  Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with
the exponential growth rates of vectors under the action of a linear cocycle on
R^d. When the linear actions are invertible, the MET guarantees an
almost-everywhere pointwise splitting of 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 that a splitting over R^d is guaranteed even 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.
