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 \({\mathcal {M}}\) in Hardy module \(H^2({\mathbb {D}}^n)\) on the unit polydisc in \(\mathbb {C}^{n}\), we define the \(n-1\) tuple of fringe operators \(\textbf{F}=(F_{1},F_{2},\ldots ,F_{n-1})\) and the n tuple of restriction operators \(\textbf{R}=(R_{z_{1}},R_{z_{2}},\ldots , R_{z_{n}})\) with respect to \({\mathcal {M}}\). In this paper, for the case \(n=3\), it is shown that the fringe operators \(\textbf{F}\) are Fredholm if and only if the tuple \(\textbf{R}-\lambda \) is Fredholm, where \(\lambda \in {\mathbb {D}}^3\), and moreover \(ind(\textbf{F})=-ind(\mathbf{R-\lambda })\), 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 \(n=2\). Finally, we also discuss the difference quotient operators in \(H^2({\mathbb {D}}^n)\), and apply them to explore the relationship between the fringe operators and compression operators on quotient module.