单位球Dirichlet空间上以拟齐次函数为符号的Toeplitz算子

In this paper, we study some algebraic properties of Toeplitz operators with quasihomogeneous symbols on the Dirichlet space of the unit ball Bn. First, we describe commutators of a radial Toeplitz operator and characterize commuting Toeplitz operators with quasihomogeneous symbols. Then we show that finite raak product of such operators only happens in the trivial case. Finally, some necessary and sufficient conditions are given for the product of two quasihomogeneous Toeplitz operators to be a quasihomogeneous Toeplitz operator.     

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.     

Bicommutants of Toeplitz operators

In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant. Received: 11 March 2008     

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...     

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.     

Bergman空间上拟齐次Toeplitz算子的乘积

使用Mellin变换作为工具,讨论了Bergman空间上以拟齐次函数为符号的Toeplitz算子的乘积问题,得出了当拟齐次函数的度处于三种不同情况时两个Toeplitz算子乘积仍是Toeplitz算子的充分必要条件.     

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.     

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.     

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.     

Finite Rank Commutators and Semicommutators of Quasihomogeneous Toeplitz Operators

We study finite rank semicommutators and commutators of Toeplitz operators on the Bergman space with quasihomogeneous symbols. We show that in this context, the situation is different from the case of harmonic Toeplitz operators. Submitted: July 23, 2007. Accepted: December 4, 2007.     

离散交换群上Toeplitz 算子的代数性质

广泛的意义下定义 Toeplitz 算子, 给出了Toeplitz 算子乘积仍为Toeplitz 算子的充分必要条件, Toeplitz算子是正规算子的充分必要条件以及 Toeplitz 算子可交换的一个必要条件,从而推广了经典 Toeplitz 算子的相应结果.     

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.     

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.     

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.     

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.     

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.     

多重调和Bergman空间上多重调和符号的Topelitz算子

研究多重调和Bergman空间上的Topelitz算子.对多重调和符号的Topelitz算子,给出了乘积性质、交换性质的符号描述.     

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.     

Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.     

解析Toeplitz代数的本质换位及其相关问题

在本文中，我们决定出多复变Ｈａｒｄｙ空间Ｈ２上解析Ｔｏｅｐｌｉｔｚ代数的本质换位．即一个算子与所有解析Ｔｏｅｐｌｉｔｚ算子本质可换，当且仅当它是符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的紧扰动．由此，符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（Ａｃ）在Ｃａｌｋｉｎ代数中的像是极大可换闭代数，这导致了Ｌ．Ｃｏｂｕｒｎ正合列的极大扩充．从这个事实，证明了符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的本质谱是连通的，这大大改进了Ｃ－Ｓ最近的工作．从本文的主要定理，证明了Ｔｏｅｐｌｉｔｚ代数Ｆ（Ｌ∞）的本质换位和本质中心是由符号属于ＱＣ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（ＱＣ），这些结果又导致了对代数Ｆ（Ｈ∞）＋Ｋ自同构群的确定．