发表机构
College of Mathematical Science, Chongqing Univeristy of Technology（重庆理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多圆盘Hardy空间中由特定内函数族生成的商模上的压缩乘法算子，刻画其自交换子与交叉交换子的Hilbert-Schmidt性质，并给出范数的显式公式。
AI 中文摘要
设$\eta_{1},\cdots,\eta_{d}$是非常数单变量内函数，满足对所有$1\leq i\leq d$有$\eta_{i}(0)=0$。记$M_{\eta}$为Hardy空间$H^{2}(\mathbb{D}^{d})$中由函数族$\{\eta_{i}(z_{i})-\eta_{i+1}(z_{i+1})\mid 1\leq i\leq d-1\}$生成的子模。对于相应的商模$N_{\eta}=H^{2}(\mathbb{D}^{d})\ominus M_{\eta}$，我们定义压缩乘法算子$S_{z_{i}}=P_{N_{\eta}}M_{z_{i}}\vert_{N_{\eta}}$，其中$P_{N_{\eta}}$是从$H^{2}(\mathbb{D}^{d})$到$N_{\eta}$的正交投影。本文刻画了自交换子$[S_{z_{i}}^{\ast},S_{z_{i}}]$和交叉交换子$[S_{z_{i}}^{\ast},S_{z_{j}}]$（$i\neq j$）的Hilbert-Schmidt性质。此外，当这些交换子为Hilbert-Schmidt算子时，建立了其Hilbert-Schmidt范数的显式公式。
英文摘要
Let $η_{1},\cdots,η_{d}$ be nonconstant one-variable inner functions satisfying $η_{i}(0)=0$ for all $1\leq i\leq d$. Denote by $M_η$ the submodule of the Hardy space $H^{2}(\mathbb{D}^{d})$ generated by the function family $\{η_{i}(z_{i})-η_{i+1}(z_{i+1})\mid 1\leq i\leq d-1\}$. For the corresponding quotient module $N_η=H^{2}(\mathbb{D}^{d})\ominus M_η$, we define the compressed multiplication operators $S_{z_{i}}=P_{N_η}M_{z_{i}}\vert_{N_η}$, where $P_{N_η}$ is the orthogonal projection from $H^{2}(\mathbb{D}^{d})$ onto $N_η$. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators $[S_{z_{i}}^{\ast},S_{z_{i}}]$ and cross-commutators $[S_{z_{i}}^{\ast},S_{z_{j}}]$ with $i\neq j$. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.
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