On polynomially bounded operators with shift-type invariant subspaces

TL;DR

This paper extends a previous result to polynomially bounded operators, showing that if their unitary asymptote contains a bilateral shift, then they have an invariant subspace similar to a unilateral shift, leading to reflexivity.

Contribution

It generalizes the shift-invariant subspace result from contractions to polynomially bounded operators and proves their reflexivity.

Findings

Existence of invariant subspaces similar to unilateral shifts

Reflexivity of certain polynomially bounded operators

Corollaries extending previous shift-invariant results

Abstract

A particular case of [07] was generalized from contractions to polynomially bounded operators in [G19]. Namely, it is proved in [G19] that if the unitary asymptote of a polynomially bounded operator $T$ contains the bilateral shift of multiplicity $1$, then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of multiplicity $1$. In the present paper, some corollaries of this result are given. In particular, reflexivity of polynomially bounded operators described above is proved.