WEIGHTED SHIFT-OPERATORS, SPECTRAL THEORY OF LINEAR EXTENSIONS, AND THE MULTIPLICATIVE ERGODIC THEOREM

The author studies the weighted shift operator (T(a)f)(x) = rho-1/2(x)a(alpha-1 x)f(alpha-1 x), acting in the space L2(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 = 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 alpha-tripple-overdot(x, upsilon) = (alpha-x, a(x)upsilon), x is-an-element-of X, upsilon is-an-element-of H can be obtained from the spectrum sigma-(T(a)) in L2 by taking the logarithm of \sigma-(T(a))\. Using the Riesz projections for T(a), the spectral subbundles for alpha-tripple-overdot 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.