  We consider group-valued cocycles over dynamical systems. The base system is
a homeomorphism $f$ of a metric space satisfying a closing property, for
example a hyperbolic dynamical system or a subshift of finite type. The cocycle
$A$ takes values in the group of invertible bounded linear operators on a
Banach space and is H\"older continuous. We prove that upper and lower Lyapunov
exponents of $A$ with respect to an ergodic invariant measure $\mu$ can be
approximated in terms of the norms of the values of $A$ on periodic orbits of
$f$. We also show that these exponents cannot always be approximated by the
exponents of $A$ with respect to measures on periodic orbits. Our arguments
include a result of independent interest on construction and properties of a
Lyapunov norm for infinite dimensional setting. As a corollary, we obtain
estimates of the growth of the norm and of the quasiconformal distortion of the
cocycle in terms of the growth at the periodic points of $f$.
