Let F_K denote the set of infinite-dimensional cocycles over a μ-ergodic flow φ^t : M → M and with fiber dynamics given by a compact semiflow on a Hilbert space. We prove that there exists a residual subset R of F_K such that for φ ∈ R and for μ-almost every x ∈ M, either:
(i) the limit operator lim (φ^t(x)^*φ^t(x))^(1/2t) when t→∞ is the null operator or else
(ii) the Oseledets–Ruelle splitting of along the φ^t -orbit of x has a dominated splitting.
