Let $\mathcal{L}$ be a cocycle on a separable Banach space $X$, and suppose that

for a constant $\kappa \in [-\infty, \infty]$. Then $\mathcal{L}$ is quasicompact if and only if $\kappa < \lambda_1$ almost everywhere. Combined with the Oseledets decomposition, the decomposition is nontrivial ($L \ge 2$) exactly when the growth rate of $\|\mathcal{L}^{(n)}_\omega\|_c$ is strictly smaller than that of $\|\mathcal{L}^{(n)}_\omega\|$.