Positivity of Toeplitz operators via Berezin transform

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of |z|$|z|$ on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive.     

On C*-Algebras of Toeplitz Operators on the Harmonic Bergman Space

Commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk have been completely classified in terms of geometric properties of the symbol class. The question when two Toeplitz operators acting on the harmonic Bergman space commute is still open. In some papers, conditions on the symbols have been given in order to have commutativity of two Toeplitz operators. In this paper, we describe three different algebras of Toeplitz operators acting on the harmonic Bergman space: The C*-algebra generated by Toeplitz operators with radial symbols, in the elliptic case; the C*-algebra generated by Toeplitz operators with piecewise continuous symbols, in the parabolic and hyperbolic cases. We prove that the Calkin algebra of the first two algebras are commutative, like in the case of the Bergman space, while the last one is not.     

有界多连通域的Schatten类Toeplitz算子

本文讨论了多连通域的Bergman空间上的以正测度为符号的Toeplitz算子.用符号测度的Berezin 变换和平均函数刻画了Toeplitz算子为Schatten类算子的充要条件.     

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.     

高维加权Bergman空间上的Toeplitz算子

本文研究了高维加权Bergman空间Ap(Bn,dVpφ)(1＜p＜∞).上的Toeplitz算子.利用Toeplitz算子的Berezin变换,获得了Ap(Bn,dVp)(1＜p＜∞)上具有L∞(Bn)符号的Toeplitz算子的有限乘积的有限和是紧算子的一些等价刻画,推广了加权Bergman空间Ap(D,dmpφ)上的Toeplitz算子的有限乘积的有限和是紧的当且仅当它的Berezin变换在单位圆盘的边界消失为0的结论     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

Commutants of Analytic Toeplitz Operators on the Harmonic Bergman Space

We study the commuting problem for Toeplitz operators on the harmonic Bergman space of the unit disk. We show that an analytic Toeplitz operator and a co-analytic Toeplitz operator with certain noncyclicity hypothesis can commute only when one of their symbols is constant. We also obtain analogous results for semi-commutants.     

Bergman空间上的斜Toeplitz算子

本文讨论了Bergman空间上斜Toeplitz算子的若干性质,证明了:如果线性算子S在每个Lap(1相似文献   

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.     

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.     

加权Bergman空间上的紧算子

本文讨论了加权Ｂｅｒｇｍａｎ空间上的Ｔｏｅｐｌｉｔｚ算子,证明了Ｔｏｐｌｉｔｚ算子的有限乘积的有限和是紧的当且仅当它的Ｂｅｒｅｚｉｎ变换在边界上趋向于零．     

Centralizers of Toeplitz Operators with Polynomial Symbols

In this note we describe centralizers of Toeplitz operators with polynomial symbols on the Bergman space. As a consequence it is shown that if an element of the norm closed algebra generated by all Toeplitz operators commutes with a Toeplitz operator of a nonconstant polynomial, then this element is a Toeplitz operator of a holomorphic function.     

Commuting Quasihomogeneous Toeplitz Operators on the Harmonic Bergman Space

In this paper, we obtain a symmetry number for the commutator of quasihomogeneous Toeplitz operators on the harmonic Bergman space. Then we use it to characterize the commuting Toeplitz operators with quasihomogeneous symbols. Also, we show that a Toeplitz operator with an analytic or co-analytic monomial symbol commutes with another Toeplitz operator only in the trivial case.     

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.     

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.     

调和Bergman空间上的Toeplitz的零积及有限秩性质

本文研究了调和Bergman空间上的Toeplitz的Brown-Halmos型零乘积问题以及有限秩的相关性质,并得到了以解析和余解析符号的Toeplitz算子乘积的有限和的零积有限秩的等价条件.     

Commuting Toeplitz Operators with Harmonic Symbols on A2（D,dμ）

For any given symmetric measure μ on the closed unit disk D, we apply the Berezin transform to characterizing semi-commuting and commuting Toeplitz operators with bounded harmonic symbols on A2（D, dμ）.     

重调和Hardy空间上的Toeplitz 算子

与调和Bergman 空间相对应, 本文研究重调和Hardy 空间h²(D²) 上的Toeplitz 算子. 本文发现, h²(D²) 上的Toeplitz 算子与经典的Hardy 空间、Bergman 空间及调和Bergman 空间上的Toeplitz算子的性质都有很大的差异. 例如解析Toeplitz 算子可以不是半可换及可交换的. 即使半可换, 其中任何一个符号可以不为常数; 即使可交换, 两个符号的非平凡线性组合也不一定是常数. 本文得到了h²(D²) 上两个解析Toeplitz 算子半可换和可换的充分必要条件.     

Berezin versus Mellin

We study zero products of two Bergman space Toeplitz operators, where one symbol is harmonic. Our results point in the direction of the zero product problem having only a trivial solution. The techniques we use are based on a formula that connects the Berezin and Mellin transform.     

Ranks of commutators of Toeplitz operators on the harmonic Bergman space

We study the rank of commutators of two Toeplitz operators on the harmonic Bergman space of the unit disk. We first show that the commutator of any two Toeplitz operators with general symbols can’t have an odd rank. But, given any integer n ≥ 0, we also show that there are two symbols for which the corresponding Toeplitz operators induce the commutator with rank 2n exactly.