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Mathematics of the USSR-Sbornik, 1991, Volume 70, Issue 1, Pages 143–163
DOI: https://doi.org/10.1070/SM1991v070n01ABEH002120
(Mi sm1137)
 

This article is cited in 9 scientific papers (total in 9 papers)

Weighted shift operator, spectral theory of linear extensions, and the Multiplicative Ergodic Theorem

Yu. D. Latushkina, A. M. Stepinb

a Sea Gidrophysical Institute Academy of Sciences of UkSSR
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
References:
Abstract: The author studies the weighted shift operator $(T_af)(x)=\rho^{1/2}(x)a(\alpha^{-1}x)f(\alpha^{-1}x)$, acting in the space $L_2(X,\mu;H)$ of functions on a compact metric space $X$ with values in a separable Hilbert space $H$. Here $\alpha$ is a homeomorphism of $X$ with a dense set of nonperiodic points, the measure $\mu$ is quasi-invariant with respect to $\alpha$, $\rho=\dfrac{d\mu\alpha^{-1}}{d\mu}$, and $a$ is a continuous function on $X$ with values in the algebra of bounded operators on $H$. It is established that the dynamic spectrum of the extension $\hat\alpha(x,v)=(\alpha x,a(x)v)$, $x\in X$, $v\in H$ can be obtained from the spectrum $\sigma(T_a)$ in $L_2$ by taking the logarithm of $|\sigma(T_a)|$. Using the Riesz projections for $T_a$, the spectral subbundles for $\hat\alpha$ are described. In the case that $a$ takes compact values, the dynamic spectrum can be computed in terms of the exact Lyapunov exponents of the cocycle constructed from $a$ and $\alpha$, corresponding to measures ergodic for $\alpha$ on $X$.
Received: 31.01.1989
Bibliographic databases:
UDC: 517.9
MSC: Primary 47B37, 47A35, 47A10; Secondary 28D99, 34C35
Language: English
Original paper language: Russian
Citation: Yu. D. Latushkin, A. M. Stepin, “Weighted shift operator, spectral theory of linear extensions, and the Multiplicative Ergodic Theorem”, Math. USSR-Sb., 70:1 (1991), 143–163
Citation in format AMSBIB
\Bibitem{LatSte90}
\by Yu.~D.~Latushkin, A.~M.~Stepin
\paper Weighted shift operator, spectral theory of linear extensions, and the Multiplicative Ergodic Theorem
\jour Math. USSR-Sb.
\yr 1991
\vol 70
\issue 1
\pages 143--163
\mathnet{http://mi.mathnet.ru//eng/sm1137}
\crossref{https://doi.org/10.1070/SM1991v070n01ABEH002120}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1072294}
\zmath{https://zbmath.org/?q=an:0777.47022}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?1991SbMat..70..143L}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1991GG78300010}
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  • https://doi.org/10.1070/SM1991v070n01ABEH002120
  • https://www.mathnet.ru/eng/sm/v181/i6/p723
  • This publication is cited in the following 9 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник - 1989–1990 Sbornik: Mathematics
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    Abstract page:681
    Russian version PDF:203
    English version PDF:27
    References:86
    First page:3
     
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