单位球Dirichlet空间上以拟齐次函数为符号的Toeplitz算子

Toeplitz Operators with Quasihomogeneous Symbols on the Dirichlet Space of ${\mathbb{B}}_n$

中文关键词:  Dirichlet空间  单位球  Toeplitz算子  拟齐次符号.

英文关键词:Dirichlet space  unit ball  Toeplitz operators  quasihomogeneous symbols.

中文摘要:

      本文主要研究单位球Dirichlet空间上以拟齐次函数为符号的Toeplitz算子的代数性质.首先,我们讨论了以径向函数为符号的Toeplitz算子的换位子,给出了可交换的以拟齐次函数为符号的Toeplitz算子.接下来,我们证明了以拟齐次函数为符号的Toeplitz算子的乘积如果是有限秩的,则该乘积只能是零算子.最后,我们给出了两个以拟齐次函数为符号的Toeplitz算子的乘积依然是一个以拟齐次函数为符号的Toeplitz算子的一些充要条件.

英文摘要:

      In this paper, we study some algebraic properties of Toeplitz operators with quasihomogeneous symbols on the Dirichlet space of the unit ball ${\mathbb{B}}_n$. First, we describe commutators of a radial Toeplitz operator and characterize commuting Toeplitz operators with quasihomogeneous symbols. Then we show that finite rank 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.