Toeplitz operators on the polydisk

In this paper it is shown that two analytic Toeplitz operators essentially doubly commute if and only if they doubly commute on the Bergman space of the polydisk.

     

Compact Toeplitz operators on the Segal-Bargmann space

In this note, we show that Miao and Zheng's characterization of compact operators on the Bergman space of the unit disk that are finite sums of finite products of Toeplitz operators (with each one of the symbols belonging to BT) also holds for the Segal-Bargmann space of Cn.     

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.

     

Isometries and Toeplitz operators of Bergman space of bounded symmetric domains

In this paper we completely characterize the isometries of Bergman space (0?p<∞, p≠2) of bounded symmetric domains. We also prove that a pair of Toeplitz operators Tf and Tg on (0<p<∞, p≠2) is isometric equivalence if and only if there is a τ∈Aut(Ω), such that g=f○τ, where Aut(Ω) is the automorphism group of Ω.     

Commuting Toeplitz operators on the polydisk

We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.

     

Hyponormal Toeplitz operators on the polydisk

By bounded vector-valued functions and block matrix representations of Hankel operators, we completely characterize the hyponormality of Toeplitz operators on the Hardy space of the polydisk.     

Hyponormality of Toeplitz operators with generalized circulant symbols

In this paper we obtain a necessary and sufficient condition for Toeplitz operators with generalized circulant symbols to be hyponormal.     

On operators which commute with analytic Toeplitz operators modulo the finite rank operators

It is shown that an operator on the Hardy space (or ) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if . Here is a finite rank operator, and in the case , is a sum of a rational function and a bounded analytic function, and in the case , is a bounded analytic function.

     

On a class of jointly hyponormal Toeplitz operators

We characterize when a pair of Toeplitz operators is jointly hyponormal under various assumptions--for example, is analytic or is a trigonometric polynomial or is analytic. A typical characterization states that is jointly hyponormal if and only if an algebraic relation of and holds and the single Toeplitz operator is hyponormal, where is a combination of and . More general results for an -tuple of Toeplitz operators are also obtained.

     

A note on Toeplitz operators on discrete groups

We study Toeplitz algebras associated to partially-ordered and quasi-partially ordered discrete groups.

     

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

Toeplitz and composition operators on H2 (B n)

The composition operators with closed rangc on H²(B _n) are characterized, and the Fredholmness of products of Toeplitz and composition operators discussed. Moreover, using composition operators, the spectra of Toeplitz operators are studied. Project supported by the National Natural Sciencc Foundation of China and the Postdoctoral Science Foundation of China.     

Compact Toeplitz operators on the weighted Dirichlet space

In this paper, we completely characterize the compactness of Toeplitz operators with continuous symbol on the weighted Dirichlet space.     

Algebraic properties of Toeplitz operators on the Dirichlet space

We study some algebraic properties of Toeplitz operators on the Dirichlet space. We first characterize (semi-)commuting Toeplitz operators with harmonic symbols. Next we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. As an application, we show that the zero product of two Toeplitz operators with harmonic symbol has only a trivial solution. Also, the corresponding compact product problem is studied.     

Trace class Toeplitz operators with unbounded symbols on weighted Bergman spaces

In this note we construct a function φ in L2（Bn,dμ） which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on weighted Bergman space Lα2（Bn,dμ） for several complex variables.     

Commutativity of kth‐order slant Toeplitz operators

In this paper, commutativity of k^th‐order slant Toeplitz operators are discussed. We show that commutativity and essential commutativity of two slant Toeplitz operators are the same. Also, we study k^th‐order slant Toeplitz operators on the Bergman space L²_a(D) and give some commuting properties, algebraic and spectral properties of k^th‐order slant Toeplitz operators on the Bergman space (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Toeplitz operators with unbounded symbols of several complex variables

In this note we construct a function φ in L²(Bn,dA) which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on Bergman space for several complex variables. In addition, we also discuss the compactness of Toeplitz operators with L¹ symbols.     

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<∞$0<p<\infty$.     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.