Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on $N_{\eta }$-type Quotient Modules in Polydisc

Let ${\eta }_1,\cdots ,{\eta }_d$ be nonconstant one-variable inner functions satisfying ${\eta }_i(0)=0$ for all $1\le i\le d$. Denote by $M_{\eta }$ the submodule of the Hardy space $H^2({\mathbb{D}}^d)$ generated by the function family $\{{\eta }_i(z_i)-{\eta }_{i+1}(z_{i+1})\mid 1\le i\le d-1\}$. For the corresponding quotient module $N_{\eta }=H^2({\mathbb{D}}^d)\ominus M_{\eta }$, we define the compressed multiplication operators $S_{z_i}=P_{N_{\eta }}{M}_{z_i}{\mathrm{\mid }}_{N_{\eta }}$, where $P_{N_{\eta }}$ is the orthogonal projection from $H^2({\mathbb{D}}^d)$ onto $N_{\eta }$. 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 $i\mathrm{\ne }j$. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.