  We prove that for semi-invertible and H\"older continuous linear cocycles $A$
acting on an arbitrary Banach space and defined over a base space that
satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of
$A$ with respect to an ergodic invariant measure for base dynamics can be
approximated with Lyapunov exponents of $A$ with respect to ergodic measures
supported on periodic orbits. Our result is applicable to a wide class of
infinite-dimensional dynamical systems.
