Algebraic Properties of Toeplitz Operators on the Harmonic Dirichlet Space

We study some algebraic properties of Toeplitz operators on the harmonic Dirichlet space of the unit disk. We first give a characterization for boundedness of Toeplitz operators. Next we characterize commuting Toeplitz operators. Also, we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. The corresponding problems for compactness are also studied.     

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.     

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.     

调和Bergman空间上Toeplitz算子的代数性质

In this paper, we first investigate the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols on the harmonic Bergman space. Next, we characterize finite rank commutators and semi-commutators of two Toeplitz operators with quasihomogeneous symbols.     

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 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.     

Quasihomogeneous Toeplitz operators on the harmonic Bergman space

In this paper we study the product of Toeplitz operators on the harmonic Bergman space of the unit disk of the complex plane \mathbbC{\mathbb{C}}. Mainly, we discuss when the product of two quasihomogeneous Toeplitz operators is also a Toeplitz operator, and when such operators commute.     

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.     

Toeplitz operators on the Dirichlet space

We study algebraic properties of Toeplitz operators acting on the Dirichlet space. We first characterize two harmonic symbols of commuting Toeplitz operators. Also, we give characterizations of the harmonic symbol for which the corresponding Toeplitz operator is self-adjoint or an isometry.     

Fock空间上径向函数诱导的Toeplitz算子

本文讨论了Fock空间上以径向函数和拟齐次函数为符号的Toeplitz算子的代数性质,给出了两个以径向函数为符号的Toeplitz算子的积仍为Toeplitz算子的充分必要条件,并且研究了以拟齐次函数为符号的Toeplitz算子的交换性.     

单位多圆柱上Bergman空间中的分别准齐次ToePlitz算子

本文研究了单位多圆柱上Bergman空间中以分别准齐次函数为记号的Toeplitz算子的代数性质.我们首先得到了两个以分别准齐次函数为记号的Toeplitz算子可以写成一个Toeplitz算子的充分必要条件,然后利用L2(Dn,dV)的一个极分解式证明了,只要其中有一个Toeplitz算子是分别准齐次的,则其零乘积问题...     

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.     

Algebraic properties of Toeplitz operators on weighted Bergman spaces

Czechoslovak Mathematical Journal - We study algebraic properties of two Toeplitz operators on the weighted Bergman space on the unit disk with harmonic symbols. In particular the product property...     

Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.     

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.     

Products of Toeplitz Operators on the Bergman Space

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern and Cucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).     

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.     

截断调和Bergman空间上Toeplitz算子的代数性质

本文首先讨论调截断和Bergman空间 $b_{n}^{2}$ 上拟齐次函数为符号的Toeplitz算子的有限秩乘积问题,其次考察两个以拟齐次函数为符号的Toeplitz算子的换位子与半换位子的有限秩问题.     

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.     

截断调和Bergman空间上的Toeplitz算子

截断调和Bergman空间b_n~2=L_a~2{w,w~2,…,w~n}~v是Hilbert空间L~2的闭子空间.研究了单位圆盘上的截断调和Bergman空间上的Toeplitz算子的乘积问题,完整地刻画了具有有界调和符号的两个Toeplitz算子的有限秩与紧的半换位子或换位子.