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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
$X$.
UDC:
517.9
MSC: Primary
47B37,
47A35,
47A10; Secondary
28D99,
34C35 Received: 31.01.1989
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