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Asymptotic limits of operators similar to normal operators


Author: György Pál Gehér
Journal: Proc. Amer. Math. Soc. 143 (2015), 4823-4834
MSC (2010): Primary 47B40; Secondary 47A45, 47B15
DOI: https://doi.org/10.1090/proc/12632
Published electronically: April 2, 2015
MathSciNet review: 3391040
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Abstract: Sz.-Nagy's famous theorem states that a bounded operator $ T$ which acts on a complex Hilbert space $ \mathcal {H}$ is similar to a unitary operator if and only if $ T$ is invertible and both $ T$ and $ T^{-1}$ are power bounded. There is an equivalent reformulation of that result which considers the self-adjoint iterates of $ T$ and uses a Banach limit $ L$. In this paper first we present a generalization of the necessity part in Sz.-Nagy's result concerning operators that are similar to normal operators. In the second part we provide a characterization of all possible strong operator topology limits of the self-adjoint iterates of those contractions which are similar to unitary operators and act on a separable infinite-dimensional Hilbert space. This strengthens Sz.-Nagy's theorem for contractions.


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Additional Information

György Pál Gehér
Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary – and – MTA-DE “Lendület” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
Email: gehergy@math.u-szeged.hu

DOI: https://doi.org/10.1090/proc/12632
Keywords: Power bounded Hilbert space operators, similarity to normal operators, asymptotic behaviour
Received by editor(s): May 7, 2014
Received by editor(s) in revised form: August 25, 2014
Published electronically: April 2, 2015
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2015 American Mathematical Society