Hilbert-Schmidt Norms of Self-Commutator and Cross-Commutator on $N_η$-type Quotient Modules in Polydisc

Abstract

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.