On certain commuting isometries, joint invariant subspaces and C*-algebras

TL;DR

This paper investigates the structure of commuting isometries and their invariant subspaces, establishing a unitary equivalence of associated C*-algebras for certain finite-codimension subspaces in multivariable Hardy spaces.

Contribution

It extends the theory of commuting isometries by analyzing their invariant subspaces and demonstrating the unitary equivalence of generated C*-algebras in multivariable Hardy spaces.

Findings

C*-algebra generated by restricted n-shift is unitarily equivalent to the original

Invariant subspaces of finite codimension retain algebraic structure

Analytic representations of commuting n-isometries are characterized

Abstract

In this paper, motivated by the Berger, Coburn and Lebow and Bercovici, Douglas and Foias theory for tuples of commuting isometries, we study analytic representations and joint invariant subspaces of a class of commuting $n$-isometries and prove that the $C^*$-algebra generated by the $n$-shift restricted to an invariant subspace of finite codimension in $H^2(\mathbb{D}^n)$ is unitarily equivalent to the $C^*$-algebra generated by the $n$-shift on $H^2(\mathbb{D}^n)$.