Asymptotic limits of operators similar to normal operators
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- by György Pál Gehér PDF
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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.References
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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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4823-4834
- MSC (2010): Primary 47B40; Secondary 47A45, 47B15
- DOI: https://doi.org/10.1090/proc/12632
- MathSciNet review: 3391040