Journal of Operator Theory

Volume 87, Issue 2, Spring 2022 pp. 251-270.

On polynomially bounded operators with shift-type invariant subspaces

Authors: Maria F. Gamal
Author institution:St. Petersburg Branch, V.A. Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023, Russia

Summary: Generalizing a particular case of a result by Kerchy (2007) for contractions, the following was proved by the author: if $T$ is a polynomially bounded operator and there exists a transformation with dense range which intertwines $T$ with 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, several corollaries of this result are given. In particular, reflexivity of polynomially bounded operators described above is proved.

DOI: http://dx.doi.org/10.7900/jot.2020aug23.2302
Keywords: similarity, unilateral shift, invariant subspaces, unitary asymptote, intertwining relation, polynomially bounded operator, reflexivity

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