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

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 $X$.