Hankel operators in the Bergman space and Schatten

K. Zhu proved in Amer. J. Math. 113 (1991), 147-167, that, for , the Hankel operators and on the Bergman space belong to the Schatten class if and only if the mean oscillation MO belongs to . In this paper we prove that the same result also holds when .

     

Schatten p -class Toeplitz operators with unbounded symbols on pluriharmonic Bergman space

In this paper, we construct the function u in L2（Bn, dA） which is unbounded on any neighborhood of each boundary point of Bn such that Tu is the Schatten p-class （0 〈 p 〈 ∞） operator on pluriharmonic Bergman space h2（Bn, dA） for several complex variables. In addition, we also discuss the compactness of Toeplitz operators with L1 symbols.     

Schatten class composition operators on weighted Bergman spaces of bounded symmetric domains

Summary We obtain trace ideal criteria for 0A ² () of a Bounded symmetric diomain in _n.     

Schatten class weighted composition operators on Bergman spaces of bounded strongly pseudoconvex domains

We characterize the Schatten class weighted composition operators on Bergman spaces of bounded strongly pseudoconvex domains in terms of the Berezin transform.     

Schatten class Toeplitz operators on generalized Fock spaces

In this paper we characterize the Schatten p   class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range 0<p<∞$0<p<\infty$.     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

Subnormality and composition operators on the Bergman space

Following the work of C. Cowen and T. Kriete on the Hardy space, we prove that under a regularity condition, all composition operators with a subnormal adjoint on A²(D) have linear fractional symbols of the form. Moreover, we show that all composition operators on the Bergman space having these symbols have a subnormal adjoint, with larger range for the parameterr than found in the Hardy space case.     

Semi-commutators of Toeplitz operators on the Bergman space

In this paper several necessary and sufficient conditions are obtained for the semi-commutator of Toeplitz operators andT _g with bounded pluriharmonic symbols on the unit ball to be compact on the Bergman space. Using -harmonic function theory on the unit ball we show that with bounded pluriharmonic symbolsf andg is zero on the Bergman space of the unit ball or the Hardy space of the unit sphere if and only if eitherf org is holomorphic.The author was supported in part by the National Science Foundation.     

Quasihomogeneous Toeplitz operators on the harmonic Bergman space

In this paper we study the product of Toeplitz operators on the harmonic Bergman space of the unit disk of the complex plane \mathbbC{\mathbb{C}}. Mainly, we discuss when the product of two quasihomogeneous Toeplitz operators is also a Toeplitz operator, and when such operators commute.     

Necessary conditions for Schatten class localization operators

We study time-frequency localization operators of the form , where is the symbol of the operator and are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order , is that belongs to the modulation space and the window functions to the modulation space . Here we prove a partial converse: if for every pair of window functions with a uniform norm estimate, then the corresponding symbol must belong to the modulation space . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For and , we recapture earlier results, which were obtained by different methods.

     

Schatten class membership of Hankel operators on the unit sphere

Let H²(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H²(S), where the symbol function f is allowed to be arbitrary in L²(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if f−Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf∈C2n implies f−Pf=0, i.e., Hf=0.     

Schatten class weighted composition operators on weighted Fock spaces

We describe the Schatten class weighted composition operators on Fock–Sobolev spaces and a large class of weighted Fock spaces, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy certain mild smoothness condition. To prove our main results, we characterize the Schatten class membership of the Toeplitz operators T _μ induced by nonnegative measures μ on the complex space ${\mathbb{C}^n}$ .     

多重调和Bergman空间上的Toeplitz算子和Hankel算子

完全刻画多重调和Bergman空间上Toeplitz算子和Hankel算子的紧性.运用紧Toeplitz算子这个结果,建立了Toeplitz代数和小Hankel代数的短正合列,推广了单位圆盘上相应的结果.     

Commutant of analytic Toeplitz operators on the Bergman space

In this paper, we characterize the commutant of Toeplitz operators on weighted Bergman space with symbol polynomial by using algebraic curves theory.     

Compact operators on the Bergman space of multiply-connected domains

If is a smoothly bounded multiply-connected domain in the complex plane and where we show that is compact if and only if its Berezin transform vanishes at the boundary.

     

Commutants of analytic Toeplitz operators on the Bergman space

In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.

     

Products of Bergman space Toeplitz operators on the polydisk

Motivated by recent works of Ahern and uković on the disk, we study the generalized zero product problem for Toeplitz operators acting on the Bergman space of the polydisk. First, we extend the results to the polydisk. Next, we study the generalized compact product problem. Our results are new even on the disk. As a consequence on higher dimensional polydisks, we show that the generalized zero and compact product properties are the same for Toeplitz operators in a certain case.The first three authors were partially supported by KOSEF(R01-2003-000-10243-0) and the last author was partially supported by the National Science Foundation.     

Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.