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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
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
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
Linking options:
https://www.mathnet.ru/eng/sm1137https://doi.org/10.1070/SM1991v070n01ABEH002120 https://www.mathnet.ru/eng/sm/v181/i6/p723
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Abstract page: | 681 | Russian version PDF: | 203 | English version PDF: | 27 | References: | 86 | First page: | 3 |
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