Commutativity of Toeplitz operators on the harmonic Dirichlet space

In this paper, we completely characterize (semi-)commutativity of Toeplitz operators with harmonic symbols on harmonic Dirichlet space.     

Finite sums of Toeplitz products on the Dirichlet space

On the Dirichlet space of the unit disk, we consider a class of operators which contain finite sums of products of two Toeplitz operators with harmonic symbols. We give characterizations of when an operator in that class is zero or compact. Also, we solve the zero product problem for products of finitely many Toeplitz operators with harmonic symbols.     

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.     

多圆柱上Dirichlet空间中Toeplitz算子的交换性

本文讨论了多圆柱上Dirichlet空间中的正规Toeplitz算子以及全纯指标和反全纯指标的两个Toeplitz算子的交换性.我们证明具有多圆柱上Dirichlet空间中反全纯指标的两个Toeplitz算子可交换当且仅当两个指标是线性相关的,同时证明全纯指标和反全纯指标的两个Toeplitz算子可交换当且仅当两个指标有一个是常数.     

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.     

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.     

Product of Toeplitz operators on the harmonic Dirichlet space

In this paper, we study Toeplitz operators with harmonic symbols on the harmonic Dirichlet space, and show that the product of two Toeplitz operators is another Toeplitz operator only if one factor is constant.     

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.     

GALERKIN-PETROV METHODS OF TOEPLITZ OPERATORS ON DIRICHLET SPACE

The convergence of several Galerkin-Petrov methods, including polynomial collocation and analytic element collocation methods of Toeplitz operators on Dirichlet space, is established. In particular, it is shown that such methods converge if the basis and test function own certain circular symmetry.     

Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.     

A Coburn type theorem for Toeplitz operators on the Dirichlet space

A celebrated theorem of Coburn asserts that, on the setting of the Hardy space, if a Toeplitz operator is nonzero, then either it is one-to-one or its adjoint operator is one-to-one. In this paper, we show that an analogous result holds for Toeplitz operators acting on the Dirichlet space.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

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.     

双圆盘加权Dirichlet空间上Toeplitz算子的约化子空间

刻画了双圆盘加权Dirichlet空间D_α(D~2)上Toeplitz算子T_Z_1N(或T_z_2N)和T_z_1N_z_2N的约化子空间.结果表明Toeplitz算子T_z_1N(或T_z_2N)的约化子空间结构与权系数α无关,而Toeplitz算子T_2_1N_z_2N的约化子空间结构与权系数α有关.     

A question of Toeplitz operators on the harmonic Bergman space

Choe and Lee [B.R. Choe, Y.J. Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J. 46 (1999) 163-174] put the question: If an analytic Toeplitz operator and a co-analytic Toeplitz operator on the harmonic Bergman space commute, then is one of their symbols constant? If one of their symbols is bounded, then we will show that the answer is yes.     

Dirichlet空间上Bergman型Toeplitz算子的代数性质

本文讨论了Dirichlet空间上由调和函数诱导的Bergman型Toeplitz算子的基本性质和代数性质,包括此类算子的自伴性、乘积性质、交换性及可逆性,并计算了算子的谱.     

Level sets and composition operators on the Dirichlet space

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.     

Toeplitz算子在Dirichlet空间上的亚正规性

In this paper,we prove that the necessary and sufficient condition for a Toeplitz operator Tu on the Dirichlet space to be hyponormal is that the symbol u is constant for the case that the projection of u in the Dirichlet space is a polynomial and for the case that u is a class of special symbols,respectively.We also prove that a Toeplitz operator with harmonic polynomial symbol on the harmonic Dirichlet space is hyponormal if and only if its symbol is constant.