A Note on $\Gamma_n$-isometries

TL;DR

This paper characterizes the boundary of the symmetrized polydisc and develops a model theory for $ abla_n$-isometries, extending classical results and providing a Beurling-Lax-Halmos type representation for invariant subspaces.

Contribution

It introduces a new model theory for $ abla_n$-isometries and extends Beurling-Lax-Halmos theorems to this setting, generalizing previous work for $n=2$.

Findings

Characterization of the distinguished boundary of the symmetrized polydisc

Development of a model theory for $ abla_n$-isometries

Beurling-Lax-Halmos type representation for invariant subspaces

Abstract

In this note we characterize the distinguished boundary of the symmetrized polydisc and thereby develop a model theory for $\Gamma_n$-isometries along the lines of \cite{AY}. We further prove that for invariant subspaces of $\Gamma_n$-isometries, similar to the case $n=2$ \cite{S}, Beurling-Lax-Halmos type representation holds.