, 1) 0 ≤ d(a, α(t)) ≤ td(a, α(0)) + (1 − t)d(a, α(1)) = 0. Deﬁnition 4.2. A subset K of ∆1 is called convex if for all a, b ∈ K the geodesic γa,b , joining a and b, is contained in K. Corollary 4.1. Let a, b, c ∈ ∆1 . Then for all t ∈ [0,1] d(at b, at c) ≤ td(b, c).

(4.7)

In particular, d(bt , ct ) ≤ td(b, c). There is a clear interpretation of the corollary above. In a Riemannian manifold M , the sectional curvature is nonpositive if and only if d(ρs (x), ρs (y)) ≤ sd(x, y), for all x, y ∈ M and all s ∈ [0, 1], where ρs (x) = expp (s exp−1 p (x)) and p ∈ M is ﬁxed (see [4]). This expression reduces, in our (non Riemannian) case to d(p s x, p s y) ≤ sd(x, y), which is (4.7). Corollary 4.2. Let a ∈ ∆1 , a fixed. Then f (γ(t)) ≤ (1 − t)f (γ(0)) + tf (γ(1)), where f (x) = d(a, x) and γ(t) is a geodesic. In particular, geodesics spheres are convex sets. 4.1. The Metric Increasing Property of the Exponential Map In this section we provide a proof of the metric increasing property (MIP) of the exponential map (Theorem 4.3) which is based on the exposition in Corach, Porta and Recht [8]. We begin with a lemma of approximation. Lemma 4.1. Let γ(t) be a curve in ∆1 , then log(γ(t)) can be approximated uniformly by polynomials for t ∈ [t0 , t1 ].

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Proof. Throughout the proof Hol(G) and S 2 denote the set of all complex analytic functions deﬁned in G, with G an open set of complex plane and the Riemann sphere, respectively. Let σ(t) be the spectrum of γ(t), σ(γ) = σ(t) be the t∈[t0 ,t1 ]

spectrum of γ in the algebra C([0, 1], L1 ) and G ⊆ C − {z : Im(z) ≤ 0} an open neighbourhood of σ(γ). Since σ(γ) is compact, S 2 − σ(γ) is connected and log(z) ∈ Hol(G) then there is a sequence Pn of polynomials such that Pn (z) → log(z) uniformly on σ(γ) ([20], Theorem 13.7). Since Pn (z) are all analytic on G, σ(γ) ⊆ G, and Pn (z) → log(z) uniformly on compact subsets of G, then Pn (γ(t)) − log(γ(t))1 → 0 as n → ∞. The Finsler structure of ∆1 is not Riemannian. However ∆1 shares some property with Riemannian manifolds of non-positive sectional curvature. For instance, the following Theorem 4.3. The exponential map in ∆1 increases distances, i.e. for all a ∈ ∆1 , X, Y ∈ B1 (H)h we have d(expa (X), expa (Y )) ≥ X − Y a .

(4.8)

Proof. Let γ1 (t) = etX ,γ2 (t) = etY and f (t) = d(γ1 (t), γ2 (t)). By Theorem 4.2, f is convex, with f (0) = 0. Hence f (t) t ≤ f (1) for all t ∈ (0, 1]. Note that 1 f (t) 1 = log(etX/2 e−tY etX/2 ) = tr log(etX/2 e−tY etX/2 ) . t t t 1 Taking limits we have f (t) ≤ f (1). t Observe next that by the previous lemma, log x can be approximated on any interval [x0 , x1 ] with 0 < x0 < x1 uniformly by polinomials Pn (x). In particular lim

t→0+

lim Pn (x) = log x

n→∞

and

1 lim P˙n (x) = . x

n→∞

(in morm .1 ). Then

1 tX/2 −tY tX/2 lim log(e e e ) = |X − Y | . + t t→0 From this inequality and convexity we conclude that f (t) ≥ t X − Y 1 . This implies that d(expa (tX), expa (tY )) ≥ t X − Y a for all a ∈ ∆1 , and all X, Y ∈ B1 (H)h . Remark 4.3. For a = 1, from the theorem above we get X X X − Y 1 ≤ log(e− 2 eY e− 2 ) , 1

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for X, Y ∈ B1 (H)h , which can also be written 1 1 log x − log y1 ≤ log(x− 2 yx− 2 ) , 1

465

(4.9)

with x, y ∈ ∆1 . Proposition 4.1. ∆1 is a complete metric space with the geodesic distance. Proof. Consider a Cauchy sequence Xn ⊂ ∆1 . By (4.9) Yn = log(Xn ) is a Cauchy .

sequence in B1 (H)h . Then there exists an operator Y ∈ B1 (H)h such that Yn −→1 Y . Hence Y Y d(Xn , eY ) = log(e 2 e−Yn e 2 ) → 0, 1

when n → ∞.

Now, we investigate when d(expa (X), expa (Y )) = X − Y a (see Corollary 3.1). The method of the proof that follows is based on a similar argument used in [2]. In this work the authors studied the occurrence of the equality for a C*-algebra A with trace, in the 2-norm and the operator norm. Theorem 4.4. Let a ∈ ∆1 , X, Y ∈ B1 (H)h . Then we have d(expa (X), expa (Y )) = X − Y a , if and only if a

− 12

Xa

− 12

and a

− 12

Ya

− 12

(4.10)

commute.

Proof. Suppose that a = 1. Clearly, d(eX , eY ) = X − Y 1 if X and Y commute. Let γb,c be the geodesic joining b = eX with c = eY , and let α(t) ∈ B1 (H)h such that γb,c (t) = eα(t) . Then α, that joins X with Y , satisﬁes the inequality X − Y 1 ≤ l(α) ≤ d(eX , eY ) = X − Y 1 . This implies, that X + t(Y − X) = α(t) = log(γb,c (t)) = log(bt c). Since the map f (t) = d(etX , etY ) is convex with f (0) = 0 and f (1) = d(eX , eY ) = X − Y 1 and sX, sY satisfy the hypothesis of this theorem for s ∈ [0, 1], we have for t, s ∈ [0, 1] sX + st(Y − X) = log(bs t cs ). If one compute

∂3 ∂t3

of this equality at s = 0, one obtains

1 2 (t + t)[[X, Y ], X − Y ]. 2 Then, [[X, Y ], X] = [[X, Y ], Y ]. Therefore, by the properties of the trace, 0=

tr([[X, Y ], X]X) = tr([[X, Y ], Y ]X) = 0, this implies that tr(X 2 Y 2 ) = tr(XY XY ).

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This means that we have equality in the Cauchy-Schwarz inequality 1

tr(XY XY ) = tr((Y X)∗ XY )

1

≤ [tr((Y X)∗ Y X)] 2 [tr((XY )∗ XY )] 2 = tr(X 2 Y 2 ) = tr(XY XY ).

So XY is a multiple of Y X, i.e. XY = βY X, β ∈ C. Replacing this condition in the above equality we get tr(X 2 Y 2 ) = tr(XY XY ) = βtr(XY Y X) = βtr(X 2 Y 2 ). This implies that β ≥ 0. If β = 0, then X and Y commute. Otherwise β = 1. This complete the proof for a = 1. For the general case, we observe that by the invariance under the action d(expa (X), expa (Y )) = X − Y a , is equivalent to −1 2

d(ea

1

Xa− 2

−1 2

, ea

1

Y a− 2

(4.11)

1 1 1 1 ) = a− 2 Xa− 2 − a− 2 Y a− 2 . 1

Hence, it follows from the case a = 1, that the equality (4.11) holds if and only if 1 1 1 1 a− 2 Xa− 2 and a− 2 Y a− 2 commute.

5. Non-positive Curvature 5.1. Metrics spaces of non-positive curvature It would be very interesting to understand the relations between the geodesic distance and general metric spaces with non-positive curvature. In this section, we will brieﬂy review some basic facts about these spaces. About ﬁfty years ago Alexandrov showed that the notions of upper and lower curvature bounds make sense for a more general class of metric spaces than Riemannian manifolds, namely, for geodesic spaces. One of the ﬁrst papers on non-positively curved spaces was written by Busemann in 1948. For more details on metric spaces with non-positive curvature we refer to [14]. We now introduce the notion of midpoint maps and Busemann’s notion of nonpositive curvature in a metric space (X, d). Deﬁnition 5.1. Let (X, d) be a metric space. A symmetric map M : X × X −→ X is called a midpoint map for (X, d) if for all x, y ∈ X d(M (x, y), x) =

1 d(x, y) = d(M (x, y), y). 2

Deﬁnition 5.2. A complete metric space (X, d) is called a geodesic length space, or simply a geodesic space, if for any two points x, y ∈ X, there exists a shortest geodesic joining them, i.e. a continuous curve such that γ : [0, 1] → X with γ(0) = x, γ(1) = y, and d(x, y) = ld (γ).

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Here, ld (γ) denotes the length of γ (respect to the metric d) and it is deﬁned as n d(γ(ti−1 ), γ(ti )) : 0 = t0 < t1 < ... < tn = 1, n ∈ N}. ld (γ) := sup{ i=1

A curve γ : [0, 1] → X is called a geodesic if there exists > 0 such that ld (γ [t,t ] ) = d(γ(t), γ(t )) whenever |t − t | < . Finally, a geodesic γ : [0, 1] → X is called a shortest geodesic if ld (γ) = d(γ(0), γ(1)). In particular, for the metric space (∆1 , d) the geodesics γa,b (in the sense of the equation (2.1)) are also shortest geodesic, since ld (γa,b ) = =

sup{ sup{

n i=1 n

d(γa,b (ti−1 ), γa,b (ti )) : 0 = t0 < t1 < ... < tn = 1, n ∈ N} l(γa,b [ti−1 ,ti ] ) : 0 = t0 < t1 < ... < tn = 1, n ∈ N}

i=1

=

sup{

n i=1

=

ti

ti−1

γ˙ a,b (t)γa,b (t) dt : 0 = t0 < t1 < ... < tn = 1, n ∈ N}

l(γa,b ).

this equality implies that γa,b is also a geodesic in the metric sense

ld (γa,b [t,t ] = l(γa,b [t,t ] ) = d(γa,b (t), γa,b (t )). Then ld (γa,b ) = l(γa,b ) = d(γa,b (0), γa,b (1)). By the above argument, we have the following statement Proposition 5.1. The metric space (∆1 , d) is a geodesic space and m( . , . ) is a midpoint point corresponding to the shortest geodesic γa,b for all a, b ∈ ∆1 . Deﬁnition 5.3. Let (X, d) be a metric space and m : X × X −→ X be a midpoint map for (X, d). Then (X, d) is said to be a m-global Busemann non-positive curvature space (m − global BN P C) if for all x, y, z ∈ X 1 d(m(x, y), m(x, z)) ≤ d(y, z). (5.1) 2 Remark 5.1. The m-global BNPC condition is equivalent to m is a convex midpoint map, i.e. for all x1 , x2 , y1 , y2 ∈ X 1 d(m(x1 , y1 ), m(x2 , y2 )) ≤ (d(x1 , x2 ) + d(y1 , y2 )). 2 Theorem 5.1. The space (∆1 , d) is a m-global Busemann NPC space. Here, we introduce the notion of convex hull of a subset K.

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Deﬁnition 5.4. Let (X, d) be a geodesic length space. The convex hull C(K) of a subset K of X is the smallest convex subset of X containing K. In general, the convex hull of a set K as deﬁned above need not exist, because the intersection of convex subsets of X need not be convex. Nevertheless, for (∆1 , d) there is a constructive approach to compute C(K) due to its Busemann NPC structure. Proposition 5.2. For any K ⊆ ∆1 , the convex hull C(K) exists and can be obtained as follows ∞ C(K) = Kn . where K0 = K and Kn :=

n=0 {a t b : a, b ∈ Kn−1 }.

Proof. Lemma 3.3.1 ([14]).

5.2. An alternative deﬁnition of sectional curvature In this section, we shall see that it is possible to give an alternative deﬁnition of sectional curvature in ∆1 . For this, we remember that in [17] Milnor recalls that the sectional curvature, sa (X, Y ), can be obtained by the following limit r X − Y a − d(expa (rX), expa (rY )) . r2 d(expa (rX), expa (rY )) where X, Y are tangent vectors at a point a. We will see that this limit makes sense in our context. Suppose that r > 0 is close to 0 such that e−rX/2 erY e−rX/2 lies within the radius of convergence of the series log(u). Them sa (X, Y ) = 6 lim+ r→0

log(e−rX/2 erY e−rX/2 ) = r(Y − X) + r3 κ(X, Y ) + o(r3 ), where 1 1 1 1 Y XY + XY X − (XY 2 + Y 2 X) − (X 2 Y + Y X 2 ). 6 12 12 24 Before stating the existence of the limit above we need the following deﬁniton and lemmas. κ(X, Y ) =

Deﬁnition 5.5. Let V a vector space and f be a function from V to R ∪ {+∞}. We shall say that Df (x0 )(v) is the right derivate of f at x0 in the direction v if the limit f (x0 + tv) − f (x0 ) Df (x0 )(v) = lim+ . t t→0 exists. In this case, we denote by v → Df (x0 )(v) the right derivate of f at x0 . Remark 5.2. ([1], Proposition 4.1) Let V a vector space and f be a nontrivial convex function from V to R ∪ {+∞}. Suppose x0 ∈ Dom(f ) and v ∈ V . Then the limit Df (x0 )(v) exists in R and satisﬁes f (x0 ) − f (x0 − v) ≤ Df (x0 )(v) ≤ f (x0 + v) − f (x0 ).

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For a ∈ ∆1 , we denote by Pa : (T ∆1 )a → R+ ,

Pa (X) = Xa

Lemma 5.1. For a ∈ ∆1 , Pa is a convex function. Moreover, Pa is right diﬀerentiable on B1 (H)h and satisﬁes Xa − X − Y a ≤ DPa (X)(Y ) ≤ X + Y a − Xa . Proof. By the remark 5.2 it suﬃces to prove that Pa is convex. Clearly this is obvious for the usual properties of a norm, since for all λ ∈ (0, 1) Pa (λX + (1 − λ)Y ) ≤ λPa (X) + (1 − λ)Pa (Y ). Theorem 5.2. Let a ∈∆1 and X, Y ∈ B1 (H)h . The limit sa (X, Y ) = lim+ r→0

r X − Y a − d(expa (rX), expa (rY )) r2 d(expa (rX), expa (rY ))

exists and verifies 1−

Y − X + κ(X, Y )1 ≤ sa (X, Y ) ≤ 0. X − Y 1

Proof. Since the metric on B1 (H)h and the geodesic distance are invariant by the action of Gl(H, B1 (H)), it suﬃces to consider the case a = 1. Note that 1 lim d(erX , erY ) = lim Y − X + r2 κ(X, Y ) + o(r2 )1 = Y − X1 . r→0+ r r→0+ Them it is enough to show the existence of the following limit 1 lim+ 3 (r X − Y 1 − r(Y − X) + r3 κ(X, Y ) + o(r3 )1 ), r→0 r which is equivalent to the existence of the limit 1 lim (X − Y 1 − (Y − X) + r2 κ(X, Y )1 ). r→0+ r2 But this exists and is equal to −DP1 (Y − X)(κ(X, Y )) and therefore s1 (X, Y ) =

−DP1 (Y − X)(κ(X, Y )) . Y − X1

By the MIP property this limit is non positive. On the other hand, −DP1 (Y − X)(κ(X, Y )) ≥ Y − X1 − Y − X + κ(X, Y )1 and therefore s1 (X, Y ) ≥ 1 −

Y −X+κ(X,Y )1 . X−Y 1

Acknowledgment I would like to thank Prof. E. Andruchow for aﬀording me the opportunity to study Geometry of Operators, a subject which interest me immensely. I gratefully acknowledge and thank him for his supervision, generous help and advice not only throughout my work on this project but also when I arrived to Buenos Aires.

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References [1] J. Aubin, Optima and Equilibria - An introduction to nonlinear analysis. 2nd Edition, Springer Verlag, 1998. [2] E. Andruchow and L. Recht, Sectional curvature and commutation of pairs of seladjoint operators. Journal of Operator Theory(to appear). [3] T, Ando; R. Horn and Ch. Johnson, The Singular Values of a Hadamard product: a basic inequality. Linear Multinear Algebra 21 (1987), 345–365. [4] W. Ballmann; M. Gromov and V. Schroeder, Manifolds of non positive curvature. Birkhuser Verlag, 1985. [5] R. Bhatia, On the exponential metric increasing property. Linear Algebra and its applications 375 (2003), 211–220. [6] G. Corach; H. Porta and L. Recht, A geometric interpretation of Segals inequality. Proc. Amer. Math. Soc. 115 (1992), 229–231. [7] G. Corach; H. Porta and L. Recht, The geometry of spaces of selfadjoint invertible elements of a C ∗ -algebra. Integral Equations and Operator Theory 16 (1993), 333– 359. [8] G. Corach; H. Porta and L. Recht, Convexity of the geodesic distance on spaces of positive operators. Illinois Journal of Mathematics 38 N 1 (1994), 87–94. [9] Erdelyi, A. et al.: Tables of integral transforms. (The Bateman manuscript), Volumen 1, Mc Graw Hill, 1954. [10] M. Fujii; R. Furuta and R. Nakamoto, Norm inequalities in the Corach-Porta-Recht theory and operator means. Illinois Journal of Mathematics 40 N 4 (1996), 1–8. [11] J. Fujii and E. Kamei, Relative operator entropy in nonconmutative information theory. Mathematica Japonica 34 (1989), 341–348. [12] P. de la Harpe, Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space. Lecture Notes in Mathematics 285, Springer-Verlag, 1972. [13] F. Hiai and H. Kosaki, Comparison of various Means of operators. Journal of Functional Analysis 163 (1999), 300–323. [14] J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Birkhuser Verlag, 1997. [15] S. Lang, Diﬀerential and Riemannian Manifolds, Springer, 1995. [16] L. Mata Lorenzo and L. Recht, Inﬁnite Dimensional Homogeneous Reductive Spaces. Acta Cientﬁca Venezolana 43 No 2 (1992), 76–90. [17] J. Milnor, Morse Theory Annals of Mathematical Studies 54, Princeton University Press, 1963. [18] G. Mostow, Some new decomposition theorems for semi-simple groups, Mem. Amer. Math. Soc. 14 (1955), 31-54. [19] W. Pusz and S. Woronowicz, Functional Calculus for sesquilinear forms and the puriﬁcation map. Rep. Math. Phys. 8 (1975), 159–170. [20] W. Rudin, Real and Complex Analysis, Tata Mc Graw Hill, 1974.

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[21] A. Uhlmann, Density operators as an arena for diﬀerential geometry. Rep. Math. Phys. 36 (1993), 253–263. Cristian Conde Saavedra 15, 3o Piso 1083, Buenos Aires Argentina e-mail: [email protected] Submitted: December 21, 2005 Revised: November 2, 2006

Integr. equ. oper. theory 57 (2007), 473–490 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040473-18, published online December 26, 2006 DOI 10.1007/s00020-006-1464-y

Integral Equations and Operator Theory

Bounded Reflexivity of Operator Subspaces II Don Hadwin, Jiankui Li and Zhidong Pan Abstract. Several classes of operators are shown to be boundedly reﬂexive; including bilateral operator-weighted shifts, weak contractions, and operators of class (SM). The commutants of many of these operators are shown to be boundedly reﬂexive. We also show that symmetric pattern subspaces with constant main diagonals are boundedly reﬂexive, and we provide some necessary and suﬃcient conditions for refb (S) to be boundedly reﬂexive. Mathematics Subject Classification (2000). Primary 47L30, 47L75. Keywords. Bounded reﬂexivity, commutant, separating vector, weak contraction.

1. Introduction Let X be a separable complex Banach space and B(X) be the set of bounded linear operators on X. If E is a subset of B(X), for any r > 0, let Er = {T ∈ E : T ≤ r}; and for any x ∈ X, let Ex = {Ax : A ∈ E} and [Ex] be the norm closure of Ex. For any subspace S of B(X), let refb (S) = {T ∈ B(X) : ∃ MT > 0 such that T x ∈ [SMT x], ∀ x ∈ X}, and let ref (S) = {T ∈ B(X) : T x ∈ [Sx], ∀ x ∈ X}. S is called boundedly reflexive if refb (S) = S and S is called reflexive if ref (S) = S. When X is a Hilbert space, we replace it with H. Clearly, if S is reﬂexive then it is weakly closed and boundedly reﬂexive. Example 3.3 of [19] shows that a boundedly reﬂexive subspace of B(H) need not be weakly closed, but a boundedly reﬂexive subspace of B(H) must be weak∗ closed, by [19, Corollary 2.9]. A boundedly reﬂexive subspace S of B(H) is called hereditarily boundedly reflexive if every weak∗ closed subspace of S is boundedly reﬂexive. Similarly, let refab (S) = {T ∈ B(X) : ∃ MT > 0 such that T x ∈ SMT x, ∀ x ∈ X} and refa (S) = {T ∈ B(X) : T x ∈ Sx, ∀ x ∈ X}. S is called algebraically boundedly reflexive if refab (S) = S and S is called algebraically reflexive if refa (S) = S. It is clear that if S is algebraically reﬂexive then it is algebraically boundedly reﬂexive. This work was completed with the support of NSF of China.

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If E is weak∗ closed subspace of B(H), then [Er x] = Er x. In this case, bounded reﬂexivity and algebraic bounded reﬂexivity coincide. In [7], many versions of reﬂexivity are uniﬁed as special cases of a general version based on dual pairs of vector spaces, however, bounded reﬂexivity deﬁned above does not belong to this general version. It does coincide with the notion of reﬂexivity of convex sets of operators, ﬁrst studied in [2], [3], and [21]; our motivation to study bounded reﬂexivity came from the study of complete positivity of elementary operators on C ∗ -algebras, see [17], [18], and [19] for more details. If E is a subset of B(X), we use E and E to denote the commutant and the double commutant of E, respectively. A vector x ∈ X is called a separating vector of E if the map T → T x, T ∈ E is injective from E to X. For T ∈ B(X), let WT denote the weakly closed algebra generated by T and I. For convenience, when WT has certain property (e.g. reﬂexive, boundedly reﬂexive, separating vectors, etc.), we say that T has the property (reﬂexive, boundedly reﬂexive, separating vectors, respectively). This paper is a continuation of [19]. Some deﬁnitions and notation can be found in [19]. The following simple proposition is repeatedly used in this paper: Proposition 1.1. If S is algebraically boundedly reflexive in B(X) and S has a separating vector, then every subspace of S is algebraically boundedly reflexive in B(X). If X is a Hilbert space, then every weak∗ closed subspace of S is boundedly reflexive. Proof. The ﬁrst statement follows directly from the deﬁnition and the second statement follows from [19, Corollary 2.9]. This paper is organized as follows: In section 2, we present some necessary and suﬃcient conditions for refb (S) to be boundedly reﬂexive. In section 3, we show bounded reﬂexivity for several classes of operators, including weak contractions and operators of class (SM ); in particular, C0 -contractions. We also show that the commutants of these operators are boundedly reﬂexive. In section 4, we show that symmetric pattern subspaces with constant main diagonals are boundedly reﬂexive. In section 5, the bounded reﬂexivity of direct sums and graphs of subspaces of B(H) is discussed.

2. On refb (S) Suppose S is a subspace of B(X). Contrary to the fact that ref (S) is always reﬂexive, refb (S) need not be boundedly reﬂexive in general. In fact, Example 2.1 shows that refb (S) may not even be norm-closed. Several necessary and suﬃcient conditions for refb (S) to be boundedly reﬂexive are given below. First, for any

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T ∈ (refb (S))1 , the unit ball of refb (S), we deﬁne k(T ) = inf {MT > 0 : T x ∈ [SMT x], ∀x ∈ X} and k(S) = sup{k(T ) : T ∈ (refb (S))1 }. a11 a12 Example 2.1. For each n, set Vn = { ∈ M2 (C) : a11 − n2 a22 = 0}, a21 a22 where M2 (C) denotes the set of all 2 × 2 complex matrices. It can be verified easily ∞ ∞ that refb (Vn ) = M2 (C). Let H = ⊕C(2) and S = { ⊕Vn ∈ B(H) : Vn ∈ n=1

Vn }. It is ∞ ⊕An { n=1 0 xn = 1 0 = 1

n=1

not hard to see that S is weakly closed in B(H); moreover refb (S) = 0 0 ∈ B(H) : sup{k(An ) : n ≥ 1} < ∞ }. Let An = and 0 n1 a b . Choose S = ∈ Vn such that An xn = Sxn . It follows that c d b . Thus, b = 0, d = n1 and a = n2 d = n. Therefore, k(An ) ≥ n. d n Take Tn = A1 ⊕ · · · ⊕ An ⊕ 0 ⊕ · · · , then Tn ∈ refb (S) and Tn → T in the norm ∞ topology, where T = ⊕An . Clearly T ∈ / refb (S). This shows refb (S) is not

norm-closed.

n=1

Theorem 2.2. If S ⊆ B(X) then the following are equivalent. (i) refb (S) is boundedly reflexive. (ii) refb (S) is norm-closed. (iii) k(S) < ∞. Proof. (i)⇒(ii). Suppose An ∈ refb (S) and An → A in the norm topology. It follows that An is bounded and An x → Ax, ∀ x ∈ X; so A ∈ refb (refb (S)) = refb (S), by (i). (ii)⇒(iii). Suppose refb (S) is norm-closed. For each natural number n, set En = {A ∈ B(X) : Ax ∈ [Sn x], for all x ∈ X}. Clearly, refb (S) = ∪∞ 1 En . If Ak ∈ En and Ak → A in the norm topology then Ak x → Ax, ∀ x ∈ X. Now Ak x ∈ [Sn x] implies Ax ∈ [Sn x], i.e. A ∈ En , so En is norm-closed. By the Baire Category Theorem, there exist a natural number n0 , 0 > 0, and A0 ∈ En0 such that {A ∈ refb (S) : ||A − A0 || ≤ 0 } ⊆ En0 . It follows, ∀ T ∈ (refb (S))1 , 0 T + A0 ∈ {A ∈ 0 refb (S) : ||A − A0 || ≤ 0 } ⊆ En0 . Let N0 = 2n 0 . For any T ∈ (refb (S))1 , x ∈ X, then (0 T + A0 )x ∈ [Sn0 x]. Thus, T x ∈ 10 A0 x + [S n 0 x] ⊆ [S 2n0 x] = [SN0 x]. 0

0

Therefore, k(S) ≤ N0 . (iii)⇒(i). For any T ∈ refb (refb (S)) and x ∈ X, multiplying T by a scalar if necessary, we can assume T x ∈ [(refb (S))1 x]. Now (iii) implies T x ∈ [SN0 x], for some N0 , i.e. T ∈ refb (S), so refb (S) is boundedly reﬂexive. For Hilbert space operators, we have the following:

Theorem 2.3. If S ⊆ B(H) then the following are equivalent. (i) refb (S) is boundedly reflexive.

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(ii) refb (S) is weak∗ closed. (iii) refb (S) is norm-closed. (iv) k(S) < ∞. Proof. (i)⇒(ii) follows from [19, Corollary 2.9], and (ii)⇒(iii) is clear. The rest is the same as the proof of Theorem 2.2. Corollary 2.4. If S is a finite-dimensional subspace of B(X), then refb (S) is boundedly reflexive. Proof. If S is ﬁnite-dimensional, then, so is ref (S), by [14, Lemma 2.7]. Since refb (S) ⊆ ref (S), refb (S) is also ﬁnite-dimensional; thus it is closed. The conclusion now follows from Theorem 2.2. It is shown in [20] that if T is a triangular operator, then T and {T } are boundedly reﬂexive. The idea in the next two propositions was one of the key ingredients utilized in [20] in proving that the commutant of a triangular operator is boundedly reﬂexive. Proposition 2.5. Suppose S is a weakly closed commutative subalgebra of B(X). Let {Eλ , λ ∈ Λ} be a family of hyperinvariant subspaces of S whose linear span is dense in X. If S|Eλ is boundedly reflexive for all λ ∈ Λ, then refb (S) ⊆ S ; in particular, refb (S) is commutative. Proof. Suppose A ∈ refb (S). It follows that A|Eλ ∈ refb (S|Eλ ). Since S|Eλ is boundedly reﬂexive, there exists a B ∈ S so that A|Eλ = B|Eλ . Take any C ∈ S , since Eλ is hyperinvariant, AC|Eλ = BC|Eλ = CB|Eλ = CA|Eλ . Since X is the closed linear span of {Eλ , λ ∈ Λ}, we conclude AC = CA, i.e. A ∈ S . Similarly, we can prove the following: Proposition 2.6. Suppose S is a weakly closed subalgebra of B(X). Let {Eλ , λ ∈ Λ} be a family of invariant subspaces of S whose linear span is dense in X. If S|Eλ is boundedly reflexive for all λ ∈ Λ, then refb (S) ⊆ S .

3. On singly-generated subalgebras To study an operator on a Banach space, it is very often more convenient to study the unital algebra generated by the operator. It follows immediately from the definitions that reﬂexive operators are boundedly reﬂexive. It is shown in [20] that two large classes of operators, which are not reﬂexive in general, are boundedly reﬂexive. More precisely, if T is a triangular operator on a Hilbert space or a cyclic operator with “thin” approximate point spectrum on a Banach space, then T is boundedly reﬂexive. In this section, we provide some additional classes of boundedly reﬂexive operators, which are not reﬂexive in general. These include bilateral operator-weighted shifts, operators that are quasisimilar to a normal operator, n-normal operators, weak contractions, and operators of class (SM ). We also show that if T is a weak contraction or an operator of class (SM ), then {T }

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is boundedly reﬂexive. In particular, if T is a C0 -contraction, then T and {T } are boundedly reﬂexive. For a subset E of B(H), let lat E denote the set of closed subspaces of H that are invariant under every operator in E, and alglat E denote the set of all operators T ∈ B(H) such that lat E ⊆ lat T . It is easy to verify that if E is a unital subalgebra of B(H), then alglat E = ref (E). A Hilbert space operator is called triangular if its matrix representation has an upper triangular form with respect to some orthonormal basis {en }∞ 1 ; an operator is called co-triangular if it is the adjoint of a triangular operator. It is shown in [20] that triangular and co-triangular operators are boundedly reﬂexive; in particular, unilateral operatorweighted shifts are boundedly reﬂexive. Theorem 3.1. If T is a bilateral operator-weighted shift on a Hilbert space, then T is boundedly reflexive. Proof. We proceed by showing a somewhat stronger result, that is, {T } ∩ alglat T is hereditarily boundedly reﬂexive. We will achieve this by showing {T } ∩ alglat T is boundedly reﬂexive and has a separating vector. ∞ ⊕Hi . For any uniformly Let Hi = H for each integer i and K = i=−∞

bounded sequence of operators Ti ∈ B(H), let T be the bilateral operator-weighted shift with weights Ti . For any A ∈ refb ({T } ∩ alglat T ), let Pn be the orthogonal n ⊕Hi . Then it is easy to check that Pn K are invariant projection of K onto i=−∞

subspaces of A and ({T } ∩ alglat T )|Pn K ⊆ {T |Pn K } ∩ alglat(T |PnK ). It follows that A|Pn K ∈ refb ({T } ∩alglat T )|Pn K ⊆ refb ({T |PnK } ∩alglat(T |PnK )). Since T |Pn K is co-triangular, {T |Pn K } ∩ alglat(T |PnK ) is boundedly reﬂexive by [20, Corollary 3.10]. Thus A|Pn K ∈ {T |PnK } ∩ alglat(T |PnK ); in particular, A|Pn K ∈ {T |PnK } , i.e. A|Pn K T |Pn K = T |Pn K A|Pn K . It follows (AT )|Pn K = (T A)|Pn K . Since the union of Pn K is dense in K, AT = T A. Clearly, A ∈ refb ({T } ∩ alglat T ) ⊆ refb (alglat T ) = alglat T . Thus A ∈ {T } ∩ alglat T . Again, since T |Pn K is co-triangular, {T |PnK } ∩alglat(T |PnK ) has a dense set of separating vectors by [20, Theorem 3.9]. Thus ({T } ∩ alglat T )|Pn K has a dense set of separating vectors. An appeal to [6, Theorem 11] yields that {T } ∩ alglat T has a dense set of separating vectors. Since WT is a weak∗ closed subspace of {T } ∩ alglat T , it is boundedly reﬂexive by Proposition 1.1. Even though, in general, the direct sum of two boundedly reﬂexive operators may not be boundedly reﬂexive, see [19, Example 3.17], we have the following theorem about direct sums, which provides a way of constructing new boundedly reﬂexive operators. Theorem 3.2. If T ∈ B(H) is boundedly reflexive and E ∈ lat T , then T ⊕ T |E is boundedly reflexive in B(H ⊕ E).

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Proof. Let T˜ = T ⊕T |E . Since WT˜ is weakly closed, it follows that ∀ M > 0, x ∈ H, [(WT˜ )M x] = (WT˜ )M x. Suppose A ∈ B(H ⊕ E) and ∃ MA > 0 such that Ax ∈ [(WT˜ )MA x] = (WT˜ )MA x, for any x ∈ H.

(3.1)

Clearly, WT˜ ⊆ WT ⊕ B(E) and WT ⊕ B(E) is boundedly reﬂexive; thus, A ∈ WT ⊕ B(E). Let A = C ⊕ B, where C ∈ WT and B ∈ B(E). Replacing A with A − (C ⊕ C|E ), if necessary, we may assume A = 0 ⊕ B. We will complete the proof by showing B = 0. Write A in the form 0 0 A= ⊕B 0 0 ⊥ withrespect  to the decomposition H ⊕ E = (E ⊕ E) ⊕ E. By (3.1), we have for x1 any  x2  in (E ⊥ ⊕ E) ⊕ E, x3     x1 x1 0 0 ( ⊕ B)  x2  ∈ (WT˜ )MA  x2  . (3.2) 0 0 x3 x3

Take any x ∈ E, by (3.2), there exists A1 0 ⊕ A3 ∈ (WT˜ )MA A2 A3 such that 0 ( 0

0 0



 0 A1   x ⊕ B) =( A2 x

0 A3



 0 ⊕ A3 )  x  . x

It follows that Bx = A3 x = 0. Hence A = 0 and T ⊕T |E is boundedly reﬂexive. Corollary 3.3. Suppose that T ∈ B(H) is boundedly reflexive and Ei ∈ lat T , 1 ≤ i ≤ m. Then T ⊕ T |E1 ⊕ · · · ⊕ T |Em is boundedly reflexive in B(H ⊕ E1 ⊕ · · · ⊕ Em ). A quasiaﬃnity is an injective operator with a dense range. We say two operators A and B are quasisimilar if there exist quasiaﬃnities C and D such that AC = CB and DA = BD. Lemma 3.4. ([20, Lemma 3.5]) If operators A and B are quasisimilar, and {A} is boundedly reflexive, then {B} is boundedly reflexive. If an operator A is quasisimilar to a normal operator then the commutant of A is boundedly reﬂexive by the above lemma and the fact that the commutant of a normal operator is reﬂexive. However, A itself may not be reﬂexive. The operator T ⊕ 0 constructed in [15] is a non-reﬂexive operator that is quasisimilar to a normal operator (a diagonal operator, indeed). The situation is diﬀerent for bounded reﬂexivity.

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The proof of the next lemma is similar to that of [6, Theorem 11], so we omit it. Lemma 3.5. Suppose A is a weakly closed unital subagebra of B(H). If A has an increasing sequence En ⊆ En+1 of invariant subspaces with closed union H such that each An := A|En has a dense set of separating vectors, then A has a dense set of separating vectors. m A ﬁnite or countable system {F basic k }1 of closed subspaces of H is called if for each i, the subspace Fi and k=i Fk are complementary, where denotes the closed linear span, and 1≤n ( k≥n Fk ) = {0} if m = ∞. For each n, let En = F1 + · · · + Fn . It follows that if {Fk }m 1 is a basic system then each En is a closed subspace.

Lemma 3.6. If T ∈ B(H) is quasisimilar to a normal operator then alglat T has a dense set of separating vectors. Proof. Set A = alglat T . Since T is quasisimilar to a normal operator, there exists a basic system {Fk }m 1 of invariant subspaces, where m can be ﬁnite or inﬁnite, such that T |Fk is similar to a normal operator, by [1]. Let En = F1 + · · · + Fn . It follows that T |En is similar to a normal operator for each n. Since every normal operator has a dense set of separating vectors, see [16, Lemma 8], it follows that T |En has a dense set of separating vectors. Since normal operators are reﬂexive, it follows that T |En is reﬂexive; thus A|En ⊆ alglat (T |En ) = WT |En . Hence A|En has a dense set of separating vectors. An appeal to Lemma 3.5 yields the conclusion. Theorem 3.7. If T ∈ B(H) is quasisimilar to a normal operator then T is boundedly reflexive. Proof. Clearly, alglat T is reﬂexive, so it is boundedly reﬂexive. Now, WT is boundedly reﬂexive by Lemma 3.6 and Proposition 1.1. The remainder of this section deals with contractions on Hilbert spaces and algebras related to these contractions. We say a contraction T ∈ B(H) is absolutely continuous if the canonical decomposition T = T1 ⊕ T2 , where T1 is a completely non-unitary contraction and T2 is a unitary operator, has the property that T2 is absolutely continuous. A completely non-unitary contraction T is said to be of class C0 if there exists a u ∈ H ∞ , u ≡ 0, such that u(T ) = 0. A contraction T is called a weak contraction if (i) σ(T ) = D, and (ii) I − T ∗ T has ﬁnite trace. We say that an absolutely continuous contraction T is in the class (SM ), if its characteristic function admits a two-sided scalar multiple; this class contains all C0 -contractions, and all absolutely continuous weak contractions. The interested reader should see [4] and [22] for more details on contractions. Suppose S is the unilateral shift on the Hardy space H 2 , and θ ∈ H ∞ is an inner function. Let H(θ) = H 2 θH 2 and S(θ) be the operator deﬁned by S(θ) = PH(θ) S|H(θ) , where PH(θ) is the projection from H 2 onto H(θ). Then Lemma 3.8. {S(θ)} is boundedly reflexive.

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Proof. By [4, p.39], S(θ) has cyclic vectors and its spectrum does not contain any interior points. Thus {S(θ)} is boundedly reﬂexive by [20, Theorem 2.1]. Lemma 3.9. If T is an operator of class C0 , then {T } is boundedly reflexive. Proof. By Lemma 3.4, we may assume that ∞

T = ⊕S(θk ), k=1

where {θk } is a sequence of inner functions and θk+1 divides θk for each k. ∞ ⊕Hk onto the n-th Let Hk = H(θk ) and let Pn be the projection from coordinate of

k=1

∞

⊕Hk . Clearly, Pn ∈ {T } . Now we deﬁne the operators Rij as

k=1 ∞

∞

k=1

k=1

follows Rij (

⊕hk ) =

⊕lk , where lk = 0 if k = i, and

PH(θi ) hj , whenever i > j, li = (θi /θj )hj , whenever i ≤ j.

It follows from [4, Theorem 3.1.16] that R ij ∈ {T } . Suppose A ∈ refb ({T } ). Since A = Pi APj in the strong operator topology, i,j

to show A ∈ {T } , it suﬃces to show Pi APj ∈ {T }. First note that Rji Pi APj = ∞ ∞ ⊕Tk with Tk = 0 for k = j and Pi APj Rji = ⊕Vk with Vk = 0 for k = i. k=1

k=1

It can be easily checked that refb ({T } ) is an algebra. Since A ∈ refb ({T } ), and Rji , Pi , Pj ∈ {T } ⊆ refb ({T } ), we have Rji Pi APj , and Pi APj Rji belong to ∞ ∞ ⊕Tk ∈ refb ({T } ) and ⊕Vk ∈ refb ({T }). Thus, Tj ∈ refb ({T } ); i.e. k=1

k=1

Pj refb ({T })|Hj ⊆ refb (Pj {T }|Hj )= refb ({S(θj )} ) = {S(θj )} , the last equality follows from Lemma 3.8; so Tj commutes with S(θj ). Similarly, Vi commutes with S(θi ). Therefore, Rji Pi APj , and Pi APj Rji commute with T . Hence, Rji (Pi APj T − T Pi APj ) = 0

(3.3)

and (3.4) (Pi APj T − T Pi APj )Rji = 0 is one to one on the range of Pi , it follows from (3.3) that

If j < i, Rji Pi APj ∈ {T }. If i ≤ j, the range of Rji contains the range of Pj , It follows from (3.4) that Pi APj ∈ {T }. Corollary 3.10. If T is an operator of class C0 then T is boundedly reflexive.

Proof. By [4, Theorem 4.1.2], we have {T } ∩ alglat T = WT . Clearly, alglat T is reﬂexive, therefore boundedly reﬂexive. By Lemma 3.9, {T } is boundedly reﬂexive. It follows that T is boundedly reﬂexive.

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If T = 0 is an algebraic operator, then T /2||T || is an operator of class C0 . Thus, by Lemma 3.9 and Corollary 3.10, we have Corollary 3.11. If T is an algebraic operator, then T and {T } are boundedly reflexive. Suppose that (X, Ω, µ) is a σ-ﬁnite measure space. For every φ ∈L∞ (µ, B(H)), we denote Mφ the operator from L2 (µ, B(H)) into itself deﬁned by (Mφ f )(x) = φ(x)f (x). Suppose that S(λ) ⊆ B(H) for λ ∈ X. The direct integral of S(λ)’s over X, ⊕ denoted by X S(λ)dµ(λ), is the set {Mφ : φ ∈ L∞ (µ, B(H)), φ(λ) ∈ S(λ) a.e.}. Similar to that of [9, Lemma 3.4], we have the following: Lemma 3.12. If {A(λ)} is a family of hereditarily boundedly reflexive subalgebras ⊕ of Mn (C), then X A(λ)dµ(λ) is boundedly reflexive. An operator T is called n-normal if T is unitarily equivalent to an n × n operator matrix with commuting normal entries; equivalently, if T is unitarily equivalent to a decomposable operator on L2 (µ, Mn (C)) for some σ-ﬁnite measure space (X, Ω, µ). N-normal operators may not be reﬂexive in general. The main theorem of [8] gives a necessary and suﬃcient condition for a binormal operator to be reﬂexive. Corollary 3.13. If A is an n-normal operator on H, then A is boundedly reflexive. ⊕ Proof. Let A = X A(λ)dµ(λ) and A(λ) = WA(λ) . Since A(λ) is algebraic, A(λ) has a separating vector. By Proposition 1.1 and Corollary 3.11, A(λ) is hereditarily ⊕ boundedly reﬂexive. By Lemma 3.12, X A(λ)dµ(λ) is boundedly reﬂexive. Since ⊕ WA is unitarily equivalent to X A(λ)dµ(λ), we have that A is boundedly reﬂexive. Let A be any subalgebra of B(H) and T ∈ A. We say that T generates A if the weakly closed ideal generated by T is equal to A. Definition 3.14. Let T1 ∈ B(H1 ) and T2 ∈ B(H2 ). T1 and T2 are said to be pseudosimilar if there exist U ∈ B(H1 , H2 ) and V ∈ B(H2 , H1 ) such that (1) U T1 = T2 U , V T2 = T1 V , and (2) V U generates WT1 and U V generates WT2 . The pseudosimilarity of T1 and T2 is denoted by T1 T2 , where U and V U,V

are called the intertwining operators. It is easy to show that pseudosimilarity is an equivalence relation and the intertwining operators are necessarily quasiaﬃnities. The next lemma shows pseudosimilarity preserves bounded reﬂexivity of operators.

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Lemma 3.15. If T1 T2 and refb (WT2 ) = WT2 then refb (WT1 ) = WT1 . U,V

Proof. We only need to show refb (WT1 ) ⊆ WT1 . Let A ∈ refb (WT1 ). By [12, Lemma 4.4], we have that U WT1 V ⊆ WT2

and

V WT2 U ⊆ WT1 .

Hence V U AV U ∈ V U refb (WT1 )V U ⊆ V refb (U WT1 V )U ⊆ V refb (WT2 )U = V WT2 U ⊆ WT1 .

(3.5)

Deﬁne JA = {B ∈ WT1 : V U AB ∈ WT1 }. We can easily check JA is a weakly closed ideal of WT1 . By (3.5), we have that V U ∈ JA . By part (2) of Deﬁnition 3.14, JA = WT1 , so I ∈ JA . It follows that V U A ∈ WT1 .

(3.6)

Similarly, deﬁne IA = {B ∈ WT1 : BA ∈ WT1 }. Then IA is a weakly closed ideal of WT1 and V U ∈ IA , by (3.6). By part (2) of Deﬁnition 3.14 again, IA = WT1 , so I ∈ IA . Therefore, A ∈ WT1 . Theorem 3.16. If T is an operator of class (SM ), then T and {T } are boundedly reflexive. Proof. By [13, Proposition 6.1 and Theorem 6.2], there exist an operator T0 of class C0 and an absolutely continuous unitary operator U such that T is pseudosimilar to T0 or T0 ⊕ U . Hence T is quasisimilar to T0 or T0 ⊕ U . First, we show {T } is boundedly reﬂexive. By Lemma 3.4, we may assume T = T0 or T = T0 ⊕ U . If T = T0 then {T } is boundedly reﬂexive, by Lemma 3.9. Now, suppose T = T0 ⊕ U . Since every operator of class C0 is also of class C00 , {T0 ⊕ U } = {T0 } ⊕ {U } , by [12, Proposition 3.10]. By Lemma 3.9, {T0 } is boundedly reﬂexive. Since U is a unitary operator, {U } is reﬂexive, thus, it is boundedly reﬂexive. Hence {T } is boundedly reﬂexive. Next, we show T is boundedly reﬂexive. By Lemma 3.15, we may assume T = T0 or T = T0 ⊕ U . Since T0 is boundedly reﬂexive by Corollary 3.10, it remains to show T0 ⊕ U is boundedly reﬂexive. If U has a bilateral shift summand then there exists an invariant subspace M such that U |M is a unilateral shift. By [23, Theorem 1], T0 ⊕ U is reﬂexive, thus, it is boundedly reﬂexive. If U has no bilateral shift summand then WT0 ⊕U = WT0 ⊕WU , by [23, Lemma 5]. Therefore, T0 ⊕ U is boundedly reﬂexive. Corollary 3.17. If T is a weak contraction, then T and {T } are boundedly reflexive.

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Proof. Let T = Ta ⊕ Ts , where Ta is the absolutely continuous part of T and Ts is the singular unitary part of T . Since T is a weak contraction, Ta is an operator of class (SM). Thus, by Theorem 3.16, Ta and {Ta } are boundedly reﬂexive. By [24, Theorem 2.1], we have the decomposition W(T ) = W(Ta ) ⊕ W(Ts ); from this, one can easily check that {Ta ⊕ Ts } = {Ta } ⊕ {Ts } . It follows from the above decompositions that T and {T } are boundedly reﬂexive. Corollary 3.18. If T is a contraction such that I − T ∗ T has finite trace then T is boundedly reflexive. Proof. If T is a weak contraction then T is boundedly reﬂexive, by Corollary 3.17. If T is not a weak contraction then T is reﬂexive, by [11, Theorem 2]; thus T is boundedly reﬂexive. We conclude this section with a proposition about algebraic bounded reﬂexivity of an algebra, not weak∗ closed, generated by an arbitrary Banach space operator. Proposition 3.19. For any T ∈ B(X) and open subset O of C that contains σ(T ), let H(O) be the algebra of all analytic functions on O and A = {ψ(T ) : ψ ∈ H(O)}. Then A is algebraically boundedly reflexive and every subspace of A is also algebraically boundedly reflexive. Proof. Suppose O1 , . . . , Om are the components of O that intersect σ(T ). Let χOk be the indicator function of Ok , Ek be the range of χOk (T ), and Tk = T |Ek . Then the algebra A is the algebraic sum of the algebras Ak = {ψ(Tk ) : ψ ∈ H(Ok )}. It is not hard to check that A is algebraically boundedly reﬂexive if and only if every Ak is algebraically boundedly reﬂexive. If Tk is algebraic, Ak is algebraically boundedly reﬂexive by [19, Corollary 3.7]. If Tk is not algebraic, Ak is algebraically reﬂexive by the proof of [10, Lemma 3]. Hence Ak is algebraically boundedly reﬂexive. By [10, Lemma 3], A has a separating vector. Thus, every subspace of A is algebraically boundedly reﬂexive by Proposition 1.1.

4. Pattern Subspaces In this section we study the pattern subspaces of Mn = Mn (C). Up to similarity, the class of pattern subspaces contains all singly-generated unital algebras, all semi-simple algebras, and all CSL algebras. Identify Mn with the set of all functions f : {1, 2, . . . , n} × {1, 2, . . . , n} → C such that each function f corresponds to the matrix (f (i, j)). A pattern is a disjoint collection P of nonempty subsets of cartesian product {1, 2, . . . , n} × {1, 2, . . . , n}, the pattern subspace SP associated with P is the linear span of the characteristic functions of the sets in P. We say that a pattern subspace of Mn is symmetric if it consists of symmetric matrices and we say S has constant main diagonals if

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every matrix in S has a constant main diagonal. By [5, Lemma 4.1], every pattern subspace of Mn is 2-reﬂexive. Nevertheless, a pattern subspace does not have to be boundedly reﬂexive, see [19, Example 3.4]. However: Theorem 4.1. If S is a symmetric pattern subspace with constant main diagonals then S is boundedly reflexive. We need the following two simple lemmas. The ﬁrst one is a special case of [19, Theorem 5.1]. Lemma 4.2. If dim S ≤ 2 then S is boundedly reflexive. For any U, V ∈ Mn , deﬁne U SV = {U T V : T ∈ S}. The following lemma follows directly from the deﬁnition of bounded reﬂexivity. Lemma 4.3. If U and V are invertible then U SV is boundedly reflexive if and only if S is boundedly reflexive. Proof of Theorem 4.1. Identify Mn as matrices of linear transformations on Cn with respect to the standard orthonormal basis {ei }n1 of Cn . For any matrix T ∈ Mn , we use tij to denote the (i, j)-entry of T . Suppose A ∈ refb (S). First we show A is symmetric and it has a constant main diagonal, i.e. aij = aji and aii = ajj , ∀ i < j. For any i < j, let E be the linear span of {ei , ej }, PE be the orthogonal projection of Cn onto E, then aii aij . PE A|E = aji ajj Since S consists of symmetric matrices,

a b : a, b ∈ C . PE S|E ⊆ b a It follows from A ∈ refb (S) that PE A|E ∈ refb (PE S|E ). By Lemma 4.2, PE S|E is boundedly reﬂexive. Hence, aii aij = PE A|E ∈ refb (PE S|E ) = PE S|E . aji ajj Thus, aij = aji and aii = ajj . To complete the proof, it suﬃces to show that if skm = srs for all S = (sij )n×n ∈ S then akm = ars . Since A and S are symmetric, we only need to show if skm = srs , where k ≤ m and r ≤ s, for all S = (sij )n×n ∈ S then akm = ars . Fix k, m, r, s with k ≤ m and r ≤ s. Assume skm = srs for all S = (sij )n×n ∈ S.

(4.1)

Let F be the linear span of {ek , em , er , es }. First suppose dim F < 4. Since we can interchange the role of skm and srs , there are 5 distinct cases:

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(1) k = r. Suppose m = s, otherwise, there is noting to prove. Let Pk be the orthogonal projection of Cn onto Cek , then akm −ars = akm −aks = Pk A(em −es ) ∈ Pk S(em − es ) = {0}. (2) m = s. By the symmetry of S, smk = ssr for all S = (sij )n×n ∈ S. The same argument as that of (1) gives amk = asr . Now, the symmetry of A implies akm = ars . (3) m = r. Use smk = srs for all S = (sij )n×n ∈ S, then the argument of (2) applies. (4) k = m and r = s. This is clear since A has a constant main diagonal. (5) k = m and r = s. Since all matrices in S have constant main diagonals, assumption (4.1) implies srr = srs for all S = (sij )n×n ∈ S. By (1), arr = ars . Since A has a constant main diagonal, akm = akk = arr = ars . Next, suppose dimF = 4. In this case ek , em , er , es are all distinct, in particular, k < m, r < s. If n = 4, then C4 = F . There are three possibilities: i) (k, m) = (1, 2) and (r, s) = (3, 4): In this case, S has the form:  a b    b a S⊆   c e    d f

c e a b

  d    f   : a, b, c, d, e, f ∈ C .  b    a

(Note that certain entries of S may be all zeros.) Suppose A ∈ refb (S). A is symmetric and has a constant main diagonal, we can write   a11 a12 a13 a14  a12 a11 a23 a24   A=  a13 a23 a11 a34  . a14 a24 a34 a11 Let



1 1  1 −1 U =  0 0 0 0

 0 0 0 0  . 1 1  1 −1

Then U is invertible, U SU ⊆

2a + 2b 0 c+e+d+f c+e−d−f

0 2a − 2b c+d−e−f c+f −d−e

c+e+d+f c+d−e−f 2a + 2b 0

c+e−d−f c+f −d−e 0 2a − 2b

: a, b, c, d, e, f ∈ C ,

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a11  a12 U AU = U   a13 a14 2a + 2a 11

=

a13 + a23 a13 + a23

a12 a11 a23 a24

a13 a23 a11 a34

12

0 + a14 + a24 − a14 − a24

a13 a13

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 a14 a24  U a34  a11

0 2a11 − 2a12 + a14 − a23 − a24 + a24 − a14 − a23

a13 + a23 + a14 + a24 a13 + a14 − a23 − a24 2a11 + 2a34 0

a13 + a23 − a14 − a24 a13 + a24 − a14 − a23 0 2a11 − 2a34

.

Let G be the linear span of {e1 , e3 } and PG be the orthogonal projection of C4 onto G. Then

2a + 2b c+e+d+f PG U SU |G ⊆ : a, b, c, d, e, f ∈ C c+e+d+f 2a + 2b

λ µ = : λ, µ ∈ C , µ λ which is boundedly reﬂexive, by Lemma 4.2. It follows 2a11 + 2a12 a13 + a23 + a14 + a24 a13 + a23 + a14 + a24 2a11 + 2a34

λ µ = PG U AU |G ∈ refb (PG U SU |G ) = PG U SU |G ⊆ : λ, µ ∈ C . µ λ Thus 2a11 + 2a12 = 2a11 + 2a34 , yielding a12 = a34 . ii) (k, m) = (1, 3) and (r, s) = (2, 4): In this case, S has the form:    a b c d       b a e c    S⊆  . : a, b, c, d, e, f ∈ C c e a f        d c f a Let



1  0 V =  0 0 V SV has the form:

 a c    c a V SV ⊆   c e    d f

b e a c

0 0 1 0

0 1 0 0

 0 0  . 0  1

  d    f   : a, b, c, d, e, f ∈ C ,  c    a

which is boundedly reﬂexive by i). Thus S is boundedly reﬂexive, by Lemma 4.3.

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iii) (k, m) = (2, 3) and (r, s) = (1, 4): In this case, S has the form:    a b c d       b a d e    . S⊆  : a, b, c, d, e, f ∈ C c d a f        d e f a  0 1 0 0  1 0 0 0   W =  0 0 1 0 . 0 0 0 1 Then W SW has the same form as that of S in ii). Thus S is boundedly reﬂexive, by ii) and Lemma 4.3. If n > 4, PF be the orthogonal projection of Cn onto F . Thus PF S|F is a symmetric pattern subspace of M4 with constant main diagonals; and PF A|F and PF S|F have the forms of those in i)–iii) above. In fact, with respect to our orthonormal basis with the ﬁxed order {ek , em , er , es }, they have the forms of those in i). The conclusion follows from i). Let



Remark. The “constant main condition in Theorem 4.1 can not be

diagonal” a b : a, b, c ∈ C . It can be veriﬁed directly that S is dropped: Let S = b c 0 1 not boundedly reﬂexive; or let Q = and note that QS is the same as 1 0 the subspace in [19, Example 3.4], which is not boundedly reﬂexive. Therefore S is not boundedly reﬂexive by Lemma 4.3.

5. Direct Sums and Graphs An operator T = T1 ⊕ · · · ⊕ Tm in B(H1 ) ⊕ · · · ⊕ B(Hm ) is said to be supported on Hk if Ti = 0 for all i = k. Proposition 5.1. Suppose S(k) is a weak∗ closed subspace of B(Hk ) and S(k) has a separating vector xk , for each k = 1, . . . , m, where m could be finite or infinite. Let J be a weak∗ closed subspace of S(1) ⊕ · · · ⊕ S(m) and J(k) = {T ∈ J : T is supported on Hk }. Then J is boundedly reflexive if and only if each J(k) is boundedly reflexive. Proof. First, we prove the case when m is ﬁnite. In this case, set x = x1 ⊕ · · ·⊕ xm . Suppose that J is boundedly reﬂexive. Since xk is a separating vector of S(k) , we have that x is a separating vector of S(1) ⊕ · · · ⊕ S(m) . Hence x is a separating vector of J . Since J(k) is a weak∗ closed subspace of J , J(k) is boundedly reﬂexive, by Proposition 1.1. Conversely, suppose A = A1 ⊕ · · · ⊕ Am ∈ refb (J ). Since J is weak∗ closed, there exists an M > 0 such that Ah ∈ JM h, for all h ∈ H1 ⊕· · ·⊕Hm ; in particular,

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Ax ∈ JM x. Thus, Ax = Bx for some B ∈ JM . Replacing A with A − B and M with 2M , if necessary, we may assume Ax = 0. By symmetry, we only need to show A1 = 0. Let A¯ = A2 ⊕ · · · ⊕ Am , x ¯ = x2 ⊕ · · · ⊕ xm . For any y ∈ H1 , there exists an Sy ⊕ Ty ∈ JM such that ¯ y = (Sy ⊕ Ty ) y . (A1 ⊕ A) x ¯ x ¯ ¯ Thus A1 y = Sy y and Ty x ¯ = A¯ x = 0. Since x ¯ is a separating vector for S(2) ⊕ · · · ⊕ S(m) , Ty = 0. Hence Sy ⊕ 0 ∈ J(1) and A1 ⊕ 0 ∈ refb (J(1) ). The bounded reﬂexivity of J(1) implies A1 ⊕ 0 ∈ J(1) . Since x1 is a separating vector of S(1) , we have A1 = 0. ∞ ⊕λk xk , where λk are nonzero scalars with ||x|| < ∞. If m = ∞, set x = k=1

The rest is similar to the case when m is ﬁnite.

Proposition 5.2. Let S be a boundedly reflexive subspace of B(H1 ) such that S has a separating vector. Suppose π: S → B(H2 ) is a continuous linear map with respect to the weak∗ topologies. Then the graph {S ⊕ π(S) : S ∈ S} is boundedly reflexive in B(H1 ⊕ H2 ). Proof. Let A = {S ⊕ π(S) : S ∈ S} and A ∈ refb (A). Then there exists an M > 0 such that Ax ∈ [AM x], for any x ∈ H1 ⊕ H2 . Since S is boundedly reﬂexive, it is weak∗ closed, by [19, Corollary 2.9]. Since ρ is weak∗ continuous, we have that A is weak∗ closed. Hence (5.1) Ax ∈ [AM x] = AM x. Since A ⊆ S ⊕ B(H2 ) and S ⊕ B(H2 ) is boundedly reﬂexive, A has the form A = S ⊕ C, where S ∈ S. Replacing A with A − (S ⊕ π(S)), if necessary, we assume S = 0. It remains to show C = 0. Let x1 be a separating vector of S. For any y ∈ H2 , by (5.1), there exists an S1 ∈ S such that x1 x1 (0 ⊕ C) = (S1 ⊕ π(S1 )) . y y Since x1 is a separating vector of S, we have S1 = 0 and Cy = 0. Since y is arbitrary, C = 0. Acknowledgment The authors wish to thank the referee for useful suggestions.

References [1] C. Apostol Operators quasisimilar to a normal operator Proc. Amer. Math. Soc. 53 (1975), 104-106 [2] E. Azoﬀ and Shehada From algebras of normal operators to intersecting hyperplanes Proc. Sympos. Pure. Math. 51 (1990), 11-16

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[3] E. Azoﬀ and Shehada On separation by families of linear functionals J. Funct. Anal. 51 (1991), 96-116 [4] H. Bercovici Operator Theory and Arithmetic in H∞ Mathematical Surveys and Monographs, 26. American Mathematical Society, Providence, RI, 1988 [5] K. Choi, Don Hadwin and B. Kim General reflexivity and pattern subspaces Linear and Multilinear Algebra 47 (2000), 259-280 [6] W. Gong, D. Larson and W. Wogen Two results on separating vectors Indiana Univ. Math. J. 43 (1994), 1159-1165 [7] Don. Hadwin A general view of reflexivity Trans. Amer. Math. Soc. 344 (1994), 325-260 [8] Don Hadwin and C. Laurie Reflexive binormal operators J. Funct. Anal. 123 (1994), 99-108 [9] D. Hadwin and E. Nordgren Subalgebras of reflexive algebras J. Operator Theory 7 (1982) 3-23 [10] Don Hadwin and S. C. Ong On algebraic and para-reflexivity J. Operator Theory 17 (1987), 23-31 [11] V. V. Kapustin A criterion for reflexivity of contractions with defect operator of Hilbert-Schmidt class Soviet Dokl. 43 (1991), 919-922 [12] V. V. Kapustin and A. V. Lipin Operator algebras and lattices of invariant subspaces I J. Soviet Math. 61 (1992), 1963-1981 [13] V. V. Kapustin and A. V. Lipin Operator algebras and lattices of invariant subspaces II J. Math. Sci. (New York) 71 (1994), 2240-2262 [14] D. Larson Reflexivity, algebraic reflexivity and linear interpolation Amer. J. Math. Soc. 110 (1988), 283–299 [15] D. Larson and W. Wogen Reflexivity properties of T ⊕ 0 J. Funct. Anal. 92 (1990), 448-467 [16] D. Larson and W. Wogen Extensions of normal operators Integral Equations Oper. Theory 20 (1994) 325-334 [17] J. Li A remark on complete positivity of elementary operators Integral Equations Operator Theory 28 (1997), 110–115 [18] J. Li and Z. Pan Reflexivity of a finite dimensional subspace of operators J. Operator Theory 46 (2001), 381-389 [19] J. Li and Z. Pan Bounded reflexivity of operator subspaces Integral Equations Oper. Theory 48 (2004), 41-59 [20] Z. Pan Commutants and hyporeflexive closure of operators J. Operator Theory 53 (2005), 367-380 [21] H. Shehada Reflexivity of convex subsets of L(H) and subspaces of lp Internat. J. Math. Math. Sci. 14 (1991), 55–67 [22] B. Sz.-Nagy and C. Foias Harmonic Analysis of Operators on Hilbert Space NorthHolland, Amsterdam, 1970 [23] K. Takahashi On the reflexivivity of contractions with isometric parts Acta. Sci. Math. 53 (1989), 147-152 [24] M. Zajac On the singular unitary part on a contraction Rev. Roumaine Math. Pures Appl. 75 (1990), 379-384

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Don Hadwin Department of Mathematics University of New Hampshire Durham, NH 03824 USA e-mail: donmath.unh.edu Jiankui Li Department of Pure Mathematics East China University of Science and Technology Shanghai 200237 P.R. China e-mail: jiankuiliyahoo.com Zhidong Pan Department of Mathematics Saginaw Valley State University University Center, MI 48710 USA e-mail: [email protected] Submitted: December 12, 2005 Revised: June 15, 2006

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Integr. equ. oper. theory 57 (2007), 491–512 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040491-22, published online December 26, 2006 DOI 10.1007/s00020-006-1471-z

Integral Equations and Operator Theory

Closed Projections and Peak Interpolation for Operator Algebras Damon M. Hay Abstract. The closed one-sided ideals of a C ∗ -algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ∗ -algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B ∗∗ which also lies in A⊥⊥ . Although this seems quite natural, the proof requires a set of new techniques which may be viewed as a noncommutative version of the subject of peak interpolation from the theory of function spaces. Thus, the right ideals with left approximate identity are closely related to a type of peaking phenomena in the algebra. In this direction, we introduce a class of closed projections which generalizes the notion of a peak set in the theory of uniform algebras to the world of operator algebras and operator spaces. Mathematics Subject Classification (2000). Primary 46H10, 46L85, 47L30; Secondary 32T40. Keywords. Peak sets, noncommutative peak sets, peak interpolation, nonselfadjoint operator algebras, ideals.

1. Introduction Let K be a compact Hausdorﬀ space and let C(K) denote the C ∗ -algebra of all complex-valued continuous functions on K. It is well known that closed ideals in C(K) consist of all functions which vanish on a ﬁxed closed subset of K. If instead, A is a uniform algebra contained in C(K), then a closed subspace J of A is a closed ideal with contractive approximate identity if and only if it consists of all functions which vanish on a ‘p-set’ for A. A subset E of K is said to be a peak set for A if there exists a function f in A such that f (x) = 1 for all x ∈ E and |f (x)| < 1 for all x ∈ E c . A p-set is the intersection of a family of peak sets. This characterization of closed ideals with contractive approximate identity in terms of

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p-sets can be attributed to Hirsberg [19] and Smith (see [26] and the references therein). The p-sets for a uniform algebra A were characterized by Glicksberg [16] as those closed subsets E ⊂ K such that µ ∈ A⊥ implies µE ∈ A⊥ . Here µE denotes the restriction of µ to E. See [15], [16], or [20] for more information on peak sets. For a general C ∗ -algebra A, a closed right ideal of A consists of the elements a in A for which qa = 0 for a closed projection q in the second dual of A. In other words, a subspace J is a right ideal of A if and only if J = (1 − q)A∗∗ ∩ A, for a closed projection q in A∗∗ . Here we are viewing A as being canonically embedded in its second dual, which is a W ∗ -algebra. In fact, 1 − q will be a weak*limit point for any left contractive approximate identity of J. Indeed all closed projections arise in this manner. Turning to the nonselfadjoint case, let A be a closed subalgebra of a unital C ∗ -algebra B, such that A contains the identity of B. We characterize the right ideals of A with left contractive approximate identity as those subspaces J of the form J = (1 − q)A∗∗ ∩ A, for a closed (with respect to B ∗∗ ) projection q in A∗∗ . However natural this may appear, the tools available in the selfadjoint theory are not applicable here. Thus, a portion of this paper develops some technical tools from which this characterization follows. Incidentally, these generalize some peak interpolation results in the theory of uniform algebras. The above mentioned characterization is a reﬁnement of the characterization in [6], which is in terms of right M -ideals. In particular, it appears to open up a new area in the theory of nonselfadjoint operator algebras, allowing for the generalization of certain important parts of the theory of C ∗ -algebras. This will be explored more fully in the sequel [7] where, for example, we apply the main result of this paper to develop a theory of hereditary subalgebras of not necessarily selfadjoint operator algebras. As is the case in the selfadjoint theory, we demonstrate that these hereditary subalgebras are connected to the facial structure of the state space. Additionally, we also give a solution there to a more than ten year old problem in the theory of operator modules. In our noncommutative setting, the peak and p-sets described above are replaced with a certain class of projections in the second dual of B, which we call the peak or p-projections for A. In the commutative case this class of projections can be identiﬁed with the characteristic functions of peak or p-sets for A. When A = B, the p-projections are exactly the closed projections in A∗∗ . The theory of these projections brings another tool from the classical theory to the world of operator spaces. The paper is organized as follows. In Section 2 we introduce the notation and discuss some background and preliminary results. In particular, we discuss the noncommutative topology of open and closed projections. Section 3 generalizes some interpolation results from the theory of function spaces to operator spaces, which will be used in Sections 4 and 5. Section 4 contains the main theorem and

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its proof. Finally, in Section 5 we look at closed projections in the weak*-closure of an operator algebra from the perspective of ‘peak phenomena.’ Acknowledgments. We ﬁrst thank our Ph.D. advisor, David Blecher, for the initial inspiration to pursue this project and for his continuous support and suggestions along the way. We also thank him for pointing out, and helping to correct, some errors in the proof of Proposition 3.1 in an earlier version. We also extend our thanks to Professor Charles Akemann for clarifying a number of points regarding open and closed projections. Finally, we thank the referee for helpful comments and recommendations.

2. Open and closed projections and preliminary results The theory of operator spaces and completely bounded maps has long been recognized as the appropriate setting for studying many problems in operator algebras. Basics on operator spaces may be found in [8], [13], [22], and [24]. We will make use of the following lemma which gives a criteria for when a completely bounded map is a complete isometry. Lemma 2.1. Let X be an operator space and Y a subspace of another operator space. Suppose T : X → Y is a one-to-one and surjective completely bounded map such that T ∗ is a complete isometry. Then T is a complete isometry and Y is closed. Proof. Let Z be the closure of Y and deﬁne R to be the same as T except with range Z. Since Z ∗ = Y ∗ , we have that R∗ : Z ∗ → X ∗ is simply T ∗ which is oneto-one and has closed range. By VI.6.3 in [10] this implies that R is onto. Since R is onto, Y = Z and R = T , and by the open mapping theorem T is bicontinuous. Hence, given ϕ ∈ X ∗ , then ψ := ϕ ◦ T −1 ∈ Y ∗ satisﬁes T ∗ ψ = ϕ, showing that T ∗ is surjective and thus, since it is also completely isometric, T ∗∗ is a complete isometry. Viewing X and Y as being canonically embedded in their second duals, T is just the restriction of T ∗∗ to X. Hence, T is also a complete isometry. Throughout this paper, B will denote a unital C ∗ -algebra and A will denote a unital subalgebra of B, for which by unital we mean that A contains the unit of B. We view A and B as being canonically embedded into the second dual B ∗∗ of B via the canonical isometry. The second dual of B, B ∗∗ , is a W ∗ -algebra. By the state space of B, which we denote S(B), we mean the set of positive functionals on B which have norm one. Each functional ϕ of B extends uniquely to a weak*continuous, or normal, functional on B ∗∗ , which we again denote by ϕ. Also, we denote the unit in B by 1B , or more often, simply by 1. By a projection in B or B ∗∗ we mean an orthogonal projection. The meet or intersection of any two projections p and q can be given abstractly as p ∧ q = lim (pq)n , n→∞

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where this limit is taken in the weak* topology. Similarly, the join is given by p ∨ q = lim 1 − (1 − p − q + pq)n . n→∞

Let M be a (not necessarily selfadjoint) weak*-closed unital subalgebra of a W ∗ algebra and suppose that p and q are projections in M . Then by the formula for p ∧ q above, p ∧ q is also in M . By induction, this extends to the meet of any ﬁnite collection of projections in M . More generally, if {pα } is any collection of projections, then ∧pα is the weak*-limit of the net of meets of ﬁnite subcollections of {pα }, each of which is in M . Thus ∧pα is also in M . Similarly, a join of projections in M is also in M . Now let π : M → B(H) be a weak*-continuous homomorphism of M into the bounded linear operators on a Hilbert space H. If p and q are projections in M , then π((pq)n ) = (π(p)π(q))n for each n, and so we have π(p ∧ q) = π(p) ∧ π(q). This clearly generalizes to meets of ﬁnitely many projections. By approximating by such ﬁnite meets, this in turn generalizes to arbitrary meets of projections. Similar statements apply to joins of projections. A projection p ∈ B ∗∗ is said to be open if it is the weak*-limit of an increasing net (bt ) of elements in B with 0 ≤ bt ≤ 1. A projection q ∈ B ∗∗ is said to be closed if 1−q is open. It is clear that a closed projection is the weak*-limit of a decreasing net of positive elements in B. It is well known that a projection p in B ∗∗ is open if and only if it is the support of a left (respectively, right) ideal in B. That is, there exists a left (respectively, right) ideal J in B such that J = B ∗∗ p ∩ B (respectively, J = pB ∗∗ ∩ B). In this case, the weak*-closure of J in B ∗∗ is B ∗∗ p (respectively, pB ∗∗ ). Moreover, p is a weak*-limit point of any increasing right contractive approximate identity for J. Lemma 2.2. If p ∈ B ∗∗ is a projection which is a weak* limit of a net (et ) in B such that et p = et , then p is open. Proof. Let J be the set of all b ∈ B such that bp = b. Then J contains (et ) and so p is in J ⊥⊥ , which is a weak*-closed left ideal of B ∗∗ . Thus B ∗∗ p ⊂ J ⊥⊥ , but also J ⊥⊥ ⊂ B ∗∗ p, so that J ⊥⊥ = B ∗∗ p. However, J = B ∗∗ p ∩ B, so that p is the support of a closed left ideal, making it an open projection. A similar argument using right ideals also holds if pet = et instead. In the case that B is commutative, open and closed projections correspond to characteristic functions of open and closed sets, respectively. It is this collection of open and closed projections which will act as a kind of substitute for topological arguments in the noncommutative situation. We now list, most without proof, some basic facts regarding these open and closed projections. Many of these facts can be found in Akemann’s papers [1] and [2], and some may also be found in [23] and [17]. The join of any collection of open projections is again an open projection. Hence, the meet of any collection of closed projections is again a closed projection. However, in contrast to the commutative situation, joins of ﬁnitely many closed projections are not necessarily closed (see [1]). For a general C ∗ -algebra B,

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the presence of a unit guarantees a kind of noncommutative compactness. That is, if q is a closed projection, given any collection of open projections {pα } such that q ≤ α pα , then there exists a ﬁnite subcollection {pα1 , . . . , pαk } such that k q ≤ i=1 pαi (see Proposition II.10 in [1]). We will refer to this as the ‘compactness property.’ A type of regularity also holds with respect to open and closed projections. Namely, any closed projection is the meet of all open projections dominating it. The following was communicated to us by Akemann. Proposition 2.3. Let q be a closed projection in B ∗∗ . Then q = {u | u ≥ q and open}. Proof. Assume that B and B ∗∗ are represented in the universal representation of B. Since q is closed we may ﬁnd an increasing net (at ) in the selfadjoint part of B such that 1 − at q weak* and (1 − at )q = q. Using the Borel functional calculus, for each t let rt = χ( 12 ,∞) (1 − at ), where χ( 12 ,∞) is the characteristic function of the open interval ( 12 , ∞). Then necessarily rt is an open projection such that 2(1 − at ) ≥ rt . Furthermore, we claim that each rt dominates q. To prove this claim, ﬁx t and let {fn } be an increasing sequence of positive continuous functions on the spectrum of 1 − at which converges point-wise to χ( 12 ,∞) and is such that fn (1) = 1 for each n. Now ﬁx n and suppose ξ ∈ Ran q is of norm one. Then (1 − at )ξ = (1 − at )qξ = qξ = ξ. So for any polynomial R, we have R(1−at )ξ = R(1)ξ. Let Rk be a sequence of polynomials converging uniformly to fn . Then fn (1 − at )ξ, ξ = lim Rk (1)ξ, ξ = lim Rk (1) = fn (1) = 1. k

k

By the converse to the Cauchy-Schwarz inequality, we have fn (1 − at )ξ = ξ for all ξ ∈ Ran q, which is to say that fn (1 − at )q = q. Hence χ( 12 ,∞) (1 − at )q = q, and so rt ≥ q. Let r0 = t rt and suppose that r0 does not equal q. Then there exists a state ϕ ∈ S(B) such that ϕ(r0 − q) = 1. This forces ϕ(r0 ) = 1 and ϕ(q) = 0 since r0 − q ≥ 0. Thus ϕ (2(1 − at )) → 0. However, since ϕ(r0 ) = 1 and rt ≥ r0 , it must be that ϕ(rt ) = 1. Applying ϕ to the inequality rt ≤ 2(1 − at ) and taking the weak* limit, we get 1 ≤ 0. Hence, r0 = q which proves the result. Finally, one of the most important results in basic topology is Urysohn’s lemma. Akemann has extended this result to closed projections: Theorem 2.4 ([2]). Let p and q be closed projections in B ∗∗ such that pq = 0. Then there exists an element a in B, 0 ≤ a ≤ 1, such that ap = p and aq = 0.

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We will often be working with closed projections in B ∗∗ which lie in the weak*-closure of A in B ∗∗ . The following gives some equivalent conditions for this. Lemma 2.5. Let A be a unital subalgebra of B and let q ∈ B ∗∗ be a projection. The following are equivalent: w∗

1. q ∈ A , 2. q ∈ A⊥⊥ , 3. A⊥ ⊂ (qA)⊥ . Proof. The equivalence of (1) and (2) is a standard result of functional analysis. w∗

w∗

Suppose (3) holds. Then ((qA)⊥ )⊥ ⊂ A⊥⊥ . However, ((qA)⊥ )⊥ = qA = qA which must contain q since A is unital. Hence, (2) holds. Now assume (2). By hypothesis, ψ(q) = 0 for all ψ ∈ A⊥ . Let ϕ ∈ A⊥ . Then for each a ∈ A, ϕ(·a) ∈ A⊥ . Thus ϕ(qa) = 0 for all a ∈ A. Hence ϕ ∈ (qA)⊥ . A class of operators which will play a role here are the completely non-unitary, or c.n.u., operators on a Hilbert space H. A contraction T is said to be completely non-unitary if there exists no reducing subspace for T on which T acts unitarily. It is well known that if T is completely non-unitary, then T n → 0 in the weak operator topology on B(H) as n → ∞. See [14] and [21] for details. If Bsa denotes the selfadjoint part of B, then Kadison’s ‘function representation’ says that Bsa may be represented as continuous aﬃne functions on S(B) ∗∗ via an order preserving linear isometry which extends weak* continuously to Bsa , ∗∗ is represented as bounded aﬃne functions on S(B). We in such a way that Bsa ∗∗ say that an element b of Bsa is lower semi-continuous if its image under this representation is a lower semi-continuous function on S(B) ([23]). Lemma 2.6. Let b be a positive, lower semi-continuous contraction in B ∗∗ and suppose ϕ0 (b) = 0 for some ϕ0 ∈ S(B). Then there exists a pure state of B which is zero at b. Proof. Let K = {ϕ ∈ S(B) | ϕ(b) = 0}. The set K is nonempty by hypothesis, and since φ and b are positive, we also have that K = {ϕ ∈ S(B) | ϕ(b) ≤ 0}. Thus K is the complement of the set {ϕ ∈ S(B) | ϕ(b) > 0} which is open in the weak* topology by the semi-continuity property of b. Thus K is weak* closed in S(B) and hence weak* compact. It is also convex by a straight-forward calculation. Thus, K is well supplied with extreme points by the Krein-Milman theorem. Now suppose that ϕ1 , ϕ2 ∈ S(B), λ is a scalar in (0, 1) and that λϕ1 + (1 − λ)ϕ2 ∈ K. Then λϕ1 (b) + (1 − λ)ϕ2 (b) = 0. However, by positivity, this forces ϕ1 (b) = ϕ2 (b) = 0 and so ϕ1 and ϕ2 are in K. In other words, K is a face of S(B) and hence must contain an extreme point of S(B). However, the extreme points of S(B) are the pure states of B.

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3. Noncommutative peak interpolation The following sequence of propositions and lemmas are the keys to the main result and generalize some classical results from the theory of function spaces (see Section II.12 of [15]). Proposition 3.1. Let X be a closed subspace of B and let q ∈ B ∗∗ be a closed projection such that ϕ ∈ (qX)⊥ for all ϕ ∈ X ⊥ . Let I = {x ∈ X : qx = 0}. Then qX is completely isometric to X/I via the map x + I → qx. Similarly, if I is defined to be {x ∈ X : xq = 0}, then Xq is completely isometric to X/I via the map x + I → xq. Proof. We will be using standard operator space duality theory, as may be found in [8], for example. First note that I is the kernel of the completely contractive map x → qx on X, so that this map factors through the quotient X/I: S

T

X → X/I → qX, where S is the natural quotient map and T is the induced linear isomorphism. Taking adjoints, we have (T ◦ S)∗ = S ∗ ◦ T ∗ and if ϕ ∈ (qX)∗ and x ∈ X, then (S ∗ ◦ T ∗ )(ϕ)(x) = S ∗ (T ∗ ϕ)(x) = (T ∗ ϕ)(Sx) = ϕ(T Sx) = ϕ(T (x + I)) = ϕ(qx), so that S ∗ ◦ T ∗ is given by ϕ → ϕ(q·), for each ϕ ∈ (qX)∗ . Identifying (qX)∗ with (qB)∗ /(qX)⊥ and X ∗ with B ∗ /X ⊥ , the map S ∗ ◦ T ∗ takes an element ϕ + (qX)⊥ to the element ϕ(q·) + X ⊥ . To show that T is a complete isometry, by Lemma 2.1 it suﬃces to show that T ∗ is a complete isometry, since T is one-to-one, surjective, and completely bounded. Since S ∗ is completely contractive, if S ∗ ◦ T ∗ is completely isometric, then ϕ = (S ∗ ◦ T ∗ )(ϕ) ≤ T ∗ ϕ ≤ ϕ, for ϕ ∈ (qX)∗ . Similar statements also hold for each matrix level. Hence, if S ∗ ◦ T ∗ is completely isometric, then so is T ∗ . Thus, in order to show that T ∗ is completely isometric, it is suﬃcient to show that S ∗ ◦ T ∗ is a complete isometry. Note that since T ◦ S is completely contractive, so is S ∗ ◦ T ∗. We let (et ) be a decreasing net in the unit ball of B, such that et → q weak* and qet = q for all t. Let ϕ ∈ (qB)∗ and ψ ∈ X ⊥ . Then ψ ∈ (qX)⊥ ⊂ (qX)⊥ . If J is the right ideal in B supported by 1 − q, then for qb ∈ Ball(qB), we have qb = b + J. Since right ideals are proximinal in a C ∗ -algebra, it follows that there exists a ∈ J such that qb = b + J = b + a. Since q(b + a) = qb and b + a ≤ 1, by replacing b with b + a it follows that ϕ + ψ(qB)∗

= sup{|ϕ(qb) + ψ(qb)| : b ∈ Ball(B)}.

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However, for b ∈ Ball(B), we have |ϕ(qb) + ψ(qb)|

= lim |ϕ(qet b) + ψ(et b)| t

≤ ϕ(q·) + ψB ∗ et b ≤ ϕ(q·) + ψB ∗ . Hence, ϕ + ψ(qB)∗ ≤ ϕ(q·) + ψB ∗ , and, thus, ϕ + (qX)⊥ ≤ ϕ(q·) + ψB ∗ . Now taking the inﬁmum over all ψ ∈ X ⊥ , we get ϕ + (qX)⊥ ≤ ϕ(q·) + X ⊥ . The matricial case is almost identical, using operator space duality principles, and is left to the reader to ﬁll in the details. The last statement of the proposition follows by a completely analogous proof. Since qX ⊂ qB and qB can be identiﬁed with a quotient B/J, where J is the right ideal of B corresponding to q, then the result above shows that the set {x + J|x ∈ X} is closed in B/J. Proposition 3.2. Let X be a closed subspace of B and let q ∈ B ∗∗ be a closed projection such that ϕ ∈ (qX)⊥ whenever ϕ ∈ X ⊥ . Let p be a strictly positive element in B and let a ∈ X such that a∗ qa ≤ p. Given > 0 there exists b ∈ X such that qb = qa and b∗ b ≤ p + 1B . Proof. First assume p = 1. Let I = {x ∈ X : qx = 0} and let δ > 0 such that 2δ + δ 2 < . Then by the previous lemma there exists an h ∈ I such that a + h ≤ qa + δ. Let b = a + h and note that qb = qa. Also, since a∗ qa ≤ 1, we have qa ≤ 1. Then b∗ b

≤

b∗ b1B = b21B

≤

(qa + δ)2 1B ≤ (1 + δ)2 1B

=

(1 + 2δ + δ 2 )1B

≤

(1 + )1B = p + 1B .

In the case that p is not necessarily 1, note that a∗ qa ≤ p is equivalent to p−1/2 a∗ qap−1/2 ≤ 1. Furthermore, note that p−1/2 a∗ qap−1/2 = (ap−1/2 )∗ q(ap−1/2 ). Now suppose that ϕ ∈ (Xp−1/2 )⊥ ⊂ B ∗ . Then ϕ(·p−1/2 ) ∈ X ⊥ . Hence, by hypothesis, ϕ(·p−1/2 ) ∈ (qX)⊥ , and thus ϕ ∈ (qXp−1/2 )⊥ . So by the p = 1 case, there exists bp−1/2 ∈ Xp−1/2 such that qbp−1/2 = qap−1/2 , and p−1/2 b∗ bp−1/2 ≤ 1 + p−1. Pre- and post- multiplying by p1/2 yields b∗ b ≤ p + p−1 p ≤ p + .

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Proposition 3.3. Let X be a unital subspace of B and suppose q is a closed projection in B ∗∗ such that ϕ ∈ (qX)⊥ for every ϕ ∈ X ⊥ . Let p be a strictly positive element in B. If a ∈ X with a∗ qa ≤ p, then there exists b in the unit ball of {x ∈ X : qx = qa}⊥⊥ such that b∗ b ≤ p. Moreover, qb = qa. Proof. As in the previous proposition, we ﬁrst show that the lemma holds in the case p = 1. Suppose p = 1. By the previous lemma, for each n > 1 there is a bn ∈ X such that qbn = qa and b∗n bn ≤ 1 + n1 . By the weak* compactness of Ball(X ⊥⊥ ), (bn ) has a weak* limit point b in Ball(X ⊥⊥ ). Thus b∗ b ≤ 1. Let (bnt ) be a subnet of (bn ) converging to b. Then by weak* continuity we must also have that qb = qa. For general p, as before we note that a∗ qa ≤ p is equivalent to p−1/2 a∗ qap−1/2 ≤ 1, and that p−1/2 a∗ qap−1/2 = (ap−1/2 )∗ q(ap−1/2 ). Now let ϕ ∈ (Xp−1/2 )⊥ . Then ϕ(·p−1/2 ) ∈ X ⊥ . Thus, by hypothesis, ϕ(·p−1/2 ) ∈ (qX)⊥ , and so ϕ ∈ (qXp−1/2 )⊥ . Hence, by the p = 1 case, there exists bp−1/2 ∈ (Xp−1/2 )⊥⊥ = X ⊥⊥ p−1/2 such that qbp−1/2 = qap−1/2 , and p−1/2 b∗ bp−1/2 ≤ 1. We pre- and post- multiply by p1/2 to get b∗ b ≤ p.

Remarks. 1.) The preceding two propositions have matricial variants. For instance, the conclusion to Proposition 3.2 can be generalized to read ‘for every strictly positive contraction p ∈ Mn (B) and a ∈ Mn (X) with a∗ (In ⊗ q)a ≤ p, there exists b ∈ Mn (X) such that (In ⊗ q)a = (In ⊗ q)b and b∗ b ≤ p + In .’ Here In denotes the identity matrix in Mn . 2.) If X is a reﬂexive unital subspace of B and q is such that ϕ ∈ (qX)⊥ for every ϕ ∈ X ⊥ , then for every strictly positive contraction p ∈ B with q ≤ p, there exists a ∈ Ball(X) such that qa = q and a∗ a ≤ p. Variants of Propositions 3.2 and 3.3 in the commutative case are related to further results in the subject of peak interpolation from the theory of function algebras (see e.g. [15]). For example, we have the following proposition. Proposition 3.4. Let q ∈ B ∗∗ be a closed projection such that ϕ ∈ A⊥ implies that ϕ(q·) = 0, where we view ϕ(q·) as an element of (qB)∗ . If > 0 and p is a strictly positive element of B, then given a ∈ B with a∗ qa ≤ p, there exists b ∈ A such that qb = qa and b∗ b ≤ p + . Proof. Suppose that ψ0 ∈ (qA)⊥ , viewing (qA)⊥ as a subspace of (qB)∗ . Now deﬁne ψ ∈ B ∗ by ψ(b) = ψ0 (qb) for all b ∈ B. Then ψ ∈ A⊥ and, so ψ(q·) = 0 as a functional on qB, by hypothesis. Hence, we also have ψ0 (qb) = 0 for all b ∈ A. Thus (qA)⊥ = {0} and so qA is norm dense in qB. However, by Proposition 3.1, qA is norm closed, and so qA = qB. Thus, given > 0, p ∈ B strictly positive,

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and a ∈ B with a∗ qa ≤ p, by Proposition 3.2 there exists b ∈ A such that qb = qa and b∗ b ≤ p + . We close this section with the deﬁnition of what it means for a contraction to ‘peak’ at a projection, and we give several lemmas which are required in the remainder of the paper. The ﬁrst lemma describes the weak* limits of powers of certain types of contractions. The last one is a useful tool for generating ‘peak projections,’ which we now deﬁne. Definition 3.5. Let a ∈ B ∗∗ be a contraction. We say that a peaks at a projection q ∈ B ∗∗ if a and q satisfy the following conditions: 1. aq = q (or equivalently, qa = q) and 2. ϕ(a∗ a) < 1 for all ϕ ∈ S(B) such that ϕ(q) = 0. In the case that a ∈ B and q is closed, we refer to q as a peak projection. We will see in Theorem 5.1 that the second condition in the deﬁnition has many equivalent variations. Lemma 3.6. If a contraction a ∈ B ∗∗ peaks at a projection q ∈ B ∗∗ , then (an ) and ((a∗ a)n ) converge weak* to q as n → ∞. Consequently, if a ∈ B, then q is a closed projection. Proof. We have that B is contained non-degenerately in B(H), where H is the Hilbert space associated with the universal representation of B. We may also view B ∗∗ as a von Neumann algebra in B(H). Let K be the range of q so that B(H) = B(K ⊕ K ⊥ ). With respect to this decomposition we may write IK 0 , a= 0 x where IK is the identity operator on K and x ∈ B(K ⊥ ). Let ξ ∈ K ⊥ be a unit vector. Let ϕ be the vector state corresponding to 0 ⊕ ξ. Then ϕ(q) = 0, so that ϕ(a∗ a) < 1. Thus xξ, xξ < 1 for any unit vector ξ in K ⊥ . If x had a reducing subspace on which x acted unitarily, then there would be a unit vector η ∈ K ⊥ such that xη, xη = 1, which is a contradiction. Thus x must be completely nonunitary. This implies that xn → 0 in the weak operator topology as n → ∞. Now let η1 and η2 be vectors in K and let ξ1 and ξ2 be vectors in K ⊥ . Then, an (η1 ⊕ ξ1 ), (η2 ⊕ ξ2 ) = =

(η1 ⊕ xn ξ1 ), (η2 ⊕ ξ1 ) η1 , η2 + xn ξ1 , ξ2 → η1 , η2 .

Thus (an ) converges to q. In order to show that ((a∗ a)n ) also converges to q, it will suﬃce to show that x∗ x is also c.n.u. For in this case, the same argument as above will also work. Suppose x∗ x has a reducing subspace V on which it acts unitarily and let ξ ∈ V be a unit vector. Then xξ2 = x∗ xξ, ξ < 1, which contradicts x∗ x acting unitarily on V . In the case that a ∈ B , then (a∗ a)n is a decreasing sequence of selfadjoint elements of B, and so q must be closed.

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It should be noted that peak projections are the same thing as the elements u(x) of Edwards and R¨ uttiman ([11], [12]). Given a contraction x ∈ B (or more generally, in a ternary ring of operators), u(x) is deﬁned to be the weak* limit of the sequence of ‘odd’ powers xx∗ x, xx∗ xx∗ x, xx∗ xx∗ xx∗ x, etc. Lemma 3.7. Let a ∈ B ∗∗ be a contraction such that aq = q for a closed projection q ∈ B ∗∗ . If b := 12 (a + 1), then b peaks at a projection r ∈ B ∗∗ such that q ≤ r, and 1 − r is the support of the weak* closed left ideal generated by 1 − b in B ∗∗ . Moreover, if a ∈ B, then r is a peak projection and 1 − r is the support of the closed left ideal generated by 1 − b in B. Proof. With b = 12 (a + 1) and aq = a, it is clear that bq = q. This implies that (1−b)(1−q) = 1−b. Hence, 1−b ∈ B ∗∗ (1−q) and this contains the intersection, J, of all weak* closed left ideals in B ∗∗ containing 1 − b. Let p ∈ B ∗∗ be the support projection for J. Then (1 − b)(1 − p) = 0, so that b(1 − p) = 1 − p. Letting r = 1 − p we have that br = r. Now suppose that ϕ is a state of B such that ϕ(r) = 0. Then surely ϕ(b∗ b) ≤ 1, but suppose that ϕ(b∗ b) = 1. Then ϕ(a∗ a) + 2Re ϕ(a) + 1 = 4, which forces ϕ(a∗ a) = ϕ(a) = 1, and hence, ϕ(b) = 1. Now let L denote the left kernel in B ∗∗ associated with ϕ. Then L is a weak* closed left ideal and we claim that 1 − b ∈ L. To see this, note that ϕ((1 − b)∗ (1 − b)) = ϕ(b∗ b) − 2Re ϕ(b) + 1 = 0. Hence, 1 − b ∈ L and consequently, J ⊂ L. If pL denotes the support projection of L, then p ≤ pL , and so 1 − pL ≤ 1 − p. Applying ϕ to this last inequality yields 1 ≤ 0, an obvious contradiction. Thus we conclude that ϕ(b∗ b) < 1. Furthermore, since J ⊂ B ∗∗ (1 − q), it follows that q ≤ 1 − p = r. In the case that a ∈ B, the closed left ideal generated by 1 − b in B is J ∩ B and the weak* closure of J ∩ B is J. It follows that the support of J is 1 − r, so that r is closed.

4. Right ideals with left contractive approximate identity We now have everything needed to prove the main theorem. The following theorem gives the diﬃcult direction. Theorem 4.1. Let A be a unital subalgebra of B. Let q ∈ B ∗∗ be a closed projection such that q ∈ A⊥⊥ . Then 1 − q is in the weak* closure of the right ideal J = {a ∈ A : (1 − q)a = a}. Proof. First recall that A⊥ ⊂ (qA)⊥ is equivalent to q ∈ A⊥⊥ by Lemma 2.5. Thus q satisﬁes the hypotheses of Propositions 3.2 and 3.3. Let u be an open projection dominating q. Then, by the noncommutative Urysohn’s lemma there exists p ∈ B, 0 ≤ p ≤ 1, such that pq = q and p(1 − u) = 0. For each integer n ≥ 0, let pn =

1 n p+ . n+1 n+1

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Then pn is strictly positive and pn q = q, so that q ≤ pn . Each pn also has the 1 property that pn (1 − u) = n+1 (1 − u). By Proposition 3.3, for each n there exists ∗∗ an ∈ Ball(A ) such that qan = q and a∗n an ≤ pn . The net (an ) is contained in the unit ball of A∗∗ , which is weak* compact. Let (ant ) be a subnet converging to an element a in the unit ball of A∗∗ . Since (1 − u)a∗nt ant (1 − u) ≤ nt1+1 , the net (1 − u)a∗nt ant (1 − u) converges to zero in norm, and hence, by the C ∗ -identity, ant (1 − u) also converges to zero in norm. It follows that a(1 − u) = 0 and au = a. Similarly, qa = q. Let b = 12 (a + 1). By Lemmas 3.7 and 3.6 we know that there is a projection r ∈ B ∗∗ with r ≥ q such that bk → r weak*. We now show that r ≤ u. To do this, we ﬁrst observe that 1 1 a+ bu = u 2 2 1 1 a+ u = 2 2 1 1 1 1 = a + u + (1 − u) − (1 − u) 2 2 2 2 1 = b − (1 − u), 2 and so 1 bk u = bk − bk−1 (1 − u). 2 Thus in the weak* limit, as k → ∞, we get 1 ru = r − r(1 − u). 2 Hence, 2ru = 2r − r(1 − u) = r + ru, and therefore ru = r. For each an , Lemma 3.7 gives rise to projections qn in B ∗∗ with qn ≥ q such that bn := 12 (an + 1) peaks at qn . This implies that each 1 − bn is in J and that ∗∗ each qn lies in A . Let Qu = qn , so that Qu ∈ A∗∗ and Qu ≤ qn for all n. Since qn ≥ q we also have Qu ≥ q. The containment B ∗∗ (1 − qn ) ⊂ B ∗∗ (1 − Qu ) follows from Qu ≤ qn . However, 1 − bn ∈ B ∗∗ (1 − qn ) for each n. Hence, 1 − bn ∈ B ∗∗ (1 − Qu ) for all n. However, 1 − bnt → 1 − b, and so 1 − b is in B ∗∗ (1 − Qu ). By the construction of r (recall 1 − r is the support projection for the weak* closed left ideal generated by 1 − b) this implies that B ∗∗ (1 − r) ⊂ B ∗∗ (1 − Qu ). Therefore, q ≤ Qu ≤ r ≤ u. As u varies over all open projections dominating q, we get q≤ Qu ≤ u = q. u≥q

u≥q

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Hence, q=

Qu =

and =

qn ,

u≥q n

u≥q

1−q

1−

qn

u≥q n

=

1−

u≥q

=

503

qn

n

(1 − qn ) .

u≥q n

If u is ﬁxed, then for each qn associated with u, bn q = q, where bn ∈ A∗∗ , as above. From Proposition 3.3, each an is a weak* limit of elements y in A satisfying qy = q. So each bn is the weak* limit of a net, (ct ) say, in A such that qct = q. Thus (1 − q)(1 − ct ) = 1 − ct and therefore the net (1 − ct ) is contained in J. Hence, its weak* limit, 1 − bn , is in J ⊥⊥ . Also, for any integer k > 0, 1 − bkn is in J ⊥⊥ . Hence 1 − qn = w∗ -limk 1 − bkn is in J ⊥⊥ . Combining this last fact with the last displayed equation we see that 1 − q is in J ⊥⊥ . As a consequence, we have our main theorem characterizing right ideals with left contractive approximate identity. Theorem 4.2. Let A be a unital subalgebra of B. A subspace J of A is a right ideal with left contractive approximate identity if and only if J = (1 − q)A∗∗ ∩ A for a closed projection q ∈ A⊥⊥ . Proof. The forward implication is the easy direction and is essentially in [6]. Suppose J is a right ideal with left contractive approximate identity (et ). Then J ⊥⊥ has a left identity p such that et → p weak* (see e.g. 2.5.8 in [8]). Since p is a contractive idempotent, it is an orthogonal projection. Since pet = et , p is an open projection by Lemma 2.2. So q = 1−p is closed. Also, J ⊂ (1−q)A∗∗ ∩A. However, if a ∈ A such that (1 − q)a = a, then et a ∈ J and so a = (1 − q)a ∈ J ⊥⊥ ∩ A = J. Thus (1 − q)A∗∗ ∩ A ⊂ J. Now let ϕ ∈ A⊥ . Then 0 = ϕ(1 − et ) → ϕ(q), and so ϕ(q) = 0. Hence q ∈ A⊥⊥ . On the other hand, suppose J is a subspace of A such that J = (1 − q)A∗∗ ∩A for a closed projection q ∈ A⊥⊥ . The subspace J is a right ideal of A and J = {a ∈ A : qa = 0}. By Theorem 4.1, J ⊥⊥ contains 1 − q, which is a left identity for J ⊥⊥ . Thus J possesses a left contractive approximate identity (see e.g. 2.5.8 in [8]).

5. Peak projections From the last section we see the role that closed projections in A⊥⊥ play in determining the right ideal structure of an operator algebra A. In this section we further

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study the closed projections in A⊥⊥ from the view point of ‘peak phenomena’ in A. Indeed, this idea was already playing a role in the proof of Theorem 4.1. Theorem 5.1. Let a be a contraction in B and let q be a closed projection in B ∗∗ such that aq = q (or equivalently, qa = q). Then the following are equivalent: 1. a peaks at q, 2. ϕ(a∗ a(1 − q)) < 1 for all ϕ ∈ S(B), 3. ϕ(a∗ a(1 − q)) < ϕ(1 − q) for all ϕ ∈ S(B) such that ϕ(1 − q) = 0, 4. ϕ(a∗ a) < 1 for every pure state ϕ of B such that ϕ(q) = 0, 5. pa < 1 for any closed projection p ≤ 1 − q, 6. ap < 1 for any closed projection p ≤ 1 − q, and 7. ap < 1 for any minimal projection p ≤ 1 − q. Proof. (3) ⇒ (2) Assume (3) holds. If ϕ ∈ S(B) is such that ϕ doesn’t vanish on 1 − q, we have ϕ(a∗ a(1 − q)) < ϕ(1 − q) = 1 − ϕ(q) ≤ 1. In the case that ϕ vanishes on 1 − q, (2) follows by the Cauchy-Schwarz inequality for positive linear functionals. (2) ⇒ (3) Suppose (2) holds. Let ϕ be a state which does not vanish on 1 − q and deﬁne ϕ(·(1 − q)) . ψ(·) = ϕ(1 − q) Because ψ is contractive and unital, it is a state on B. Applying ψ to both sides of a∗ a(1 − q) < 1 we get ϕ(a∗ a(1 − q)(1 − q)) < 1, ϕ(1 − q) which implies that ϕ(a∗ a(1 − q)) < ϕ(1 − q). So (3) holds. (1) ⇒ (4) This is immediate from the deﬁnition of ‘peak.’ (2) ⇒ (1) Assume (2) holds and let ϕ ∈ S(B) be a state such that ϕ(q) = 0. Then ϕ(a∗ a) = ϕ(a∗ a) − ϕ(q) = ϕ(a∗ a(1 − q)) < 1. Hence, a peaks at q. (4) ⇒ (2) Assume (4) and let ϕ ∈ S(B) and suppose that ϕ(a∗ a(1 − q)) = 1. It follows that ϕ(a∗ a) = 1 and ϕ(q) = 0. Consequently, ϕ(1 − a∗ a) = 0. Now let J be the left ideal B ∗∗ (1 − q) ∩ B of B, so that J ∩ J ∗ = (1 − q)B ∗∗ (1 − q) ∩ B is a hereditary subalgebra of B. Note that 1 − a∗ a = (1 − a∗ a)(1 − q) is an element of J ∩ J ∗ . Let (et ) ⊂ J ∩ J ∗ be an increasing contractive approximate identity for J ∩ J ∗ . Then (et ) is an increasing left contractive approximate identity for J, However, since 1 − q is the support projection for J, then necessarily et → 1 − q weak*, and so ϕ(et ) → ϕ(1 − q) = 1. However, ϕ|J∩J ∗ = lim ϕ|J∩J ∗ (et ) t

and so ϕ|J∩J ∗ = 1, making ϕ|J∩J ∗ a state of J ∩J ∗ such that ϕ|J∩J ∗ (1−a∗ a) = 0. By Lemma 2.6 there exists a pure state ψ of J ∩ J ∗ which annihilates 1 − a∗ a. We can then extend ψ to a pure state ψ˜ of B. We then have ˜ − a∗ a) = ψ(1) ˜ − ψ(a ˜ ∗ a) = 1 − ψ(a ˜ ∗ a). 0 = ψ(1

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˜ ∗ a) = 1. We also have Thus ψ(a 1

=

˜ ˜ + ψ(1 ˜ − q) ψ(1) = ψ(q) ˜ t) ˜ + lim ψ(e ψ(q)

=

˜ + lim ψ(et ) ψ(q)

=

˜ + ψ = ψ(q) ˜ + 1, ψ(q)

=

t t

˜ which forces ψ(q) = 0. Thus we have found a pure state on B which annihilates q, but takes the value 1 at a∗ a. This contradicts our assumption of (4). Thus, the supposition that ϕ(a∗ a(1 − q)) = 1 must be rejected, and therefore (2) holds. (6) ⇒ (7) This is immediate from the fact that any minimal projection is automatically closed. (7) ⇒ (4) Assume (7) and let ϕ be a pure state of B which annihilates q. Let L be the left-kernel of ϕ. Then L = B ∗∗ (1 − p) ∩ B for some minimal closed projection p ∈ B ∗∗ , and the weak* closure of L in B ∗∗ will be B ∗∗ (1 − p) (3.13.6 in [23]). Now viewing ϕ as a normal state on B ∗∗ , let L be the left-kernel of ϕ with respect to B ∗∗ . This will be a weak* closed left ideal and L = B ∗∗ r for some projection r ∈ B ∗∗ with q ∈ L . The containment B ∗∗ (1 − p) ∩ B ⊂ B ∗∗ r is obvious. Passing to the weak* closure we get B ∗∗ (1 − p) ⊂ B ∗∗ r. Hence, 1 − p ≤ r, or rather 1 − r ≤ p, which by minimality of p forces p = 1 − r. We conclude that w∗ L = L and therefore q ∈ B ∗∗ (1 − p) . Hence q ≤ 1 − p, or equivalently, p ≤ 1 − q. Thus by condition (7), ap < 1. Now decompose a∗ a as a∗ a = pa∗ ap + pa∗ a(1 − p) + (1 − p)a∗ ap + (1 − p)a∗ a(1 − p) and apply ϕ to get ϕ(a∗ a) = ϕ(pa∗ ap) + ϕ(pa∗ a(1 − p)) + ϕ((1 − p)a∗ ap) + ϕ((1 − p)a∗ a(1 − p)). The last 3 terms vanish by the Cauchy-Schwarz inequality and the fact that ϕ(1 − p) = 0. Thus we have ϕ(a∗ a) = ϕ(pa∗ ap) ≤ pa∗ ap = ap2 < 1, establishing (4). (2) ⇒ (5) Assume (2) and let p ≤ 1 − q be closed. Since p is closed, a∗ pa is upper semi-continuous on S(B). Therefore, a∗ pa attains its maximum on S(B) at some state ϕ, and hence pa2 = ϕ(a∗ pa). Since p ≤ 1 − q we have a∗ pa ≤ a∗ (1 − q)a, and so ϕ(a∗ pa) ≤ ϕ(a∗ (1 − q)a) = ϕ(a∗ a(1 − q)) < 1, by (2) and the fact that a and q commute. Thus, the norm of pa must be strictly less than 1. (5) ⇒ (6) Apply the implication (6) ⇒ (5), which has already been established, to a∗ .

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Remarks. First, the condition ap < 1 is equivalent to pa∗ < 1, which means that a∗ a in conditions (1)-(4) can be replaced with aa∗ . Second, similarly to the equivalence of (6) and (7), condition (5) is equivalent to the statement that pa < 1 for any minimal projection p ≤ 1 − q. Definition 5.2. If X is a subspace of B, a closed projection q ∈ B ∗∗ is called a peak projection for X if there exists a contraction a ∈ X which peaks at q. If q is an intersection of peak projections for X, we refer to q as a p-projection for X. If B = C(K), the continuous complex-valued functions on a compact space K, then the peak projections for X correspond to the characteristic functions of peak sets. It is also easy to see that if E is a peak set and a peaks on E, then an converges point-wise to χE . By Lemma 3.6, the same is true for peak projections. For a compact Hausdorﬀ space K, any closed set in K will be a p-set for C(K) by Urysohn’s lemma. The same thing holds in the noncommutative case as well. Proposition 5.3. Any closed projection in B ∗∗ is a p-projection for B. Proof. Let q ∈ B ∗∗ be a closed projection. Then for any open projection u ≥ q, there exists au ∈ B with 0 ≤ au ≤ 1 such that au q = q and au (1 − u) = 0. Now let qu be the weak* limit of anu as n → ∞. Since multiplication is separately weak* continuous, it follows that au qu = qu au = qu . From this property and again the separate weak* continuity of multiplication, it follows that qu is a projection. Since au (1 − u) = 0, we also have qu ≤ u. We now claim that qu is a peak projection for B with peak au . We only need to check that ϕ(a2u ) < 1 for any pure state ϕ ∈ B ∗ such that ϕ(qu ) = 0. However, since anu → qu weak*, it follows that ϕ(anu ) → 0. Suppose that ϕ(a2u ) = 1. Then representing B concretely on a Hilbert space so that ϕ is a vector state ϕ(·) = π(·)ξ, ξ , we see that π(a2u )ξ, ξ = 1. So by the converse to the Cauchy-Schwarz inequality, we must have that π(a2u )ξ = ξ, which contradicts that ϕ(an ) → 0. So qu is a peak projection. Note also that the equation anu q = q implies that q ≤ qu . Now we take the intersection qu of all such qu as u varies over all open projections dominating q. We now show that q = qu . To see this, note that since q ≤ qu , we have that q ≤ qu . By Proposition 2.3, q≤ qu ≤ u = q. Thus, q = qu . This next proposition describes a peak projection in terms of a support projection associated with its peak. Proposition 5.4. A projection q ∈ B ∗∗ is a peak projection for a unital subspace X of B if and only if there exists a contraction a ∈ X such that 1 − q is the right support projection for 1 − a.

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Proof. We assume that B and B ∗∗ are acting on the universal Hilbert space Hu for B. First suppose that q is a peak projection for A with peak a ∈ A. Let ξ ∈ Hu and let r denote the right support projection of 1 − a. Since an → q weak*, aq = q implies that ξ ∈ Ran q if and only if aξ = ξ. This in turn is equivalent to saying ξ ∈ Ran q if and only if ξ ∈ Ker (1−a). Since r is the projection onto [Ker (1−a)]⊥ , this last statement is equivalent to 1 − q = r. Now suppose that a ∈ A is a contraction and q is a closed projection such that the range projection r, of 1−a, is equal to 1−q. Then 1−a = (1−a)r = (1−a)(1−q) implies aq = q. Now deﬁne b = (a + 1)/2, so that bq = q. Let ϕ be a state on B such that ϕ(q) = 0. We may assume that ϕ(·) = ·η, η for a unit vector η ∈ Hu . Then qη = 0, and so rη = η. Now suppose that ϕ(b∗ b) = 1. Then 1 (ϕ(a∗ a) + 2Re ϕ(a) + 1). 4 Hence, 1 = ϕ(a∗ a) = aη, η , and so by the converse to the Cauchy-Schwarz inequality, aη = η. Thus ξ ∈ Ker (1 − b), and so rη = 0, which is a contradiction. Thus ϕ(b∗ b) < 1 and so q is a peak projection with peak b. 1 = ϕ(b∗ b) =

In the commutative case, it is easy to see that if E and F are peak sets for a uniform algebra A with peaks f and g, respectively, in A, then E ∩ F will be a peak set with peak 12 (f + g). For general C ∗ -algebras, we have the following generalization. Proposition 5.5. Let q1 and q2 be two peak projections with peaks a1 and a2 , respectively. Then q1 ∧ q2 is also a peak projection with peak 12 (a1 + a2 ). Proof. That q1 ∧ q2 and 12 (a1 + a2 ) satisfy the ﬁrst condition in the deﬁnition of peak projection is immediate since q1 ∧ q2 is dominated by both q1 and q2 . To show the second condition, let ϕ be a pure state of B which annihilates q1 ∧ q2 . Let b = 12 (a1 + a2 ). We wish to show ϕ(b∗ b) < 1. This is equivalent to showing ϕ(a∗1 a1 ) + ϕ(a∗1 a2 ) + ϕ(a∗2 a1 ) + ϕ(a∗2 a2 ) < 4, for which it suﬃces to show that either ϕ(a∗1 a1 ) < 1 or ϕ(a∗2 a2 ) < 1. So suppose ϕ(a∗1 a1 ) = ϕ(a∗2 a2 ) = 1. Let π : B ∗∗ → B(H) be the weak* continuous cyclic representation associated with ϕ and let ξ ∈ H be the corresponding cyclic vector. Then π(a∗1 a1 )ξ, ξ = ϕ(a∗1 a1 ) = 1. Thus, by the converse to the Cauchy-Schwarz inequality, we must have π(a∗1 a1 )ξ = ξ. However, since (a∗1 a1 )n → q1 weak*, and so π(a∗1 a1 )n → π(q1 ) in the weak operator topology, we must have ϕ(q1 ) = 1. The same argument shows that ϕ(q2 ) = 1 and so ξ must be in the range of both π(q1 ) and π(q2 ). Hence, ϕ(q1 ∧ q2 ) = π(q1 ) ∧ π(q2 )ξ, ξ = 1, which is a contradiction. The next result shows that p-projections must be in the weak* closure of A in B ∗∗ when A is a subalgebra.

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Proposition 5.6. Let A be a unital subalgebra of B, and let q be a p-projection for A. Then ϕ(q) = 0 for all ϕ in A⊥ . Consequently, ϕ is in (qA)⊥ and q is in A⊥⊥ . Proof. First assume q is a peak projection and let a ∈ A peak on q. For any ϕ in A⊥ and any integer n > 0, ϕ(an ) = 0. However, an converges weak* to q. Thus ϕ(q) = 0. Now suppose q = i qi and let ϕ ∈ A⊥ . Let > 0. By Proposition II.3 in [1], there exists an open projection p ∈ B ∗∗ such that p ≥ q and |ϕ|(p − q) < , where |ϕ| is obtained from the polar decomposition of ϕ (3.6.7 in [23]). By hypothesis q ≤ p. Hence 1 − p ≤ i i i (1 − qi ), and so by the compactness property of closed projections there, exist ﬁnitely many projections q1 , q2 , . . . , qn in the n nfamily {qi } n such that 1 − p ≤ i=1 (1 − qi ). Thus q ≤ i=1 qi ≤ p. Now let Q = i=1 qi , which is again a peak projection. By the last paragraph it follows that ϕ(Q) = 0, and so |ϕ(Q − q)| = |ϕ(q)| for all ϕ ∈ A⊥ . The functional |ϕ| has the property that |ϕ(x)|2 ≤ ϕ|ϕ|(x∗ x) for all x ∈ B ∗∗ . Thus we have |ϕ(Q − q)|2

≤ ϕ|ϕ|((Q − q)∗ (Q − q)) = ϕ|ϕ|(Q − q)) ≤ ϕ|ϕ|(p − q)) < ϕ.

Since was arbitrary, this shows that ϕ(q) = 0. If x ∈ A, the map ϕ(·x) is also in A⊥ . Thus ϕ(qx) = 0 for every x in A. This shows that ϕ is in (qA)⊥ . By Lemma 2.5, this implies that q is in A⊥⊥ . It is natural to ask if the notion of a peak or p-projection is dependent on the particular C ∗ -algebra in which we view A as residing. That is, if we have embeddings of A into two diﬀerent C ∗ -algebras, can the peak projections arising from the each embedding be identiﬁed in some way? By an embedding of A into a C ∗ -algebra, we mean a completely isometric homomorphism of A. The following proposition shows that the notion of a peak or p-projection is indeed independent of the particular embedding. Proposition 5.7. Let A be a unital subalgebra of B. Let π : A → B1 be a unital completely isometric homomorphism of A into another C ∗ -algebra B1 . If q is a p-projection for A in B ∗∗ , then π ∗∗ (q) is a p-projection for π(A) inside B1∗∗ . Proof. We assume that B1 is acting on its universal Hilbert space Hu , that is B1 ⊂ B(Hu ). Assume that q is a peak projection with peak a ∈ A. Clearly, π(a)π ∗∗ (q) = π ∗∗ (q). Now suppose that ϕ ∈ S(B1 ) such that ϕ(π ∗∗ (q)) = 0 and extend ϕ to a vector state ϕ˜ on B(Hu ). Viewing π as a unital completely positive map into B(Hu ), by Arveson’s extension theorem we may extend π to a completely positive map ρ : B → B(Hu ). This in turn can be extended to a weak* continuous map ρ˜ on B ∗∗ . Since π ∗∗ is the unique weak* continuous extension of π to A∗∗ , we must have ρ˜|A∗∗ = π ∗∗ . We next observe that ϕ˜ ◦ ρ is a state on B which extends uniquely to a weak* continuous state on B ∗∗ , which by uniqueness must be ϕ˜ ◦ ρ˜.

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Hence, (ϕ◦ ˜ ρ)(q) = ϕ(˜ ˜ ρ(q)) = ϕ(π ∗∗ (q)) = 0. Thus, since q is a peak projection, we ∗ must have that ϕ(ρ(a ˜ a)) < 1. By the Kadison-Schwarz inequality for completely positive maps, we have ∗ ∗ ˜ ρ(a)) ≤ ϕ(ρ(a ˜ a)) < 1. ϕ(π(a)∗ π(a)) = ϕ(ρ(a)

Thus π ∗∗ (q) is a peak projection for π(A). Now if q is just a p-projection with q = ∧qi for peak projections qi , then q is the weak* limit of the net of meets for ﬁnitely many qi . Thus, by the discussion about meets in Section 2, it follows that π ∗∗ (q) = ∧π ∗∗ (qi ). Thus π ∗∗ (qi ) is a p-projection. Minimal projections which are also p-projections correspond to p-points (singleton p-sets) in the commutative case. The closure of the set of p-points for a uniform algebra is the Shilov boundary (see e.g. [15] and [25]). Let A and B be as before, but assume A generates B as a C ∗ -algebra. Then there exists a largest closed two-sided ideal J of B such that the canonical quotient map B → B/J restricts to a complete isometry on A ([3]). The ideal J is the so-called ‘Shilov ideal’ for A. Let p be the closed projection in B ∗∗ corresponding to the Shilov boundary ideal, then p dominates all minimal projections which are also p-projections for A. Moreover, p dominates all minimal projections in A⊥⊥ (see [7]). Unfortunately, however, the join of the orthogonal complements of all such minimal projections does not in general equal the support of the Shilov ideal. As a simple counterexample take the algebra α β A= : α, β ∈ C , 0 α which contains no minimal projections. One of the interesting aspects of p-projections is their relationship to approximate identities. For instance, we have the following proposition ([18],[7]). Proposition 5.8. If A is a unital subalgebra of B and p ∈ B ∗∗ is the support projection for a right ideal of A with a left approximate identity of the form (1−xt ) for xt ≤ 1, then 1 − p is a p-projection for A. Proof. Let J be a right ideal of A with left approximate identity (et ) with et = 1 − xt and xt in the unit ball of A. Let p be the support projection of J and deﬁne q = 1 − p. Then q is necessarily closed, et → 1 − q weak*, J = {a ∈ A : (1 − q)a = a}, and (1 − et )q = q. For each t let Jt be the intersection of all right ideals of B containing et . Then there exists a unique closed projection qt in B ∗∗ such that Jt = (1 − qt )B ∗∗ ∩ B. By Lemma 3.7 qt is a peak projection on which 1 1 2 [1 + (1 − et )] = 1 − 2 et peaks, and such that qt ≥ q. Now set r = ∧qt . We have that r ≥ q, but suppose r − q = 0. Then (r − q)et → (r − q)(1 − q) = r − q, and r − q ≤ qt . Thus (1 − qt )B ∗∗ ⊂ (1 − (r − q))B ∗∗ , so that (et ) ⊂ (1 − (r − q))B ∗∗ . Hence (1−(r−q))et = et for each t, and so (r−q)et = 0. However, (r−q)et → r−q. Thus r − q = 0 and so r = q, making q an intersection of peak projections.

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Remark. It can also be shown that if an ideal J of a unital operator algebra A has a left contractive approximate identity, then it has a left approximate identity of the form (1 − xt ), where xt ∈ A and limt xt = 1 ([7]). Moreover, if we can choose the xt in Ball(A), for every such ideal, then the p-projections are exactly the orthogonal complements of the support projections for right ideals with left contractive approximate identity. It is natural to make the following deﬁnition. Definition 5.9. Let A be a unital subalgebra of B. A projection q ∈ B ∗∗ is said to be an approximate p-projection for A if q is closed and q ∈ A⊥⊥ . The following shows that approximate p-projections possess peaking properties. Theorem 5.10. Let A be a unital subalgebra of B and let q ∈ B ∗∗ be a closed projection. The following are equivalent: 1. q is an approximate p-projection, 2. for every > 0 and for every open projection u ≥ q, there exists a ∈ (1 + )Ball(A) such that qa = q and a(1 − u) ≤ , and 3. for every > 0 and strictly positive p ∈ B with p ≥ q, there exists a ∈ A such that qa = q and a∗ a ≤ p + . Proof. (1) ⇒ (3) This is essentially Proposition 3.2. Let > 0 and let p ∈ B be a strictly positive element of B such that p ≥ q. Since q is in A⊥⊥ , by Lemma 2.5, q satisﬁes the hypothesis of Proposition 3.2. Thus there exists a ∈ A such that qa = q and a∗ a ≤ p + . (3) ⇒ (2) Let > 0 and suppose u ≥ q is open. Let δ = 2 /2. As in the ﬁrst part of the proof of Theorem 4.1, by the noncommutative Urysohn’s lemma, there exists a strictly positive contraction p ∈ B such that pq = q and p(1 − u) = δ(1 − u). By (3) there exists a ∈ A such that qa = q and a∗ a ≤ p + δ. Hence, (1 − u)a∗ a(1 − u) ≤ 2δ(1 − u), and so a(1 − u) ≤ . (2) ⇒ (1) Let u ≥ q be an open projection. By (2), for each natural number n there exists an ∈ (1 + 1/n)Ball(A) such that qan = q and a(1 − u) ≤ n1 . The net (an ) has a weak* limit point a ∈ Ball(A⊥⊥ ). Since (1 − u)an ≤ n1 for each n, we must also have a(1 − u) = 0, and hence au = a. Let b = 12 (a + 1), which is in A⊥⊥ . By Lemma 3.7, there exists a projection qu ∈ A⊥⊥ such that q ≤ qu and bk → qu . As in the proof of Theorem 4.1 we also that q ≤ qu ≤ u for each u. However, by Proposition 2.3, this implies that q = ∧qu as u varies over all open projections dominating q. However, each qu is in A⊥⊥ . Thus, so is q, by the discussion in Section 2. For a uniform algebra A ⊂ C(K), Glickberg’s peak set theorem says that a closed set E is a p-set for A if and only if µ ∈ A⊥ implies µE ∈ A⊥ . With this and Lemma 2.5 in mind, it is then natural to ask whether or not the p-projections are precisely the approximate p-projections, in the noncommutative setting. Certainly any p-projection is an approximate p-projection. The reverse implication holds for

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unform algebras by the classical Glicksberg theorem, and it holds when A = B is a C ∗ -algebra by Proposition 5.3. It is also true for operator algebras which are also reﬂexive Banach spaces, as the following simple observation shows. In particular, it is true for ﬁnite dimensional algebras. Proposition 5.11. Let A be a unital subalgebra of B such that A is also a reflexive Banach space. Let q ∈ B ∗∗ be a closed projection. The following are equivalent: 1. q is a p-projection for A, 2. q is an approximate p-projection for A, and 3. q ∈ A. Proof. The implication (1) ⇒ (2) follows from Proposition 3.6 and the fact that A is an algebra. If (2) holds, by reﬂexivity, q is in A, establishing (3). If q is in A, then it is trivially a p-projection. So (3) ⇒ (1) holds. Approximate p-projections enjoy some of the properties of p-projections. For example, by some observations in Section 2, if A is a unital subalgebra of a unital C ∗ -algebra B, then the meet of a collection of approximate p-projections for A is also an approximate p-projection. It can also be shown that if the join of a collection of approximate p-projections happens to be closed, then it is also an approximate p-projection (see [18]). By Corollary 5.5 of [7], if q ∈ B ∗∗ is a closed projection and X is a unital subspace of B such that for every strictly positive contraction p ∈ B with q ≤ p there exists a ∈ X such that aq = q and a∗ a ≤ p, then q is a p-projection for X. As described above, for many algebras the class of p-projections is the same as the class of approximate p-projections. The most tantalizing remaining question here is whether or not these two notions are the same for a general unital operator algebra. Nonetheless, the correspondence between right ideals with left approximate identity and approximate p-projections will be key in importing some results from C ∗ -algebra theory to general operator algebras. Indeed, we have begun this in [7].

References [1] C. A. Akemann, The general Stone-Weierstrass problem, J. Functional Analysis 4 (1969), 277-294. [2] C. A. Akemann, Left ideal structure of C ∗ -algebras, J. Functional Analysis 6 (1970), 305-317. [3] W. B. Arveson, Subalgebras of C ∗ -algebras, Acta Math. 123 (1969), 141-224. [4] D. P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), 15-30. [5] D. P. Blecher, One-sided ideals and approximate identities in operator algebras, J. Australian Math. Soc. 76 (2004), 425-447. [6] D. P. Blecher, E. G. Eﬀros, and V. Zarikian, One-sided M -ideals and multipliers in operator spaces, I, Pacific J. Math. 206 (2002), 287-319.

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[7] D. P. Blecher, D. M. Hay, and M. Neal, Hereditary subalgebras of operator algebras, J. Operator Theory, to appear. [8] D. P. Blecher and C. Le Merdy, Operator Algebras and their Modules, Oxford Univ. Press, 2004. [9] D. P. Blecher and V. Zarikian, The calculus of one-sided M -ideals and multipliers in operator spaces, to appear, Mem. Amer. Math. Soc. 842 (2006). [10] N. Dunford and J. Schwartz, Linear Operators. I. New York: Interscience. 1958. [11] C. M. Edwards and G. T. R¨ uttiman, On the facial structure of the unit balls in a JBW*-triple and its predual, J. London Math. Soc. 38 (1988), 317-332. [12] C. M. Edwards and G. T. R¨ uttiman, Compact tripotents in bi-dual JB*-triples, Math. Proc. Camb. Philos. Soc., 120 (1996), 155-173. [13] E. Eﬀros and Z. J. Ruan, Operator Spaces, Oxford Univ. Press, 2000. [14] C. Foia¸s and B. Sz.-Nagy, Harmonic Analysis of Operators on Hilbert Space, NorthHolland Publishing Company, Amsterdam, 1970. [15] T. W. Gamelin, Uniform Algebras, Prentice-Hall (1969). [16] I. Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415-435. [17] R. Giles and H. Kummer, A non-commutative generalization of topology, Indiana University Mathematics Journal, Vol. 21, No. 1 (1971). [18] D. M. Hay, Noncommutative topology and operator algebras, Ph.D. thesis, University of Houston, 2006. [19] B. Hirsberg, M -ideals in complex function spaces and algebras, Israel J. Math. 12 (1972), 133-146. [20] K. Jarosz, A characterization of weak peak sets for function algebras, Bul. Austral. Math. Soc. 29 (1984), 129-135. [21] C. S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhauser, Boston, 1997. [22] V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press (2002). [23] G. Pedersen, C ∗ -algebras and their Automorphism Groups, Academic Press (1979). [24] G. Pisier, Introduction to Operator Space Theory, Cambridge University Press (2003). [25] R. R. Phelps, Lectures on Choquet’s Theorem, Springer-Verlag, Berlin, 2001. [26] R. R. Smith and J. D. Ward, Applications of convexity and M -ideal theory to quotient Banach algebras, Quart. J. Math. Oxford (2), 30 (1979), 365-384. Damon M. Hay Department of Mathematics and Statistics, University of North Florida 4567 St. Johns Bluﬀ Road, South Jacksonville, FL 32224-2645 USA e-mail, Damon Hay: [email protected] Submitted: March 17, 2006 Revised: May 12, 2006

Integr. equ. oper. theory 57 (2007), 513–520 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040513-8, published online December 26, 2006 DOI 10.1007/s00020-006-1472-y

Integral Equations and Operator Theory

The Brauer–Ostrowski Theorem for Matrices of Operators Gerd Herzog and Christoph Schmoeger Abstract. The classical Brauer-Ostrowski Theorem gives a localization of the spectrum of a matrix by a union of Cassini ovals. In this paper we prove a corresponding result for operator matrices. Mathematics Subject Classification (2000). Primary 47A10; Secondary 47A05. Keywords. Operator matrices, localization of the spectrum, Cassini ovals.

1. Introduction In [5] and [1] Ostrowski and Brauer independently observed that each eigenvalue of a matrix A = (ajk ) ∈ Cn×n , n ≥ 2 is contained in a Cassini oval     n n   |ark |  |ask | λ ∈ C : |λ − arr | |λ − ass | ≤    r=k=1

s=k=1

with r = s. In [2] Feingold and Varga obtained the corresponding result for block matrices. In several cases these results lead to a better localization of the spectrum of a matrix than Gershgorin’s Theorem, compare [8] and the references given there. Aﬀected by Gil”s and Salas’ devolvements of Gershgorin’s Theorem [3],[7], we study in this paper the Brauer-Ostrowski Theorem in the frame of operator matrices.

2. Notations Let X be a complex Banach space, and T : X → X linear and bounded. In the sequel we consider: the spectrum σ(T ) = {λ ∈ C : λI − T is bijective},

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the resolvent set ρ(T ) = C \ σ(T ), the point spectrum σp (T ) = {λ ∈ C : ∃x ∈ X : x = 0, (λI − T )x = 0}, the continuous spectrum σc (T ) = {λ ∈ C : λ ∈ / σp (T ), (λI − T )(X) = X, (λI − T )(X) = X}, the residual spectrum / σp (T ), (λI − T )(X) = X}, σr (T ) = {λ ∈ C : λ ∈ the approximate point spectrum σap (T ) = {λ ∈ C : ∃(xn ) ⊆ X : xn = 1 and (λI − T )xn → 0 (n → ∞)}, and the compression spectrum σcom (T ) = {λ ∈ C : (λI − T )(X) = X}. Note, that σp (T ), σc (T ) and σr (T ) are pairwise disjoint, that σ(T ) = σp (T ) ∪ σc (T ) ∪ σr (T ), and that σr (T ) = σcom (T ) \ σp (T ). ∗

Moreover let X denote the dual space of X, let T ∗ denote the adjoint of T , and note that σ(T ) = σ(T ∗ ), σcom (T ) = σp (T ∗ ) and T = T ∗.

3. Matrices of operators Let n ∈ N, n ≥ 2, and (X1 , · 1 ), . . . , (Xn , · n ) complex Banach spaces. We consider the complex Banach space X = X1 × · · · × Xn ,

n

x∞ = max xi i (x = (x1 , . . . , xn ) ∈ X). i=1

Now, let A : X → X be linear and bounded. Then  A11 A12 · · ·  .. .. .. A = (Ajk ) =  . . . An1

An2

···

 A1n ..  , .  Ann

where Ajk : Xk → Xj is linear and bounded (j, k = 1, . . . , n). For each j ∈ {1, . . . , n} we set, pj (A) =

n j=k=1

Ajk ,

qj (A) =

n j=k=1

Akj .

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For r, s ∈ {1, . . . , n} with r = s we deﬁne the following sets, corresponding to the ovals of Cassini: (p) Crs (A) = σ(Arr ) ∪ σ(Ass ) ∪ λ ∈ ρ(Arr ) ∩ ρ(Ass ) : ((λI − Arr )−1 (λI − Ass )−1 )−1 ≤ pr (A)ps (A) , and (q) (A) = σ(Arr ) ∪ σ(Ass ) Crs ∪ λ ∈ ρ(Arr ) ∩ ρ(Ass ) : ((λI − Arr )−1 (λI − Ass )−1 )−1 ≤ qr (A)qs (A) . Since A∗ = (Ajk )∗ = (A∗kj ) we have (p) (q) Crs (A∗ ) = Crs (A).

(3.1)

Next, let C

(p)

(A) :=

n

(p) Crs (A),

C

(q)

(A) :=

r,s=1,r=s

n

(q) Crs (A),

r,s=1,r=s (p)

∗

(q)

and note that by means of (3.1), C (A ) = C (A). Moreover, observe that if (p) (q) n = 2, then C12 (A) = C12 (A), hence C (p) (A) = C (q) (A).

4. Localization of the spectrum Theorem 4.1. Let A = (Ajk ) : X → X be linear and bounded. Then σap (A) ⊆ C (p) (A). (m)

(m)

Proof. Let λ ∈ σap (A). For each m ∈ N there exists x(m) = (x1 , . . . , xn ) ∈ X such that 1 x(m) ∞ = 1, λx(m) − Ax(m) ∞ ≤ . m For m ∈ N let r(m), s(m) ∈ {1, . . . , n} be such that r(m) = s(m), and (m)

(m)

(m)

xr(m) r(m) ≥ xs(m) s(m) ≥ xi

i

(i ∈ {1, . . . , n} \ {r(m), s(m)}).

By means of the pigeon hole principle we can assume without loss of generality ∞ that the sequences (r(m))∞ m=1 and (s(m))m=1 are constant. Hence let r = r(m) (m) and s = s(m). Then r = s and, since x ∞ = 1, (m)

xi We deﬁne

(m) i ≤ x(m) / {r, s}, m ∈ N). s s ≤ xr r = 1 (i ∈ (m)

z (m) = (z1 , . . . , zn(m) ) := λx(m) − Ax(m) (m ∈ N), and consider the following cases: (p) 1. Let λ ∈ σ(Arr ) ∪ σ(Ass ). Then λ ∈ Crs (A), thus λ ∈ C (p) (A). 2. Let λ ∈ ρ(Arr ) ∩ ρ(Ass ). From (4.2) we get zr(m) = (λI − Arr )x(m) − r

n r=k=1

(m)

Ark xk

(4.1) (4.2)

(4.3)

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and

n

zs(m) = (λI − Ass )x(m) − s

s=k=1

By means of (4.3) we have



IEOT

n

= (λI − Arr )−1 zr(m) + x(m) r

r=k=1

thus, according to (4.1),



−1  (m) zr r + 1 = x(m) r r ≤ (λI − Arr )

−1

≤ (λI − Arr ) From (4.4) we get

(4.4) 

(m) Ark xk  ,



n

(m)

r=k=1

Ark xk k 

(4.5)

1 (m) + xs s pr (A) . m



n

= (λI − Ass )−1 zs(m) + x(m) s

s=k=1

hence

(m)

Ask xk .

 −1  (m) x(m) zs s + s s ≤ (λI − Ass )

 (m) Ask xk  ,

n s=k=1

 (m)

Ask xk k 

(4.6)

(m) 1 m + xr r ps (A) 1 + ps (A) . Ass )−1 m

≤ (λI − Ass )−1

= (λI − We proceed by proving that there exist α > 0 and m0 ∈ N such that x(m) s s ≥ α If not, then there is a subsequence thus (4.1) gives (mν )

xi

(m ≥ m0 ).

(m ) (xs ν )

of

(m) (xs )

(4.7) with

(m ) xs ν

→ 0 (ν → ∞),

→ 0 (ν → ∞) (i ∈ {1, . . . , n} \ {r}),

and, by (4.3), Since

(m ) xr ν

(λI − Arr )xr(mν ) → 0 (ν → ∞). = 1 for all ν ∈ N, we get the contradiction λ ∈ σap (Arr ) ⊆ σ(Arr ).

Thus (4.7) holds. According to (4.5) and (4.6) we have (m) x(m) s s = 1 · xs s 1 1 −1 −1 (m) + xs s pr (A) + ps (A) , ≤ (λI − Arr ) (λI − Ass ) m m

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The Brauer–Ostrowski Theorem

and therefore, by (4.7),

517

1 1 ≤ (λI − Arr ) (λI − Ass ) + ps (A) + pr (A) (m) m mxs s 1 1 −1 −1 + pr (A) + ps (A) ≤ (λI − Arr ) (λI − Ass ) αm m −1

1

−1

(p)

for m ≥ m0 . With m → ∞ we derive λ ∈ Crs (A) ⊆ C (p) (A).

In particular we have: Corollary 4.2. σp (A) ∪ σc (A) ⊆ C (p) (A). Proof. Follows from Theorem 4.1 and σp (A) ∪ σc (A) ⊆ σap (A). Corollary 4.3. σcom (A) ⊆ C

(q)

(A).

Proof. Since σcom (A) = σp (A∗ ), Corollary 4.2 shows that σcom (A) ⊆ C (p) (A∗ ) = C (q) (A).

In case n = 2 we have seen that C (p) (A) = C (q) (A). Hence we have: Corollary 4.4. If n = 2, then σ(A) ⊆ C (p) (A).

5. Weighted norms Let w1 , . . . , wn > 0. We deﬁne equivalent norms on X1 , . . . , Xn , respectively, by setting |||ξ|||i = wi ξi (ξ ∈ Xi , i = 1, . . . , n). For the operators Ajk : Xk → Xj we have Ajk = sup Ajk ξj , ξk =1

hence |||Ajk ||| :=

sup |||ξ|||k =1

|||Ajk ξ|||j =

wj Ajk (j, k = 1, . . . , n). wk

By application of Theorem 4.1 to this situation we obtain: Theorem 5.1. Let w1 , . . . , wn > 0. Then n σ(Arr ) ∪ σ(Ass ) σap (A) ⊆ r,s=1,r=s

−1 ∪ λ ∈ ρ(Arr ) ∩ ρ(Ass ) : (λI − Arr )−1 (λI − Ass )−1    n n w w r s ≤ Ark   Ask  . wk wk r=k=1

s=k=1

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Remark 5.2. Theorem 5.1 can be extended to the case that Wi : Xi → Xi is linear, bounded and invertible (i = 1, . . . , n) and |||ξ|||i = Wi ξi

(ξ ∈ Xi , i = 1, . . . , n).

Then

|||Ajk ||| = Wj Ajk Wk−1 (j, k = 1, . . . , n), and the corresponding inclusion for σap (A) is valid. Now, consider the scalar matrix  0 A12 A13  A21 0 A23  B= .. .. ..  . . .

... ... .. .

A1n A2n .. .

An1

An(n−1)

0

...

...

   , 

and its spectral radius r(B) = max |µ|. For each τ ≥ 0 we deﬁne µ∈σ(B)

Cτ :=

n

σ(Arr ) ∪ σ(Ass )

r,s=1,r=s

−1 ∪ λ ∈ ρ(Arr ) ∩ ρ(Ass ) : (λI − Arr )−1 (λI − Ass )−1 ≤ τ2 .

Theorem 5.3. Let B and r(B) be as above. Then σ(A) ⊆ Cr(B) . Proof. Let P denote the n×n matrix with each entry equals 1. Let ε > 0. According to [6, Theorem 10.20] there exists δ > 0 such that r(B + δP ) ≤ r(B) + ε. Now, B + δP is irreducible and therefore has a strictly positive Perron eigenvector v = (v1 , . . . , vn ) ∈ (0, ∞)n . Set wk = vk−1 (k = 1, . . . , n) and let j ∈ {1, . . . , n}. From (B + δP )v = r(B + δP )v ≤ (r(B) + ε)v (coordinatewise) we derive n (r(B) + ε)vj ≥ δvj + (Ajk + δ)vk , j=k=1

hence

n n wj vk Ajk = Ajk ≤ r(B) + ε. wk vj

j=k=1

j=k=1

Now Theorem 5.1 shows that σap (A) ⊆ Cr(B)+ε , and with ε → 0+ we obtain σap (A) ⊆ Cr(B) . By replicating this proof with A∗ instead of A we obtain σcom (A) = σp (A∗ ) ⊆ Cr(B ) = Cr(B) , since B and its transposed B have the same spectral radius. So, ﬁnally σ(A) = σap (A) ∪ σcom (A) proves σ(A) ⊆ Cr(B) .

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6. Examples In order to apply Theorem 4.1,5.1 or 5.3 it is comfortable if the expressions (λI − Ajj )−1 −1

(j = 1, . . . , n)

have a simple structure. If T is a normal operator on a complex Hilbert space, then (λI − T )−1 −1 = dist(λ, σ(T ))

(λ ∈ ρ(T ))

see [4, p. 277]. If T is a multiplication operator on a space of complex valued continuous function C(K) (K a compact metric space, say, and C(K) endowed with the maximum norm), that is (T ξ)(t) = g(t)ξ(t) (t ∈ K) for some g ∈ C(K), then σ(T ) = g(K) and likewise (λI − T )−1 −1 = dist(λ, σ(T )) = dist(λ, g(K))

(λ ∈ ρ(T )).

For example, let X = C(K)n , and let A = (Ajk ) : X → X be such that (Ajj ξ)(t) = gj (t)ξ(t) with gj ∈ C(K) (j = 1, . . . , n). Let B be as in section 5. Then, according to Theorem 5.3 σ(A) ⊆

n

{λ : dist(λ, gr (K))dist(λ, gs (K)) ≤ r(B)2 }.

r,s=1,r=s

In the following example let X3 = C([0, 1]) be endowed with the maximum norm · 3 , and X2 = C 1 ([0, 1]), X1 = C 2 ([0, 1]) endowed with the norms ξ2 = max{ξ3 , ξ 3 } and ξ1 = max{ξ3 , ξ 3 , ξ 3 }, respectively. Let α ≥ 0, and let A : X → X be deﬁned by   1 x1 (t) + α 0 cos(ts)x3 (s)ds . (Ax)(t) =  x1 (t) − x2 (t) x1 (t) + x2 (t) + exp(2πit)x3 (t) Note that σ(A) = {λ : |λ| = 1} if α = 0. Application of Theorem 5.3 proves that σ(A) ⊆ Cr(B) = {λ : |λ2 − 1| ≤ r(B)2 } ∪{λ : |λ − 1| ||λ| − 1| ≤ r(B)2 } ∪ {λ : |λ + 1| ||λ| − 1| ≤ r(B)2 } = {λ : |λ − 1| ||λ| − 1| ≤ r(B)2 } ∪ {λ : |λ + 1| ||λ| − 1| ≤ r(B)2 }, with



0 0 B= 1 0 1 1

 α 0 . 0

It is easy to check that r(B) = 1 if α = 1/2, and if α < 1/2, then r(B) < 1 and 0∈ / Cr(B) . Thus A is invertible in this case. Figure 1 shows Cr(B) with r(B) ≈ 0.915 for α = 0.4, and r(B) ≈ 0.231 for α = 0.01.

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2

2

1

1

0

0

-1

-1

-2

-2

-2

-1

0

1

2

-2

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-1

0

1

2

Figure 1. α = 0.4, α = 0.01

References [1] A. Brauer, Limits for the characteristic roots of a matrix. II. Duke Math. J. 14 (1947), 21-26. [2] D.G. Feingold; R.S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem. Pacific J. Math. 12 (1962), 1241-1250. [3] M.I. Gil’, Invertibility conditions and bounds for spectra of operator matrices. Acta Sci. Math. (Szeged) 67 (2001), 299-314. [4] T. Kato, Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966. ¨ [5] A. Ostrowski, Uber die Determinanten mit u ¨berwiegender Hauptdiagonale. Comment. Math. Helv. 10 (1937), 69-96. [6] W. Rudin, Functional analysis. McGraw-Hill Series in Higher Mathematics. McGrawHill Book Co., New York-D¨ usseldorf-Johannesburg, 1973. [7] H.N. Salas, Gershgorin’s theorem for matrices of operators. Linear Algebra Appl. 291 (1999), 15-36. [8] R.S. Varga, Gerˇsgorin and his circles. Springer Series in Computational Mathematics, 36. Springer-Verlag, Berlin, 2004. Gerd Herzog and Christoph Schmoeger Institut f¨ ur Analysis, Universit¨ at Karlsruhe D-76128 Karlsruhe Germany e-mail: [email protected] [email protected] Submitted: April 19, 2006 Revised: November 17, 2006

Integr. equ. oper. theory 57 (2007), 521–565 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040521-45, published online December 26, 2006 DOI 10.1007/s00020-006-1473-x

Integral Equations and Operator Theory

C ∗ -Algebras of Bergman Type Operators with Piecewise Continuous Coeﬃcients Yu. I. Karlovich and Lu´ıs V. Pessoa Abstract. The C ∗ -algebra An,m generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L2 (Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C ∗ -algebras associated with points z ∈ Π ∪ ∂Π and pairs (z, λ) ∈ ∂Π × R. Applying a symbol calculus for the abstract unital C ∗ -algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points z ∈ ∂Π being the discontinuity points of coeﬃcients. A symbol calculus for the C ∗ -algebra An,m is constructed and a Fredholm criterion for the operators A ∈ An,m is obtained. Mathematics Subject Classiﬁcation (2000). Primary 47A53, 47L15; Secondary 33C05, 47G10, 47L30. Keywords. Poly-Bergman and anti-poly-Bergman projections, local principle, limit operators, C ∗ -algebra, symbol calculus, Fredholmness, Gauss hypergeometric function.

1. Introduction Let Π = {z ∈ C : Im z > 0} be the open upper half-plane of the complex plain ˙ and C equipped with the Lebesgue area measure dA(z) = dxdy, Π = Π ∪ R ˙ R = R ∪ {∞}. Given a ﬁnite union L of Lyapunov curves in Π such that the set ˙ is ﬁnite, we denote by P C(L) the C ∗ -subalgebra of L∞ (Π) consisting of all L∩R continuous functions on Π \ L which have one-sided limits at the points of L. All the authors were partially supported by FCT project POCTI/MAT/59972/2004 (Portugal). The first author was also supported by PROMEP (M´exico). The second author was also supported by FCT (Portugal) and FSE for his stay and joint work with the first author in M´exico.

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Let B := B(L2 (Π)) be the C ∗ -algebra of all bounded linear operators on the Hilbert space L2 (Π) = L2 (Π, dA), K := K(L2 (Π)) the closed two-sided ideal of all compact operators in B, and let B π := B/K be the quotient C ∗ -algebra. Given numbers n, m ∈ N, let An,m be the C ∗ -subalgebra of B(L2 (Π)) generated by the poly-Bergman projections BΠ,1 , . . . , BΠ,n , by the anti-poly-Bergman proΠ,m , and by the multiplication operators aI (a ∈ P C(L)). In Π,1 , . . . , B jections B the present paper we construct a symbol calculus for the C ∗ -algebra An,m and obtain a Fredholm criterion for the operators A ∈ An,m . According to [19, Theorem 1.1], the poly-Bergman and anti-poly-Bergman projections are represented via the two-dimensional singular integral operator SΠ over the upper half-plane Π by the following formulas: n ∗ n ∗ n n Π,n = I − SΠ BΠ,n = I − SΠ (n ∈ N). (1.1) SΠ , B SΠ Π,n are orthogonal projections onto the poly-Bergman The operators BΠ,n and B 2 spaces An (Π) and anti-poly-Bergman spaces A2n (Π), respectively, where A2n (Π) and A2n (Π) are the Hilbert subspaces of L2 (Π) that consist of n-diﬀerentiable functions such that, respectively, (∂/∂z)n f = 0 and (∂/∂z)n f = 0 (see, e.g., [2], [7], [22]). According to [22, Theorem 4.5], the space L2 (Π) admits the following orthogonal decomposition: ∞ ∞ 2 2 2 A(k) (Π) L (Π) = A(k) (Π) (1.2) k=1

k=1

where the true poly-Bergman spaces of order n are deﬁned as ⊥ A2(n) (Π) = A2n (Π) ∩ A2n−1 (Π) for n > 1, A2(1) (Π) = A21 (Π) := A2 (Π), (1.3) and the true anti-poly-Bergman spaces of order n are deﬁned by ⊥ A2(n) (Π) = A2n (Π) ∩ A2n−1 (Π) for n > 1, A2(1) (Π) = A21 (Π) := A2 (Π). (1.4) The C ∗ -algebra generated by the Bergman projection of a bounded domain G with a smooth boundary ∂G and by multiplications by piecewise continuous functions having one-sided limits at the points of a ﬁnite union of curves intersecting ∂G at distinct points was investigated in [21]. A generalization of this paper to piecewise continuous coeﬃcients admitting more than two one-sided limits at points of ∂G was elaborated in [13]. In [9] we constructed a symbol calculus for the C ∗ -algebra A1,1 generated by the Bergman projection BΠ , by the anti-Bergman Π , and by the operators of multiplication by piecewise continuous projection B ˙ functions in P C(L) admitting two or more one-sided limits at the points z ∈ L∩ R, and also obtained a Fredholm criterion for the operators A ∈ A1,1 . The present paper generalizes the results of [9] to much more complicated C ∗ -algebras An,m . Making use of the Allan-Douglas local principle [5, Theorem 1.34], limit operators techniques (see, e.g., [4], [18]), and the Plamenevsky results [17] (also see [8]) on two-dimensional convolution operators with symbols admitting homogeneous discontinuities, we reduce the study to simpler C ∗ -algebras associated with the ˙ ∩ L) × R. While in [13] the study of local points z ∈ Π and pairs (z, λ) ∈ (R

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C ∗ -Algebras of Bergman Type Operators

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algebras is based on a symbol calculus constructed in [20] for the unital C ∗ -algebras generated by N orthogonal projections sum of which equals the unit and by 1 one-dimensional orthogonal projection, we apply a symbol calculus for the unital C ∗ -algebras generated by N orthogonal projections sum of which equals the unit and by M one-dimensional orthogonal projections, which was constructed in [9]. Such algebras are models of local algebras for An,m at points z ∈ ∂Π being the discontinuity points of coeﬃcients. The study of these local algebras essentially based on the relations [11] for the Gauss hypergeometric function

F − n, 1 − iλ; m; 1 − e

−2iθ

=

n (−n)k (1 − iλ)k k=0

(m)k k!

1 − e−2iθ

k

,

which is related in the case m = 2 to the Meixner-Pollaczek [1, p. 348] polynomials Pn1 (−λ; θ) = (n + 1)einθ F − n, 1 − iλ; 2; 1 − e−2iθ . The paper is organized as follows. In Section 2 following [8] we apply the Plamenevsky decomposition of the two-dimensional Fourier transform to convolution type operators with homogeneous data, introduce the poly-Bergman and anti-poly-Bergman spaces and projections, and also prove that the operator SΠ is a unitary isomorphism of the space A2(k) onto A2(k+1) and of the space A2(k+1) onto A2 for every k ∈ N, with SΠ A2 = {0}. (k)

(1)

In Section 3 we state the Allan-Douglas local principle and apply it to the quotient C ∗ -algebra Aπn,m = An,m /K, reducing the study to local C ∗ -algebras Aπn,m,z associated with the points z ∈ Π. In Section 4, by analogy with [13] and [9], we study the local algebras Aπn,m,z ˙ ∩ L we apply the limit operafor all z ∈ Π. In particular, in the case z ∈ R

α−1 and

α BΠ,n W tors techniques [4], [18] and study the form of the operators W −1

Wα BΠ,n Wα on the basis of [9] and (1.1), where α is a quasiconformal [12] shift

α is a unitary shift operator given by W

α f = |Jα |1/2 (f ◦ α), of Π onto itself, W and Jα is the Jacobian of α. Finally, we establish that the local algebras Aπn,m,z ˙ ∩ L are isomorphic to C ∗ -algebras of the form for z ∈ R Π,j , χR I : i = 1, 2, . . . , n; j = 1, 2, . . . , m; k = 1, 2, . . . , N , On,m,ω = alg BΠ,i , B k where ω = (ω1 , ω2 , . . . , ωN ), ωk are the angles between the rays outgoing from the origin and deﬁning sectors Rk ⊂ Π, and χRk are the characteristic functions of Rk . In Section 5, applying the results of Section 2 on convolution type operators ∗ with homogeneous data, we prove that the C -algebra On,m,ω is isomorphic to the ∗ 2 C -subalgebra An,m,N of Cb R, B(L (T+ )) generated by the operator functions (j) (λ), λ → χk I (i = 1, . . . , n; j = 1, . . . , m; k = 1, . . . , N ) λ → B(i) (λ), λ → B (j) (λ) are one-dimensional orwhere for every λ ∈ R the operators B(i) (λ) and B 2 thogonal projections acting on the space L (T+ ) and associated with the true polyBergman projections BΠ,(i) of L2 (Π) onto A2(i) (Π) and true anti-poly-Bergman

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Π,(j) of L2 (Π) onto A2 (Π), and χk are the characteristic functions projections B (j) of the arcs T+ ∩ Rk . The remaining part of this section is devoted to the veriﬁcation of applicability of the abstract symbol calculus established in [9, Theorem 8.1, Lemma 8.3] to the C ∗ -algebras An,m,N,λ ⊂ B(L2 (T+ )) (λ ∈ R) generated by the (j) (λ), χk I (i = 1, . . . , n; j = 1, . . . , m; k = 1, . . . , N ). In operators B(i) (λ), B this connection the crucial role belongs to the description of the norm one gener ators gλ,n (t) and gλ,n (t) of the spaces Im B(n) (λ) and Im B (n) (λ) in terms of the −2 , which is obtained in Gauss hypergeometric function F 1 − k, 1 − iλ; 1; 1 − t Theorem 5.4. In Section 6 we study the continuity on R and the asymptotics at ±∞ of the inner products χk gλ,j , χk gλ,r and χk gλ,j , χk gλ,r , prove the continuity of the matrix functions Mj (λ) on R = [−∞, +∞] and calculate their limits at ±∞, where Mj (λ) for j = 1, 2, . . . , n and j = n + 1, . . . , n + m describe the matrix parts (j) (λ), respectively. Here we also of the symbols of the operators B(j) (λ) and B calculate the inner products χk gλ,j , χk gλ,1 and χ 2,3, . . . k gλ,j , χk gλ,1 for j = −2iθ interms of the Gauss hypergeometric functions F 2 − j, 1 − iλ; 2; 1 − e and F 1 − j, 1 − iλ; 2; 1 − e−2iθ , respectively, and prove that for every j = 2, 3, ... −2iθ and and every θ ∈ (0, π), the obtained polynomials F 1 − j, 1 − iλ; 2; 1 − e F 2 − j, 1 − iλ; 2; 1 − e−2iθ do not have common zeros λ ∈ R. The latter allows us to prove in Section 7 that for every λ ∈ R the C ∗ -algebra An,m,N,λ is isomorphic to the C ∗ -algebra CN ⊕ CMN ×MN where M = n + m. Sections 7 contains the main results of the paper. Making use of the results of previous sections we construct a symbol calculus for the initial C ∗ -algebra An,m and obtain a Fredholm criterion for the operators A ∈ An,m .

2. Preliminaries and notation 2.1. C ∗ -algebra of convolution type operators with homogeneous data Let B(H) be the C ∗ -algebra of bounded linear operators acting on a Hilbert space H. Let H∞ be the C ∗ -subalgebra of essentially bounded positively homogeneous functions of order zero on R2 , that is, the functions a ∈ L∞ (R2 ) such that a|T ∈ L∞ (T) and a(tτ ) = a(τ ) for almost all τ ∈ T and all t > 0. Let R stand for the C ∗ -subalgebra of B(L2 (R2 )) generated by the multiplication operators A = aI (a ∈ H∞ ) and by the two-dimensional convolution operators F −1 bF (b ∈ H∞ ). Here F ∈ B(L2 (R2 )) is the Fourier transform deﬁned by 1 (F u)(x) = 2π

R2

u(t)e−ix·t dt,

x ∈ R2 ,

(2.1)

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where x · t is the scalar product of vectors x, t ∈ R2 , and F −1 is the inverse Fourier transform. We also consider the Mellin transform and its inverse given by

1 v(r)r−iλ dr, M : L2 (R+ , rdr) → L2 (R), (M v)(λ) = √ 2π R+

1 u(λ)riλ−1 dλ. M −1 : L2 (R) → L2 (R+ , rdr), (M −1 u)(r) = √ 2π R For an operator-valued R → B(L2 (T)), λ → L(λ), we denote by I ⊗λ L(λ) 2 function 2 the operator in B L (R) ⊗ L (T) given by the formula [(I ⊗λ L(λ))f ] (λ, t) = [L(λ)f (λ, ·)] (t),

(λ, t) ∈ R × T.

∗

(2.2)

2

Given λ ∈ R, we introduce the C -algebra Ωλ ⊂ B(L (T)) generated by the operators a(t)I and E(λ)−1 b(w)E(λ) where a, b ∈ L∞ (T) and E(λ) ∈ B(L2 (T)) are unitary operators deﬁned in [17] (see also [8] and [9]). Note that below we do not use the explicit form of the operators E(λ)±1 . Following [8], we consider the C ∗ -algebra Ω of bounded norm-continuous operator-valued functions U : R → B(L2 (T)), λ → U (λ) ∈ Ωλ with U = sup{U (λ) : λ ∈ R} and use the notation (U (λ))λ∈R for the function λ → U (λ). Passing to polar coordinates in the plane, we obtain the decomposition L2 (R2 ) = L2 (R+ , rdr) ⊗ L2 (T).

(2.3)

By [8, Proposition 2.4] (also see [17, Proposition 2.1]), we have the decomposition F = (M −1 ⊗ I)(V ⊗ I)(I ⊗λ E(λ))(M ⊗ I) where the tensor products M ±1 ⊗ I are taken relatively to (2.3) and V ∈ B(L2 (R)) is given by (V f )(λ) = f (−λ) for λ ∈ R. Taking in account the relations (M ⊗ I)(a(x)I)(M −1 ⊗ I) (M ⊗ I)(F −1 b(ξ)F )(M −1 ⊗ I)

= I ⊗ a(t)I, = I ⊗λ E(λ)−1 b(w)E(λ) ,

(2.4)

where x, ξ ∈ R2 and t, ω ∈ T, one can obtain the following. Proposition 2.1. [8, Proposition 2.5] The C ∗ -algebra R is isomorphic to a C ∗ subalgebra of Ω, and the isomorphism is given on the generators of R by aI → a(t)I λ∈R , F −1 bF → E(λ)−1 b(w)E(λ) λ∈R . 2.2. Poly-Bergman and anti-poly-Bergman spaces and projections Let U be a domain in C, and let A2n (U ) and A2n (U ) be the normed linear subspaces of L2 (U ) that consist of n-diﬀerentiable functions such that, respectively, (∂/∂z)n f = 0 and (∂/∂z)n f = 0, where 1 ∂ ∂ ∂ 1 ∂ ∂ ∂ = +i , = −i . ∂z 2 ∂x ∂y ∂z 2 ∂x ∂y The structure of the functions in A2n (U ) and A2n (U ) is as follows.

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Lemma 2.2. [2, p. 11] Let f be an n-diﬀerentiable function on a domain U ⊂ C. (i) If (∂/∂z)n f = 0, then there exist analytic functions ψk (k = 0, 1, . . . , n − 1) n−1 such that f = k=0 z k ψk ; (ii) If (∂/∂z)n f = 0, then there exist analytic functions ψk (k = 0, 1, . . . , n − 1) n−1 such that f = k=0 z k ψ k . From [7, Lemma 3.5.1] it follows that if K ⊂ U is a compact set, then there exists a constant CK < ∞ such that sup f (z) ≤ CK f L2 (U) , f ∈ A2n (U ) ∪ A2n (U ). (2.5) z∈K

If a sequence {fi } ⊂ A2n (U ) converges in L2 (U ) to a function f ∈ L2 (U ) then, by (2.5), it converges uniformly on all compact subsets of U , and therefore its limit f is an n-analytic function (see [2, Corollary 1.8]). Consequently A2n (U ) are closed subspaces of L2 (U ). As A2n (U ) = f : f ∈ A2n (U ) , the spaces A2n (U ) are also closed in L2 (U ). Thus A2n (U ) and A2n (U ) are Hilbert subspaces of L2 (U ), with the inner product induced by L2 (U ). Let U = Π. From (1.3) and (1.4) it follows that A2n (Π) =

n

A2n (Π) =

A2(k) (Π),

k=1

n

A2(k) (Π).

(2.6)

k=1

where the true poly-Bergman spaces A2(k) (Π) and true anti-poly-Bergman spaces A2(k) (Π) are deﬁned by (1.3) and (1.4), respectively, and their orthogonal sum equals L2 (Π). Π,n The poly-Bergman projections BΠ,n and anti-poly-Bergman projections B 2 2 2 are the orthogonal projections of L (Π) onto its subspaces An (Π) and An (Π), Π,1 are usual Bergman and anti-Bergman respectively. If n = 1, then BΠ,1 and B projections BΠ and BΠ . The true poly-Bergman projections BΠ,(n) and true antiΠ,(n) are the orthogonal projections of L2 (Π) onto the poly-Bergman projections B 2 2 spaces A(n) (Π) and A(n) (Π). By (2.6), BΠ,n =

n k=1

BΠ,(k) ,

Π,n = B

n k=1

Π,(k) , B

(2.7)

Π,(1) = B Π,1 , and whence BΠ,(1) = BΠ,1 , B BΠ,(n) = BΠ,n − BΠ,n−1 ,

Π,(n) = B Π,n − B Π,n−1 B

(n > 1).

(2.8)

The pairwise orthogonality of the spaces in (1.2) implies the following. Π,m = 0 holds, and for all Lemma 2.3. For all n, m ∈ N, the equality BΠ,n B Π,(m) = 0. Π,(n) B n, m ∈ N (n = m) we have the equalities BΠ,(n) BΠ,(m) = 0, B

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2.3. Relations between diﬀerent poly-Bergman and anti-poly-Bergman spaces Given a domain U ⊂ C, let SU and SU∗ be the two-dimensional singular integral operators acting on the space L2 (U ) and given by

f (w) f (w) 1 1 ∗ dA(w), (SU f )(z) = − dA(w). (SU f )(z) = − π U (w − z)2 π U (w − z)2 From the formula for the Fourier transform of the kernels of multidimensional singular integral operators (see, e.g., [14, Chapter X, p. 249]) it follows that SR2 = F −1 (ξ/ξ)F,

SR∗ 2 = F −1 (ξ/ξ)F,

(2.9)

where F is the two-dimensional Fourier transform acting on L2 (R2 ) by (2.1). Let us relate the spaces A2(k) , A2(k) to the spaces A2(k+1) , A2(k+1) . Theorem 2.4. For every k ∈ N, the operator SΠ is a unitary isomorphism of the ∗ is a space A2(k) onto A2(k+1) and of the space A2(k+1) onto A2(k) , the operator SΠ 2 2 2 unitary isomorphism of the space A(k+1) onto A(k) and of the space A(k) onto 2 ∗ A(1) = {0}. A2(k+1) , and SΠ A2(1) = {0}, SΠ Π BΠ,k = 0 by Lemma 2.3, we obtain Proof. Let k ∈ N and f, g ∈ A2(k) . Since B ∗ Π )f, g = f, g − B Π BΠ,k f, g = f, g . SΠ f, SΠ g = SΠ SΠ f, g = (I − B

Thus the operator SΠ is a unitary isomorphism of A2(k) onto its image. If f ∈ A2(k) , then Π )f = SΠ BΠ,(k) S ∗ SΠ f = BΠ,(k+1) SΠ f, SΠ f = SΠ BΠ,(k) f = SΠ BΠ,(k) (I − B Π 2 2 and consequently SΠ A(k) ⊂ A(k+1) . Conversely, if f ∈ A2(k+1) , then ∗ f ∈ SΠ A2(k) , f = BΠ,(k+1) f = SΠ BΠ,(k) SΠ whence A2(k+1) ⊂ SΠ A2(k) . Thus, (2.10) SΠ A2(k) = A2(k+1) , and, therefore, SΠ : A2(k) → A2(k+1) is a unitary operator. It is easily seen that the ∗ ∗ : A2(k+1) → A2(k) if and operator SΠ |A2(k) : A2(k+1) → A2(k) coincides with SΠ only if 2 ∗ (2.11) A(k+1) ⊂ A2(k) . SΠ

∗ , Let C denote the operator of complex conjugation, Cϕ = ϕ. Since CSΠ C = SΠ we conclude from (1.1) that Π,(k) and A2 = f : f ∈ A2 . CBΠ,(k) C = B (2.12) (k) (k) 2 Hence (2.11) is equivalent to the relation SΠ A(k+1) ⊂ A2(k) . Pick f ∈ A2(k+1) . Then ∗ Π,(k+1) f = SΠ SΠ Π,(k) SΠ f = B Π,(k) SΠ f, SΠ f = SΠ B BΠ,(k) SΠ f = (I − BΠ )B

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∗ which implies that SΠ A2(k+1) ⊂ A2(k) . Consequently, the adjoint operator SΠ maps the space A2(k+1) onto the space A2(k) , that is, 2 ∗ SΠ A(k+1) = A2(k) . (2.13) Furthermore, from (2.10), (2.13), and (2.12) it follows that 2 2 ∗ A(k) = CSΠ A2(k) = A2(k+1) , SΠ A2(k+1) = CSΠ A(k+1) = A2(k) . (2.14) Π f = I − S ∗ SΠ f, it Finally, let f ∈ A2(1) . Then from the equalities f = B Π ∗ follows that SΠ SΠ f = 0. Consequently, SΠ f = 0 because ∗ SΠ

∗ SΠ f 2L2 (Π) = SΠ f, SΠ f = SΠ SΠ f, f = 0. 2 2 ∗ Thus, SΠ A(1) = {0} and hence, by analogy with (2.14), SΠ A(1) = {0}.

From (1.2) and Theorem 2.4 we immediately deduce the following. Corollary 2.5. The space L2 (Π) admits the next orthogonal decomposition ∞ ∞ k 2 ∗ k 2 L2 (Π) = SΠ A (Π) . SΠ A (Π) k=0

k=0

∗ k 2 : A(j+k) → A2(j) Corollary 2.6. For every k ∈ N and every j ∈ N, the operators SΠ ∗ k k and SΠ : A2(j) → A2(j+k) are adjoint to the unitary operators SΠ : A2(j) → A2(j+k) 2 k A = {0} if and S k : A2 → A2 , respectively; and S k A2 = {0}, S ∗ Π

k ≥ j.

(j+k)

(j)

Π

(j)

Π

(j)

3. The Allan-Douglas local principle and the C ∗ -algebra An,m Let A be a unital C ∗ -algebra and Z a central C∗ -subalgebra of A containing the identity of A. Let M(Z) denote the maximal ideal space of Z. With every x ∈ M(Z) we associate the closed two-sided ideal Ix of A generated by the ideal x of Z. Consider the quotient C ∗ -algebra Ax := A/Ix and the canonical projection πx : A → Ax . Below we need the next part of the Allan-Douglas local principle (see, e.g., [6, Theorem 7.47], [5, Theorem 1.34]). Theorem 3.1. [Allan/Douglas] Let A be a unital C ∗ -algebra satisfying the conditions mentioned above. If a ∈ A, then a is invertible in A if and only if for every x ∈ M(Z) the coset ax := πx (a) is invertible in Ax . For every a ∈ A, a = max{ax : x ∈ M(Z)}. 2 ∗ ∞ Given operators B1 , . . . , Bn ∈ B(L (Π)) and a C -subalgebra A ⊂ L (Π), let alg B1 , .. . , Bn denote the C ∗ -algebra generated by the operators B1 , . . . , Bn , and let alg B1 , . . . , Bn ; A denote the C ∗ -algebra generated by the operators B1 , . . . , Bn and aI (a ∈ A). In case A = P C(L) we will write alg B1 , . . . , Bn ; L .

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Fix n, m ∈ N and consider the C ∗ -algebra Π,1 , . . . , B Π,m ; L ⊂ B(L2 (Π)). (3.1) An,m := alg BΠ,1 , . . . , BΠ,n , B The C ∗ -algebra An,m contains the C ∗ -algebra alg BΠ ; C(Π) generated by the Bergman projection BΠ and by the multiplications by continuous functions on Π = Π ∪ R ∪ {∞}, and thus, by [9, Remark 2.7], An,m contains the ideal K of all compact operators in B(L2 (Π)). To obtain a Fredholm criterion for the operators A ∈ An,m we need to study the invertibility of the cosets Aπ := A + K in the quotient C ∗ -algebra Aπn,m := An,m /K. To this end we will apply the Allan-Douglas local principle to the algebra Aπn,m over some its central C ∗ -subalgebra. From [14, Chapter X, Theorem 7.1] it follows that the singular integral operators SΠ and ∗ are of local type, that is, they commute with all operators aI a ∈ C(Π) to SΠ within compact operators. Thus, (1.1) immediately implies the following. Proposition 3.2. For every n ∈ N and every a ∈ C(Π), the commutators aBΠ,n − Π,n − B Π,n aI are compact on the space L2 (Π). BΠ,n aI and aB Proposition 3.2 shows that the poly-Bergman projec and anti-poly-Bergman tions are operators of local type and that Z π := cI + K : c ∈ C(Π) is a central subalgebra of the C ∗ -algebra Aπn,m . Obviously, the commutative C ∗ -algebra Z π is (isometrically) *-isomorphic to the C ∗ -algebra C(Π), and therefore the maximal ideal space of Z π can be identiﬁed with Π. For every point z ∈ Π, let Jzπ denote the closed two-sided ideal of the C ∗ -algebra Aπn,m generated by the maximal ideal Izπ := cI + K : c ∈ C(Π), c(z) = 0 ⊂ Z π . (3.2) According to [3, Proposition 8.6], the ideal Jzπ has the form Jzπ = (cA)π : c ∈ C(Π), c(z) = 0, A ∈ An,m .

(3.3)

Hence, with every z ∈ Π we associate the quotient C ∗ -algebra Aπn,m,z := Aπn,m /Jzπ . Applying Theorem 3.1, we immediately obtain the following criterion. Theorem 3.3. An operator A ∈ An,m is Fredholm on the space L2 (Π) if and only π π π ∗ if for every z ∈ Π the coset Azπ := A + J z is invertible in the quotient C -algebra π π An,m,z , and A = max Az : z ∈ Π . We say that cosets Aπ , B π ∈ Aπn,m are locally equivalent at a point z ∈ Π if z Aπ − B π ∈ Jzπ , and in that case we write Aπ ∼ B π . π π are locally equivalent to Lemma 3.4. For every n ∈ N, the cosets BΠ,n and B Π,n zero at every point z ∈ Π.

Proof. Fix z ∈ Π and choose a continuous function η with a compact support n−1 k ∗ k K ⊂ Π such that η(z) = 1. Since BΠ,n = k=0 SΠ BΠ,1 (SΠ ) in view of (1.1) and ∗ since the operators SΠ and SΠ are of local type, we infer that π z n−1 k π π ∗ k π z π SΠ ηBΠ,1 ηI (SΠ ) . ∼ ηBΠ,n ηI ∼ (3.4) BΠ,n k=0

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π z π As ηBΠ,1 ηI = 0π (see, e.g., [9, Lemma 3.3]), we deduce from (3.4) that BΠ,n ∼ Π,n can be treated in the same 0π . The case of anti-poly-Bergman projections B way. The further study of the local algebras requires to describe the set L of discontinuities for the coeﬃcients of operators A ∈ An,m . We assume that L satisﬁes the conditions: (L1) for each z ∈ Π ∩ L there exist numbers rz > 0 and nz ∈ N such that every disk D(z, r) of radius r ∈ (0, rz ) centered at z is divided by L into nz domains with z as a common limit point; (L2) for each z ∈ L∩R there exists a neighborhood Vz of z such that Vz ∩L consists of a ﬁnite number of Lyapunov arcs having only the point z in common and forming at this point pairwise distinct non-zero angles with R+ less than π; (L3) if ∞ ∈ L, then there exists a neighborhood V∞ of the point z = ∞ such that the set {−1/ζ : ζ ∈ V∞ ∩ L} consists of a ﬁnite number of Lyapunov arcs having only the origin in common and forming at this point pairwise distinct non-zero angles with R+ less than π. Thus, for a suﬃciently small neighborhood Vz of a z ∈ Π, the set Vz ∩ (Π \ L) consists of a ﬁnite number, say nz , of connected components Ωk (z) whose closures contain z.

4. Local algebras 4.1. Local algebras (easy cases) ˙ ∩ L). First we study the local algebras Aπn,m,z associated with points z ∈ Π \ (R π ˙ ∩ L. These algebras are organized simpler than the local algebras An,m,z for z ∈ R Consider the dilations dy (y > 0) and the translations th (h ∈ R) of Π given by dy (w) = yw, th (w) = w + h for all w ∈ Π. (4.1) With these shifts of Π onto itself we associate the unitary shift operators Wdy and Wth in B(L2 (Π)), acting on functions f ∈ L2 (Π) by the rules (Wdy f )(w) = yf (yw), Wth f (w) = f (w + h) for all w ∈ Π. (4.2) Π,n commutes From (1.1) it follows that every orthogonal projection BΠ,n and B with all dilation operators Wdy (y > 0) and all translation operators Wth (h ∈ R): = BΠ,n , Wdy BΠ,n Wd−1 y Wth BΠ,n Wt−1 = BΠ,n , h

Π,n W −1 = B Π,n (n = 1, 2, . . .); Wdy B dy −1 Π,n W = B Π,n (n = 1, 2, . . .). Wth B th

Given A ∈ B(L2 (Π)) such that the strong limits A0 := s-lim Wdy AWd−1 , A∞ := s-lim Wdy AWd−1 y y y→+0

y→+∞

(4.3)

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exist, we say that A0 and A∞ are the limit operators of A (with respect to the family Wdy y>0 as y → 0 and y → ∞, respectively). From the Banach-Steinhaus theorem it follows that A0 , A∞ ∈ B(L2 (Π)) and = A, A0 ≤ lim inf Wdy AWd−1 y y→+0 (4.4) = A. A∞ ≤ lim inf Wdy AWd−1 y y→+∞

Let Λ be the C ∗ -algebra of all operators of local type in B(L2 (Π)) and let Λ := Λ/K. To every point z ∈ Π we assign the closed two-sided ideal Jzπ of Λπ generated by the maximal ideal Izπ of Z π , where Izπ is given by (3.2). By analogy with (3.3), Jπ := (cA)π : c ∈ C(Π), c(z) = 0, A ∈ Λ . (4.5) π

z

⊂ We also introduce the quotient C ∗ -algebras Λπz := Λπ /Jzπ . Obviously, If an operator A ∈ B(L2 (Π)) commutes with all dilation operators Wdy for y > 0, then A = A0 = A∞ . In that case from [9, Proposition 7.5] it follows that if Aπ ∈ Jτπ for τ = 0 or τ = ∞, then Aτ = 0 and hence A = 0. Thus, we obtain the following. Jzπ

Jzπ .

Proposition 4.1. If an operator A ∈ B(L2 (Π)) commutes with all dilation operators Wdy (y > 0) and Aπ ∈ Jτπ for τ = 0 or τ = ∞, then A = 0. Let δi,j stand for the Kronecker symbol and let Ck denote the C ∗ -algebra of complex-valued vectors x = (x1 , . . . , xk ) with the usual operations of addition and multiplication by complex scalars, with the entry-wise multiplication, the adjoint x1 , . . . , x ¯k ), and the norm x = max |x1 |, . . . , |xk | . x∗ = (¯ If two C ∗ -algebras A1 and A2 are (isometrically) *-isomorphic, we write ∼ A1 = A2 . The easy cases of local algebras are described in the next lemma. Lemma 4.2. For the C ∗ -algebra An,m given by (3.1), the following holds: (i) if z ∈ Π \ L, then Aπn,m,z ∼ = C; (ii) if z ∈ Π ∩ L, then Aπn,m,z ∼ = Cnz , where nz ∈ N is given by condition (L1); π ∼ Cn+m+1 . ˙ \ L, then A (iii) if z ∈ R = n,m,z

Proof. Cases (i) and (ii) are considered similarly to [9, Lemma 4.2] by applying Lemma 3.4. Fix z ∈ Π. By (L1), L divides the disk D(z, r) of radius r ∈ (0, rz ) centered at z into nz domains Ωk (z) (k = 1, 2, . . . , nz ) having the common limit point z, where nz = 1 and Ω1 (z) = D(z, r) in case (i). The C ∗ -algebra isomorphism of Aπn,m,z onto Cnz is given by (aI)πz → (a1 (z), a2 (z) . . . , anz (z)) (a ∈ P C(L)), Π,j )πz → (0, . . . , 0) (j = 1, . . . , m), (BΠ,i )πz → (0, . . . , 0) (i = 1, . . . , n), (B lim a(ζ) (k = 1, 2, . . . , nz ). ak (z) := ζ→z, ζ∈Ωk (z)

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˙ \ L. In this case any function a ∈ P C(L) is continuous at z, (iii) Let z ∈ R π z and thus (aI) ∼ a(z)I π . From relations (2.7) it follows that Π,(1) )π , . . . , (B Π,(m) )π , (I − P )π Aπn,m,z = alg (BΠ,(1) )πz , . . . , (BΠ,(n) )πz , (B z z z where P :=

n i=1

BΠ,(i) +

m j=1

Π,(j) . B

(4.6)

By Lemma 2.3, the cosets Π,(j) )π (j = 1, . . . , m) Pi := (BΠ,(i) )πz (i = 1, . . . , n), Pn+j := (B z and Pn+m+1 := (I − P )πz are self-adjoint idempotents satisfying the relations: Pi Pj = δi,j Pi (i, j = 1, . . . , n + m + 1),

P1 + P2 + . . . + Pn+m+1 = I.

π π π π , . . . , B π Let us check that the cosets BΠ,(1) , . . . , BΠ,(n) ,B are not Π,(1) Π,(m) , P equivalent to zero at the points z ∈ R ∪ {∞}. On the contrary, suppose that z π there is a point z ∈ R such that BΠ,(i) ∼ 0π for some i = 1, . . . , n. Since every projection BΠ,(i) commutes with all translation operators Wth (h ∈ R) given by π 0 z π π (4.2), the relation BΠ,(i) ∼ 0 implies that BΠ,(i) = Wtz BΠ,(i) Wt∗z ∼ 0π . Because BΠ,(i) commutes with all dilation operators Wdy (y > 0), Proposition 4.1 implies ∞ π π that BΠ,(i) = 0, which is impossible. Analogously, if BΠ,(i) ∼ 0 , then again by Proposition 4.1 we arrive at the contradiction BΠ,(i) = 0. Thus, none of the cosets π ˙ Obviously, the same is true BΠ,(i) is locally equivalent to zero at some point z ∈ R. π . Finally, since the projection I − P , with P given by (4.6), for the cosets B Π,(j)

commutes with all dilation operators Wdy (y > 0) and all translation operators Wtz (z ∈ R), we deduce from Proposition 4.1 that P = I if the coset (I − P )π is ˙ which contradicts (1.2). locally equivalent to zero at a point z ∈ R, π ∼ ˙ \ L, and the correThus (see [9, Lemma 4.1]), An,m,z = Cn+m+1 for z ∈ R sponding C ∗ -algebra isomorphism is deﬁned on the generators of the C ∗ -algebra Aπn,m,z by (BΠ,(i) )πz Π,(j) )πz (B

→ (δi,s )n+m+1 s=1

(I −

→

P )πz

(i = 1, 2, . . . , n),

→ (δn+j,s) )n+m+1 s=1

(j = 1, 2, . . . , m),

(δn+m+1,s) )n+m+1 , s=1

which completes the proof.

4.2. Local algebras at the points of R ∩ L Let B = B(L2 (Π)), K = K(L2 (Π)), and B π = {Aπ = A + K : A ∈ B}. Given an analytic bijection ϕ of Π onto itself, we consider the shift operator Wϕ ∈ B deﬁned by (Wϕ f )(z) = f (ϕ(z))ϕ (z), z ∈ Π. Obviously, µϕ : B → B, A → Wϕ AWϕ∗

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is a C ∗ -algebra isomorphism, and An,m,ϕ := µϕ (An,m ) is a C ∗ -algebra. As µϕ (K) = K, we can deﬁne the quotient C ∗ -algebra isomorphism π µπϕ : B π → B π , Aπ → Wϕ AWϕ∗ . ˙ and assume that ϕ(0) = z. In that case µπϕ (Jzπ ) = J0π in view of Fix z ∈ R (4.5). Since the C ∗ -algebras An,m and An,m,ϕ consist of local type operators, the sets π π π π π π A A n,m,z := A + Jz : A ∈ An,m , n,m,ϕ,0 := A + J0 : A ∈ An,m,ϕ are C ∗ -subalgebras of Λπz and Λπ0 , respectively. Analogously to [9, Lemma 7.1], applying the relation Jzπ ∩ Aπn,m = Jzπ , we obtain the following. ˙ Lemma 4.3. Let ϕ : Π → Π be an analytic bijection and ϕ(0) = z where z ∈ R. π π Then An,m,z ∼ = A n,m,ϕ,0 and the corresponding isomorphism has the form π π : Aπn,m,z → A σϕ,z n,m,ϕ,0 ,

Aπ + Jzπ → µπϕ (Aπ ) + J0π .

(4.7)

Fix a point z ∈ R ∩ L and choose a neighborhood Vz ⊂ Π satisfying condinz tion (L2) imposed on L. Then Vz \ L = k=1 Ωk (z) where nz − 1 is the number of curves in L with the endpoint z and Ωk (z) are the connected components of Vz \ L. Let Lz denote the set of rays in Π outgoing from the origin and being the images (under the translation tangents to the curves of L at the point zζ →zζ − z) of the Rk where Rkz are sectors of Π with vertex at the origin z. Hence Π \ Lz = nk=1 which correspond to the domains Ωk (z) (the sets Ωk (z) and Rkz are numerated counterclockwise). Along with the C ∗ -algebra An,m we consider the C ∗ -algebra Π,1 , . . . , B Π,m ; Lz ⊂ B. Azn,m := alg BΠ,1 , . . . , BΠ,n , B z Since Z π is a central subalgebra of the C ∗ -algebra Az,π n,m := An,m /K, we conclude π z,π z,π ∗ π z π that An,m is a C -subalgebra of Λ and therefore A n,m,0 := A + J0 : A ∈ An,m is a C ∗ -subalgebra of Λπ0 = Λπ /J0π . Given a domain U ⊂ C and a homeomorphism α of U onto itself with piecewise continuous partial derivatives ∂α/∂z and ∂α/∂z, we deﬁne the unitary shift

α ∈ B(L2 (U )) by W

α f = |Jα |1/2 (f ◦ α) where Jα is the Jacobian of α. operator W Consider the closed upper semi-disk Dr+ := z ∈ C : |z| ≤ r, Im z ≥ 0 . Below we will use the following result established in [9, Lemmas 5.1 and 6.2].

Lemma 4.4. Given r > 0, let α be a bijection of Π onto itself such that α(z) = z for z ∈ Π \ Dr+ , α|Dr+ is a quasiconformal diﬀeomorphism of Dr+ onto itself, α(z) = z,

(∂α/∂z)(z) = 1,

and the derivatives ∂α/∂z, ∂α/∂z satisfy

α SΠ W

α−1 − SΠ π ∈ J0π , W

(∂α/∂z)(z) = 0

for

z ∈ R, Dr+ .

the H¨ older condition in ∗ −1 ∗ π

α SΠ W Wα − SΠ ∈ J0π .

(4.8) Then (4.9)

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If condition (L2) is fulﬁlled, then there exist an r > 0 and a bijection α : Π → Π possessing the properties mentioned above and satisfying the relations α t−z (Ωk (z)) ∩ Dr+ = Rkz ∩ Dr+ (k = 1, 2, . . . , nz ). (4.10) Note that in Lemma 4.4 the Jacobian Jα is positive and separated from zero in Dr+ . Similarly to [9, Lemma 7.2], applying Lemmas 4.3 and 4.4, we obtain the following. z,π and Lemma 4.5. If z ∈ R ∩ L and condition (L2) is fulﬁlled, then Aπn,m,z ∼ =A n,m,0 z,π is given by the isomorphism νzπ : Aπn,m,z → A n,m,0 π + Jzπ ) νzπ (BΠ,i π + J π ) ν π (B

az =

nz

z Π,j νzπ ((aI)π

k=1

z

= =

+ Jzπ ) =

ak (z)χRzk ,

π BΠ,i + J0π π + Jπ B 0

Π,j

(az I)π + J0π

ak (z) =

lim

ζ→z, ζ∈Ωk (z)

(i = 1, 2, . . . , n), (j = 1, 2, . . . , m),

(4.11)

(a ∈ P C(L)), a(ζ) (k = 1, 2, . . . , nz ).

(4.12)

4.3. Local algebra at the point ∞ ∈ L Let now z = ∞. Choose a neighborhood V∞ ⊂ Π satisfying condition (L3) imposed ∞ −1 lk where lk are Lyapunov arcs of L given by the on L. Then V∞ ∩ L = nk=1 equations ζ = fk (t), t ∈ (M, ∞), with |fk | = 1, and having the only common endpoint fk (∞) = ∞. Let L∞ denote the union of the rays γk := ζ = bk x : x ∈ [0, ∞] ⊂ Π (k = 1, 2, . . . , n∞ − 1) outgoing from the origin, where bk = fk (∞) := lim t−1 fk (t) = 0. Hence Π \ L∞ = t→∞ n∞ ∞ ∞ R where R are sectors of Π with vertex at the origin which correspond to k k k=1 connected components Ωk (∞) of V∞ \ L (the sets Ωk (∞) and Rk∞ are numerated counterclockwise). Along with the C ∗ -algebra An,m given by (3.1), we consider the C ∗ -algebra A∞ n,m := alg BΠ,1 , . . . , BΠ,n , BΠ,1 , . . . , BΠ,m ; L∞ ⊂ B. ∞ Since Z π is a central subalgebra of the C ∗ -algebra A∞,π n,m := An,m /K, we conclude ∞,π π that An,m ⊂ Λ and hence the set ∞,π := Aπ + Jπ : A ∈ A∞ A (4.13) n,m,∞ ∞ n,m π is a C ∗ -subalgebra of Λπ∞ = Λπ /J∞ .

Lemma 4.6. If ϕ : Π → Π is the analytic involution z → −1/z and h(z) = z 2 /z 2 , then ∗ ∗ Wϕ SΠ Wϕ = SΠ hI, Wϕ SΠ Wϕ = hSΠ . (4.14)

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Proof. Obviously, Wϕ = Wϕ−1 = Wϕ∗ . Let f be an inﬁnitely diﬀerentiable function with a compact support in Π. Direct computation shows that

1 h(w)f (w) dA(w) =: (If )(z) Wϕ SΠ Wϕ f (z) = − lim ε→0 π Π\σ (z) (w − z)2 ε where σε (z) := ϕ−1 Dε (−1/z) and Dε (z) = w ∈ C : |w − z| < ε . Fix δ > 0 such that Dδ (z) ⊂ Π. Because hf is a Lipschitz function within Π, we obtain

1 h(w)f (w) h(w)f (w) − h(z)f (z) 1 dA(w) − dA(w) (If )(z) = − 2 π Π\Dδ (z) (w − z)2 π Dδ (z) (w − z)

1 dA(w) − h(z)f (z) lim . ε→0 π D (z)\σ (z) (w − z)2 ε δ iθ Since ∂σε (z) : |w − z| < ε|wz| = z + ρε (θ)e : 0 ≤ θ ≤ 2π and = ∂ w ∈iθC ρε (θ) = εz z + ρε (θ)e , integrating in polar coordinates with center z gives

2π

2π dA(w) −2iθ = − e ln ρ (θ)dθ = − e−2iθ ln εz z + ρε (θ)eiθ dθ, ε 2 0 0 Dδ (z)\σε (z) (w − z) where the latter integral tends to zero as z → 0. Thus

1 h(w)f (w) h(w)f (w) − h(z)f (z) 1 (If )(z) = − dA(w) − dA(w) π Π\Dδ (z) (w − z)2 π Dδ (z) (w − z)2

1 h(w)f (w) = − lim dA(w) = SΠ (hf ) (z), ε→0 π Π\D (z) (w − z)2 ε which gives the ﬁrst formula in (4.14). Passing to adjoint operators gives the second formula in (4.14). ∞,π and the isomorphism Lemma 4.7. If ∞ ∈ L and (L3) holds, then Aπn,m,∞ ∼ =A n,m,∞ π π ∞,π is given by ν∞ : An,m,∞ → A n,m,∞ π π π ν∞ (BΠ,i + J∞ ) π π π ν (B + J )

∞ Π,j π ν∞ ((aI)π

a∞ =

n∞ k=1

+

∞ π J∞ )

ak (∞)χR∞ , k

=

π π BΠ,i + J∞ π + Jπ B

=

π (a∞ I) + J∞

=

∞

Π,j

ak (∞) =

π

lim

(i = 1, 2, . . . , n), (j = 1, 2, . . . , m),

(4.15)

(a ∈ P C(L)),

ζ→∞, ζ∈Ωk (∞)

a(ζ) (k = 1, 2, . . . , n∞ ). (4.16)

Proof. Consider the analytic bijection of Π onto itself given by ϕ(z) = −1/z. Since π ϕ(0) = ∞ and µπϕ (J∞ ) = J0π , from Lemma 4.3 it follows that π π π (4.17) Aπn,m,∞ ∼ =A n,m,ϕ,0 = A + J0 : A ∈ µϕ (An,m )

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π π where the isomorphism σϕ,∞ : Aπn,m,∞ → A n,m,ϕ,0 is given by π π π π σϕ,∞ (BΠ,i + J∞ ) = Wϕ BΠ,i Wϕ + J0π (i = 1, 2, . . . , n), π π π Π,j Π,j Wϕ π + J0π (j = 1, 2, . . . , m), (B + J∞ ) = Wϕ B σϕ,∞ π π π ((aI)π + J∞ ) = (a ◦ ϕ)I + J0π (a ∈ P C(L)). σϕ,∞

(4.18)

By (1.1) and (4.14), for every n = 1, 2, . . ., Wϕ BΠ,n Wϕ

∗ n = I − (SΠ hI)n (hSΠ ) ,

Π,n Wϕ Wϕ B

∗ n = I − (hSΠ ) (SΠ hI)n ,

(4.19)

where h(z) = z 2 /z 2 . Hence the C ∗ -algebra Aπn,m,ϕ has the form Π,j Wϕ (j = 1, . . . , n; j = 1, . . . , m); ϕ−1 (L) K. alg Wϕ BΠ,i Wϕ , Wϕ B Consider a bijection α : Π → Π given by Lemma 4.4 and satisfying the relations (4.20) α ϕ(Ωk (∞)) ∩ Dr+ = Rk∞ ∩ Dr+ (k = 1, 2, . . . , n∞ ). + + The restriction of α to Dr is a quasiconformal diﬀeomorphism onto Dr and α(0) =

−1 ∈ Jπ for every Y π ∈ Jπ . Since ϕ(L) satisﬁes condition (L2),

α Y π W 0. Hence W α 0 0 we infer from (4.20) that

α (a ◦ ϕ)W

−1 π + Jπ = (a∞ ◦ ϕ)I π + Jπ (a ∈ P C(L)), W (4.21) α 0 0 where a∞ is given by (4.16). Furthermore, (4.8) and Taylor’s formula with remainder in the Peano form imply that α(z) = z +ε(z)z for z ∈ Dr+ where lim ε(z) = 0. |z|→0

Therefore,

2 2 2 1 + ε(z) α(z) h(z) − (h ◦ α)(z) = z = 1− → 0 as |z| → 0. − z α(z) 1 + ε(z)

Thus, the function h − h ◦ α is continuous in Dr+ and vanishing at the origin, which π implies that (h − h ◦ α)I ∈ J0π . Hence, from (4.19) and (4.9) it follows that

α Wϕ BΠ,n Wϕ W

−1 − Wϕ BΠ,n Wϕ π ∈ Jπ , W α 0 (4.22)

α Wϕ B Π,n Wϕ W

α−1 − Wϕ B Π,n Wϕ π ∈ J0π W for every n = 1, 2, . . .. Taking into account (4.22) and (4.21), we conclude that the map π

α AW

α−1 π + J0π , ψα,∞ : Aπ + J0π → W π acting on the generators of the C ∗ -algebra A n,m,ϕ,0 by the rule π π π Wϕ BΠ,i Wϕ + J0π = Wϕ BΠ,i Wϕ + J0π (i = 1, 2, . . . , n), ψα,∞ π Π,j Wϕ π + J0π = Wϕ B Π,j Wϕ π + J0π (j = 1, 2, . . . , m), ψα,∞ Wϕ B π π ψα,∞ ((a ◦ ϕ)I)π + J0π = (a∞ ◦ ϕ)I + J0π (a ∈ P C(L)), (4.23)

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∞,π is an isomorphism onto the C ∗ -algebra A Aπ + J0π : A ∈ A∞ n,m,ϕ , n,m,ϕ,0 := where −1 A∞ B := alg W B W , W W (j = 1, . . . , n; j = 1, . . . , m); ϕ (L ) . ϕ Π,i ϕ ϕ Π,j ϕ ∞ n,m,ϕ Hence, ∼ ∞,π π A n,m,ϕ,0 = An,m,ϕ,0 .

(4.24)

π ) = J0π that the map It remains to observe from (4.19) and the equality µπϕ (J∞

∞,π → A ∞,π , µπϕ,∞ : A n,m,∞ n,m,ϕ,0

π Aπ + J∞ → µπϕ (Aπ ) + J0π

∞,π given by (4.13) onto A ∞,π , and is an isomorphism of the C ∗ -algebra A n,m,∞ n,m,ϕ,0 π π π µπϕ,∞ (BΠ,i + J∞ ) = Wϕ BΠ,i Wϕ + J0π (i = 1, 2, . . . , n), Π,j Wϕ π + Jπ (j = 1, 2, . . . , m), (4.25) π + Jπ ) = Wϕ B µπϕ,∞ (B Π,j ∞ 0 π π ) = (a∞ ◦ ϕ)I + J0π (a ∈ P C(L)). µπϕ,∞ ((a∞ I)π + J∞ Thus, ∼ ∞,π ∞,π A n,m,ϕ,0 = An,m,∞ .

(4.26)

Finally, from (4.17), (4.24), and (4.26) it follows that the C ∗ -algebras Aπn,m,∞ −1 π π ∞,π are isomorphic, and ν π := µπ and A ◦ ψα,∞ ◦ σϕ,∞ is the C ∗ -algebra n,m,∞ ∞ ϕ,∞ π ∞,π isomorphism of An,m,∞ onto An,m,∞ that acts by (4.15) according to (4.18), (4.23), (4.25). ˙ ∩L 4.4. Canonical form of local algebras related to R Let χU be the characteristic function of a set U . By Lemmas 4.5 and 4.7, z,π = alg B π + Jπ , B π + Jπ , (χRz I)π + Jπ : Aπn,m,z ∼ = A 0 0 0 n,m,0 Π,i Π,j k i = 1, . . . , n; j = 1, . . . , m; k = 1, . . . , nz , z ∈ R ∩ L, (4.27) π + Jπ , (χR∞ I)π + Jπ : ∞,π = alg B π + Jπ , B Aπn,m,∞ ∼ = A n,m,∞ ∞ ∞ ∞ Π,i Π,j k i = 1, . . . , n; j = 1, . . . , m; k = 1, . . . , n∞ , ∞ ∈ L. We introduce the operator C ∗ -algebras Π,1 , . . . , B Π,m , χR I : k = 1, . . . , N On,m,ω = alg BΠ,1 , . . . , BΠ,n , B k

(4.28)

where ω = (ω1 , ω2 , . . . , ωN ) and ωk are the angles of the sectors Rk with vertex z = 0. ˙ ∩ L, then the C ∗ -algebra Aπn,m,z is isomorphic to the C ∗ Theorem 4.8. If z ∈ R algebra On,m,ω of the form (4.28) where N = nz and Rk = Rkz (k = 1, 2, . . . , N ).

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˙ ∩ L. By (4.27), it is suﬃcient to prove that the C ∗ -algebra (4.28) Proof. Let z ∈ R z,π if z ∈ R and to the C ∗ -algebra A ∞,π if is isomorphic to the C ∗ -algebra A n,m,∞ n,m,0 z = ∞. z,π , A → Aπ + Jπ . Fix z ∈ R ∩ L and consider the map Ψ0 : On,m,ω → A 0 n,m,0 z,π . Suppose Obviously, Ψ0 is a C ∗ -algebra homomorphism of On,m,ω onto A n,m,0 z,π , that is, there exists a B ∈ On,m,ω such that Ψ0 (A) is invertible in A n,m,0

(AB − I)π ∈ J0π

and (BA − I)π ∈ J0π .

Then Proposition 4.1 implies the invertibility of A because all operators in On,m,ω commute with each dilation operator Wdy (y > 0). Thus A and Ψ0 (A) have the same spectral radius and consequently Ψ0 is isometric. ∞,π ∼ z,π ∼ Hence A n,m,0 = On,m,ω if z ∈ R ∩ L, and analogously An,m,∞ = On,m,ω .

5. C ∗ -algebra On,m,ω 5.1. An isomorphism of the C ∗ -algebra On,m,ω By Theorem 4.8 and Lemma 4.2(iii), the local algebras Aπn,m,z for z ∈ R ∪ {∞} are isomorphic to the C ∗ -algebras On,m,ω of the form (4.28). Let χΠ be the characteristic function of the upper half-plane Π. According Π,j , respectively, with the operators to (1.1) we identify the operators BΠ,i and B i j j i (5.1) χΠ I − χΠ SR2 χΠ I χΠ SR∗ 2 χΠ I , χΠ I − χΠ SR∗ 2 χΠ I χΠ SR2 χΠ I acting on the space L2 (R2 ). Then we conclude from (5.1) that On,m,ω is the C ∗ subalgebra of B(L2 (R2 )) generated by the two-dimensional singular integral operator SR2 and by the operators of multiplication by the functions χΠ , χRk ∈ H∞ (k = 1, 2, . . . , N ), where H∞ is the C ∗ -algebra of essentially bounded positively homogeneous functions of order zero on R2 . Since the functions ξ/ξ and ξ/ξ also are in H∞ , it follows from (2.9) that On,m,ω is a C ∗ -subalgebra of the C ∗ -algebra R deﬁned in Subsection 2.1. Hence, by Proposition 2.1, On,m,ω is isomorphic to a C ∗ -subalgebra of Ω. So, we infer from Proposition 2.1 and from formulas (5.1), (2.9) and (2.4) that n (λ) (5.2) Π,n (M −1 ⊗I) = I ⊗λ B (M ⊗I)BΠ,n (M −1 ⊗I) = I ⊗λ Bn (λ), (M ⊗I)B n (λ) ∈ B(L2 (T)) (n = 1, 2, . . .) are given by where the operators Bn (λ), B n ∗ n (λ)(SΠ ) (λ), Bn (λ) = χ+ I − SΠ

n (λ) = χ+ I − (S ∗ )n (λ)S n (λ), B Π Π

(5.3)

where χ+ is the characteristic function of the upper half-circle T+ and SΠ (λ) = χ+ E(λ)−1 (w/w)E(λ)χ+ I,

∗ SΠ (λ) = χ+ E(λ)−1 (w/w)E(λ)χ+ I. (5.4)

n (λ) By Proposition 2.1, the operator functions λ → Bn (λ) and λ → B are bounded and norm-continuous. The sectors Rk of the partition of Π by rays

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γ1 , . . . , γN −1 outgoing from the origin have the form Rk = z ∈ Π : θk−1 < arg z < θk (k = 1, 2, . . . , N ) where θ0 = 0 < θ1 < . . . < θN −1 < θN = π. Since (M ⊗ I)(χRk I)(M −1 ⊗ I) = I ⊗ χk I,

k = 1, 2, . . . , N,

(5.5)

where χk is the characteristic function of the arc T ∩ Rk , from (5.2) and (5.5) it follows that the C ∗ -algebra On,m,ω is isomorphic to the C ∗ -subalgebra of Ω generated by the following operator functions with values in B(L2 (T)): j (λ) Bi (λ) , B , χk I (i = 1, . . . , n; j = 1 . . . , m; k = 1, . . . , N ) λ∈R

λ∈R

λ∈R

This isomorphism is given on the generators of On,m,ω by the rule: → (Bi (λ))λ∈R (i = 1, 2, . . . , n), j (λ))λ∈R (j = 1, 2, . . . , m), → (B (5.6) (k = 1, 2, . . . , N ). χRk I → (χk I)λ∈R Now we return to the L2 (Π) context. Let Cb R, B(L2 (T+ )) denote the C ∗ algebra of bounded continuous operator functions R → B(L2 (T+ )). Thus we arrive at the following. BΠ,i Π,j B

∗ ∗ Theorem 5.1. Due to (5.6),2 the C -algebra On,m,ω is isomorphic to the C -subalgebra An,m,N of Cb R, B(L (T+ )) generated by the operator functions

j (λ), λ → χk I (i = 1, . . . , n; j = 1, . . . , m; k = 1, . . . , N ). λ → Bi (λ), λ → B Thus, for every ﬁxed λ ∈ R, we need to study the C ∗ -subalgebra j (λ), Qk : i = 1, . . . , n; j = 1, . . . , m; k = 1, . . . , N An,m,N,λ = alg Bi (λ), B (5.7) of B(L2 (T+ )) where Qk := χk I ∈ B(L2 (T+ )). To this end we apply the following result. Theorem 5.2. [9, Theorem 8.1, Lemma 8.3] Let H be a Hilbert space and let Qk , Pj (k = 1, 2, . . . , N ; j = 1, 2, . . . , M ) be orthogonal projections in B(H) such that (i) Qk Ql = 0 (k, l = 1, 2, . . . , N ; k = l), N (ii) k=1 Qk = I, (iii) Pj (j = 1, 2, . . . , M ) are one-dimensional projections, (iv) Pi Pj = 0 (i, j = 1, 2, . . . , M ; i = j), M (v) j=1 (Im Pj )⊥ ∩ Im Qk = {0} (k = 1, 2, . . . , N ), (vi) if v1 , . . . , vM are norm one generators of the spaces Im P1 , . . . , Im PM , respectively, then the vectors Qk v1 , . . . , Qk vM are linearly independent for every k = 1, 2, . . . , N . Let A denote the C ∗ -subalgebra of B(H) generated by the projections Qk and Pj , N let S = diag Sk k=1 where Sk are invertible matrices in CM×M that transform the systems νk = {Qk v1 , Qk v2 , . . . , Qk vM } of linearly independent vectors in H

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onto orthonormal systems, and let S be the C ∗ -subalgebra of CMN ×MN generated by the matrices k = diag δk,l IM N and S Pj S −1 (k = 1, 2, . . . , N ; j = 1, 2, . . . , M ), Q l=1 N M where Pj = diag δj,i i=1 El,k l,k=1 and 

El,k

Qk v1 , Qk v1

Qk v2 , Qk v1

···

  Qk v1 , Qk v2 Qk v2 , Qk v2 · · · :=   .. .. ..  . . . Qk v1 , Qk vM Qk v2 , Qk vM · · ·

Qk vM , Qk v1

Qk vM , Qk v2

.. .

    ∈ CM×M .  

Qk vM , Qk vM

Then the map σ, deﬁned on generators of A by k Qk → (δk,1 , δk,2 , . . . , δk,N ) ⊕ Q Pj → (0, 0, . . . , 0) ⊕ S Pj S −1

(k = 1, 2, . . . , N ), (j = 1, 2, . . . , M ),

extends to a C ∗ -algebra isomorphism of the C ∗ -algebra A onto the C ∗ -algebra CN ⊕ S. Clearly, for all k = 1, 2, . . . , N , the operators Qk = χk I are orthogonal pro jections on L2 (T+ ) satisfying the conditions: Qk Ql = δkl Ql , N k=1 Qk = I. Since Π,j = 0, we conclude that Π,j are projections on L2 (Π) for which BΠ,i B BΠ,i and B j (λ) are orthogonal projections on L2 (T+ ) such that Bi (λ)B j (λ) = 0. Bi (λ) and B According to (2.8) we put B(j) (λ) := Bj (λ) − Bj−1 (λ), B(1) (λ) := B1 (λ) := B(λ),

(j) (λ) := B j (λ) − B j−1 (λ) B (1) (λ) := B 1 (λ) := B(λ). B

(j > 1),

This and (5.3) imply that j−1 ∗ j−1 B(j) (λ) = SΠ (λ)B(λ)(SΠ ) (λ),

j−1 ∗ j−1 (j) (λ) = (SΠ B ) (λ)B(λ)S Π (λ), (5.8)

∗ where SΠ (λ), SΠ (λ) are given by (5.4) and all the operators are considered in 2 B(L (T+ )). By [13, Theorem 3.8] and [9, Lemma 9.3], B(λ) and B(λ) are one-dimensional orthogonal projections. Hence, by (5.8), B(j) (λ) and B(j) (λ) are one- or zerodimensional orthogonal projections. Since the operator functions λ → B(j) (λ) and (j) (λ) are norm-continuous due to Proposition 2.1, from [10, Chapter 1, λ → B (j) (λ) do not depend on Lemma 4.10] it follows that dim Im B(j) (λ) and dim Im B (j) (λ) are one-dimensional projections for all λ ∈ R λ, and thus B(j) (λ) and B Π,(j) would be the zero and all j ∈ N, because otherwise the operators BΠ,(j) or B projections for some j ∈ N, which is impossible.

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Setting M = n + m, Qk = χk I (k = 1, 2, . . . , N ), Pj = B(j) (λ) (j = (j−n) (λ) (j = n + 1, n + 2, . . . , M ), we see that the C ∗ 1, 2, . . . , n), and Pj = B algebra An,m,N,λ given by (5.7) coincides with the C ∗ -algebra A studied in Theorem 5.2. Obviously, the ﬁrst four conditions of these theorem are fulﬁlled. To apply Theorem 5.2 to the C ∗ -algebra An,m,N,λ it remains to verify the fulﬁllment of conditions (v) and (vi). Given j ∈ N and λ ∈ R, let gλ,j and gλ,j denote norm one generators of the (j) (λ), respectively. Fix k = 1, 2, . . . , N . Since the space spaces Im B(j) (λ) and Im B Im (χk I) is inﬁnite dimensional, there is a function g ∈ Im (χk I) such that ⊥ 0 = g ∈ χk gλ,i , χk gλ,j : i = 1, 2, . . . , n; j = 1, 2, . . . , m . Therefore, the relations gλ,j = g, χk gλ,j = 0, g, gλ,i = χk g, gλ,i = g, χk gλ,i = 0, g, gλ,j = χk g, M imply that 0 = g ∈ j=1 (Im Pj )⊥ ∩ Im Qk , and hence condition (v) is fulﬁlled. To check the fulﬁllment of condition (vi), we need to obtain explicit formulas (j) (λ). gλ,j of the images of operators B(i) (λ) and B for the generators gλ,j and (k) (λ) 5.2. Images of the operators B(k) (λ) and B Theorem 5.3. The functions fn (z) = (n + 1)(z + i)n (z + i)−n−2

(z ∈ Π, n = 0, 1, 2, . . .)

(5.9)

satisfy the relations SΠ f n

=

−fn+1 + f0

k SΠ f0

=

(−1)k (fk − fk−1 )

(n = 0, 1, 2, . . .),

(5.10)

(k = 1, 2, . . .). (5.11) Proof. Given z ∈ Π, we set Πε = Π \ Dε (z), where Dε (z) = w ∈ C : |w − z| < ε and 0 < ε < Im z. From (5.9) it follows that

fn (w) 1 dA(w) = I1 (z) + I2 (z), (5.12) (SΠ fn )(z) = − lim π ε→0 Πε (w − z)2

∂ fn (w) 1 lim dA(w), π ε→0 Πε ∂w w − z

(5.13) ∂ fn+1 (w) 1 lim I2 (z) := dA(w). π ε→0 Πε ∂w w−z To calculate the integrals in (5.13), we will apply, respectively, the Green formulas

1 ∂u (w) dA(w) = − u(ξ) dξ, u ∈ C 1 (U ), (5.14) 2i ∂U U ∂w

1 ∂u (w) dA(w) = u(ξ) dξ, u ∈ C 1 (U ), (5.15) ∂w 2i U ∂U where

I1 (z) :=

which are valid for bounded domains U with piecewise smooth boundaries ∂U .

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Calculating the derivative and passing to polar coordinates (r, θ) centered at z we conclude that 1 ∂ fn (w) ∂ fn (w) as w → ∞, =O 4 ∈ L1 (Πε , dA(w)). (5.16) ∂w w − z r ∂w w − z Deﬁne Πε,R = Πε ∩ DR (z) for R > 0. In view of (5.16), the Lebesgue dominated convergence theorem guarantees that

∂ fn (w) ∂ fn (w) dA(w) = lim dA(w). (5.17) R→+∞ Π ∂w w − z Πε ∂w w − z ε,R Applying (5.14) we obtain

∂ fn (w) dA(w) Πε,R ∂w w − z

1 1 fn (x) fn (w) fn (w) 1 dx − dw + dw =− 2i IR x − z 2i ΓR w − z 2i ∂Dε (z) w − z

(5.18)

where IR ∪ ΓR and ∂Dε (z) are oriented counterclockwise and IR = x ∈ R : |x − z| ≤ R , ΓR = w ∈ Π : |w − z| = R . The change of variable w → z + Reiθ and (5.9) imply that

2π fn (w) |z + Re−iθ + i|n 2π(n + 1)(R + |z| + 1)n dw dθ ≤ , ≤ (n + 1) |z + Reiθ + i|n+2 (R − |z| − 1)n+2 0 ΓR w − z whence

lim

R→+∞

ΓR

fn (w) dw = 0. w−z

(5.19)

Since fn (x) = (n + 1)f0 (x) for x ∈ R and since f0 ∈ L1 (R), we conclude that

fn (x) f0 (x) lim dx = (n + 1) dx. (5.20) R→+∞ I x − z x −z R R Thus, by (5.17) to (5.20), we get

1 ∂ fn (w) f0 (x) fn (w) n+1 dx + dw. dA(w) = − 2i 2i ∂Dε (z) w − z Πε ∂w w − z R x−z

(5.21)

Applying the Lebesgue dominated convergence theorem we obtain

2π

2π fn (w) dw = −i lim lim e−2iθ fn (z+εeiθ ) dθ = −i e−2iθ fn (z) dθ = 0, ε→0 ∂D (z) w − z ε→0 0 0 ε which together with (5.13) and (5.21) gives

n+1 f0 (x) dx. I1 (z) = − 2πi R x − z

(5.22)

Since the function ϕz (w) := f0 (w)/(w −z) = (w −z)−1 (w +i)−2 is analytic in a domain containing Π, with exception of the point w = z where ϕz has a pole of

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order 1, and since ϕz |R ∈ L1 (R) and |ϕz (w)| = O |w|−3 as w → ∞, we conclude that

f0 (x) f0 (w) dx = lim dw, + w − z R→+∞ x − z R [−R,R]∪TR where T+ R = z ∈ Π : |z| = R . For suﬃciently large R > 0, the latter integral is constant and, by the residue theorem, equals 2πif0 (z). Thus,

f0 (x) dx = 2πif0 (z), (5.23) R x−z and therefore we deduce from (5.22) that I1 (z) = −(n + 1)f0 (z).

(5.24)

By analogy with (5.21), applying (5.15) to integral I2 (z), we obtain

1 ∂ fn+1 (w) f0 (x) fn+1 (w) n+2 dx − dw. (5.25) dA(w) = ∂w w − z 2i x − z 2i Πε R ∂Dε (z) w − z

Applying again the Lebesgue dominated convergence theorem we get

2π

fn+1 (w) lim dw = i fn+1 (z) dθ = 2πifn+1 (z). ε→0 ∂D (z) w − z 0 ε

(5.26)

Combining this with (5.23) we infer from (5.25) and (5.13) that I2 (z) = (n + 2)f0 (z) − fn+1 (z).

(5.27)

Finally, (5.24), (5.27), and (5.12) give (5.10), which in its turn implies (5.11). Theorem 5.4. For every k ∈ N and every λ ∈ R, the functions k−1 (λ)gλ,1 gλ,k = SΠ

and

∗ k−1 gλ,k = (SΠ ) (λ) gλ,1

(5.28)

(k) (λ), respectively, and are norm one generators of the spaces Im B(k) (λ) and Im B gλ,k (t)

= (−1)k−1 G(λ) tiλ−1 F (1 − k, 1 − iλ; 1; 1 − t−2 ),

(5.29)

gλ,k (t)

= (−1)k−1 G(−λ) t−iλ+1 F (1 − k, 1 − iλ; 1; 1 − t2 ),

(5.30)

where t ∈ T+ , F (−m, b; c; z) is the (2, 1)-hypergeometric function given for b, z ∈ C, c ∈ C \ {0, −1, −2, . . . , −m + 1} and m = 0, 1, 2, . . . by F (−m, b; c; z) =

m (−m)n (b)n n z , (c)n n! n=0

with (a)0 = 1 and (a)n = a(a + 1) . . . (a + n − 1) for n = 1, 2, . . ., and 1/2 2λ G(λ) := if λ ∈ R \ {0}, G(0) := lim G(λ) = π −1/2 . λ→0 1 − e−2πλ

(5.31)

(5.32)

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Proof. By [13] and [9, Lemma 9.3], for every λ ∈ R the functions gλ,1 (t) = G(λ)tiλ−1 ,

gλ,1 (t) = g−λ,1 (t) = G(−λ)t−iλ+1 ,

t ∈ T+ ,

(5.33)

(1) (λ), respectively. are norm one generators of the spaces Im B(1) (λ) and Im B ∗ As the operators SΠ (λ) and SΠ (λ) are, respectively, isometries of the spaces (k) (λ) onto their images (cf. Theorem 2.4), the functions gλ,k Im B(k) (λ) and Im B and gλ,k given by (5.28) have norm one for all k ∈ N. Let us show that these func(k) (λ), respectively. Since tions are generators of the spaces Im B(k) (λ) and Im B B(λ)u = B(λ)u, gλ,1 gλ,1 = u, gλ,1 gλ,1

for every

u ∈ L2 (T+ ),

for functions g ∈ L2 (T+ ) we obtain B(k) (λ)g

= =

" ! k−1 k−1 ∗ k−1 ∗ k−1 SΠ (λ)B(λ)(SΠ ) (λ)g = SΠ (λ) (SΠ ) (λ)g, gλ,1 gλ,1 ! k−1 ! k−1 k−1 ∗ k−1 ) (λ)g, gλ,1 SΠ (λ)gλ,1 = g, SΠ (λ)gλ,1 SΠ (λ)gλ,1 (SΠ

and, analogously, ! ∗ k−1 (k) (λ)g = g, (S ∗ )k−1 (λ) B gλ,1 (SΠ ) (λ) gλ,1 . Π Thus, for every k ∈ N, k−1 Im B(k) (λ) = span SΠ (λ)gλ,1 ,

(k) (λ) = span (S ∗ )k−1 (λ) Im B gλ,1 , Π

so, the functions (5.28) are norm one generators of the spaces Im B(k) (λ) and (k) (λ). Im B As F 1 − k, 1 − iλ, 1, 1 − t2 = 1 for k = 1, we see from (5.33) that the formulas (5.29) and (5.30) are valid in case k = 1. To prove (5.29) and (5.30) for k > 1, we will apply the following formula (see [16, p. 18, (2.29)]):

(rt−1 + i)n −iλ n+1 (M ⊗ I)fn (λ, t) = √ r dr 2π R+ (rt + i)n+2 (5.34) (n + 1)tiλ−1 −2 = √ , B(1 − iλ, 1 + iλ)F − n, 1 − iλ; 2; 1 − t 2π iiλ+1 where (λ, t) ∈ R × T+ , the functions fn (n = 0, 1, . . .) are given by (5.9), M ⊗ I : L2 (R+ , rdr) ⊗ L2 (T+ ) → L2 (R) ⊗ L2 (T+ ), B(z, w) is the Beta function, and F −n, 1−iλ; 2; 1−t−2 is the (2, 1)-hypergeometric function (5.31). If n = 0, then from (5.33) and (5.34) it follows that gλ,1 (t) = C(λ) (M ⊗ I)f0 (λ, t), t ∈ T+ , (5.35) √ iλ+1 2π i C(λ) := G(λ), λ ∈ R, (5.36) B(1 − iλ, 1 + iλ)

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where G(λ) is given by (5.32). If the integer k > 1, then, taking into account (5.35), (2.2) and (5.11), for every (λ, t) ∈ R × T+ , we obtain " k−1 k−1 (λ) (M ⊗ I)f0 (λ, ·) (t) SΠ (λ)gλ,1 (t) = C(λ) SΠ k−1 = C(λ) (M ⊗ I)SΠ (5.37) f0 (λ, t) k−1 C(λ) (M ⊗ I) fk−1 − fk−2 (λ, t). = (−1) Hence, (5.34), (5.36) and (5.37) imply that k−1 C(λ)tiλ−1 B(1 − iλ, 1 + iλ)Ψk (λ, 1 − t−2 ) SΠ (λ)gλ,1 (t) = (−1)k−1 √ 2π iiλ+1 = (−1)k−1 G(λ)tiλ−1 Ψk (λ, 1 − t−2 ) (5.38) for k > 1, where

Ψk (λ, z) := kF 1 − k, 1 − iλ; 2; z − (k − 1)F 2 − k, 1 − iλ; 2; z ,

(5.39)

and the Gauss hypergeometric functions in (5.39) are deﬁned for all z ∈ C because they are polynomials for k = 2, 3, . . .. It follows from the formula (γ − α − 1)F α, β; γ; z + αF α + 1, β; γ; z = (γ − 1)F α, β; γ − 1; z (see (9.2.5) in [11, p. 242]) with α = 1 − k, β = 1 − iλ and γ = 2 that kF 1 − k, 1 − iλ; 2; z − (k − 1)F 2 − k, 1 − iλ; 2; z = F (1 − k, 1 − iλ; 1; z). (5.40) Combining (5.38), (5.39) and (5.40) we obtain (5.29) in case k > 1. ∗ = CSΠ C where Cϕ = ϕ and since gλ,1 = g−λ,1 by (5.33), Finally, since SΠ we get ∗ k−1 k−1 (λ) gλ,1 = CSΠ g−λ,1 = g−λ,k . (5.41) gλ,k = SΠ Hence, by (5.41) and (5.29), gλ,k (t) = = which gives (5.30).

(−1)k−1 G(−λ)t−iλ−1 F 1 − k, 1 + iλ; 1; 1 − t−2 (−1)k−1 G(−λ)t1−iλ F 1 − k, 1 − iλ; 1; 1 − t2 , t ∈ T+ ,

Suppose that for some k = 1, 2, . . . , N and some constants ai , bj ∈ C, n m χk (t) ai gλ,i (t) + bj gλ,j (t) = 0 for all t ∈ T+ . i=1 j=1 gλ,j admits an In view of Theorem 5.4, the linear combination ni=1 ai gλ,i + m j=1 bj analytic extension to the whole half-plane Π. Since this function identically equals zero on the arc eiθ : θk−1 < θ < θk ⊂ T+ , it equals zero for all t ∈ T+ . Therefore ai = bj = 0 for all i, j because the set gλ,i , gλ,j : i = 1, 2, . . . , n; j = 1, 2, . . . , m is gλ,j (i = 1, 2, . . . , n; j = 1, 2, . . . , m orthonormal in L2 (T+ ). Thus, the vectors gλ,i , are linearly independent on every arc T+ ∩ Rk , and hence condition (vi) is fulﬁlled.

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6. Properties of inner products 6.1. Continuity and asymptotics of inner products Fix n, m ∈ N, M = n+m, and N = 2, 3, . . .. In this section we study the properties of the inner products αkj,r (λ) = χk vλ,j , χk vλ,r

(k = 1, 2, . . . , N ; j, r = 1, 2, · · · , M ) (6.1) on L2 (T+ ), where χk is the characteristic function of the arc eiθ : θk−1 ≤ θ ≤ θk of T+ , 0 ≤ θk−1 < θk ≤ π for all k = 1, 2, . . . , N , with θ0 = 0 and θN = π, and vλ,j = gλ,j (j = 1, 2, . . . , n), vλ,j = gλ,j−n (j = n + 1, n + 2, . . . , n + m). (6.2) Proposition 6.1. For every k ∈ N and every λ ∈ C \ {−ik, −i(k − 1), · · · , 0},

k 1 k (−1)s −2(λ+is)θ −2λθ −2iθ k e 1−e dθ = − , (6.3) e 2i s=0 s s − iλ k k! 1 k (−1)s −2(λ+is)θ 1 e − = − e−2λθ if θ = 0, π, (6.4) 2i s=0 s s − iλ 2i (−iλ)k+1 k k λ k (−1)s −2isθ e lim = 1 − e−2iθ = 0 if θ ∈ (0, π). (6.5) λ→∞ i s s − iλ s=0 Proof. The formulas (6.3) and (6.5) are evident. We claim that k k (−1)s k! = . s x+s (x)k+1 s=0

(6.6)

Clearly, (6.6) holds for k = 0. Assuming that (6.6) is valid for k, in the case k + 1 we get k+1 k k+1 k + 1 (−1)s k (−1)s k (−1)s = + s s x + s s=1 s − 1 x + s x+s s=0 s=0 k k k (−1)s k (−1)s k! k! (k + 1)! = − = − = , s x + s s=0 s x + s + 1 (x)k+1 (x + 1)k+1 (x)k+2 s=0 which proves (6.6). Clearly, (6.6) immediately implies (6.4).

Proposition 6.2. For j, r = 1, 2, . . ., j+r

(−1)

# j−1 r−1 1 if j = r, (1 − j)µ (1 − r)ν (µ + ν)! = 2 2 (µ!) (ν!) 0 if j = r. µ=0 ν=0

(6.7)

Proof. For every µ = 0, 1, . . . , j − 1, it follows that (1 − j)µ (2 − j)µ (2 − j)µ−1 = − µ! µ! (µ − 1)!

(6.8)

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where (2 − j)j−1 = 0 and, by deﬁnition, (2 − j)−1 /(−1)! = 0. Hence, for 0 ≤ ν ≤ j − 1, j−1 j−2 j−1 (1 − j)µ (µ + 1)ν (2 − j)µ (µ + 1)ν (2 − j)µ−1 (µ + 1)ν = − µ! ν! µ! ν! (µ − 1)! ν! µ=0 µ=0 µ=1 j−2 j−2 (2 − j)µ (µ + 1)ν − (µ + 2)ν (2 − j)µ (µ + 2)ν−1 =− = µ! ν! µ! (ν − 1)! µ=0 µ=0

where (µ + 2)−1 /(−1)! = 0 if ν = 0. Applying this recurrent formula, we infer that # j−1 j−1−ν (ν + 1 − j)µ (−1)j−1 (1 − j)µ (µ + 1)ν ν = (−1) = µ! ν! µ! 0 µ=0 µ=0

if ν = j − 1, if ν < j − 1.

Let r ≤ j. Applying the latter relation for each ν = 0, 1, . . . , r − 1, we obtain (−1)j+r

j−1 r−1 (1 − j)µ (1 − r)ν (µ + ν)! (µ!)2 (ν!)2 µ=0 ν=0

= (−1)j+r

j−1 r−1 (1 − j)µ (µ + 1)ν (1 − r)ν (1 − r)r−1 = (−1)r−1 = δj,r , µ! ν! ν! (r − 1)! µ=0 ν=0

which completes the proof because of the symmetry of the cases r ≥ j and r ≤ j. Lemma 6.3. The functions αkj,r given by (6.1)–(6.2) become continuous on R = [−∞,+∞] for all k = 1, 2, · · · , N and all j, r = 1, 2, · · · , M if to put αkj,r (±∞) = lim αkj,r (λ).

λ→±∞

Proof. Without loss of generality we assume that n ≥ m. Since gλ,j = g−λ,j according to (5.41), we conclude that χk gλ,j , χk gλ,r = χk g−λ,r , χk g−λ,j ,

χk gλ,p , χk gλ,j = χk gλ,j , χk gλ,p . (6.9)

Therefore, we only need to check the continuity on R for the functions λ → αkj,r (λ) = χk gλ,j , χk gλ,r ,

λ → αkj,n+p (λ) = χk gλ,j , χk gλ,p

for j, r = 1, 2, . . . , n; p = 1, 2, . . . , m and k = 1, 2, . . . , N . By (5.29) and by the relation tiλ−1 = tiλ+1 (t ∈ T), we get

θk gλ,j (t) gλ,r (t) |dt| αkj,r (λ) = χk gλ,j , χk gλ,r = θk−1

θk = (−1)j+r G2 (λ) e−2λθ F (1 − j, 1− iλ; 1; 1− e−2iθ )F (1 − r, 1 + iλ; 1; 1− e2iθ )dθ. θk−1

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Since the integrated function is continuous in λ ∈ R for every θ ∈ [0, π], it is suﬃcient to check the existence of ﬁnite limits of the integral as λ → ±∞. By (5.31) and (6.3), αkj,r (λ)

j+r

= (−1)

× (−1)ν

j−1 r−1 (1 − j)µ (1 − iλ)µ (1 − r)ν (1 + iλ)ν G (λ) (µ!)2 (ν!)2 µ=0 ν=0 2

θk

e−2(λ−iν)θ (1 − e−2iθ )µ+ν dθ

θk−1

= (−1)j+r G2 (λ) ×

j−1 r−1 (1 − j)µ (1 − iλ)µ (1 − r)ν (1 + iλ)ν (µ!)2 (ν!)2 µ=0 ν=0

µ+ν " θk (−1)s (−1)ν µ + ν e−2(λ+i(s−ν))θ . s −2i s=0 (s − ν) − iλ θk−1

(6.10)

First we assume that k = 2, 3, . . . , N − 1, that is, 0 < θk−1 < θk < π. Then we infer from (5.32) that for every θ ∈ (0, π), (1 − iλ)µ (1 + iλ)ν G2 (λ) 2λ(1 − iλ)µ (1 + iλ)ν e−2λθ = lim = 0. λ→±∞ λ→±∞ (s − ν) − iλ e2λθ (s − ν) − iλ 1 − e−2λπ (6.11) Thus, if 0 < θk−1 < θk < π, then we get lim χk gλ,j , χk gλ,r = 0. lim

λ→±∞

If k = 1, then applying (6.4) we deduce that (6.10) takes the form α1j,r (λ) = (−1)j+r G2 (λ) ×

j−1 r−1 (1 − j)µ (1 − iλ)µ (1 − r)ν (1 + iλ)ν (−1)ν (µ!)2 (ν!)2 −2i µ=0 ν=0

$ µ+ν % µ + ν (−1)s (µ + ν)! e−2(λ+i(s−ν))θ1 − . s (s − ν) − iλ (−iλ − ν)µ+ν+1 s=0

(6.12)

By (5.32), (6.7), (6.11) and (6.12), we obtain α1j,r (λ)

=

(−1)

lim α1j,r (λ)

=

0.

lim

λ→+∞ λ→−∞

j+r

j−1 r−1 (1 − j)µ (1 − r)ν (µ + ν) = δj,r , (µ!)2 (ν!)2 µ=0 ν=0

(6.13)

Similarly, if k = N , then we infer from (6.10) and (6.4) that j+r 2 αN G (λ) j,r (λ) = (−1)

$

j−1 r−1 (1 − j)µ (1 − iλ)µ (1 − r)ν (1 + iλ)ν (−1)ν (µ!)2 (ν!)2 −2i µ=0 ν=0

% µ+ν µ + ν (−1)s (µ + ν)! e−2λπ −2(λ+i(s−ν))θN −1 e − × . s (−iλ − ν)µ+ν+1 s=0 (s − ν) − iλ

(6.14)

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Consequently, (5.32), (6.7), (6.11) and (6.14) imply that lim αN j,r (λ)

=

(−1)j+r

lim αN j,r (λ)

=

0.

λ→−∞ λ→+∞

j−1 r−1 (1 − j)µ (1 − r)ν (µ + ν) = δj,r , (µ!)2 (ν!)2 µ=0 ν=0

(6.15)

By (5.29) and (5.30), for j = 1, 2, . . . , n; p = 1, 2, . . . , m and k = 1, 2, . . . , N , we get αkj,n+p (λ) = χk gλ,j , χk gλ,p = (−1)j+p G(λ)G(−λ)

θk e−2iθ F (1 − j, 1 − iλ; 1; 1 − e−2iθ )F (1 − p, 1 + iλ; 1; 1 − e−2iθ )dθ. × θk−1

Hence from (5.31) it follows that αkj,n+p (λ) = (−1)j+p G(λ)G(−λ)

j−1 p−1 (1 − j)µ (1 − iλ)µ (1 − p)ν (1 + iλ)ν θk × e−2iθ (1 − e−2iθ )µ+ν dθ 2 2 (µ!) (ν!) θ k−1 µ=0 ν=0 j−1 p−1 (1 − j)µ (1 − iλ)µ (1 − p)ν (1 + iλ)ν (µ!)2 (ν!)2 µ=0 ν=0 µ+ν+1 "θk 1 − e−2iθ .

= (−1)j+p G(λ)G(−λ) ×

1 2i(µ + ν + 1)

(6.16)

θk−1

By (6.16), the functions λ → αkj,n+p (λ) are continuous on R. Since due to (5.32), lim G(λ)G(−λ)(1 − iλ)µ (1 + iλ)ν = lim

λ→±∞

λ→±∞

λ(1 − iλ)µ (1 + iλ)ν = 0, sinh(λπ)

we conclude from (6.16) that lim αkj,n+p (λ) = 0 for all required k, j, p. Thus, all λ→±∞

the functions αkj,r (j, r = 1, . . . , M ; k = 1, . . . , N ) are continuous on R. In what follows we will write f (λ) ∼ g(λ) as λ → ±∞ if f (λ) = o g(λ) as λ → ±∞ if

f (λ)/g(λ) = 1, λ→±∞ lim f (λ)/g(λ) = 0. lim

λ→±∞

Lemma 6.4. For every j, r = 1, 2, . . . , n and every p = 1, 2, . . . , m, ∼

j+r−2 −2λθk−1 c+ e j,r,k λ

αkj,r (λ)

∼

αkj,n+p (λ)

∼

j+r−2 −2λ(θk −π) c− e j,r,k λ cj,p,k λj+p−1 sinh−1 (πλ)

α1j,r (λ)

∼

δj,r + o(1) as λ → +∞,

αkj,r (λ)

as

λ → +∞

(k = 2, 3, . . . , N ),

as

λ → −∞

(k = 1, 2, . . . , N − 1),

λ → ±∞

(k = 1, 2, . . . , N ),

as

αN j,r (λ)

∼ δj,r + o(1) as λ → −∞,

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where δj,r is the Kronecker symbol and j−1 r−1 (−1)j ij+r−2 1 − e−2iθk−1 1 − e2iθk−1 , (j − 1)!(r − 1)! j−1 r−1 (−1)j−1 ij+r−2 1 − e−2iθk c− 1 − e2iθk , j,r,k = (j − 1)!(r − 1)! j+p−1 "θk (−1)j ij+p−1 1 − e−2iθ cj,p,k = . 2(j − 1)!(p − 1)!(j + p − 1) θk−1

c+ j,r,k =

Proof. First assume that j, r = 1, 2, . . . , n and k = 2, 3 . . . , N − 1. Put θk− := min{θk−1 , θk },

θk+ := max{θk−1 , θk }.

Since 0 < θk−1 < θk < π, from (6.5) it follows that µ+ν "θk (−1)s (−1)ν µ + ν e−2(λ+i(s−ν))θ s −2i s=0 (s − ν) − iλ θk−1 ∓ µ+ν (−1)ν −2(λ−iν)θ∓ µ + ν (−1)s e−2isθk k e ∼ s −2i s − i(λ − iν) s=0 ∓ µ+ν (−1)ν −2(λ−iν)θ∓ k 1 − e−2iθk e −2λ ∓ ∓ µ ∓ ν 1 ∼ e−2λθk 1 − e−2iθk 1 − e2iθk −2λ

∼

as λ → ±∞.

On the other hand, since (−1)n−1 (1 − n)n−1 = (n − 1)!, we conclude that (−1)j+r G2 (λ) ∼

j−1 r−1 (1 − j)µ (1 − iλ)µ (1 − r)ν (1 + iλ)ν (µ!)2 (ν!)2 µ=0 ν=0

(−1)j−1 ij+r−2 2λj+r−1 (j − 1)!(r − 1)! 1 − e−2πλ

as λ → ±∞.

Then we infer from (6.10) that for all k = 2, 3, . . . , N − 1, j+r−2 −2λθk−1 αkj,r (λ) ∼ c+ e j,r,k λ

αkj,r (λ) ∼

as

j+r−2 −2λ(θk −π) c− e j,r,k λ

λ → +∞,

as λ → −∞.

If k = 1, then from (6.12) and (6.13) it follows that α1j,r (λ) ∼ δj,r + o(1) α1j,r (λ)

∼

j+r−2 −2λ(θ1 −π) c− e j,r,1 λ

as λ → +∞, as λ → −∞.

Analogously, if k = N , then (6.14) and (6.15) imply that + j+r−2 −2λθN −1 αN e j,r (λ) ∼ cj,r,n λ

αN j,r (λ)

∼ δj,r + o(1)

as

λ → +∞,

as

λ → −∞.

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Finally, we deduce from (6.16) and (5.32) that αkj,n+p (λ) ∼ cj,p,k

λj+p−1 sinh(πλ)

as λ → ±∞,

for all k = 1, 2, . . . , N ; j = 1, 2, . . . , n and p = 1, 2, . . . , m.

6.2. Matrices Mj (λ) Let n, m ∈ N, M = n + m, and N = 2, 3, . . .. According to Theorem 5.2 for every λ ∈ R we introduce the Gram matrices El,k (λ) ∈ CM×M given by   αkn+1,1 (λ) . . . αkn+m,1 (λ) αk1,1 (λ) . . . αkn,1 (λ)   .. .. .. .. .. ..   . . . . . .   k k  αk1,n (λ) . . . αkn,n (λ) αn+1,n (λ) . . . αn+m,n (λ)    El,k (λ) =    αk1,n+1 (λ) . . . αkn,n+1 (λ) αkn+1,n+1 (λ) . . . αkn+m,n+1 (λ)      .. .. .. .. .. ..   . . . . . . k k k k α1,n+m (λ) . . . αn,n+m (λ) αn+1,n+m (λ) . . . αn+m,n+m (λ) (6.17) for all l, k = 1, 2, . . . , N , where, by (6.1) and (6.2), αkj,r (λ) = χk gλ,j , χk gλ,r ,

αkj,n+p (λ) = χk gλ,j , χk gλ,p ,

αkn+p,j (λ) = χk gλ,p , χk gλ,j ,

αkn+p,n+q (λ) = χk gλ,p , χk gλ,q

for j, r = 1, 2, . . . , n and p, q = 1, 2, . . . , m. We also consider the matrices k = diag δk,l IM N Q (k = 1, 2, . . . , N ), l=1 M N Pj (λ) = diag δj,i i=1 El,k (λ) l,k=1 (j = 1, 2, . . . , M ).

(6.18)

(6.19) (6.20)

As we know from Subsection 5.2, for every k = 1, 2, . . . , N and every λ ∈ R, the set (6.21) χk vλ,1 , χk vλ,2 , . . . , χk vλ,M , with vλ,j given by (6.2), is a system of linearly independent vectors in L2 (T+ ). Applying the Gram-Schmidt orthogonalization process to the set (6.21) we obtain the orthonormal set ekλ,1 , ekλ,2 , . . . , ekλ,M ⊂ L2 (T+ ). Let S(λ) = diag{Sk (λ)}N k=1 where Sk (λ) ∈ CM×M are invertible matrices that transform the systems (6.21) onto the orthonormal systems ekλ,1 , ekλ,2 , . . . , ekλ,M . For every j = 1, 2, . . . , M , we also deﬁne the matrix functions Mj (λ) := S(λ)Pj (λ)S −1 (λ)

(λ ∈ R).

(6.22)

k S −1 (λ) for all k = 1, 2, . . . , N in virtue of (6.19). k = S(λ)Q Note that Q Let us check that Mj (·) extend to matrix functions continuous on R. First we will show that the functions λ → ekλ,s , vλ,j have ﬁnite limits at ±∞.

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Lemma 6.5. The functions λ → ekλ,s , vλ,j are continuous on R for all k = 1, 2, . . . , N and all s, j = 1, 2, . . . , M (= n + m); and # 1 if 1 ≤ s = j ≤ n, 1 N lim eλ,s , vλ,j = lim eλ,s , vλ,j = λ→+∞ λ→−∞ 0 otherwise, # 1 if n < s = j ≤ M, lim e1λ,s , vλ,j = lim eN λ,s , vλ,j = λ→−∞ λ→+∞ 0 otherwise, lim ekλ,s , vλ,j = 0

λ→−∞

if k = 2, 3, . . . , N − 1.

Proof. As is known (see, e.g., [15, Chapter I, § 5]), the elements ekλ,s ∈ L2 (T+ ) for all k = 1, 2, . . . , N and s = 1, 2, . . . , M can be represented in the form αk1,1 (λ) αk2,1 (λ) ... αks,1 (λ) .. .. .. .. 1 . . . . ekλ,s (t) = & k k k α (λ) α (λ) . . . α (λ) k k 1,s−1 2,s−1 s,s−1 Ds−1 (λ)Ds (λ) χk (t)vλ,1 (t) χk (t)vλ,2 (t) . . . χk (t)vλ,s (t) where t ∈ T+ , D0k (λ) = 1 and

αk1,1 (λ) .. . k k Dsk (λ) = Ds,s (λ) = 0, Ds,j (λ) = k α 1,s−1 (λ) αk (λ) 1,j

for s, j = 1, 2, . . . , M . Hence  0    k k Ds,j (λ) eλ,s , vλ,j = &   k  D (λ)Dk (λ) s−1

αk2,1 (λ) .. . αk2,s−1 (λ) αk2,j (λ)

... αks,1 (λ) .. .. . . . . . αks,s−1 (λ) ...

if

j = 1, 2, . . . , s − 1,

if

j = s, s + 1, . . . , M.

αks,j (λ) (6.23)

(6.24)

s

In particular, by (6.23) and (6.24), for every s = 1, 2, . . . , M , k e , vλ,s 2 = Dk (λ)/Dk (λ) = 0. λ,s s s−1

(6.25)

Clearly, by Lemma 6.3 and by (6.23) and (6.24), the functions λ → ekλ,s , vλ,j

are continuous on R for all k = 1, 2, . . . , N and all s, j = 1, 2, . . . , M . Let us study the asymptotics of the determinants Dsk (λ) (s = 1, 2, . . . , M ) as λ → ±∞. From Lemma 6.4 it follows that the diagonal blocks Ak1 (λ) = (αkr,j (λ))nj,r=1 ,

max{s,n}

Ak2 (λ) = (αkr,j (λ))j,r=n+1

are dominated in Dsk (λ) for all k = 1, 2, . . . , N as λ → ±∞. Moreover, the main diagonal of Ak1 (λ) is dominated in Ak1 (λ) if k = 1 and λ → +∞ or if k = N and λ → −∞, and the main diagonal of Ak2 (λ) is dominated in Ak2 (λ) if k = 1 and

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λ → −∞ or if k = N and λ → +∞. Therefore, if n < s ≤ M , then taking into account (6.9) we conclude that k Dsk (λ) ∼ Dnk (λ)Ds−n (−λ) as λ → ±∞.

(6.26)

Let τ = 1, 2 . . . , n and k (λ) := (−1)ν βµ,ν

"θk µ+ν (−1)s e−2(λ+i(s−ν))θ . s (s − ν) − iλ θk−1 s=0

µ+ν

Fix k = 2, 3, . . . , N − 1. Since αkj,r (λ) = χk gλ,j , χk gλ,r , from (6.23) and (6.10) it follows that     1 1 O 1 O   1 1   .. .. τ     k . . (λ) α     .. .. r,j=1    j,r . . 1 1  (6.27) O 1 O 1 1 τ −1 τ −1 τ −1 k (λ) ν,µ=0 diag (1 − iλ)r . βµ,ν = diag (−2i)−1 G2 (λ)(1 + iλ)r r=0

On the other hand, setting   µ−ν µ if (−1) γµ,ν := ν  0 if

r=0

µ ≥ ν,

0 = 1, 0

where

µ < ν,

we obtain τ −1 k τ −1 τ −1 γµ,ν µ,ν=0 βµ,ν (λ) ν,µ=0 γν,µ µ,ν=0 "θk τ −1 = (−1)µ−ν (µ − ν − iλ)−1 e−2(λ+i(µ−ν)θ)

θk−1 µ,ν=0

=: Mτk (λ).

(6.28)

τ −1 Since det γµ,ν µ,ν=0 = 1, we deduce from (6.27) and (6.28) that Dτk (λ) =

G2 (λ) −2i

τ det Mτk (λ)

τ+ −1

(1 + iλ)r (1 − iλ)r .

(6.29)

r=0

Finally, Lemma 6.4 and (6.29) imply that  , −1  (−1)τ τr=0 (r!)2 + λPτ,k (λ) e−2λτ θk−1 Dτk (λ) ∼  ,τ −1 (r!)2 + λP (λ) e2λτ (θk −π) τ,k r=0

as λ → +∞, as λ → −∞,

(6.30)

where Pτ,k (λ) are polynomial in λ. The same holds if k = 1 and λ → −∞ or k = N and λ → +∞. If k = 1 and λ → +∞ or k = N and λ → −∞, then Dτk (λ) → 1.

(6.31)

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Thus, we infer from (6.26), (6.30) and (6.31) that for every suﬃciently small ε > 0 and for every s = 1, 2, . . . , M ,  2λ(θk−1 +ε) min{s,n} −2λ(θk −π−ε) max{s−n,0}  o e if k = 2, 3, . . . , N − 1, e       1 + o(1) if k = 1, 1 ≤ s ≤ n,   1 −2λ(θ −π−ε)(s−n) 1 = o e if k = 1, n < s ≤ M, Dsk (λ)      o e2λ(θN −1 +ε)s if k = N, 1 ≤ s ≤ n,     N N if k = N, n ≤ s ≤ M, 1/Dn (λ) + o 1/Dn (λ) (6.32) as λ → +∞,  2λ(θk −π−ε) min{s,n} −2λ(θk−1 +ε) max{s−n,0}  o e if k = 2, 3, . . . , N − 1, e      2λ(θ −π−ε)s 1  ) if k = 1, 1 ≤ s ≤ n,   o(e 1 = 1/Dn1 (λ) + o 1/Dn1 (λ) if k = 1, n ≤ s ≤ M, Dsk (λ)      1 + o(1) if k = N, 1 ≤ s ≤ n,     −2λ(θN −1 +ε)(s−n) o e if k = N, n < s ≤ M, (6.33) as λ → −∞. From Lemma 6.4 it follows that for every suﬃciently small ε > 0, every s, j = 1, 2, . . . , M and every k = 2, 3, . . . , N − 1,   o e−2λ(θk−1 −ε) min{s,n} e2λ(θk −π+ε) max{s−n,0} as λ → +∞, k Ds,j (λ) =  o e−2λ(θk −π+ε) min{s,n} e2λ(θk−1 −ε) max{s−n,0} as λ → −∞. Analogously, for every suﬃciently small    δs,j + o(1)  1 Ds,j (λ) = o e−λ(π−ε)    2λ(θ1 −π+ε)(s−n) o e

ε > 0, if

1 ≤ s ≤ j ≤ n,

if

1 ≤ s ≤ n < j ≤ M,

if

n < s ≤ j ≤ M,

as λ → +∞,

 o e−2λ(θ1 −π+ε)s    1 (λ) = Ds,j o e−2λ(θ1 −π+ε)(s−1)+λ(π−ε)    δs,j Dn1 (λ) + o Dn1 (λ)

if

1 ≤ s ≤ j ≤ n,

if

1 ≤ s ≤ n < j ≤ M,

if

n < s ≤ j ≤ M,

as λ → −∞,

 o e−2λ(θN −1 −ε)s if    N Ds,j (λ) = o e−2λ(θN −1 −ε)(s−1)−λ(π−ε) if   N  N if δs,j Dn (λ) + o Dn (λ)

1 ≤ s ≤ j ≤ n, 1 ≤ s ≤ n < j ≤ M, n < s ≤ j ≤ M,

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as λ → +∞,  δsj + o(1) if 1 ≤ s ≤ j ≤ n,    N if 1 ≤ s ≤ n < j ≤ M, o eλ(π−ε) Ds,j (λ) =    o e2λ(θN −1 −ε)(s−n) if n < s ≤ j ≤ M, as λ → −∞. Therefore, we deduce from (6.24), (6.26), (6.32), (6.33) and from the formulas k for Ds,j (λ) that for every suﬃciently small ε > 0 and for all 1 ≤ s ≤ j ≤ M and k = 2, 3, . . . , N − 1, - −λ(θ −(4s−1)ε) k−1 if s = 1, 2, . . . , n, o e k eλ,s , vλ,j = −λ(π−θ −(4s−1)ε) k o e if s = n + 1, . . . , M, as λ → +∞, ekλ,s , vλ,j

- λ(π−θ −(4s−1)ε) k o e = λ(θ −(4s−1)ε) o e k−1

if

s = 1, 2, . . . , n,

if

s = n + 1, . . . , M,

as λ → −∞. Furthermore,

 δs,j + o(1) if    1 −λ(π−ε) o e if eλ,s , vλ,j =    −λ(π−θ1 −(4(s−n)−1)ε) if o e

1 ≤ s ≤ j ≤ n, 1 ≤ s ≤ n < j ≤ M, n < s ≤ j ≤ M,

as λ → +∞,

 o eλ(π−θ1 −(4s−1)ε)    e1λ,s , vλ,j = o e−λ(θ1 −(4s−2)ε)    δs,j + o(1)

if

1 ≤ s ≤ j ≤ n,

if

1 ≤ s ≤ n < j ≤ M,

if

n < s ≤ j ≤ M,

as λ → −∞,

 o e−λ(θN −1 −(4s−1)ε)    eN o e−λ(π−θN −1 −(4s−2)ε) λ,s , vλ,j =    δs,j + o(1)

if

1 ≤ s ≤ j ≤ n,

if

1 ≤ s ≤ n < j ≤ M,

if

n < s ≤ j ≤ M,

as λ → +∞,

 δs,j + o(1) if    N λ(π−ε) if o e eλ,s , vλ,j =   λ(θ  o e N −1 −(4(s−n)−1)ε) if

1 ≤ s ≤ j ≤ n, 1 ≤ s ≤ n < j ≤ M, n < s ≤ j ≤ M,

as λ → −∞, which gives the assertion of the lemma if to choose an appropriate small ε > 0.

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Theorem 6.6. For every j = 1, 2, . . . , M , the matrix function Mj (·) belongs to the space C(R, CMN ×MN ) if to put Mj (±∞) = lim Mj (λ) where

lim Mj (λ) = diag δj,s

λ→+∞

MN s=1

λ→±∞

,

MN lim Mj (λ) = diag δM(N −1)+j,s s=1 . (6.34)

λ→−∞

Proof. By (6.22) and [9, (8.19)], for η, τ = 1, 2, · · · , M N , the (η, τ )-entries of the matrices Mj (λ) are given by (6.35) Mj (λ) η,τ = S(λ)Pj (λ)S −1 (λ) η,τ = eλ,η , vλ,j eλ,τ , vλ,j , where eλ,M(k−1)+s = ekλ,s for all k = 1, 2, . . . , N and s = 1, 2, . . . , M . Therefore, from (6.24) and (6.35) we deduce that N Mj (λ) = Bjk,r (λ) k,r=1 (j = 1, 2, . . . , M ) (6.36) where the M × M blocks Bjk,r (λ) are  ekλ,1 , vλ,j erλ,1 , vλ,j

 ..  .   k r  Bjk,r (λ) =  eλ,j , vλ,j eλ,1 , vλ,j

 0   ..  . 0

given by · · · ekλ,1 , vλ,j erλ,j , vλ,j 0 .. .. .. . . . r k · · · eλ,j , vλ,j eλ,j , vλ,j 0 ··· 0 0 .. .. .. . . . ··· 0 0

··· .. . ··· ··· .. .

0 .. . 0 0 .. .

···

0

      .     (6.37)

The latter and Lemma 6.5 immediately imply (6.34).

6.3. Inner products: continuation Fix λ ∈ R and consider the characteristic function χ of an arc of T+ . Let us relate gλ,1 for j = 1, 2, . . . , M to the Gauss the inner products χgλ,j , χgλ,1 and χgλ,j , χ hypergeometric function. Straightforward calculations gives the following. Lemma 6.7. Let χ be the characteristic function of the arc eiθ : θ ∈ [θ1 , θ2 ] of T+ where 0 ≤ θ1 < θ2 ≤ π and θ2 − θ1 < π. Then for every j ∈ N,  −1  if j = 1,  π (θ2 − θ1 ) = 0 −2i(j−1)θ −2i(j−1)θ 2 1 χg0,j , χg0,1 = (6.38) −e j −1 e  if j = 2, 3, . . . ,  (−1) π 2i(j − 1) χg0,j , χ g0,1 = (−1)j π −1

e−2ijθ2 − e−2ijθ1 . 2ij

(6.39)

If j = 1, then from (6.38) and (6.39) it follows that χg0,j , χg0,1 = 0

⇐⇒ θ2 − θ1 = kπ/(j − 1) for k ∈ Z,

χg0,j , χ g0,1 = 0

⇐⇒ θ2 − θ1 = k π/j

for k ∈ Z.

(6.40)

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Hence, if 0 < θ2 − θ1 < π, then (6.40) implies that the relations χg0,j , χg0,1 = 0 and χg0,j , χ g0,1 = 0 hold simultaneously if and only if 0 < k/(j − 1) = k /j < 1 or, equivalently, k − k /j = k for some k, k ∈ Z \ {0} with 0 < k /j < 1, which is impossible. Thus, taking into account (6.38) for j = 1, we obtain the following. Corollary 6.8. If 0 ≤ θ1 < θ2 ≤ π and θ2 − θ1 < π, then for every j ∈ N, the inner products χg0,j , χg0,1 and χg0,j , χ g0,1 are not equal to zero simultaneously. Lemma 6.9. Let χ be the characteristic function of the arc eiθ : θ ∈ [θ1 , θ2 ] of T+ , where 0 ≤ θ1 < θ2 ≤ π and θ2 − θ1 < π. Then, for every λ ∈ R \ {0}, "θ2 (−1)j−1 G2 (λ) −2λθ e 1 − e−2iθ F 2 − j, 1 − iλ; 2; 1 − e−2iθ 2i θ1 (6.41) if j = 2, 3, . . .; and χgλ,j , χgλ,1 =

"θ2 (−1)j−1 G(λ)G(−λ) 1 − e−2iθ F 1 − j, 1 − iλ; 2; 1 − e−2iθ 2i θ1 (6.42) if j ∈ N, where G(λ) is given by (5.32). gλ,1 = χgλ,j , χ

Proof. Let λ ∈ R \ {0}. Applying (6.10) we deduce that for j = 2, 3, . . ., χgλ,j , χgλ,1

j−1 k (−1)j−1 G2 (λ) (1 − j)k (1 − iλ)k k (−1)s −2(λ+is)θ "θ2 e s s − iλ −2i (k!)2 θ1 s=0 k=0 j−1 j−1 " j−1 2 θ2 (1 − j) (1 − iλ) (−1) G (λ) k k . = (−1)s e−2(λ+is)θ −2i k!s!(k − s)!(s − iλ) θ1 s=0

=

(6.43)

k=s

On the other hand, setting (2 − j)k (1 − iλ)k /(k!) = 0 for k = −1,

(6.44)

we obtain "θ2 (−1)j−1 G2 (λ) −2λθ e 1 − e−2iθ F 2 − j, 1 − iλ; 2; 1 − e−2iθ 2i θ1 j−2 k+1 "θ2 j−1 2 k+1 (2 − j)k (1 − iλ)k (−1) G (λ) = (−1)s e−2(λ+is)θ s 2i (k + 1)!k! θ1 s=0 k=0

j−1

=

(−1)

2i

j−2 "θ2 (2 − j)k (1 − iλ)k −2(λ+is)θ . (−1) e k!s!(k + 1 − s)! θ1 s=0

G (λ) 2

j−1

s

k=s−1

(6.45)

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Since (2 − j)j−1 = 0 and (6.44) holds, from (6.8) it follows that j−1 (1 − j)k (1 − iλ)k = k!s!(k − s)!(s − iλ) k=s

% j−2 $ (2 − j)k (1 − iλ)k (2 − j)k (1 − iλ)k+1 − k!s!(k − s)!(s − iλ) k!s!(k + 1 − s)!(s − iλ)

k=s−1

= −

j−2 (2 − j)k (1 − iλ)k . k!s!(k + 1 − s)!

k=s−1

Combining the latter with (6.43) and (6.45), we obtain (6.41). Applying (6.16) we infer that for every j ∈ N, χgλ,j , χ gλ,1 =

j−1 k+1 "θ2 (−1)j−1 G(λ)G(−λ) (1 − j)k (1 − iλ)k 1 − e−2iθ , 2i (k + 1)!k! θ1 k=0

which immediately gives (6.42) in view of (5.31). Lemma 6.10. For every j = 2, 3, . . . and every θ ∈ (0, π), the polynomials F 1 − j, 1 − iλ; 2; 1 − e−2iθ and F 2 − j, 1 − iλ; 2; 1 − e−2iθ do not have common zeros λ ∈ R. Proof. Let us set Fn−j,θ (λ) := F n − j, 1 − iλ; 2; 1 − e−2iθ . By (5.40) and by γ(1 − z)F α, β; γ; z − γF α − 1, β; γ; z + (γ − β)zF α, β; γ + 1; z = 0

(see (9.2.6) in [11, p. 242]), we deduce that, respectively, F 1 − j, 1 − iλ; 1; 1 − e−2iθ (6.46) = jF1−j,θ − (j − 1)F2−j,θ , −2iθ −2iθ −2iθ e = F 1 − j, 1 − iλ; 1; 1 − e F 2 − j, 1 − iλ; 1; 1 − e (6.47) − iλ 1 − e−2iθ F2−j,θ . If j = 3, 4. . . ., then inserting (6.46) for j − 1 and j in (6.47), we get jF1−j,θ (λ) = aj,θ (λ)F2−j,θ (λ) + bj,θ F3−j,θ (λ), −2iθ

(6.48) −2iθ

(j − 1 − iλ) + (j − 1 + iλ), bj,θ = (2 − j)e . If j = 2, then F 2 − j, 1 − iλ; 2; 1 − e−2iθ = 1, and hence the assertion of the lemma holds. Let j ≥ 3 and, contrary to the mentioned assertion, suppose that aj,θ (λ) = e

F1−j,θ (λ) = 0 = F2−j,θ (λ)

for some λ ∈ R.

(6.49)

Fix this λ. Because bj,θ = 0, we deduce from (6.48) that F3−j,θ (λ) = 0. Hence, by induction, we infer from (6.48) that Fn−j,θ (λ) = 0 for all n = 1, 2, . . . , j, which is impossible because Fn−j,θ (λ) = 1 for n = j. Thus, (6.49) is not valid, which completes the proof.

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7. Symbol calculus and Fredholm criterion 7.1. The C ∗ -algebra SM,N,λ Fix λ ∈ R. Let M = n + m and let SM,N,λ be the C ∗ -subalgebra of CMN ×MN k (k = 1, 2, . . . , N ) and Mj (λ) (j = 1, 2, . . . , M ) given generated by the matrices Q by (6.19) and (6.22), respectively. Fix k = 1 and put vj = χ1 gλ,j (j = 1, 2, . . . , n),

vj = χ1 gλ,j−n (j = m + 1, m + 2, . . . , M ). (7.1)

Since the functions vj (j = 1, 2, . . . , M ) are linearly independent, applying the Gram-Schmidt orthogonalization process we obtain f1 = v1 ,

fj = vj −

j−1

vj , es es

(j = 2, 3, . . . , M ),

s=1

where ej := fj /fj for all j = 1, 2, . . . , M . As is known (8.15)]),  j−1  1   , v

− e , v

e , v

v  j i s j s i  fj s=1 . ej , vi =  f = f , f

= 0  j j j    0

(see, e.g., [9, (8.13)–

if

j < i,

if

j = i,

if

j > i.

(7.2)

Theorem 7.1. If λ ∈ R and (a) χ1 gλ,i , χ1 gλ,1 = 0 or χ1 gλ,i , χ1 gλ,1 = 0 for all i = 1, 2, . . . , n, and gλ,1 = 0 for all j = 1, 2, . . . , m, (b) χ1 gλ,j , χ1 gλ,1 = 0 or χ1 gλ,j , χ1 then the C ∗ -algebra SM,N,λ coincides with CMN ×MN , where M = n + m. Proof. Let El,j be the matrix in CMN ×MN with the only one non-zero entry (l, j). Since El,j = [E1,l ]∗ E1,j for all l, j = 1, 2, . . . , M N , we only need to prove that E1,j ∈ SM,N,λ for every j = 1, 2, . . . , M N . k ∈ SM,N,λ for every k = 1, 2, . . . , N , we conclude that Es,s ∈ SM,N,λ Since Q for all s = 1, 2, . . . , M N . On the other hand, we infer from (6.25) that ekλ,j , vλ,j erλ,j , vλ,j = 0 for all j = 1, 2, . . . , M and all k, r = 1, 2, . . . , N, whence, by (6.36) and (6.37), the entries Mj (λ) M(k−1)+j,M(r−1)+j = ekλ,j , vλ,j erλ,j , vλ,j = 0. Thus, the matrices EM(k−1)+j,M(r−1)+j belong to SM,N,λ for all j = 1, 2, . . . , M and all k, r = 1, 2, . . . , N . Therefore, the equality EM(k−1)+l,M(r−1)+j = EM(k−1)+l,l El,j Ej,M(r−1)+j implies that EM(k−1)+l,M(r−1)+j ∈ SM,N,λ for all k, r = 1, 2, . . . , N and all l, j = 1, 2, . . . , M if and only if El,j ∈ SM,N,λ for all l, j = 1, 2, . . . , M . Hence, it is

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suﬃcient to consider only the case k, r = 1, that is, to study the M × M block N N SM,N,λ 1,1 = diag δ1,ν IM ν=1 SM,N,λ diag δ1,ν IM ν=1 . Thus, it remains to prove that E1,j ∈ SM,N,λ 1,1 for every j = 1, 2, . . . , M . According to (6.35) we obtain Mj (λ) s,l = es , vj el , vj , (s, l = 1, 2, . . . , j; j = 1, 2, . . . , M ), (7.3) where vj are given by (7.1). From conditions (a)–(b) it follows that v1 , v1 = 0,

v1 , vn+1 = 0,

(7.4)

or vj , vn+1 = 0.

(7.5)

and for every j = 2, 3, . . . , M , vj , v1 = 0 −1

Since e1 , vj = v1 v1 , vj , we infer from (7.3) and (7.2) that the entry (7.6) Mj (λ) 1,j = e1 , vj fj = v1 −1 vj , v1 fj is not equal to zero if and only if vj , v1 = 0. Analogously, (7.3) implies that (7.7) Mn+1 (λ) n+1,j = en+1 , vn+1 ej , vn+1 = fn+1 ej , vn+1

if 1 ≤ j ≤ n, and Mj (λ) n+1,j = en+1 , vj ej , vj = vj , en+1 fj

(7.8)

if n + 1 ≤ j ≤ M . By (7.4) and (7.6),

E1,1 , E1,n+1 ∈ SM,N,λ 1,1 .

(7.9)

Suppose that vs , v1 = 0 for all s = 2, 3, . . . , j−1 with j ≤ n, but vj , v1 = 0. Then, by (7.5), vj , vn+1 = 0, and, by (7.6), E1,s ∈ SM,N,λ 1,1 for all s = 2, 3, . . . , j − 1. (7.10) If es , vj = 0 for some s = 2, 3, . . . , j − 1 , where j = 2, 3, . . . , n, then in view of (7.3) the (s, j)-entry es , vj fj of the matrix Mj (λ) is non-zero. Hence, in this case the matrix Es,j belongs to SM,N,λ 1,1 . Therefore, taking into account (7.10), we infer that E1,j = E1,s Es,j ∈ SM,N,λ 1,1 . If es , vj = 0 for all s = 1, 2, . . . , j − 1 with j ≤ n, then, according to (7.2), j−1 1 1 vj , vn+1 = 0. ej , vn+1 = es , vj es , vn+1 = vj , vn+1 − fj fj s=1 Hence, by (7.7), we obtain Mn+1 (λ) n+1,j = 0. Therefore, En+1,j ∈ SM,N,λ 1,1 . Consequently, taking into account (7.9), we again deduce that E1,j = E1,n+1 En+1,j ∈ SM,N,λ 1,1 .

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As a result,

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E1,j ∈ SM,N,λ 1,1

561

for all j = 1, 2, . . . , n.

(7.11)

Suppose now that vn+s , v1 = 0 for all s = 2, 3, . . . , p − 1 with p ≤ m, but vn+p , v1 = 0. Then, by (7.5), vn+p , vn+1 = 0, and, by (7.6), E1,n+s ∈ SM,N,λ 1,1 for all s = 2, 3, . . . , p − 1. (7.12) If es , vn+p = 0 for some s = 2, 3, . . . , n, then, by (7.3), the (s, n + p)-entry ofthe matrix Mn+p (λ) is non-zero. Hence, the matrix Es,n+p belongs to SM,N,λ 1,1 . Therefore, taking into account (7.11), we infer that E1,n+p = E1,s Es,n+p ∈ SM,N,λ 1,1 . If es , vn+p = 0 for all s = 1, 2, . . . , n, then, according to (7.2), n 1 es , vn+1 es , vn+p

en+1 , vn+p = vn+1 , vn+p − fn+1 s=1 =

fn+1 −1 vn+p , vn+1 = 0.

Hence, by (7.8), the (n + 1, n + p)-entry of the matrix Mn+p (λ) is non-zero. This means that the matrix En+1,n+p belongs to SM,N,λ 1,1 . Consequently, taking into account (7.9), we infer that E1,n+p = E1,n+1 En+1,n+p ∈ SM,N,λ 1,1 . As a result, E1,n+p ∈ SM,N,λ 1,1 for all p = 1, 2, . . . , m. (7.13) Finally, E1,j ∈ SM,N,λ 1,1 for all j = 1, 2, . . . , M in view of (7.12) and (7.13), which completes the proof. By the ﬁrst equality in (6.10) and by (6.38), - −1 −2λθ k − 1 − e−2πλ e if − e−2λθk−1 χk gλ,1 , χk gλ,1 = if π −1 θk − θk−1 According to (6.42) and (6.39) we also obtain  −2iθ λ  k  − e if − e−2iθk−1 2i sinh(πλ) χk gλ,1 , χk gλ,1 =   −(2iπ)−1 e−2iθk − e−2iθk−1 if

λ = 0,

(7.14)

λ = 0.

λ = 0,

(7.15)

λ = 0.

By (7.14), (7.15) and (6.9) (see also [9]), we conclude that for all k = 1, 2, . . . , N and all λ ∈ R, χk gλ,1 , χk gλ,1 = 0,

χk gλ,1 , χk gλ,1 = 0,

χk gλ,1 , χk gλ,1 = 0.

(7.16)

Lemma 7.2. If 0 = θ0 < θ1 < π, then conditions (a) and (b) of Theorem 7.1 are fulﬁlled for every λ ∈ R.

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Proof. By (7.16), conditions (a) and (b) hold for i = j = 1. Due to Corollary 6.8, these conditions are fulﬁlled for λ = 0 and all i = 1, 2, . . . , n and all j = 1, 2, . . . , m. Let j = 2, 3, . . . , n and λ ∈ R \ {0}. Since θ0 = 0, (6.41) and (6.42) imply that (−1)j−1 G2 (λ) −2λθ1 e 1− e−2iθ1 F 2 −j, 1− iλ; 2; 1− e−2iθ1 , χ1 gλ,j , χ1 gλ,1 = 2i (−1)j−1 G(λ)G(−λ) 1 − e−2iθ1 F 1 − j, 1− iλ; 2; 1 − e−2iθ1 . gλ,1 = χ1 gλ,j , χ1 2i If condition (a) is violated, we conclude that the system - F 2 − j, 1 − iλ; 2; 1 − e−2iθ1 = 0, F 1 − j, 1 − iλ; 2; 1 − e−2iθ1 = 0 has a solution λ ∈ R \ {0}. But this contradicts Lemma 6.10. Hence (a) holds for all j = 1, 2, . . . , n and all λ ∈ R. The fulﬁllment of condition (b) for j = 1, 2, . . . , m can be proved by analogy or can be reduced to the case (a) with the help of (6.9). Theorem 7.1 and Lemma 7.2 immediately imply the following. Corollary 7.3. For every λ ∈ R, the C ∗ -algebras SM,N,λ and CMN ×MN coincide. 7.2. Main results By Section 5, for the C ∗ -algebra An,m,N,λ given by (5.7) all the conditions of Theorem 5.2 are fulﬁlled. Thus, Theorem 5.2 and Corollary 7.3 give the following. Theorem 7.4. For every λ ∈ R the C ∗ -algebra An,m,N,λ is isomorphic to the C ∗ -algebra CN ⊕ CMN ×MN , and the isomorphism is given on the generators of An,m,N,λ by B(i) (λ) (j) (λ) B χk I

→ (0, 0, . . . , 0) ⊕ Mi (λ) → (0, 0, . . . , 0) ⊕ Mn+j (λ)

(i = 1, 2, . . . , n),

→ (δk,1 , δk,2 , . . . , δk,N ) ⊕ diag δk,j IM

N j=1

(j = 1, 2, . . . , m), (k = 1, 2, . . . , N ).

Theorems 4.8, 5.1, 7.4, and 6.6 imply the following. ˙ ∩ L is a common endpoint of nz − 1 arcs of L, then Theorem 7.5. If z ∈ R ∗ π the local C -algebra An,m,z is isomorphic to a C ∗ -subalgebra CN ⊕ Cz of CN ⊕ C R, CMN ×MN , where N = nz , M = n + m. The isomorphism is given by π BΠ,(i) z → 0, . . . , 0 ⊕ λ → Mi (λ) (i = 1, 2, . . . , n), π Π,(j) B → 0, . . . , 0 ⊕ λ → Mn+j (λ) (j = 1, 2, . . . , m), z π N aI z → a1 (z), . . . , aN (z) ⊕ λ → diag ak (z)IM k=1 , where ak (z) (k = 1, 2, . . . , N ) are the limits of the function a ∈ P C(L) at the point z within the connected components Ωk (z) of Vz ∩ (Π \ L), and the matrix functions Mj (·) ∈ C R, CMN ×MN are deﬁned by (6.36), (6.37), (6.24), (6.23), and (6.34).

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Lemma 4.2 and Theorem 7.5 give the complete description of the local algebras Aπn,m,z for all points z ∈ Π. Collecting all these descriptions together according to Theorem 3.3 and gathering the representations at points z ∈ Π that send the Π,(j) (j = 1, 2, . . . , m) into zero we establish operators BΠ,(i) (i = 1, 2, . . . , n) and B the main result of the paper. Theorem 7.6. The C ∗ -algebra / Π,(1) , . . . , B Π,(m) ; L K Aπn,m = alg BΠ,(1) , . . . , BΠ,(n) , B is isomorphic to the C ∗ -subalgebra Φ(Aπn,m ) of the C ∗ -algebra nz C CM ⊕ Cz , Ψn,m,L := ⊕ ˙ z∈R\L

z∈Π

˙ z∈R∩L

and the isomorphism Φ : Aπn,m → Φ(Aπn,m ) is given by π λ→ Mi (λ) , Φ BΠ,(i) := ei ⊕ 0, . . . , 0 ⊕ π := Φ B Π,(j) Φ (aI)π :=

z∈Π

0, . . . , 0

⊕

z∈Π

˙ z∈R\L

˙ z∈R\L

en+j

˙ z∈R∩L

⊕

λ → Mn+j (λ) ,

˙ z∈R∩L

a1 (z), . . . , anz (z) ⊕ a(z), . . . , a(z)

z∈Π

⊕

˙ z∈R\L

nz λ → diag ak (z)IM k=1 ,

˙ z∈R∩L

where i = 1, 2, . . . , n, j = 1, 2, . . . , m, M = n + m, ej = (0, . . . ,0, 1, 0, . . . , 0) ∈ CM with the unit at the j-entry, Cz is the C ∗ -subalgebra of C R, CMnz ×Mnz determined in Theorem 7.5, nz is the number of connected components Ωk (z) of the set Vz ∩ (Π \ L) for a suﬃciently small neighborhood Vz of a point z ∈ Π, a ∈ P C(L), and ak (z) :=

lim

ζ→z, ζ∈Ωk (z)

a(ζ) (k = 1, 2, . . . , nz ).

An operator A ∈ An,m is Fredholm on the space L2 (Π) if and only if its symbol Φ(Aπ ) is invertible in the C ∗ -algebra Ψn,m,L , that is, if [Φ(Aπ )](z) k = 0 for all z ∈ Π and all k = 1, 2, . . . , nz ; ˙ \ L and all j = 1, 2, . . . , M ; [Φ(Aπ )](z) j = 0 for all z ∈ R π ˙ ∩ L and all λ ∈ R, det [Φ(A )](z) (λ) = 0 for all z ∈ R where [Φ(Aπ )](z) k for z ∈ Π are the k-entries of the vector [Φ(Aπ )](z) ∈ Cnz ˙ \ L are the j-entries of the vector [Φ(Aπ )](z) ∈ CM . and [Φ(Aπ )](z) j for z ∈ R Note that from (2.7) and Theorem 7.6 we can easily obtain the symbols of Π,j in the C ∗ -algebra An,m . the operators BΠ,i and B

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References [1] G. E. Andrews, R. Askey, and R. Roy, Special Functions. Cambridge University Press, Cambridge, 1999. [2] M. B. Balk, Polyanalytic Functions. Akademie Verlag, Berlin, 1991. [3] A. B¨ ottcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkh¨ auser, Basel, 1997. [4] A. B¨ ottcher, Yu. I. Karlovich, and V. S. Rabinovich, The method of limit operators for one-dimensional singular integrals with slowly oscillating data. J. Operator Theory 43 (2000), 171–198. [5] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators. Akademie-Verlag, Berlin, 1989 and Springer-Verlag, Berlin, 1990. [6] R. G. Douglas, Banach Algebra Techniques in Operator Theory. Academic Press, New York, 1972. [7] A. Dzhuraev, Methods of Singular Integral Equations. Longman Scientiﬁc & Technical, 1992. [8] A. N. Karapetyants, V. S. Rabinovich, and N. L. Vasilevski, On algebras of two dimensional singular integral operators with homogeneous discontinuities in symbols. Integral Equations Operator Theory 40 (2001), 278–308. [9] Yu. I. Karlovich and L. Pessoa, Algebras generated by Bergman and anti-Bergman projections and by multiplications by piecewise continuous coeﬃcients. Integral Equations Operator Theory 52 (2005), 219–270. [10] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, 1984. [11] N. N. Lebedev, Special Functions and Their Applications. Prentice-Hall, Inc. XII, Englewood Cliﬀs, N.J., 1965. [12] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane. SpringerVerlag, New York, 1973. [13] M. Loaiza, Algebras generated by the Bergman projection and operators of multiplication by piecewise continuous functions. Integral Equations Operator Theory 46 (2003), 215–234. [14] S. G. Mikhlin and S. Pr¨ ossdorf, Singular Integral Operators. Springer-Verlag, Berlin, 1986. [15] M. A. Naimark, Normed Algebras. Wolters-Noordhoﬀ Publishing, Groningen, 1972. [16] F. Oberhettinger, Tables of Mellin Transforms. Springer-Verlag, Berlin, 1974. [17] B. A. Plamenevsky, Algebras of Pseudodiﬀerential Operators. Kluwer Academic Publ., 1989. [18] V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Birkh¨ auser, Basel, 2004. [19] J. Ram´ırez and I. M. Spitkovsky, On the algebra generated by the poly-Bergman projection and a composition operator. Factorization, Singular Operators and Related Problems, Proc. of the Conf. in Honour of Professor Georgii Litvinchuk, eds. S. Samko, A. Lebre, and A. F. dos Santos, Kluwer Academic Publ., Dordrecht, 2003, 273–289.

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[20] E. Ram´ırez de Arellano and N. L. Vasilevski, Bargmann projection, three-valued functions and corresponding Toeplitz operators. Contemporary Mathematics 212 (1998), 185–196. [21] N. L. Vasilevski, Banach algebras generated by two-dimensional integral operators with a Bergman kernel and piecewise continuous coeﬃcients. I. Soviet Math. (Izv. VUZ) 30 (1986), No. 2, 14–24. [22] N. L. Vasilevski, On the structure of Bergman and poly-Bergman spaces. Integral Equations Operator Theory 33 (1999), 471–488. Yu. I. Karlovich Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa C.P. 62209 Cuernavaca, Morelos M´exico e-mail: [email protected] Lu´ıs V. Pessoa Departamento de Matem´ atica Instituto Superior T´ecnico Av. Rovisco Pais, 1049 - 001 Lisboa Portugal e-mail: [email protected] Submitted: May 14, 2006

Integr. equ. oper. theory 57 (2007), 567–582 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040567-16, published online December 26, 2006 DOI 10.1007/s00020-006-1466-9

Integral Equations and Operator Theory

Backward Aluthge Iterates of a Hyponormal Operator Have Scalar Extensions Eungil Ko Abstract. The backward Aluthge iterate (defined below) of a hyponormal operator was initiated in [11]. In this paper we characterize the backward Aluthge iterate of a weighted shift. Also we show that the backward Aluthge iterate of a hyponormal operator has an analogue of the single valued extension property for W 2k+2 (D, H). Finally, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior in the plane. Mathematics Subject Classification (2000). 47B20, 47A11. Keywords. Backward Aluthge iterates, Aluthge transforms, subscalar operators, the property (β).

1. Introduction Let H be a complex Hilbert space, and denoted by L(H) the algebra of all bounded linear operators on H. If T ∈ L(H), we write σ(T ), σap (T ), and σp (T ) for the spectrum, the approximate point spectrum, and the point spectrum of T , respectively. An arbitrary operator T ∈ L(H) has a unique polar decomposition T = U |T |, 1 where |T | = (T ∗ T ) 2 and U is the appropriate partial isometry satisfying kerU = ker|T | = kerT and kerU ∗ = kerT ∗ . Associated with T is a related operator 1 1 |T | 2 U |T | 2 , called the Aluthge transform of T , and denoted throughout this paper by T˜ . For an arbitrary T ∈ L(H), the sequence {T˜ (n) } of Aluthge iterates of T is deﬁned by T˜(0) = T and T˜ (n+1) = (T˜ (n) )∼ for n ∈ N. The Aluthge transform T˜ of an operator T has been studied extensively, most often in connection with phyponormal, log-hyponormal, and w-hyponormal operators (deﬁned in section 3). In [9] Jung-Ko-Pearcy initiated a study of the Aluthge transform of an arbitrary bounded operator. They obtained there various spectral identities and showed that The work was supported by a grant (R14-2003-006-01000-0) from Korea Research Foundation.

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if T is a quasiaﬃnity, then the invariant subspace lattice Lat(T ) is nontrivial if and only if Lat(T˜ ) is nontrivial. Recall that an operator T ∈ L(H) is hyponormal if T ∗ T ≥ T T ∗ where T ∗ is the adjoint of T . The backward Aluthge iterate of a hyponormal operator was initiated in [11] and is deﬁned by the following; Definition 1.1. An operator T ∈ L(H) is called a backward Aluthge iterate of a hyponormal operator of order k if T˜ (k) is a hyponormal operator for some k ∈ N. An operator T ∈ L(H) is said to satisfy the single-valued extension property if for any open subset U in C, the function T − z : O(U, H) −→ O(U, H) deﬁned by the obvious pointwise multiplication is one-to-one where O(U, H) denotes the Fr´echet space of H-valued analytic functions on U with respect to uniform topology. If T has the single valued extension property, then for any x ∈ H there exists a unique maximal open set ρT (x) (⊃ ρ(T ), the resolvent set) and a unique H-valued analytic function f deﬁned in ρT (x) such that (T − z)f (z) = x,

z ∈ ρT (x).

An operator T ∈ L(H) is said to satisfy the property (β) if for every open subset G of C and every sequence fn : G −→ H of H-valued analytic function such that (T − z)fn (z) converges uniformly to 0 in norm on compact subset of G, fn (z) → 0 uniformly in norm on compact subsets of G. ¯ denote the closure of D Let D be an open disk in C the complex plane. Let D ¯ denote the space of continuous complex valued functions on in C. And let C m (D) ¯ with ﬁrst, second, . . ., and mth order partials that have continuous extensions D ¯ The notation f ∞ will be used to denote the sup norm of a function from D to D. ¯ Then deﬁne for f ∈ C m (D), ¯ f on D. f ≡ f ∞ + fx ∞ + fy ∞ + fxx∞ + · · · + fyy···y ∞ where for example fx denotes the partial with respect to the coordinate variable ¯ into a Banach space. Note that the pointwise product of x. This makes C m (D) m ¯ ¯ (in fact C m (D) ¯ with this norm is a two functions in C (D) is again in C m (D) topological algebra). A bounded linear operator S on H is called scalar of order m if for some open disc D in C there exists a map ¯ −→ L(H) Φ : C m (D) such that 1) Φ is an algebra homomorphism, and Φ is continuous when the above ¯ and the operator norm is placed on L(H), and 3) Φ(z) = S, norm is used on C m (D) where as usual z stands for the identity function on C, and 4) the constant one function is mapped to the identity operator on H. The map Φ is called a spectral resolution for S.

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Definition 1.2. An operator has a scalar extension of order m, i.e., is subscalar of order m if it is similar to the restriction of a scalar operator of order m to a closed invariant subspace. This paper has been divided into six sections. Section two deals with some preliminary facts. In section three, we characterize the backward Aluthge iterate of a weighted shift. In section four, we show that the backward Aluthge iterate of a hyponormal operator has an analogue of the single valued extension property for W 2k+2 (D, H). In section ﬁve, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior in the plane. In section six, we give some applications of the results in section ﬁve.

2. Preliminaries Let dµ(z) denote the planar Lebesgue measure. Fix a complex (separable) Hilbert space H and a bounded open disk D of C. We shall denote by L2 (D, H) the Hilbert space of measurable functions f : D → H, such that 1 f 2,D = { f (z)2dµ(z)} 2 < ∞. D

2

¯ = 0) The space of functions f ∈ L (D, H) which are analytic on D (i.e. ∂f is denoted by A2 (D, H) = L2 (D, H) ∩ O(D, H). A2 (D, H) is called the Bergman space for D. Note that A2 (D, H) is complete (i.e. A2 (D, H) is a Hilbert space). We denote by P the orthogonal projection of L2 (D, H) onto A2 (D, H). Let us deﬁne now a Sobolev type space, called W m (D, H) where D is a bounded disk in C and m is a ﬁxed non-negative integer. W m (D, H) will be the ¯ , ∂¯2 f, . . . , ∂¯m f in the space of those functions f ∈ L2 (D, H) whose derivatives ∂f 2 sense of distributions still belong to L (D, H). Endowed with the norm f 2W m =

m

∂¯i f 22,D

i=0 m

W (D, H) becomes a Hilbert space contained continuously in L2 (D, H). ¯ let Mf denote on W m (D, H) given by multiplication Now for f ∈ C m (D), by f . This has a spectral resolution of order m, deﬁned by the functional calculus ¯ −→ L(W m (D, H)), ΦM (f ) = Mf . ΦM : C m (D) Therefore, Mz is a scalar operator of order m.

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3. The backward Aluthge iterates of a weighted shift In this section, we characterize the backward Aluthge iterates of a weighted shift. Recall that an operator T ∈ L(H) is said to be p-hyponormal, 0 < p ≤ 1, if (T ∗ T )p ≥ (T T ∗)p where T ∗ is the adjoint of T . If p = 1, T is called hyponormal and if p = 12 , T is called semi-hyponormal. Semi-hyponormal operators were introduced by Xia (see [21]), and p-hyponormal operators for a general p, 0 < p < 1, have been studied by Aluthge. Any p-hyponormal operators are q-hyponormal if q ≤ p by L¨ owner’s theorem (see [16]). But there are examples to show that the converse of the above statement is not true (see [1]). In particular, Aluthge proved in [1] that if T is p-hyponormal with 0 < p < 12 , then T˜ is (p + 12 )-hyponormal and T˜˜ is hyponormal. If an operator T ∈ L(H) is invertible and log(T T ∗) ≤ log(T ∗T ), then T is called a log-hyponormal operator (see [20]). Since log : (0, ∞) → (−∞, ∞) is operator monotone, every invertible p-hyponormal operator is log-hyponormal. But it is known that there is a log-hyponormal operator which is not p-hyponormal (see Example 1.2 in [20]). Also if |T˜ | ≥ |T | ≥ |T˜ ∗ |, an operator T = U |T | is called a w-hyponormal operator. Recall that an operator T ∈ L(H) is a backward Aluthge iterate of a hyponormal operator of order k if T˜(k) is a hyponormal operator for some k ∈ N. We denote this class by BAIH(k). For example, BAIH(1) contains all semi-hyponormal operators and BAIH(2) contains all p-hyponormal (0 < p < 12 ), log-hyponormal, and w-hyponormal operators, etc (see [1], [2], and [20]). The following proposition and theorem provide that BAIH(k) has some good reasons for the future study. Proposition 3.1. The class of all semi-hyponormal operators forms a proper subclass of BAIH(1). Proof. If T = U |T | (polar decomposition) is semi-hyponormal in L(H), then it is easy to show that T ∈ BAIH(1). Consider the matrices 0 1 1 0 U= and |T | = . 1 0 0 2 Then T = U |T | is not semi-hyponormal. But since U ∗ |T |U = U |T |U ∗ , T˜∗ T˜ = T˜T˜∗ . Hence T ∈ BAIH(1). Remark 3.2. In general we can easily get that if U ∗ |T |U ≥ U |T |U ∗ , then T ∈ BAIH(1). Next we study the case of normality of T when T ∈ BAIH(1). Proposition 3.3. Let T ∈ BAIH(1) with kerT ⊂ kerT ∗ . If m(σ(T )) = 0 where m is the planar Lebesgue measure, then T is normal.

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Proof. Since T˜ is hyponormal and σ(T ) = σ(T˜ ) by [9], Putnam’s inequality implies that T˜ is normal. Since |T˜| = |T | = |(T˜ )∗ |, we get T˜|T |1/2 = |T |1/2 T˜ . Since |T |1/2 T˜ = T˜ |T |1/2 = |T |1/2 T, we obtain ((T˜ )∗ − T ∗ )|T |1/2 = 0. Hence (T˜ )∗ = T ∗ on ran|T |1/2 = ran|T |. Since ker|T | = kerT , by the hypothesis (T˜ )∗ = T ∗ = 0 on ker|T |. Hence (T˜)∗ = T ∗ . Thus T˜ = T , i.e., T is normal. The following theorem characterizes a weighted shift in BAIH(k). As results, we get examples of weighted shifts in BAIH(k). Theorem 3.4. Let T = U |T | (polar decomposition) be a weighted shift with weights {αn }∞ n=0 of positive real numbers. Then T ∈ BAIH(k) if and only if (

k

C

k

αkn+jj )1/2 ≤ (

j=0

k

j=0

C

k

j αkn+j+1 )1/2

for n = 0, 1, 2, . . .. Proof. We prove this theorem by induction. Let {en }∞ n=0 be an orthonormal basis ∗ 1/2 of a Hilbert space H. Since |T |en = (T T ) en = αn en for n = 0, 1, . . . and U en = 1/2 1/2 en+1 , we get that T˜en = αn αn+1 en+1 for n = 0, 1, . . .. Hence T ∈ BAIH(1) if 1/2 and only if (αn αn+1 ) ≤ (αn+1 αn+2 )1/2 for n = 1, 2, . . .. If k = 1, it is trivial from the above result. Assume that it is true when k = s, i.e., s s s s C Cj T ∈ BAIH(s) ⇐⇒ ( αsn+jj )1/2 ≤ ( αsn+j+1 )1/2 j=0

˜ (s)

n = 0, 1, 2, . . .. Then T

j=0

is a weighted shift with weights {(

s

j=0

C

s

αsn+jj )1/2 }∞ n=0 .

Let T˜ (s) = Us |T˜ (s) | be the polar decomposition of T˜ (s) . Since |T˜(s) |en = (

s

j=0

C

s

αsn+jj )1/2 en

and Us en = en+1 for n = 0, 1, . . ., we have T˜ (s+1) en = (

s

j=0

C

s+1

αsn+jj )1/2

(

s

j=0

C

s+1

j αsn+j+1 )1/2

for n = 0, 1, . . .. Since s Cj +s Cj−1 =s+1 Cj , we obtain (

s

j=0

C

αsn+jj )(

s

j=0

C

j αsn+j+1 )=

s+1 j=0

C

j αs+1 n+j .

en+1

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Hence T˜ (s+1) is a weighted shift with weights s+1

{(

j=0

C

s+1

j 1/2 αs+1 n+j )

}∞ n=0 .

Hence s+1

T ∈ BAIH(s + 1) ⇐⇒ (

j=0

C

s+1

j 1/2 αs+1 n+j )

s+1

≤(

j=0

C

s+1

j 1/2 αs+1 n+j+1 )

n = 0, 1, 2, . . ..

The following corollary shows that BAIH(k − 1) is properly contained in BAIH(k) for k = 1, 2, . . .. The proof follows from the direct calculation of Theorem 3.4. √ Corollary 3.5. Let T be a weighted shift with weights { x, 2/3, . . . , x n + 1/n + 2, . . .}. Then Tx ∈ BAIH(k) ⇐⇒ 0 < x ≤

k i + 1 k Ci−1 −k Ci k + 2 } }. { { i+2 k+3 i=1

Furthermore, Tx ∈ BAIH(k)\BAIH(k − 1) if and only if k−1

{

i=1

k i + 1 k−1 Ci−1 −k−1 Ci k + 1 i + 1 k Ci−1 −k Ci k + 2 } }<x≤ } }. { { { i+2 k+2 i+2 k+3 i=1

4. An analogue of the single valued extension property In this section we give an analogue of the single valued extension property for W 2k+2 (D, H) and an arbitrary operator T in BAIH(k). Notation 4.1. At this point, we set forth some notations. (1) T˜ (j) = Uj |T˜(j) | is the polar decomposition of T˜ (j) for j = 0, 1, . . . , k where T˜ (1) = T˜ , T˜ (0) = T and U0 = U . 0 1 1 1 1 (2) j=s |T˜ (j) | 2 := |T˜(s) | 2 · · · |T˜ | 2 |T | 2 . (3) D denotes a bounded open disk containing σ(T ) ∪ {0}. The following theorem shows that for every T ∈ BAIH(k), supp((T − z)f ) = supp(f ) holds for every f ∈ W 2k+2 (D, H). Theorem 4.2. If T is in BAIH(k), then the operator T − z : W 2k+2 (D, H) −→ W 2k+2 (D, H) is one-to-one.

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Proof. If f ∈ W 2k+2 (D, H) is such that (T − z)f = 0, then for i = 0, 1, . . . , 2k + 2 and r = 1, 2, . . . , k, we get 0

(T˜ (r) − z)

1

|T˜ (j) | 2 ∂¯i f = 0.

(4.1)

j=r−1

By [18, Corollary 2.2] with r = k, we have, for i = 0, 1, . . . , 2k, (I − P )

0

1

|T˜ (j) | 2 ∂¯i f = 0,

(4.2)

j=k−1

where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). Hence from (4.1) we obtain for i = 0, 1, . . . , 2k 0

(T˜(k) − z)P

1

|T˜ (j) | 2 ∂¯i f = 0.

j=k−1

Since T˜ (k) has the single valued extension property, we get that for i = 0, 1, . . . , 2k 0

1

|T˜ (j) | 2 ∂¯i f = P

j=k−1

0

1

|T˜ (j) | 2 ∂¯i f = 0.

(4.3)

j=k−1

Since 0

T˜ (k−1)

1

|T˜(j) | 2 ∂¯i f = 0,

(4.4)

j=k−2

from (4.1) we obtain zP

0

1

|T˜ (j) | 2 ∂¯i f = 0

(4.5)

j=k−2

for i = 0, 1, . . . , 2k. By [18, Corollary 2.2] with T = (0), (I − P )

0

1

|T˜ (j) | 2 ∂¯i f = 0,

(4.6)

j=k−2

for i = 0, 1, . . . , 2k − 2. From (4.5) and (4.6) we have zP

0

1

|T˜ (j) | 2 ∂¯i f = 0

j=k−2

for i = 0, 1, . . . , 2k − 2. By [7, Corollary 10.7], there exists a constant c > 0 such that 0 0 1 1 |T˜ (j) | 2 ∂¯i f 2,D ≤ zP |T˜ (j) | 2 ∂¯i f 2,D = 0 cP j=k−2

j=k−2

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for i = 0, 1, . . . , 2k − 2. Hence from (4.6) we have 0

1 |T˜ (j) | 2 ∂¯i f = 0.

(4.7)

j=k−2

Continue this process through (4.3) to (4.7). Then we get f = 0. Hence T − z is one-to-one. Corollary 4.3. If T = S + N where S is similar to R ∈ BAIH(k), SN = N S, and N m = 0, then T − z is one-to-one on W 2k+2 (D, H). Proof. If f ∈ W 2k+2 (D, H) is such that (T − z)f = 0, then (S − z)f = −N f.

(4.8)

Hence (S − z)N j−1 f = −N j f for j = 1, 2, . . . , m. We prove that N j f = 0 for j = 0, 1, . . . , m − 1 by induction. Since N m = 0, (S − z)N m−1 f = −N m f = 0. Since XRX −1 = S where R ∈ BAIH(k) and X is invertible, X(R − z)X −1N m−1 f = 0. Since X and R−z are one-to-one by Theorem 4.2, N m−1 f = 0. Assume that it is true when j = k, i.e., N k f = 0. From (4–8), we get (S − z)N k−1 f = −N k f = 0, i.e., X(R − z)X −1 N k−1 f = 0. Since X and R − z are one-to-one by Theorem 4.2, N k−1 f = 0. By induction we get f = 0. Corollary 4.4. Let T ∈ BAIH(k). If S = V T V ∗ where V is an isometry, then S − z is one-to-one on W 2k+2 (D, H). Proof. If f ∈ W 2k+2 (D, H) is such that (S − z)f = 0, then (T − z)V ∗ ∂¯i f = 0 for i = 0, 1, . . . , 2k + 2. From Theorem 4.2, we get V ∗ ∂¯i f = 0, hence V T V ∗ ∂¯i f = S ∂¯i f = 0 for i = 0, 1, . . . , 2k + 2. Thus z ∂¯i f = 0 for i = 0, 1, . . . , 2k + 2. By an application of [18, Proposition 2.1] with T = (0), we have (I − P )f 2,D = 0, where P denotes the orthogonal projection of L2 (D, H onto A2 (D, H). Hence zf = zP f = 0. From [7, Corollary 10.7], there exists a constant c > 0 such that cP f 2,D ≤ zf 2,D = 0. So f = P f = 0. Thus S − z is one-to-one.

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5. Scalar extensions In this section we show that the backward Aluthge iterates of a hyponormal operator have scalar extensions. 1

1

Proposition 5.1. Let T = U |T | (polar decomposition) be in L(H). If T˜ = |T | 2 U |T | 2 satisﬁes the property (β), then so does T .

Proof. Assume that T˜ satisﬁes the property (β). Let gn ∈ O(U, H) be such that 1 (T − z)gn (z) converges uniformly to 0 on compact subsets G of U . Since |T | 2 T = 1 1 T˜ |T | 2 , (T˜ − z)|T | 2 gn (z) converges uniformly to 0 for all z ∈ G. Since T˜ satisﬁes 1 the property (β), |T | 2 gn (z) converges uniformly to 0 for all z ∈ G. So zgn (z) converges uniformly to 0 for all z ∈ G as does T gn (z). Since (0) is hyponormal and hyponormal operators satisfy the property (β), gn (z) converges uniformly to 0 for all z ∈ G. Hence T satisﬁes the property (β). Next we prove our main theorem. Theorem 5.2. Every backward Aluthge iterate of a hyponormal operator of order k has a scalar extension of order 2k + 2, i.e., is a subscalar operator of order 2k + 2. Proof. Suppose that T is an arbitrary backward Aluthge iterate of a hyponormal operator of order k. Consider an arbitrary bounded open disk D in the complex plane C which contains σ(T ) and the quotient space H(D) = W 2k+2 (D, H)/(T − z)W 2k+2 (D, H) endowed with the Hilbert space norm. The class of a vector f or an operator on ˜ Let M be the operator of multipliH(D) will be denoted by f˜, respectively A. 2k+2 (D, H). As noted at the end of section 2, M is a scalar cation by z on W ˜ . Since operator of order 2k + 2 and has a spectral distribution Φ. Let S ≡ M 2k+2 ¯ 2k+2 (T − z)W (D, H) is invariant under every operator Mf , f ∈ C (D), we ˜ infer that S is a scalar operator of order 2k + 2 with spectral resolution Φ. Consider the natural map V : H −→ H(D) deﬁned by V h = 1 ⊗ h, for h ∈ H, where 1 ⊗ h denotes the constant function sending any z ∈ D to h. Then V T = SV . In particular ranV is an invariant subspace for S. In order to complete the proof, it suﬃces to show that V is one-to-one and has closed range. Let hn ∈ H and fn ∈ W 2k+2 (D, H) be sequences such that lim (T − z)fn + 1 ⊗ hn W 2k+2 = 0.

n→∞

(5.1)

It suﬃces to show that limn→∞ hn = 0. By the deﬁnition of the norm of Sobolev space (5.1) implies lim (U |T | − z)∂¯i fn 2,D = 0 (5.2) n→∞

1 1 for i = 1, 2, . . . , 2k + 2. Since T˜ = |T | 2 U |T | 2 , from (5.2) we induce 1

lim (T˜ − z)∂¯i (|T | 2 fn )2,D = 0

n→∞

(5.3)

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1 1 for i = 1, 2, . . . , 2k + 2. Furthermore, since T˜ (j) = |T˜ (j−1) | 2 Uj−1 |T˜ (j−1) | 2 for j = 1, . . . , k, from (5.3) we get that for i = 1, 2, . . . , 2k + 2 and j = 1, . . . , k 1 1 1 lim (T˜ (j) − z)∂¯i (|T˜ (j−1) | 2 · · · |T˜ | 2 |T | 2 fn )2,D = 0.

n→∞

(5.4)

Since T˜ (k) is hyponormal and 0

lim (T˜ (k) − z)∂¯i (

n→∞

1

|T˜ (j) | 2 fn )2,D = 0

(5.5)

j=k−1

for i = 1, 2, . . . , 2k + 2, it follows from [18, Corollary 2.2] that 0

lim (I − P )∂¯i (

n→∞

1

|T˜(j) | 2 fn )2,D = 0

(5.6)

j=k−1

for i = 0, 1, . . . , 2k where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). Then (5.5) and (5.6) imply 0

lim (T˜ (k) − z)P (

n→∞

1

|T˜(j) | 2 fn )2,D = 0

(5.7)

j=k−1

for i = 0, 1, . . . , 2k. Since T˜ (k) is hyponormal, it satisﬁes the property (β). So it is easy to show that 0 1 |T˜ (j) | 2 fn )2,D = 0 (5.8) lim P ∂¯i ( n→∞

j=k−1

for i = 0, 1, . . . , 2k. Hence from (5.6) and (5.8), we obtain 0

lim ∂¯i (

n→∞

1

|T˜ (j) | 2 fn )2,D = 0

(5.9)

j=k−1

for i = 0, 1, . . . , 2k. Since T˜(k−1) = Uk−1 |T˜ (k−1) | is the polar decomposition of T˜ (k−1) , from (5.9) we have 0

lim T˜ (k−1) ∂¯i (

n→∞

1

|T˜(j) | 2 fn )2,D = 0

(5.10)

j=k−2

for i = 0, 1, . . . , 2k. Substituting (5.10) into (5.4), we have lim z ∂¯i (

n→∞

0

1

|T˜ (j) | 2 fn )2,D = 0

(5.11)

j=k−2

for i = 0, 1, . . . , 2k. Applying [18, Corollary 2.2] with T = (0), we get lim (I − P )∂¯i (

n→∞

0

j=k−2

1

|T˜(j) | 2 fn )2,D = 0

(5.12)

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for i = 0, 1, . . . , 2k − 2 where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). Then from (5.4) and (5.12), we obtain lim (T˜ (k−1) − z)P ∂¯i (

n→∞

0

1

|T˜ (j) | 2 fn )2,D = 0

(5.13)

j=k−2

for i = 1, . . . , 2k − 2. Since T˜ (k−1) satisﬁes the property (β) by Proposition 5.1, we can easily show that lim P ∂¯i (

n→∞

0

1

|T˜ (j) | 2 fn )2,D = 0

(5.14)

j=k−2

for i = 1, . . . , 2k − 2. So it follows from (5.12) and (5.14) that 0

lim ∂¯i (

n→∞

1

|T˜ (j) | 2 fn )2,D = 0

(5.15)

j=k−2

for i = 1, . . . , 2k − 2. In order to complete our proof, we need the following claim. 0 1 Claim. If limn→∞ ∂¯i ( j=r−1 |T˜ (j) | 2 fn )2,D = 0 for i = 1, . . . , 2r, then 1 0 |T˜ (j) | 2 fn )2,D = 0 for i = 1, . . . , 2r − 2 limn→∞ ∂¯i ( j=r−2

where r = 2, . . . , k. Proof of Claim. We prove this claim by induction. If r = k, it is clear from (5.15) through (5.9). Assume that it is true for r = t. If 0

lim ∂¯i (

n→∞

1

|T˜ (j) | 2 fn )2,D = 0

j=t−2

for i = 1, . . . , 2t − 2, then since T˜ (t−2) = Ut−2 |T˜ (t−2) |, 0

lim T˜ (t−2)∂¯i (

n→∞

1

|T˜ (j) | 2 fn )2,D = 0

(5.16)

j=t−3

for i = 1, . . . , 2t − 2. From (5.4) and (5.16), we have lim z ∂¯i (

n→∞

0

1

|T˜ (j) | 2 fn )2,D = 0

(5.17)

j=t−3

for i = 1, . . . , 2t − 2. By [18, Corollary 2.2] with T = (0), we get lim (I − P )∂¯i (

n→∞

0

1

|T˜ (j) | 2 fn )2,D = 0

(5.18)

j=t−3

for i = 0, 1, . . . , 2t − 4 where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). Substituting (5.18) into (5.4), we obtain lim (T˜ (t−2) − z)P ∂¯i (

n→∞

0

j=t−3

1

|T˜(j) | 2 fn )2,D = 0

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for i = 1, . . . , 2t − 4. Since T˜ (t−2) satisﬁes the property (β) by Proposition 5.1, 0

lim P ∂¯i (

n→∞

1

|T˜ (j) | 2 fn )2,D = 0

(5.19)

j=t−3

for i = 1, . . . , 2t − 4. It follows from (5.18) and (5.19) that lim ∂¯i (

n→∞

0

1

|T˜ (j) | 2 fn )2,D = 0

j=t−3

for i = 1, . . . , 2t − 4. So we complete the proof of our claim. Let us come back now to the proof of Theorem 5.2. By Claim, we get 1

lim ∂¯i |T | 2 fn 2,D = 0

n→∞

for i = 1, 2 when r = 2. Since T = U |T |, lim T ∂¯i fn 2,D = 0 n→∞

(5.20)

for i = 1, 2. By (5.2) and (5.20), we have lim z ∂¯i fn 2,D = 0 n→∞

for i = 1, 2. By [18, Corollary 2.2] with T = (0), lim (I − P )fn 2,D = 0

n→∞

(5.21)

where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). Now from (5.1) and (5.21), we obtain lim (T − z)P fn + 1 ⊗ hn 2,D = 0.

n→∞

Let Γ be a curve in D surrounding σ(T ). Then for z ∈ Γ lim P fn (z) + (T − z)−1 (1 ⊗ hn ) = 0

n→∞

uniformly. Hence by Riesz-Dunford functional calculus, 1 lim P fn (z)dz + hn = 0. n→∞ 2πi Γ 1 But since 2πi Γ P fn (z)dz = 0 by Cauchy’s theorem, limn→∞ hn = 0. Thus the map V is one-to-one and has closed range. Since BAIH(2) contains all p-hyponormal (0 < p < 12 ), log-hyponormal, and w-hyponormal operators, we obtain the following corollary from Theorem 5.2. Corollary 5.3. All p-hyponormal (0 < p < 12 ), log-hyponormal, and w-hyponormal operators are subscalar of order 6. Corollary 5.4. Let T ∈ BAIH(k). If f is a function analytic in a neighborhood of σ(T ), then f (T ) is subscalar.

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Proof. From a general property of an analytic functional calculus, we get that V f (T ) = f (S)V with the same notation of the proof of Theorem 5.2 where f → f (T ) is the functional calculus morphism. Since V is one-to-one and has closed range by the proof of Theorem 5.2, f (T ) is subscalar. Recall that if U is a nonempty open set in C and if Ω ⊂ U has the property that supλ∈Ω |f (λ)| = supβ∈U |f (β)| ∞

for every function f ∈ H (U ) (i.e. for all f bounded and analytic on U ), then Ω is said to be dominating for U . If we apply Theorem 5.2 and [8], we can generalize Scott W. Brown’s theorem (i.e. we can recapture [11, Theorem 6.2]). Corollary 5.5. ([11, Theorem 6.2]) If T ∈ BAIH(k) and the spectrum σ(T ) of T has the property that σ(T ) ∩ U is dominating for some nonempty open set U ⊂ C, then T has a nontrivial invariant subspace. 1

Recall that an operator T ∈ L(H) is said to be power regular if lim T n x n n→∞ exists for every x ∈ H. This class includes all decomposable operator, spectral operators, etc. In [4] Bourdon showed that hyponormal operators are power regular. Here we generalize the result in the following theorem. Corollary 5.6. Every backward Aluthge iterate of a hyponormal operator of order k is power regular. Proof. It is known that a scalar operator is power regular and the restriction of power regular operators to their invariant subspaces remain power regular. Since every backward Aluthge iterate of a hyponormal operator of order k is the restriction of a scalar operator to one of its invariant subspace by Theorem 5.2, it is power regular. Corollary 5.7. If T is in BAIH(k), then it has the property (β). Hence it has the single valued extension property. Proof. Since every scalar operator satisﬁes the property (β) and the property (β) is transmitted from an operator to its restriction to closed invariant subspaces, it follows from Theorem 5.2 that every backward Aluthge iterate of a hyponormal operator satisﬁes the property (β). Hence it has the single valued extension property. Corollary 5.8. Let T1 and T2 be in BAIH(k). If AT1 = T2 A, then for every closed set F ⊂ C, AHT1 (F ) ⊂ HT2 (F ) where HTi (F ) = {x ∈ H : σTi (x) ⊂ F } for i = 1, 2. Proof. Since T1 and T2 satisfy the property (β) by Corollary 5.7, if x ∈ HT1 (F ), then σT1 (x) ⊂ F . Hence F c ⊂ ρT1 (x). So there is an analytic H-valued function

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f deﬁned on F c such that (T1 − z)f (z) ≡ x,

z ∈ F c.

Now (T2 − z)Af (z) = A(T1 − z)f (z) ≡ Ax,

z ∈ F c.

Since Af : F c → H is analytic, F c ⊂ ρT2 (Ax), i.e. σT2 (Ax) ⊂ F . Hence Ax ∈ HT2 (F ).

6. Further applications C. Kitai showed (in [12]) that hyponormal operators are not hypercyclic. In the following theorem we generalize Kitai’s theorem on BAIH(k), i.e., every backward Aluthge iterates T of a hyponormal operator is not hypercyclic. Recall that if T ∈ L(H) and x ∈ H, then {T nx}∞ n=0 is called the orbit of x under T , and is denoted by orb(T, x). If orb(T, x) is dense in H, then T is called a hypercyclic operator. Also recall that an operator T is said to be semi-Fredholm if ranT is closed and either kerT or H\ ranT is ﬁnite dimensional. The semi-Fredholm spectrum σsF (T ) of T is the set {z ∈ C : T − z is not semi-Fredholm}. Theorem 6.1. If T ∈ BAIH(k), then it is not hypercyclic. Proof. If T ∈ BAIH(k) is hypercyclic, then σp (T ∗ ) = φ by [12, Corollary 2.4]. Hence T ∗ has the single valued extension property. Since T has the single valued extension property by Corollary 5.7, every point of σ(T )\σsF (T ) is an isolated point from [15, Remark 1]. Since T˜ (k) is isoloid (i.e., isoσ(T˜ (k) ) ⊂ σp (T˜ (k) )), from [9] T is isoloid. Hence σ(T )\σsF (T ) ⊂ σp (T ). Since σp (T ) = φ, σ(T ) = σsF (T ). By [3, Theorem 14.15] σ is continuous at T . By [14], σ(T ) is nowhere dense set which is the closure of its isolated points. Since T is isoloid, isoσ(T ) ⊂ σp (T ) = φ. Hence σ(T ) = φ. So we have the contradiction. We remark from Theorem 6.1 that p-hyponormal, log-hyponormal, and whyponormal operators are not hypercyclic. Corollary 6.2. If T ∈ BAIH(k) is invertible, then T and T −1 have a common nontrivial invariant closed set. Proof. Since T is not hypercyclic from Theorem 6.1, the proof follows from [12, Theorem 2.15]. Recall that an operator T ∈ L(H) is decomposable provided that, for each open cover {U, V } of C, there exist closed T -invariant subspaces Y , Z of H such that H = Y + Z, σ(T |Y ) ⊂ U , and σ(T |Z ) ⊂ V . Here, T |Y denotes the restriction of T to Y . Theorem 6.3. Let T ∈ BAIH(k) and let T = zI for all z ∈ C. If S is a decomposable quasiaﬃne transform of T , then T has a nontrivial hyperinvariant subspace.

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Proof. Assume that X is a quasiaﬃnity such that XS = T X where S is decomposable. If T has no nontrivial hyperinvariant subspace, we may assume that σp (T ) = φ and HT (F ) = {0} for each closed F proper in σ(T ) by Theorem 5.2 ¯ = φ and and [14, Lemma 3.6.1]. Let {U, V } be an open cover of C with σ(T )\U σ(T )\V¯ = φ. Then ¯ ) + XHS (V¯ ) ⊆ HT (U ¯ ) + HT (V¯ ) = {0}. XH = XHS (U So we have a contradiction.

Recall that an X ∈ L(H, K) is called a quasiaﬃnity if it has trivial kernel and dense range. An operator A ∈ L(H) is said to be a quasiaﬃne transform of an operator T ∈ L(K) if there is a quasiaﬃnity X ∈ L(H, K) such that XA = T X. Furthermore, operators A and T are quasisimilar if there are quasiaﬃnities X and Y such that XA = T X and AY = Y T . Proposition 6.4. If two backward Aluthge iterates of a hyponormal operator of order k are quasisimilar, then they have equal spectra and essential spectra, respectively. Proof. This follows from [19] and Corollary 5.7.

Proposition 6.5. Suppose T ∈ BAIH(k) and suppose S ∈ L(H) satisfy the property (β). If S and T are quasisimilar, then S satisﬁes Weyl’s theorem (i.e., σ(T ) − ω(T ) = π00 (T ), where ω(T ) denotes Weyl spectrum of T and π00 (T ) denotes the set of all eigenvalues of ﬁnite multiplicity of T ). Proof. Since T = U |T | (polar decomposition) satisﬁes the property (β) by Corollary 5.7, from [19] it suﬃces to show that T satisﬁes Weyl’s theorem. Since T˜ (k) is hyponormal, it is known that T˜ (k) satisﬁes Weyl’s theorem. Then by applications of [10] T satisﬁes Weyl’s theorem. Proposition 6.6. If T ∈ BAIH(k) and S is similar to T , then S satisﬁes Weyl’s theorem. Proof. If z ∈ σ(T ) − ω(S), then S − z is Weyl. Since S is similar to T , there exists an invertible operator X such that S = X −1 T X. Since T − z = X(S − z)X −1 is Weyl, z ∈ σ(T ) − ω(T ) = π00 (T ). Since π00 (S) = π00 (T ), z ∈ π00 (S). Conversely, assume that π00 (S) = π00 (T ). Since T satisﬁes Weyl’s theorem by applications of [10], z ∈ σ(T ) − ω(T ). Since T − z is Weyl, S − z is Weyl. Hence z ∈ σ(S) − ω(S).

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References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Int. Eq. Op. Th. 13(1990), 307-315. [2] A. Aluthge and D. Wang, w-hyponormal operators, Int. Eq. Op. Th. 36(2000), 1-10. [3] C. Apostol, L.A. Fialkow, D.A. Herrero, and D. Voiculescu, Approximation of Hilbert space operators, Volumn II, Research Notes in Math. 102, Pitman, Boston, 1984. [4] P. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44(1997), 345-353. [5] S.W. Brown, Hyponormal operators with thick spectrum have invariant subspaces, Ann. of Math. 125(1987), 93-103. [6] I. Colojoarˇ a and C. Foia¸s, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968. [7] J.B. Conway, Subnormal operators, Pitman, London, 1981. [8] J. Eschmeier, Invariant subspaces for subscalar operators, Arch. Math. 52(1989), 562-570. [9] I. Jung, E. Ko, and C. Pearcy, Aluthge transforms of operators, Int. Eq. Op. Th. 38(2000), 437-448. [10] I. Jung, E. Ko, and C. Pearcy, Spectral pictures of Aluthge transforms of operators, Int. Eq. Op. Th. 40(2001), 52-60. [11] I. Jung, E. Ko, and C. Pearcy, The iterated Aluthge transform of an operator, Int. Eq. Op. Th. 45(2003), 375-387. [12] C. Kitai, Invariant closed sets for linear operators, Ph.D. Thesis, Univ. of Toronto, 1982. [13] E. Ko, On p-hyponormal operators, Proc. Amer. Math. Soc. 128(2000), 775-780. [14] R. Lange and S. Wang, New approaches in spectral decomposition, Contem. Math. 128, Amer. Math. Soc., 1992. [15] K.B. Laursen, Essential spectra through local spectral theory, Proc. Amer. Math. Soc. 125(1997), 1425-1434. ¨ [16] K. L¨ owner, Uber monotone Matrix Funktionen, Math. Z. 38(1934), 177-216. [17] M. Martin and M. Putinar, Lectures on hyponormal operators, Op. Th.:Adv. Appl. 39, Birkh¨ auser Verlag, Boston, 1989. [18] M. Putinar, Hyponormal operators are subscalar, J. Operator Theory 12(1984), 385395. [19] M. Putinar, Quasisimilarity of tuples with Bishop’s property (β), Int. Eq. Op. Th. 15(1992), 1047-1052. [20] K. Tanahasi, On log-hyponormal operators, Int. Eq. Op. Th. 34(1999), 364-372. [21] D. Xia, Spectral theory of hyponormal operators, Op. Th.:Adv. Appl. 10, Birkh¨ auser Verlag, Boston, 1983. Eungil Ko Department of Mathematics, Ewha Women’s University, Seoul 120-750, Korea e-mail: [email protected] Submitted: November 19, 2005 Revised: July 7, 2006

Integr. equ. oper. theory 57 (2007), 583–592 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040583-10, published online December 26, 2006 DOI 10.1007/s00020-006-1469-6

Integral Equations and Operator Theory

Microlocal Analysis of the Bochner-Martinelli Integral Nikolai Tarkhanov and Nikolai Vasilevski Abstract. In order to characterise the C ∗ -algebra generated by the singular Bochner-Martinelli integral over a smooth closed hypersurfaces in Cn , we compute its principal symbol. We show then that the Szeg¨ o projection belongs to the strong closure of the algebra generated by the singular Bochner-Martinelli integral. Mathematics Subject Classification (2000). Primary 32A25; Secondary 47L15, 47G30. o projection. Keywords. Bochner-Martinelli integral, symbol, C ∗ -algebra, Szeg¨

1. Introduction The Bochner-Martinelli integral formula for holomorphic functions in a bounded domain in Cn is of great importance in complex analysis, cf. [5]. It is a generalization to many variables of the classical Cauchy formula which actually gave rise to the theory of singular integral equations. A canonical Cauchy-type singular integral known as Hilbert transform is a corner stone of harmonic analysis. To handle more reﬁned integral operators of many-dimensional complex analysis, such as Cauchy-Fantappi`e integrals or Szeg¨o projections, etc., there have been elaborated several calculi of pseudodiﬀerential operators relevant to several complex variables, cf. [7]. The Bochner-Martinelli integral does not apply to derive explicit formulas for a solution of the ∂¯ -equation, which is a fundamental equation of complex analysis. On the other hand, the singular Bochner-Martinelli integral over each smooth hypersurface satisﬁes the cancellation condition, and thus deﬁnes a singular integral operator on the hypersurface. In other words, it belongs to the algebra of pseudodiﬀerential operators of order zero with polyhomogeneous symbols. Since its kernel is very explicit the problems of complex analysis have never required, as far as we know, the knowledge of its symbol but in the case n = 1. The analysis

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of the Bochner-Martinelli singular integral so far undertaken does not go beyond the potential theory of the 1950s, cf. [4]. Since explicit kernels are more useful than the Fourier transform under the presence of singularities of the underlying hypersurface, the existing theory of the Bochner-Martinelli integral leads to the conclusion that no pseudodiﬀerential technique is required in this theory at all. However, the theory of operator algebras gives us an evidence to the contrary. To eﬀectively describe the C ∗ -algebra generated by the singular Bochner-Martinelli integral the knowledge of its explicit kernel is obviously insuﬃcient, for the composition rule for the kernels includes integration over the hypersurface and cannot be carried out explicitly. On the other hand, the composition rule for the principal symbols allows one to evaluate the principal symbol of a power of the singular Bochner-Martinelli integral explicitly. This immediately gives rise to the Calkin algebra of the algebra under study. The aim of this paper is to bring together two areas whose interaction might enrich each other. The ﬁrst of the two is the theory of algebras of pseudodiﬀerential operators with symbol structure. And the second area is the potential theory of the Bochner-Martinelli kernel, which applies to many central problems of complex analysis.

2. Singular Bochner-Martinelli integral Let S be a smooth closed hypersurface in Cn , where n ≥ 1. The surface measure ds on S is induced by the Lebesgue measure dy = dy1 ∧ . . . ∧ dy2n in R2n , where the complex structure is introduced by ζj = yj + ıyn+j , for j = 1, . . . , n. A trivial veriﬁcation shows that dy = (2ı)−n dζ¯ ∧dζ, where dζ = dζ1 ∧. . .∧dζn and similarly ¯ for dζ. If S is given in the form S = {ζ ∈ Cn : (ζ) = 0}, where ∈ C 1 (Cn ) is a real-valued function satisfying ∇(ζ) = 0 for all ζ ∈ S, ∇(ζ) standing for the real gradient of at ζ, then ∇(ζ) ν(ζ) = |∇(ζ)| is the unit normal vector of S at a point ζ ∈ S oriented in the direction of increasing of ρ. The complex vector νc = (νc,1 , . . . , νc,n ) with coordinates νc,j = νj + ıνn+j is called the complex normal of the hypersurface S. In the coordinates of Cn we obviously have ∂ ∂ ζ¯j νc,j (ζ) = |∇ζ¯(ζ)| for j = 1, . . . , n. ¯ the wedge product of all diﬀerentials dζ¯1 , . . . , dζ¯n We now denote by dζ[j] ¯ but dζj .

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¯ Lemma 2.1. For each j = 1, . . . , n, the pull-back of the diﬀerential form dζ ∧ dζ[j] under the embedding S → Cn is equal to (−1)j−1 (2ı)n−1 ı νc,j ds, where ds is the surface measure on S. Proof. An easy computation shows that the pull-back of the diﬀerential form dy[j] under the embedding S → R2n is equal to (−1)j−1 νj ds, for every j = 1, . . . , 2n. From this the lemma follows immediately. Given an integrable function f : S → C with compact support on S, the Bochner-Martinelli integral of f is deﬁned by M f (z) = f (ζ)U (ζ, z) (2.1) S

for z ∈ S, where U (ζ, z) =

n ζ¯j − z¯j ¯ (n − 1)! (−1)j−1 dζ[j] ∧ dζ n (2πı) j=1 |ζ − z|2n

is referred to as the Bochner-Martinelli kernel, cf. [5]. Obviously, M f is a harmonic function in Cn \ S, and it vanishes at inﬁnity unless n = 1. Moreover, M f is of ﬁnite order growth near S, hence M f possesses weak limit values on S both from within and without S. If z ∈ S then the integral (2.1) no longer exists, for the kernel U (ζ, z) has a point singularity at z whose order just amounts to the dimension of S. Moreover, if f is merely continuous then even the Cauchy principal value of M f may fail to exist, i.e., f (ζ)U (ζ, z) (2.2) p.v. M f (z) = lim ε→0

ζ∈S |ζ−z|≥ε

for z ∈ S.

However, if the function f satisﬁes Dini’s condition at z, i.e., ∞, where

0

1

ωθ (f, z) dθ < θ

ωθ (f, z) := sup |f (ζ) − f (z)| ζ∈S |ζ−z|<θ

is the continuity modulus of f at z, then the Cauchy principal value integral (2.2) exists at this point. Lemma 2.2. If f ∈ L1comp(S) satisﬁes Dini’s condition at a point z ∈ S then the singular integral (2.2) exists, and 1 lim M f (z ± εν(z)) = ∓ f (z) + p.v. M f (z). ε→0 2 Proof. For this and other generalizations of the classical Sokhotskii-Plemelj jump formulas we refer the reader to [5].

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Lemma 2.2 shows in particular that the cancellation condition, which is necessary and suﬃcient for the existence of a singular integral operator, is fulﬁlled for singular Bochner-Martinelli integral (2.2). From now on we restrict our discussion to this singular integral operator and will write it simply MS f . If S is of class C ∞ ∞ then MS : Ccomp (S) → C ∞ (S) is a polyhomogeneous pseudodiﬀerential operator of zero order on S.

3. Evaluation of the symbol In this section we evaluate the principal symbol of the singular Bochner-Martinelli integral (2.2). To this end, we identify the cotangent space Tz∗ S of S at a point z ∈ S with all linear forms on Tz∗ R2n which vanish on the one-dimensional subspace of Tz∗ R2n spanned by ν(z). Since Tz∗ R2n ∼ = R2n , one can actually specify ∗ 2n Tz S as the hyperplane through the origin in R which is orthogonal to the vector ν(z). Lemma 3.1. For all z ∈ S and ξ ∈ Tz∗ S, the symbol of order 0 of the operator MS is equal to n 1 ξn+j ξj σ 0 (MS )(z, ξ) = νj (z) − νn+j (z) 2 j=1 |ξ| |ξ| Proof. Using Lemma 2.1 we get n ζ¯j − z¯j 1 MS f (z) = f (ζ) νc,j (ζ) ds(ζ) σ2n S |ζ − z|2n j=1 n ∂ G2n (z − ζ) ds(ζ), f (ζ) 2 νc,j (ζ) = ∂ζj S j=1

(3.1)

where σ2n is the area of the (2n − 1) -dimensional sphere in R2n , and G2n (z) =

1 1 σ2n (2 − 2n) |z|2n−2

is the standard fundamental solution of convolution type of the Laplace operator in R2n . Let G2n stand for the operator with Schwartz kernel G2n (z − ζ) in R2n . This is a polyhomogeneous pseudodiﬀerential operator of order −2 well known as the Newton potential in R2n . Its principal symbol is −|ξ|−2 . The equality (3.1) means that n ∂ νc,j (ζ) (f σS ) , (3.2) MS f (z) = G2n 2 ∂ζj j=1 where the prime stands for the transposed operator, and σS is the surface layer on S.

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We thus see that the pseudodiﬀerential MS on S is the restriction to S of the pseudodiﬀerential operator n ∂ Ψ = G2n 2 νc,j (ζ) ∂ζj j=1 on all of R2n . This latter is of order −1 and its principal symbol is easily evaluated, namely n ı νc,j (z) (ξj − ıξn+j ) |ξ|2 j=1

σ −1 (Ψ )(z, ξ) = =

ı

n

νc,j (z)

j=1

ξj − ıξn+j |ξ|2

for z in a neighbourhood of S and ξ ∈ R2n . A familiar argument now shows that the principal symbol of MS is given by the formula ∞ 1 p.v. σ −1 (Ψ )(z, tν(z) + ξ) dt σ 0 (MS )(z, ξ) = 2π −∞ ∞ n (tνj (z) + ξj ) − ı(tνn+j (z) + ξn+j ) ı νc,j (z) p.v. dt = 2π j=1 |tν(z) + ξ|2 −∞ ∞ n tνc,j (z) + (ξj − ıξn+j ) ı νc,j (z) p.v. dt = 2 2 2π j=1 −∞ t + 2t ν(z), ξ + |ξ| for all z ∈ S and ξ ∈ R2n orthogonal to the vector ν(z). Note that the integral on the right-hand side diverges, however, its Cauchy principal value exists, which is due to the condition ν(z), ξ = 0. We ﬁnally obtain ∞ n 1 ı νc,j (z) (ξj − ıξn+j ) dt σ 0 (MS )(z, ξ) = 2 + |ξ|2 2π j=1 t −∞ n

ı νc,j (z) (ξj − ıξn+j ) , 2|ξ| j=1

= which just amounts to σ 0 (MS )(z, ξ) =

ı 1 ξn+j ξj νj (z) − νn+j (z) +

ν(z), ξ, 2 j=1 |ξ| |ξ| 2|ξ| n

showing the lemma. Note that for S = {z ∈ Cn : zn = 0} Lemma 3.1 gives σ 0 (MS )(z, ξ) = −

1 ξn , 2 |ξ|

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which obviously agrees with the symbol of the Hilbert transform on the real axis, i.e., for n = 1. Worth mentioning a very simple and transparent geometric interpretation of the symbol σ 0 (MS ). The tangent hyperplane Tz S of S at a point z contains the complex plane TC,z S of real dimension 2n − 2 determined by the equation (νc (z), ζ − z) = 0, where by (·, ·) is meant the scalar product in Cn . This plane is called the complex tangent plane of S at z. The vector νc (z) ∈ Cn spans over C a complex plane NC,z S of real dimension 2 in Cn , which is called the complex normal plane of S at z. The whole space Cn splits into the orthogonal sum TC,z S ⊕ NC,z S, if we put the origin at z. The question is now how well is S ﬁt in this complex decomposition. The complex vector νc (z) is identiﬁed under the complex structure in R2n with the real vector ν(z). Hence, the multiplication of νc (z) with ı puts it in the tangent hyperplane Tz S. Being the intersection of Tz S and NC,z S, the real vector ıνc (z) = (−νn+1 , . . . , −ν2n , ν1 , . . . , νn ) completes νc (z) to an orthonormal basis of NC,z S. The real line in Tz S determined by ıνc (z) is well known in complex analysis and is usually referred to as distinguished direction. Lemma 3.1 then shows that 1 σ 0 (MS )(z, ξ) =

ıνc (z), ξ, (3.3) 2|ξ| i.e., the value of the principal symbol of MS at (z, ξ) just amounts to half the orthogonal projection of the cotangent vector ξ ∈ Tz∗ S onto the distinguished direction.

4. C ∗ -algebra Suppose D is a bounded domain in Cn whose boundary is a smooth compact closed hypersurface S. We ﬁrst note that the operator MS is essentially selfadjoint, i.e., the diﬀerence MS − MS∗ is compact. This is a direct consequence of the fact that the principal symbol of MS is real-valued. Furthermore, it is known that the operator MS is selfadjoint if and only if S is a sphere in Cn , cf. [5]. Denote by A = A(C(S), MS ) the C ∗ -algebra generated by the BochnerMartinelli integral MS , its adjoint operator MS∗ , and by all multiplication operators aI with a ∈ C(S). For n = 1, the algebra A(C(S), MS ) coincides with the whole algebra of onedimensional singular integral operators, while for n ≥ 2 it is a proper subalgebra of zero order pseudodiﬀerential operators on S. Theorem 4.1. The C ∗ -algebra A is irreducible. Proof. In order to establish the theorem we show that the algebra A does not have any non-trivial invariant subspace. Each invariant subspace of its subalgebra {aI : a ∈ C(S)} ∼ = C(S) is obviously of the form VΣ = {χΣ f : f ∈ L2 (S)}, where χΣ is the characteristic function of a measurable subset Σ ⊂ S of positive

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measure. Now, the space VΣ is invariant for the entire algebra A if and only if χΣ MS = MS χΣ I. In this case the function χΣ · 1 1 = χΣ I + M S 1 2 1 = I + M S χΣ I 1 2 1 I + M S χΣ = 2 admits an analytic extension to the domain D which thus possesses the boundary values 1, if z ∈ Σ , χΣ (z) = 0, if z ∈ S \ Σ , χΣ

=

on S. Thus, modulo a zero measure set, either Σ = ∅ or Σ = S. Hence it follows that each invariant subspace of the algebra A is trivial, being either {0} or L2 (S), as desired. Being irreducible, the algebra A contains non-zero compact operators (for example, commutators [aI, MS ]). By Theorem 2.4.9 of [6], it contains the entire ideal K of compact operators on L2 (S). Denote by A = A/K the Calkin algebra of the algebra A. The essential S of MS in the spectrum of the operator MS , i.e., the spectrum of the image M Calkin algebra, coincides obviously with the range of its principal symbol. By (3.3), the essential spectrum of MS just amounts to the interval [−1/2, 1/2], and thus, S is by the standard functional calculus, the C ∗ -subalgebra of A generated by M isomorphic and isometric to C[−1/2, 1/2]. The description of the Calkin algebra A can be obtained either by using the familiar localization technique or by deducing the result from the well-known description of the Calkin algebra for the C ∗ -algebra of zero order pseudodiﬀerential operators on the surface S. Theorem 4.2. The Calkin algebra A of the algebra A = A(C(S), MS ) is isomorphic and isometric to C(S × [−1/2, 1/2]). Under this isomorphism, the homomorphism π : A → A is generated by the following mapping of generators of the algebra A: π: aI π : MS

→ a(z), → σ,

z ∈ S; σ ∈ [−1/2, 1/2].

Note that a zero order pseudodiﬀerential operator Ψ belongs to the algebra A if and only if its principal symbol has the form 1 σ 0 (Ψ )(z, ξ) = a z, (4.1)

ıνc (z), ξ 2|ξ| for a suitable function a ∈ C(S × [−1/2, 1/2]).

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5. The Szeg¨ o projection Let D be a strictly pseudoconvex domain in Cn , and let S be its boundary. As in Section 2, we write ds for the surface measure on S, and consider L2 (S) with respect to this measure. Moreover, we denote by H 2 (S) the Hardy space on S, that is, the subspace of 2 L (S) consisting of all functions admitting an analytic continuation to the domain D. Let PS stand for the orthogonal projection of L2 (S) onto H 2 (S), called also Szeg¨o projection. By Lemma 2.2, the limit values on S of the Bochner-Martinelli integral from within S are related to the singular integral by the formula M− = (1/2)I + MS . In [8] the iterations of M− are proved to converge in the strong topology of L(L2 (S)) to the Szeg¨o projection PS in case S is the boundary of a ball in Cn . The question arises whether this result can be extended to arbitrary strictly pseudoconvex domains D. N Using (3.3), we may readily evaluate the principal symbol of iterations M− for all N = 1, 2, . . .. This yields 1 N 1 N +

ıνc (z), ξ )(z, ξ) = σ 0 (M− 2 2|ξ| 1 ξ N 1 +

ıν = (z), (5.1) c 2N |ξ| whenever z ∈ S and ξ ∈ R2n is orthogonal to the normal vector ν(z). From (5.1) we deduce by the Schwarz inequality that 1, if ξ = t ıνc (z), t > 0, N lim σ 0 (M− )(z, ξ) = (5.2) 0, otherwise, N →∞ for all z ∈ S. Moreover, this convergence is uniform on each subset of T ∗ S of the form z ∈ S and |ξ/|ξ| − ıν(z)| ≥ ε, with ε > 0. Note that {(z, ξ) : ξ = t ıνc (z), t > 0} is just the symplectic submanifold of T ∗ S, on which the symbols of generalized Toeplitz operators live, cf. [2, § 1]. In [2], the covector ıνc (z) is written as one-form ¯ ∂(z) −ι∗ ¯ |∂(z)| where ι : S → Cn is the inclusion map. N The equality (5.2) shows that the limit of σ 0 (M− ) is exactly the principal symbol of the Szeg¨ o projection PS . Since the generalized Toeplitz operators of negative order induce compact operators on L2 (S), cf. ibid., (5.2) gives us a microlocal version of Romanov’s theorem [8] which is also valid for arbitrary strictly pseudoconvex surfaces. Theorem 5.1. Assume that S is the boundary of a strictly pseudoconvex domain. Then N lim ((1/2)I + MS ) = PS + KS N →∞

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in the strong topology of L(L2 (S)), where KS is a compact operator. As but one direct and simple consequence of Theorem 5.1 we mention that PS (2MS ) + KS

= (2MS ) PS + KS = PS (2MS ) PS + KS = PS ,

KS

KS

where and are compact operators, and KS = KS PS = PS KS . While for S being the boundary of a ball one has exactly PS (2MS ) = = =

(2MS ) PS PS (2MS ) PS PS .

In conclusion we mention as well that if Ψ is a pseudodiﬀerential operator in A with principal symbol (4.1) then PS Ψ PS = PS (a(z, 1/2)I) PS holds modulo compact operators.

References 1. L. Boutet de Monvel, On the index of Toeplitz operators of several complex variables, Inventiones Math. 50 (1979), 249–272. 2. L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators, Princeton University Press, Princeton, New Jersey, 1981. 3. V. Guillemin, Toeplitz operators in n dimensions, Integral Equations Operator Theory 7 (1984), no. 2, 145–205. 4. N.M. G¨ unter, Potential Theory and Its Application to Fundamental Problems of Mathematical Physics, Gostekhizdat, Moscow, 1953, 415 p. 5. A.M. Kytmanov, The Bochner-Martinelli Integral, and Its Applications, Birkh¨ auser Verlag, Basel et al., 1995. 6. G.I. Murphy, C ∗ -Algebras and Operator Theory, Academic Press, Inc., Boston et al., 1990. 7. A. Nagel and E.M. Stein, Lectures on Pseudo-Diﬀerential Operators, Princeton University Press, Princeton, New Jersey, 1979. 8. A.V. Romanov, Spectral analysis of Martinelli-Bochner integral for ball in Cn , and its applications, Functional Anal. Appl. 12 (1978), no. 3. Nikolai Tarkhanov Institute of Mathematics University of Potsdam PO Box 60 15 53 14415 Potsdam Germany e-mail: [email protected]

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Nikolai Vasilevski Departamento de Matem´ aticas CINVESTAV del I.P.N. Apartado Postal 14-740 07000 M´exico, D.F. M´exico e-mail: [email protected] Submitted: February 8, 2006 Revised: October 20, 2006

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Integr. equ. oper. theory 57 (2007), 593–598 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/040593-6, published online December 26, 2006 DOI 10.1007/s00020-006-1477-6

Integral Equations and Operator Theory

On The Extended Eigenvalues of Some Volterra Operators Srdjan Petrovic Abstract. We show that a large class of compact quasinilpotent operators has extended eigenvalues. As a consequence, if V is such an operator, then the associated spectral algebra BV contains its commutant {V } as a proper subalgebra. Mathematics Subject Classification (2000). Primary 47A65; Secondary 47G10, 47A62, 47A15. Keywords. Volterra operator, eigenvalie, quasinilpotent operator.

1. Introduction and Preliminaries Let A(x), B(x) be functions in L2 (0, 1), the Hilbert space of square integrable functions from [0, 1] to C. We consider the Volterra type integral operators x B(t)f (t) dt (1.1) V f (x) = A(x) 0 2

for f in L (0, 1). This paper can be regarded as a sequel to [1] where the simplest operator of the form (1.1) was studied, namely the integral operator V0 with A(x) ≡ 1, B(x) ≡ 1. The research was motivated by a question from [3]: if V is a compact quasinilpotent operator and BV is the associated spectral radius algebra, does BV always properly contain the commutant {V } ? An eﬀective way of demonstrating that {V } = BV is to establish the existence of the so-called extended eigenvalues of V . (A complex number λ is an extended eigenvalue of the operator A if there is a nonzero operator X such that AX = λXA.) Indeed (cf., [3, Corollary 2.4]), if |λ| ≤ 1 and V X = λXV for some X = 0 then X ∈ BV . On the other hand, it is easy to show that not all such extended eigenvectors X commute with V . This approach was instrumental in proving in [1] that the extended point spectrum of V0 is precisely the positive real axis. Independently, Karaev has obtained the same result in [2].

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Unfortunately, this line of attack is not universally available. Shkarin has shown in [4] that there are compact quasinilpotent operators with no extended eigenvalues except λ = 1. In fact, the set of those operators that have nontrivial extended eigenvalues forms a set of ﬁrst category in the metric space of compact quasinilpotent operators. Nevertheless, the question whether BV = {V } for each operator in this class remains open. In this paper we generalize some results of [1] to operators of the form (1.1) with A(x)B(x) ∈ R a. e. In particular, we show that operators of this class possess nontrivial extended eigenvalues. In Secton 3 we take a look at some other types of the kernel of V that allow extended eigenvalues. We are grateful to Professors Karaev and Shkarin for giving us access to the early versions of their papers that helped our research. We would also like to thank the referee for a number of valuable suggestions.

2. The main result As stated in the introduction, the central object of our study are operators V on L2 (0, 1) of the form (1.1) where A, B ∈ L2 (0, 1). It was shown in [1] that, when A(x) ≡ 1 and B(x) ≡ 1, every λ ∈ (0, ∞) is an extended eigenvalue of V . Our main result shows that this remains true in a more general setting. x Theorem 2.1. Let V f (x) = 0 A(x)B(t)f (t) dt, where A and B are functions in L2 (0, 1) such that A(x)B(x) ∈ R for a. e. x ∈ [0, 1], and let λ > 0. Then there exists a bounded linear operator Z = 0 such that V Z = λZV . x Proof. First we consider the case λ < 1. Let L0 (x) = 0 A(t)B(t) dt. The function L0 is continuous so it attains its maximum in [0, 1]. Let xmax be a point where this maximum is attained. Without loss of generality we will assume that there exist α ∈ (0, xmax ) such that L0 (α) < L0 (xmax ) and, for any > 0, L0 is not constant on (α − , α + ). Otherwise, L0 would be constant on (0, xmax ) and we would consider the operator −V instead. Of course, any extended eigenvector for −V is also an extended eigenvector for V , and in this new scenario we could have the same problem arise only if L0 were constant on [0, 1] which would imply that A(x)B(x) = 0 a. e. Now, if either A(x) = 0 a. e. or B(x) = 0 a. e. we would have V = 0. If both A and B vanish on some sets of positive measure, say A = 0 on E and B = 0 on F , then Z = χF ⊗ χE satisﬁes V Z = ZV = 0. In either case the theorem would be proved. Let m0 = min{L0 (x) : 0 ≤ x ≤ 1}, m > 0, and L(x) = m − m0 + L0 (x). Then min{L(x) : 0 ≤ x ≤ 1} = m > 0, so L(x) > 0 on [0, 1]. Let M = max{L(x) : 0 ≤ x ≤ 1}. We deﬁne h : R → R by h(x) = χ[(1−λ)M,∞) (x), and the operator Z by Zf (x) = 0

x

A(x)B(t)h(L(x) − λL(t))f (t) dt.

(2.1)

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We will demonstrate that Z is a nonzero operator satisfying V Z = λZV . Notice that t x A(x)B(t) dt A(t)B(s)h(L(t) − λL(s))f (s) ds V Zf (x) = 0 0 x x B(s)f (s) ds A(t)B(t)h(L(t) − λL(s)) dt, = A(x) 0

while

s

t A(x)B(t)h(L(x) − λL(t)) dt A(t)B(s)f (s) ds 0 0 x x B(s)f (s) ds A(t)B(t)h(L(x) − λL(t)) dt. = A(x)

x

ZV f (x) =

0

s

Thus, it suﬃces to show that x A(t)B(t)h(L(t) − λL(s)) dt = λ s

x s

A(t)B(t)h(L(x) − λL(t)) dt.

(2.2)

Clearly, L (x) = L0 (x) = A(x)B(x) a. e. Using the substitution u = L(t) − λL(s) in the ﬁrst and u = L(x) − λL(t) in the second integral, the desired equality (2.2) becomes L(x)−λL(s)

L(x)−λL(s)

h(u) du = (1−λ)L(s)

h(u) du.

(2.3)

(1−λ)L(x)

The domains of integration diﬀer by the interval with endpoints (1 − λ)L(s) and (1 − λ)L(x). Since both of these numbers are dominated by (1 − λ)M , h vanishes on this interval and (2.3) holds. Consequently, V Z = λZV . Next we show that Z = 0. Since L(α) < M there exists δ > 0 such that L(α) < M − δ/λ. Moreover, there exists > 0 so that L(t) < M − δ/λ for |t − α| < . For such t, M − δ − λL(t) > (1 − λ)M . Since L is continuous there is η > 0 such that L(x) > M − δ for |x − xmax | < η. It follows that, for such x, and for |t − α| < , L(x) − λL(t) > (1 − λ)M , hence h(L(x) − λL(t)) = 1. If Z = 0 then we would have A(t)B(t) = 0 a. e. for |t − α| < and thus L0 would be constant there, contradicting the choice of α. The case λ > 1 is proved in a similar fashion. We outline the diﬀerences: the choice between V and −V is made to ensure that L0 (t) is not constant on (0, tmin ), where tmin is a point where L0 attains its minimum m0 ; h(x) = χ[(1−λ)m,∞) (x); α is selected so that L0 (α) > m0 . The same calculations lead to (2.3) as the equality to prove. The domains of integration again diﬀer by the interval with endpoints (1 − λ)L(s) and (1 − λ)L(x). Since both of these (negative) numbers are dominated by (1−λ)m, h vanishes on this interval and (2.3) holds. Consequently, V Z = λZV . To show that Z = 0 we use the fact that L(α) > m to deduce that there are > 0 and δ > 0 so that L(x) > m + δ for |x − α| < . Also, there is η > 0 such that, if |t− tmin | < η, L(t) < m+ δ/λ. For such t and |x− α| < , L(x)− λL(t) > (1 − λ)m, so h(L(x) − λL(t)) = 1. The conclusion then follows in exactly the same manner as in the case λ < 1.

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Remark 1. It is possible to construct the operator Z in a diﬀerent way, at least when A(x)B(x) > 0 for a. e. x ∈ [0, 1]. Namely, under this assumption L(x) = x A(t)B(t) dt is increasing, hence injective and, for each λ > 1, there exists a 0 function φ on [0, 1] satisfying L(x) = λL(φ(x)). Let φ(x) B(t)g(t) dt, (2.4) Zg(x) = A(x) 0

for g ∈ L2 (0, 1). It is straightforward to establish that Z is a bounded, nonzero operator satisfying V Z = λZV . When 0 < λ < 1, the existence of the corresponding extended eigenvectors follows from the fact that V is unitarily equivalent to ˜ the adjoint V˜ ∗ of an operator V˜ of the form as in (1.1) with A(x) = B(1 − x) ˜ and B(x) = A(1 − x). Although this result is weaker than Theorem 2.1, the extended eigenvectors are not the same so we obtain an additional insight into the structure of BV . For example, under the assumption of Theorem 2.1 together with A(x)B(x) > 0, the operators in (2.1) and the adjoints of those in (2.4) have a common invariant subspace. Theorem 2.1 allows us to conclude that BV = {V } . Corollary 2.2. Let V be as in (1.1), with A(x)B(x) ∈ R for a. e. x ∈ [0, 1]. Then BV = {V } . Proof. The operator Z supplied in the proof of Theorem 2.1 (for λ < 1) belongs to BV . However, if V Z = ZV (= −λZV ) then V Z = ZV = 0. In particular, it would follow that V has a nontrivial nullspace in which case any operator of the form f ⊗ g (with V f = 0) is in BV . Yet, it belongs to {V } only if V ∗ g = 0.

3. Some open problems The ideas of Section 2 can be sometimes applied even if the integral operator is not of the form (1.1). For example, in the case of the operator deﬁned in (2.4) an essential role xis played by the function φ(x) satisfying condition L(x) = λL(φ(x)), for L(x) = 0 A(t)B(t) dt. When V is of the more general form x F (x, t)f (t) dt (3.1) V f (x) = t

0

we may deﬁne L(x, t) = 0 F (x, s) ds and look for a function φ such that L(x, t) = λL(φ(x), φ(t)). Clearly, such a function need not exist — as an example take L(x, t) = x + t2 . When it does, however, λ is an extended eigenvalue for V . The following theorem makes this more precise. t Theorem 3.1. Let V be as in (3.1), let L(x, t) = 0 F (x, s) ds, and suppose that there exists a strictly increasing, absolutely continuous function φ with the property that φ(0) = 0, L(x, t) = λL(φ(x), φ(t)), and the composition operator Cφ is bounded on L2 (0, 1). Then every λ > 1 is an extended eigenvalue for V .

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x Proof. It is easy to see that, for f ∈ L2 (0, 1), V Cφ f (x) = 0 F (x, t)f (φ(t)) dt and φ(x) F (φ(x), t)f (t) dt. Furthermore, the latter integral can be transCφV f (x) = 0 x formed to 0 F (φ(x), φ(t))φ (t)f (φ(t)) dt using the Change of Variable Theorem (cf., [5, (6.95)]). Thus, it suﬃces to establish that F (x, t) = λF (φ(x), φ(t))φ (t). Since L is an absolutely continuous function of its second argument, the equality in question can be obtained by diﬀerentiating L(x, t) = λL(φ(x), φ(t)) with respect to t. As an illustration of this method, we give the following example. x Example 3.2. Let V f (x) = 0 x(x + t)f (t) dt. Therefore F (x, t) = x(x + t). We √ 3 notice that, if φ(x) = x/ λ for some λ > 1, then λF (φ(x), φ(t))φ (t) = F (x, t). Consequently, the composition operator Cφ satisﬁes V Cφ = λCφ V . It is not hard to see that the success of this example is based on the fact that L(x, t) = x2 t + xt2 /2 is a homogeneous polynomial. On the other hand, Remark 1 shows that this is not necessary. Thus, it is natural to ask the following question: Problem 1. Characterize kernels F (x, y) to which Theorem 3.1 applies. Another class of kernels that allow the existence of extended eigenvalues are those of the form (3.2) F (x, y) = A(x)[L(x) − L(y)]n B(y) x for some positive integer n, and L(x) = 0 A(t)B(t) dt. These kernels arise when the operators of the form (1.1) are raised to the power n + 1. Indeed, one can use x induction to prove that, if V is as in (1.1), and L(x) = 0 A(t)B(t) dt, then x [L(x) − L(t)]n−1 B(t)f (t) dt. V n f (x) = A(x) (n − 1)! 0 As an illustration, the following result has Theorem 2.1 as a special case for n = 1. x Theorem 3.3. Let V f (x) = A(x) 0 [L(x) − L(t)]n−1 B(t)f (t) dt, where A and B functions in L2 (0, 1) such that A(x)B(x) ∈ R for a. e. x ∈ [0, 1], L(x) = are x A(t)B(t) dt, and let λ > 0. Then there exists a bounded linear operator Z = 0 0 such that V Z = λZV . Theorem 3.1 and Theorem 3.3 show that an integral operator of the Volterra type may have extended eigenvalues even if its kernel does not separate variables. On the other hand, we know from [4] that there are many compact quasinilpotent operators without extended eigenvalues. Thus, it would be interesting to come up with a concrete operator with this property. Since the kernel cannot separate the variables, it is natural to look for the kernel of the form F (x, y) = A(x) + B(y). The simplest case F (x, y) = x + √y cannot be used since it satisﬁes the hypotheses of Theorem 3.1 with φ(x) = x/ λ. Thus, we are naturally led to F (x, y) = x + y 2 . x Problem 2. Does the operator V on L2 (0, 1) deﬁned by V f (x) = 0 (x + t2 )f (t) dt have extended eigenvalues?

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References [1] A. Biswas, A. Lambert, and S. Petrovic, Extended Eigenvalues and the Volterra Operator, Glasg. Math. J. 44 (2002), no. 3, 521–534. [2] M. Karaev, On extended eigenvalues and extended eigenvectors of some operator classes. Proc. Amer. Math. Soc. 134 (2006), no. 8, 2383–2392. [3] A. Lambert, S. Petrovic, Beyond hyperinvariance for compact operators. J. Funct. Anal. 219 (2005), no. 1, 93–108. [4] S. Shkarin, Compact operators without extended eigenvalues, preprint. [5] K. Stromberg, Introduction to classical real analysis, Wadsworth International Mathematics Series. Wadsworth International, Belmont, Calif., 1981. Srdjan Petrovic Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA e-mail: [email protected] Submitted: September 15, 2006 Revised: October 30, 2006