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Journal of Operator Theory

Volume 84, Issue 1, Summer 2020  pp. 3-34.

On existence of shift-type invariant subspaces for polynomially bounded operators

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: A very particular case of results by K\'erchy (2007) is as follows. Let the unitary asymptote of a contraction $T$ contain 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 $1$. The proof is based on the Sz.-Nagy--Foias functional model for contractions. In the present paper this result is generalized to polynomially bounded operators. The proof is based on a result by Bourgain (1984).

Keywords: shift-type invariant subspace, polynomially bounded operator, similarity, unilateral shift

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