Abstract
The author studies the weighted shift operator acting in the space of functions on a compact metric space with values in a separable Hilbert space . Here is a homeomorphism of with a dense set of nonperiodic points, the measure is quasi-invariant with respect to , , and is a continuous function on with values in the algebra of bounded operators on . It is established that the dynamic spectrum of the extension , , can be obtained from the spectrum in by taking the logarithm of . Using the Riesz projections for , the spectral subbundles for are described. In the case that takes compact values, the dynamic spectrum can be computed in terms of the exact Lyapunov exponents of the cocycle constructed from and , corresponding to measures ergodic for on .
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