Fredholm Index of 3-Tuple of Restriction Operators and the Pair of Fringe Operators for Submodules in H2(D3)

For a submodule M in Hardy module H 2 ( D n ) on the unit polydisc in C n , we define the n - 1 tuple of fringe operators F = ( F 1 , F 2 , … , F n - 1 ) and the n tuple of restriction operators R = ( R z 1 , R z 2 , … , R z n ) with respect to M . In this paper, for the case n = 3 , it is shown that the fringe operators F are Fredholm if and only if the tuple R - λ is Fredholm, where λ ∈ D 3 , and moreover i n d ( F ) = - i n d ( R - λ ) , 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 ( D n ) , and apply them to explore the relationship between the fringe operators and compression operators on quotient module.