Invariant Subspaces Generated by a Single Function in the Polydisk

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 $H^2(U^n)$.