Bergman 空间上以三角多项式为符号的Toeplitz算子的正规性

The Normality of Toeplitz Operators withTrigonometric Polynomial Symbols on theBergman Space

摘要: 本文围绕 Bergman 空间上 Toeplitz 算子的交换性及正规性问题展开研究. 首先回顾了 Toeplitz 算子的构造(以投影算子为核心), 并梳理了该领域的已有研究成果(如Brown, Halmos, Axler等 人关于不同函数空间下 Toeplitz 算子交换性的经典结论). 基于引理 1 和引理 2, 本文重点证明了 以符号为 u(z) = |z|² + az^m + bz¯ⁿ, m ≥ n 下 Toeplitz 算子正规的充要条件. 并分别给出了不同参 数关系(m = n, m > n > 0, m > n = 0)下的具体判定条件. 此外, 还推导了 3 个推论, 揭示了该类 Toeplitz 算子正规性与自伴性的等价关系. 研究结果丰富了 Bergman 空间上 Toeplitz 算子的相 关结论, 为该领域的进一步研究提供了参考.

Abstract: This paper focuses on the commutativity and normality of Toeplitz operators on Bergman spaces. First, it reviews the construction of Toeplitz operators (centered on projection operators) and summarizes existing research results in this fleld (such as the classic conclusions of Brown, Halmos, Axler, and others on the commutativ- ity of Toeplitz operators in different function spaces). Based on Lemmas 1 and 2, this paper mainly proves the necessary and sufficient conditions for the normality of Toeplitz operators with the symbol u(z) = |z|² + az^m + bz¯ⁿ (where m ≥ n ). It also provides specific judgment conditions under different parameter relationships: m = n;m > n > 0; and m > n = 0 . In addition, 3 corollaries are derived, revealing the equivalence between the normality and self-adjointness of this type of Toeplitz operator. The research results enrich the relevant conclusions of Toeplitz operators on Bergman spaces and provide a reference for further research in this field.