Compact Hankel operators on harmonic Bergman spaces

We study Hankel operators on the harmonic Bergman spaceb ²(B), whereB is the open unit ball inR ⁿ,n2. We show that iff is in then the Hankel operator with symbolf is compact. For the proof we have to extend the definition of Hankel operators to the spacesb ^p(B), 1<p<, and use an interpolation theorem. We also use the explicit formula for the orthogonal projection ofL ²(B, dV) ontob ²(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in are compact.     

The essential norm of Hankel operator on the Bergman spaces of strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cⁿ and let fL²(D). For the Hankel operator H_f on the Bergman space A²(D), it is shown that the essential norm of H_f in L²(D) is comparable to the distance norm from H_f to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).     

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Pták defined the notion of generalized Hankel operator as follows: Given two contractions and , an operatorX: is said to be a generalized Hankel operator ifT ₂ X=XT ₁ ^* andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT ₁ andT ₂. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Pták's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.Dedicated to the memory of our master and friend Vlastimil Pták     

Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR ^2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol lies in the modulation spaceM _,1 (R ^2d ), then the corresponding pseudodifferential operator is bounded onL ²(R ^d ) and, more generally, on the modulation spacesM _p,p (R ^d ) for 1p. If lies in the modulation spaceM _2,2 ^s (R ^2d )=L _s ^/2 (R ^2d )H ^s (R ^2d ), i.e., the intersection of a weightedL ²-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Operator ranges and non-cyclic vectors for the backward shift

In this paper we look at operators on the Hardy spaceH ²(D) with range containing all of the non-cyclic vectors of the backward shift. We show several classes of such operators must be surjective, including Toeplitz, Hankel and composition operators.     

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

Toeplitz operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaceb ^p (B), whereB is the open unit ball inR ⁿ(n2), for 1<p. We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.     

Weyl’s Theorem for Algebraically Paranormal Operators

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyls theorem holds for f(T) for every f $\in$ H((T)); (ii) a-Browders theorem holds for f(S) for every S $\prec$ T and f $\in$ H((S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankel operators on the torus

In this paper we show that the theory of Hankel operators in the torus ^d , ford>1, presents striking differences with that on the circle , starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbolsL ( ) by BMOr( ^d ), a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari-AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Picktype matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked withA ₂ weights. This completes some aspects of the theory of BMO in product spaces.Sadosky was partially supported by NSF grants DMS-9205926, INT-9204043 and GER-9550373, and her visit to MSRI is supported by NSF grant DMS-9022140 to MSRI.     

Polaroid operators,SVEP and perturbed Browder,Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T ^*+A ^*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).     

Conditions for Admissibility of Observation Operators and Boundedness of Hankel Operators

We derive new necessary and sufficient conditions for admissibility of observation operators for certain C ₀-semigroups. We also prove a new sufficient criterion for admissibility for observation operators with infinite-dimensional output space on contraction semigroups. If the contraction semigroup is completely non-unitary and its co-generator has finite defect indices, then this criterion is also necessary. In the case of the right shift semigroup on L ²(0,), these conditions translate into conditions for the boundedness of Hankel operators.     

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state and letH be a finite maximal subdiagonal subalgebra ofM. LetH ² be the closure ofH in the noncommutative Lebesgue spaceL ²(M). We consider Toeplitz operators onH ² whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH ²( ) with symbol fromL ( ), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak^* continuous linear functionals on the space of Toeplitz operators.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Toeplitz operators hyponormal with the unilateral shift

For the unilateral shift operator U on the Hardy space H²(T), we describe conditions on operators T, acting on H²(T), that are necessary and sufficient for the pair (U, T) to be jointly hyponormal. One necessary condition is that T be a Toeplitz operator. Consequently, we study certain nonanalytic symbols that give rise to Toeplitz operators hyponormal with the shift, and thereby obtain examples of noncommuting, jointly hyponormal pairs.Supported in part by a research grant from NSERC     

Common properties of operatorsRS andSR andp-hyponormal operators

LetR andS be bounded linear operators on a Bananch space. We discuss the spectral and subdecomposable properties and properties concerning invariant subspaces common toRS andSR. We prove that, by these properties,p-hyponormal and log-hyponormal operators and their generalized Aluthge transformations are all subdecomposable operators;T andT(r, 1–r)(0<r<1) have same spectral structure and equal spectral parts ifT denotesp-hyponormal or dominant operator; for everyT L(H), 0<r<1,T has nontrivial (hyper-)invariant subspace ifT(r, 1–r) does.This research was supported by the National Natural Science Foundation of China.     

Cyclicity results for Jordan and compressed Toeplitz operators

A vectorx in a Hilbert spaceH iscyclic for a bounded linear operatorTHH if the closed linear span of the orbit {T ⁿ xn0} ofx underT is all ofH. Operators which have a cyclic vector are said to be cyclic.Jordan operators are the infinite direct sums of Jordan cells acting on finite- dimensional Hilbert spaces. Necessary and sufficient conditions for a Jordan operator to be cyclic are given (see Corollary 6). In this case, a dense set of cyclic vectors is exhibited (see Corollary 4). Sufficient conditions for uncountable collections of cyclic Jordan operators to have a common cyclic vector are given and, in this case, a dense set of common cyclic vectors is exhibited (see Corollary 9).Analogues of these cyclicity results for Jordan operators are obtained for compressions of analytic Toeplitz operatorsT _A FAF on the Hardy spaceH ² to subspaces (BH ²) invariant for the backward shiftT _z ^* whereB is a Blaschke product by showing that such compressions are quasisimilar to Jordan operators.     

Products of Hankel operators

This paper studies products of Hankel operators on the Hardy space. We show thatH _f ^(1)* H _f(2) H _f ^(3)* =0 for all permutation if and only if eitherH _f1 orH _f2 orH _f3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that

Generalized Browder’s Theorem and SVEP

A bounded operator a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H ₀(λI − T) as λ belongs to certain subsets of . In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators.     

Riesz Idempotent and Algebraically M-hyponormal Operators

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let be the Riesz idempotent for λ₀, where D is a closed disk of center λ₀ which contains no other points of σ (T). In this note we show that E is self-adjoint and As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every (ii) a-Browder’s theorem holds for f(S) for every and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.