Fredholm Index of 3-Tuple of Restriction Operators and the Pair of Fringe Operators for Submodules in \(H^2({\mathbb {D}}^3)\)

For a submodule \({\mathcal {M}}\) $\mathcal{M}$ in Hardy module \(H^2({\mathbb {D}}^n)\) $H^2({\mathbb{D}}^n)$ on the unit polydisc in \(\mathbb {C}^{n}\) ${\mathbb{C}}^n$ , we define the \(n-1\) $n-1$ tuple of fringe operators \(\textbf{F}=(F_{1},F_{2},\ldots ,F_{n-1})\) $\mathbf{F}=(F_1,F_2,\dots ,F_{n-1})$ and the n tuple of restriction operators \(\textbf{R}=(R_{z_{1}},R_{z_{2}},\ldots , R_{z_{n}})\) $\mathbf{R}=(R_{z_1},R_{z_2},\dots ,R_{z_n})$ with respect to \({\mathcal {M}}\) $\mathcal{M}$ . In this paper, for the case \(n=3\) $n=3$ , it is shown that the fringe operators \(\textbf{F}\) $\mathbf{F}$ are Fredholm if and only if the tuple \(\textbf{R}-\lambda \) $\mathbf{R}-\lambda$ is Fredholm, where \(\lambda \in {\mathbb {D}}^3\) $\lambda \in {\mathbb{D}}^3$ , and moreover \(ind(\textbf{F})=-ind(\mathbf{R-\lambda })\) $ind(\mathbf{F})=-ind(\mathbf{R}-\lambda )$ , which answer a question of Yang (Proc Am Math Soc 131 (2):533–541, 2003) partly and generalize a result of Luo et al. (J Math Anal Appl 465(1):531–546, 2018) in the case \(n=2\) $n=2$ . Finally, we also discuss the difference quotient operators in \(H^2({\mathbb {D}}^n)\) $H^2({\mathbb{D}}^n)$ , and apply them to explore the relationship between the fringe operators and compression operators on quotient module.