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

On the spectrum of infinite dimensional random products of compact operators