A Shimorin-type analytic model on an annulus for left-invertible operators and applications

TL;DR

This paper introduces a new analytic model for left-invertible operators that generalizes existing models, representing them as multiplication operators on a space of vector-valued analytic functions on an annulus or disc, with applications to composition operators.

Contribution

It extends Shimorin's and Gellar's models to a broader class of operators, providing a unified framework for their analysis on annuli and discs.

Findings

Modeling of left-invertible operators as multiplication operators

Extension of existing models to annuli and discs

Application to composition operators in $ ext{ell}^2$-spaces

Abstract

A new analytic model for left-invertible operators, which extends both Shimorin's analytic model for left-invertible and analytic operators and Gellar's model for bilateral weighted shift is introduced and investigated. We show that a left-invertible operator $T$, which satisfies certain conditions can be modelled as a multiplication operator $\mathscr{M}_z$ on a reproducing kernel Hilbert space of vector-valued analytic functions on an annulus or a disc. A similar result for composition operators in $\ell^2$-spaces is established.