On the spectrum of infinite dimensional random products of compact operators

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic measure which is positive on non-empty open sets, we conclude that there is a residual subset of cocycles within which, for almost every x, either the Oseledets-Ruelle's decomposition along the orbit of x is dominated or has a trivial spectrum.