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.     

Self-adjointness of Toeplitz operators on the Segal-Bargmann space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.     

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.     

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.     

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.     

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.     

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.     

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.

     

Algebraic properties of Toeplitz and small Hankel operators on the harmonic Bergman space

In this paper,we discuss some algebraic properties of Toeplitz operators and small Hankel operators with radial and quasihomogeneous symbols on the harmonic Bergman space of the unit disk in the complex plane C.We solve the product problem of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.Meanwhile,we characterize the commutativity of quasihomogeneous Toeplitz operator and quasihomogeneous small Hankel operator.     

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.     

广义Segal-Bargmann空间上无界Toeplitz算子的交换

本文给出了复平面C上广义Fock空间中两个Toeplitz算子T_u和T_v的性质.假设u是一个径向函数,两算子是可交换的.在一定的增长条件之下,我们证明出u也是一个径向函数.最后还构造了一个具有本性无界符号的S_p紧,Toeplitz算子.     

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.

     

Commuting Toeplitz operators on the Hardy space of the bidisk

On the Hardy space of the bidisk, we give a necessary and sufficient condition for a bounded symbol of a Toeplitz operator that commutes with another Toeplitz operator whose symbol is a certain type of bounded symbol.     

Berezin symbol and invertibility of operators on the functional Hilbert spaces

We give in terms of reproducing kernel and Berezin symbol the sufficient conditions ensuring the invertibility of some linear bounded operators on some functional Hilbert spaces.     

多圆盘Bergman空间上Toeplitz算子的乘积和交换性

该文给出了多圆盘Bergman空间上两个带有某种符号的Toeplitz算子的乘积等于另一个Toeplitz算子的充分必要条件,并且给出了乘积算子所带符号的公式.接下来,相应的研究了它的交换性.这些研究结果都是根据符号函数的Mellin变换.     

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.     

Powers and roots of Toeplitz operators

We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in which commute with Toeplitz operators whose symbols are of the form .

     

重调和Hardy空间Toeplitz算子的交换性

本文研究重调和Hardy空间h~2(Ω)上,Toeplitz算子的交换性,给出了h~2(T~2)上一个解析Toeplitz算子与另一个共轭解析Toeplitz算子交换的充分必要条件.     

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.