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.

     

Commuting Toeplitz operators with pluriharmonic symbols

By making use of -harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in -dimensional complex space.

     

Commuting dual Toeplitz operators with pluriharmonic symbols

In this paper we completely characterize commuting dual Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball. We show that for φ and ψ pluriharmonic, SφSψ=Sψφ on only in the trivial case. Here the trivial case is φ or holomorphic.     

多圆盘上的本质可换的Toeplitz算子

本文讨论多圆盘上Ｂｅｒｇｍａｎ空间上的具有多重调和符号的Ｔｏｅｐｌｉｔｚ算子的本质可换性与可换性．我们获得了一个解析或共轭解析Ｔｏｅｐｌｉｔｚ算子与具有多重调和符号的Ｔｏｅｐｌｉｔｚ算子本质可换的充分必要条件是它们是可换的．     

多圆盘上的交换对偶Toeplitz算子

We completely characterize commutativity of Sρ and Sψ on L2α (Dn)⊥ for bounded pluriharmonic symbols ρ and ψ on Dn, and prove that SρSψ=Sψρ if and only if ρ is analytic or ψ is analytic.     

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.

     

多圆盘上的本质交换对偶Toeplitz算子

在单位多圆盘的Bergman空间上,本文分别刻画了以有界可测函数和有界多重调和函数为符号的本质交换对偶Toeplitz算子.     

Commuting dual Toeplitz operators on weighted Bergman spaces of the unit ball

In this paper, we study the commutativity of dual Toeplitz operators on weighted Bergman spaces of the unit ball. We obtain the necessary and sufficient conditions for the commutativity, essential commutativity and essential semi-commutativity of dual Toeplitz operator on the weighted Bergman spaces of the unit ball.     

Commuting Toeplitz operators on the hardy space of the polydisk

In this paper, we show that two Toeplitz operators T_f and T_g on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [T_f, T_g] is n-harmonic.     

Commuting Toeplitz operators on the Dirichlet space

In this paper, we study the commutativity of Toeplitz operators with continuous symbols on the Dirichlet space. First, under a mild condition concerning absolute continuity we characterize (semi-)commuting Toeplitz operators. This is a generalization of the case of harmonic symbols. Also, if one of the symbol is radial or analytic, we get another characterization, which is different from the case on the Bergman space.     

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.     

Commuting Toeplitz Operators on the Pluriharmonic Bergman Space

We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.     

Commuting Toeplitz operators on the Segal-Bargmann space

Consider two Toeplitz operators Tg, Tf on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily.     

Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we characterize commuting dual Toeplitz operators with harmonic symbols on the orthogonal complement of the Dirichlet space in the Sobolev space. We also obtain the sufficient and necessary conditions for the product of two dual Toeplitz operators with harmonic symbols to be a finite rank perturbation of a dual Toeplitz operator.     

Commuting dual Toeplitz operators on the harmonic Dirichlet space

In this paper,we characterize the symbols for(semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space.We show that for φ,ψ∈W~(1,∞),S_φS_ψ=S_ψ Sφ on(D_h)~⊥ if and only if φ and ψ satisfy one of the following conditions:(1) Both φ and ψ are harmonic functions;(2) There exist complex constants α and β,not both 0,such that φ=αψ +β.     

Bergman空间上可交换的Hankel算子和Toeplitz算子

证明了Bergman空间上的两个小Hankel算子如果是可交换的且其中一个是拟齐次的小Hankel算子,则另一个也是拟齐次的.还研究了Toeplitz算子和小Hankel算子的交换性.     

单位球Dirichlet空间上的交换对偶Toeplitz算子

研究对偶Toeplitz算子（半）交换时其符号的关系.通过对其符号的分解,借助多复变函数的有关理论,得到了单位球Dirichlet空间上以多重调和函数为符号的对偶Toeplitz算子Sψ与Sψ（半）交换的充要条件.     

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.     

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.     

Schatten-Herz type positive Toeplitz operators on pluriharmonic Bergman spaces

Recently, Schatten-Herz type Toeplitz operators have been studied on the Bergman spaces and the harmonic Bergman spaces. Motivated these results, we study characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman spaces.