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Journal of Operator Theory

Volume 86, Issue 1, Summer 2021  pp. 93-123.

Hilbert space operators with two-isometric dilations

Authors:  Catalin Badea (1), Laurian Suciu (2)
Author institution: (1) Universite Lille, CNRS, UMR 8524 - Laboratoire Paul Painleve, France
(2) Department of Mathematics and Informatics, ``Lucian Blaga'' University of Sibiu, Roumania

Summary: A continuous linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. We study operators having liftings or dilations to $2$-isometries. The adjoint of an operator which admits such liftings is the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators and to operators similar to contractions. Two types of liftings to $2$-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.

Keywords: dilations, $2$-isometric lifting, concave operator, A-contraction, Dirichlet shift

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