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

This article is cited in 4 scientific papers (total in 4 papers)

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
Received: 02.04.2008

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
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\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}
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\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
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    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  mathnet  crossref  mathscinet  isi
    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  mathscinet  elib
    3. K. V. Storozhuk, “Isometries with Dense Windings of the Torus in $C(M)$”, Funct. Anal. Appl., 46:3 (2012), 232–233  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  isi  elib
  • Сибирский математический журнал Siberian Mathematical Journal
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