标题： 斜托普利茨和斜小汉克尔算子在加权伯格曼空间上的代数和谱性质 显示英文标题

标题： Algebraic and Spectral properties of slant Toeplitz and slant little Hankel Operators on weighted Bergman Space

摘要： 本文研究了加权 Bergman 空间\(\mathcal{A}_\alpha^2(\mathbb{D})\)上的\(k^{th}-\)阶斜 Toeplitz 和斜小 Hankel 算子。这些算子分别通过斜移位算子\(W_k\)与经典 Toeplitz 和 Hankel 算子的组合构造而成。我们推导了它们的矩阵表示，并建立了有界性、紧性和正规性的判据。得到了交换性条件，表明两个这样的算子交换当且仅当它们的符号线性相关。正规性被表征为斜 Toeplitz 算子只有在符号为常数时才是正规的，而斜小 Hankel 算子在符号为解析函数时才是正规的。紧性被证明恰好发生在符号消失的时候。分析了包括本质谱和特征值分布在内的谱性质。数值模拟验证了理论结果，并突出了两类算子之间的关键结构和计算差异。 显示英文摘要

摘要： This paper studies the \(k^{th}-\)order slant Toeplitz and slant little Hankel operators on the weighted Bergman space \(\mathcal{A}_\alpha^2(\mathbb{D})\). These operators are constructed using a slant shift operator \(W_k\) composed with classical Toeplitz and Hankel operators, respectively. We derive their matrix representations and establish criteria for boundedness, compactness, and normality. Commutativity conditions are obtained, showing that two such operators commute if and only if their symbols are linearly dependent. Normality is characterized as slant Toeplitz operators are normal only for constant symbols, while slant little Hankel operators are normal if the symbol is analytic. Compactness is shown to occur precisely when the symbol vanishes. Spectral properties, including essential spectra and eigenvalue distributions, are analyzed. Numerical simulations corroborate the theoretical findings and highlight key structural and computational differences between the two operator classes.