Invariant subspaces generated by a single function in the Polydisk

Abstract

In this study, we partially answer a question left open in Rudin's book "Function Theory in Polydisks" on the structure of invariant subspaces of the Hardy space H (2)(U (n) ) on the polydisk U (n) . We completely describe all invariant subspaces generated by a single function in the polydisk. Then, using our results, we prove the unitary equivalence of this type of invariant subspace and a characterization of outer functions in H (2)(U (n) ).