On the unilateral shift as a Hilbert module over the disc algebra

TL;DR

This paper investigates the unilateral shift as a Hilbert module over the disc algebra, focusing on extension groups and the triviality of polynomial extensions, using function theoretic decompositions of model spaces.

Contribution

It introduces the concept of polynomial extensions in this context and proves their triviality for a class of modules, advancing understanding of their projectivity.

Findings

Polynomial extensions of a contractive module by the adjoint of the unilateral shift are trivial.

A function theoretic decomposition of Sz.-Nagy--Foias model space is used as a key tool.

The study relates to the projectivity problem of the unilateral shift as a Hilbert module.

Abstract

We study the unilateral shift (of arbitrary countable multiplicity) as a Hilbert module over the disc algebra and the associated extension groups. In relation with the problem of determining whether this module is projective, we consider a special class of extensions, which we call "polynomial". We show that the subgroup of polynomial extensions of a contractive module by the adjoint of the unilateral shift is trivial. The main tool is a function theoretic decomposition of the Sz.-Nagy--Foias model space for completely non-unitary contractions.