Toeplitz算子的亚正规性与次正规性

The characterization of the symbols of subnormal Toeplitz operators remains an open problem in operator theory in function space. A subnormal operator is hyponormal. There is the Bram-Halmos criterion: a bounded operator in Hilbert space is subnormal if and only if is k-hyponormalityl； especially, 1-hyponormality is equivalent to the hyponormality. The study of hyponormality arised in Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic ? Our object mainly focus on the hyponormality、the difference between hyponormality and subnormality and Halmos’s Problem 5 of Toeplitz operaotrs and block Toeplitz operators in Bergman space and in vector valued Bergman space, especially:.1. applying the operator theory, the theory of Bergman spaces, the theory of limit and so on, we characterise the hyponormal Toeplitz operators and hyponormal block Toeplitz operators in the Bergman space and in the vector valued Bergman space;.2. applying the operator theory, the function theory and the factorization of the matrix valued function, we describe the gap between the hyponormal and subnormal ( block ) Toeplitz operators;.3. applying the operator theory, the function theory, the space theory and the factorization of the matrix valued function, we describe when the Halmos’s Problem 5 is answered in affirmative in Bergman space and in vector valued Bergman space.

用符号函数的性质来刻画次正规的Toeplitz算子是函数空间上算子理论的公开问题. 次正规算子一定是亚正规的，反之由Bram-Halmos标准：若一个算子是次正规的等价于其是 k-亚正规的；特别的，1-亚正规即为亚正规的. 亚正规算子的研究起源于Halmos的第五问题：“是否每一个次正规的Toeplitz算子或是正规的，或是解析的 ？”本项目研究（向量值）Bergman空间上（块）Toeplitz算子亚正规性、亚正规与次正规的间隔和Halmos第五问题：.1. 运用算子理论、Bergman空间理论和极限思想等，刻画（向量值）Bergman空间上亚正规的（块）Toeplitz算子；.2. 运用算子理论、函数论和矩阵分解等，描述亚正规（块）Toeplitz 算子与次正规（块）Toeplitz 算子的间隔；.3. 运用算子理论、函数论、空间理论和矩阵分解等，给出Halmos 第五问题的解是肯定回答的刻画.

Toeplitz算子的亚正规性起源于Halmos第五问题：是否每一个次正规算子或是正规的，或是解析的。由定义可以得出，次正规算子一定是亚正规算子，反之有很大间隔，本项目旨在研究（向量值）Bergman空间上（块）Toeplitz算子的亚正规性和次正规的若干问题。本项目给出了向量值Bergman空间上以调和多项式函数为符号的块Toeplitz算子的亚正规性的充分条件和必要条件；刻画了Bergman空间上以某类多项式函数为符号的Toeplitz算子若其平方是亚正规的则其或是正规的或是解析的；给出了Halmos第五问题在Bergman空间中的解是肯定回答的一种诠释。