A version of the theorem of Johnson, Palmer and Sell for quasicompact cocycles

The well-known theorem of Johnson, Palmer and Sell asserts that the endpoints of the Sacker–Sell spectrum of a given cocycle of invertible matrices over a topological dynamical system $(M,f)$ are realized as Lyapunov exponents with respect to some ergodic invariant probability measure for $f$. In this note we establish the version of this result for quasicompact cocycles of operators acting on an arbitrary Banach space.