Fock-Sobolev空间上的复对称加权复合算子

Complex Symmetric Weighted Composition Operators on the Fock-Sobolev Spaces

摘要: 伴随不同函数空间上的复对称加权复合算子得到广泛关注，本文致力于研究Fock-Sobolev空间上的复对称加权复合算子$\phi C_{\tau }$ 。通过引入复对称算子的概念，运用混合偏导相关公式、分类讨论、数学归纳、反证等方法，得到了$\phi \left(z\right)$ 恒不为零以及$\phi C_{\tau }$ 的核空间只含零向量，得到了$\phi C_{\tau }$ 的特征值都可表示为$\phi \left(\kappa \right){\left({\partial }^1\tau \right)}^n\left(\kappa \right)$ 形式以及其点谱的具体表达式，并给出了乘法算子$\phi$ 和复合算子$\tau$ 关于共轭算子和再生核函数的关系式。这些发现深化了对Fock-Sobolev空间上的复对称加权复合算子的理解，也为其他函数空间上复对称加权复合算子的研究奠定了理论基础。

Abstract: Complex symmetric weighted composition operators on different function spaces are widely concerned. In this paper, we study complex symmetric weighted composition operators on Fock-Sobolev spaces. By introducing the concept of complex symmetric operator, using the correlation formula of mixed partial derivative, classification discussion, mathematical induction, inverse proof and other methods, we get $\phi \left(z\right)$ is always non-zero and the kernel space of $\phi C_{\tau }$ only contains zero vector, get the eigenvalues of $\phi C_{\tau }$ can be expressed as $\phi \left(\kappa \right){\left({\partial }^1\tau \right)}^n\left(\kappa \right)$ and the specific expression of its point spectrum, and give the relations of multiplication operator $\phi$ and compound operator $\tau$ with respect to conjugate operator and the reproducing kernel. These findings deepen the understanding of complex symmetric weighted composition operators on Fock-Sobolev spaces, and also lay a theoretical foundation for the study of complex symmetric weighted composition operators on other function spaces.