Invariant subspaces generated by a single function in the polydisc

TL;DR

This paper characterizes all invariant subspaces generated by a single function in the Hardy space of the polydisc, providing a complete description and linking them to outer functions and unitary equivalence.

Contribution

It offers the first complete description of singly generated invariant subspaces in the polydisc Hardy space, addressing an open question from Rudin's work.

Findings

Complete characterization of invariant subspaces generated by a single function.

Unitary equivalence of these subspaces with specific function spaces.

Characterization of outer functions in the polydisc Hardy space.

Abstract

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