On unitarily equivalent submodules

TL;DR

This paper investigates the structure of submodules in Hardy spaces on complex domains, revealing that certain isomorphic submodules with finite or infinite codimension only occur in specific settings, and characterizing their properties.

Contribution

It demonstrates that unitarily equivalent submodules with finite codimension occur only in Hardy-like spaces over planar or finitely-connected domains, and characterizes modules with infinite codimension as subnormal and Hardy-like.

Findings

Such submodules only occur in Hardy-like spaces over planar domains.

Modules with infinite codimension are subnormal and Hardy-like on nice domains.

The phenomenon is restricted to domains in the complex plane and finitely-connected domains.

Abstract

The Hardy space on the unit ball in C^n provides examples of a quasi-free, finite rank Hilbert module which contains a pure submodule isometrically isomorphic to the module itself. For n=1 the submodule has finite codimension. In this note we show that this phenomenon can only occur for modules over domains in the complex plain and for finitely-connected domains only for Hardy-like spaces, the bundle shifts. Moreover, we show for essentially reductive modules that even when the codimension is infinite, the module is subnormal and again, on nice domains such as the unit ball, must be Hardy-like.