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 Sibirsk. Mat. Zh., 2009, Volume 50, Number 4, Pages 928–932 (Mi smj2015)

Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup

K. V. Storozhuk

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let $X$ be a Banach space and let $T\colon X\to X$ be a linear power bounded operator. Put $X_0=\{x\in X\mid T^nx\to0\}$. We prove that if $X_0\ne X$ then there exists $\lambda\in\mathrm{Sp}(T)$ such that, for every $\varepsilon>0$, there is $x$ such that $\|Tx-\lambda x\|<\varepsilon$ but $\|T^nx\|>1-\varepsilon$ for all $n$. The technique we develop enables us to establish that if $X$ is reflexive and there exists a compactum $K\subset X$ such that $\lim\inf_{n\to\infty}\rho\{T^nx,K\}<\alpha(T)<1$ for every norm-one $x\in X$ then $\operatorname{codim}X_0<\infty$. The results hold also for a one-parameter semigroup.

Keywords: operator semigroup, asymptotic finite-dimensionality.

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English version:
Siberian Mathematical Journal, 2009, 50:4, 737–740

Bibliographic databases:

UDC: 517.954+517.984.5

Citation: K. V. Storozhuk, “Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup”, Sibirsk. Mat. Zh., 50:4 (2009), 928–932; Siberian Math. J., 50:4 (2009), 737–740

Citation in format AMSBIB
\Bibitem{Sto09} \by K.~V.~Storozhuk \paper Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup \jour Sibirsk. Mat. Zh. \yr 2009 \vol 50 \issue 4 \pages 928--932 \mathnet{http://mi.mathnet.ru/smj2015} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2583631} \transl \jour Siberian Math. J. \yr 2009 \vol 50 \issue 4 \pages 737--740 \crossref{https://doi.org/10.1007/s11202-009-0084-6} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000268837600022} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70350022256} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. K. V. Storozhuk, “A condition for asymptotic finite-dimensionality of an operator semigroup”, Siberian Math. J., 52:6 (2011), 1104–1107
2. Emelyanov Eduard Yu., “Asimptoticheski konechnomernye operatory v banakhovykh prostranstvakh. nedavnie prodvizheniya i otkrytye problemy”, Matematicheskii forum (Itogi nauki. Yug Rossii), 5 (2011), 57–62
3. K. V. Storozhuk, “Isometries with Dense Windings of the Torus in $C(M)$”, Funct. Anal. Appl., 46:3 (2012), 232–233
4. A. G. Baskakov, I. I. Strukova, I. A. Trishina, “Solutions almost periodic at infinity to differential equations with unbounded operator coefficients”, Siberian Math. J., 59:2 (2018), 231–242
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