Summary: Oseledets’ celebrated Multiplicative Ergodic Theorem (MET) [V. I. Oseledets, Trans. Mosc. Math. Soc. 19, 197–231 (1968; Zbl 0236.93034); translation from Tr. Mosk. Mat. Obshch. 19, 179–210 (1968)] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on \(\mathbb{R}^d\). When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of \(\mathbb{R}^d\) into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets’ MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [Ergodic Theory Dyn. Syst. 30, No. 3, 729–756 (2010; Zbl 1205.37015)] that a splitting over \(\mathbb{R}^d\) is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.