Bergman空间上卷积形Toeplitz算子

Algebraic Properties of Convolution‑Form Toeplitz Operators on the Bergman Space

摘要: Bergman空间上Toeplitz算子是函数空间算子理论的重要研究对象。本文主要研究Bergman空间上卷积形Toeplitz算子的代数性质。首先给出该卷积形Toeplitz算子有界性的条件，借助积分三角不等式、圆盘自同构导数公式、变量替换以及柯西–施瓦茨不等式完成定理的充分性与必要性证明。其次利用Bergman空间的标准正交基，结合几何级数展开、幂级数分解与空间正交关系，推导得到卷积形Toeplitz算子的矩阵元表达式，证明该算子在标准正交基下为下三角矩阵，主对角线元素为常数，并给出不同阶数下算子矩阵的具体实例。相关结果有助于进一步理解卷积形Toeplitz算子的结构特征，丰富Bergman空间上算子理论的研究内容。

Abstract: Toeplitz operators on Bergman spaces constitute an important research subject in operator theory of function spaces. This paper mainly investigates the algebraic properties of convolutiontype Toeplitz operators on Bergman spaces. First, the condition for the boundedness of such convolutiontype Toeplitz operators is established. The necessity and sufficiency of the theorem are proved by means of the integral triangle inequality, derivative formula for disk automorphisms, variable substitution and the CauchySchwarz inequality. Second, based on the orthonormal basis of the Bergman space, combined with geometric series expansion, powerseries decomposition and orthogonality relations of the space, the expression for matrix elements of convolutiontype Toeplitz operators is derived. It is shown that the operator admits a lowertriangular matrix representation with constant maindiagonal entries under the standard orthonormal basis, and concrete examples of operator matrices of different orders are presented. The obtained results contribute to further understanding the structural features of convolutiontype Toeplitz operators and enrich the research of operator theory on Bergman spaces.