调和Bergman空间上以${\left|z\right|}^2+a{z}^m+b{\overline{z}}^n$为符号的Toeplitz算子的正规性

Normality of Toeplitz Operator with ${\left|z\right|}^2+a{z}^m+b{\overline{z}}^n$ Symbol on Harmonic Bergman Space

摘要: 本文刻画了在调和Bergman空间上以${\left|z\right|}^2+a{z}^m+b{\overline{z}}^n$ 为符号的Toeplitz算子的正规性问题。本文先对调和Bergman空间及Toeplitz算子的相关知识进行了介绍，梳理了调和Bergman空间、Toeplitz算子研究的具有代表性的相关成果。本文还给出了参数相关的以${\left|z\right|}^2+a{z}^m+b{\overline{z}}^n$ 为符号的Toeplitz算子在调和Bergman空间上的正规性的充分必要条件，在该充要条件的基础上提出了两个推论，建立及推广了该符号下Toeplitz算子的自伴性与正规性的等价关系，补充了这类算子结构性质的相关结论。

Abstract: This paper investigates the normality of Toeplitz operators with the symbol ${\left|z\right|}^2+a{z}^m+b{\overline{z}}^n$ on the harmonic Bergman space. First, it introduces the relevant knowledge of the harmonic Bergman space and Toeplitz operators, and summarizes the representative research achievements in the study of the harmonic Bergman space and Toeplitz operators. In addition, the paper presents the necessary and sufficient conditions for the normality of parameter-dependent Toeplitz operators with the symbol ${\left|z\right|}^2+a{z}^m+b{\overline{z}}^n$ on the harmonic Bergman space. Based on these necessary and sufficient conditions, two corollaries are proposed, the equivalence relationship between the self-adjointness and normality of Toeplitz operators under this symbol is established and generalized, and the relevant conclusions on the structural properties of such operators are supplemented.