Let $\mathcal{L}$ be a cocycle on a separable Banach space $X$, and suppose the growth rate of the compactness seminorm of its iterates is almost everywhere a constant $\kappa$:

Then for any Oseledets decomposition $(L, (\lambda_i), \ldots)$ of $\mathcal{L}$, the top exponent $\lambda_1 = \lim_n \frac1n \log\|\mathcal{L}^{(n)}_\omega\|$ satisfies $\kappa \le \lambda_1$, and the decomposition is nontrivial ($L \ge 2$) exactly when $\kappa \ne \lambda_1$. This is the last clause of Theorem 3 of Lee (2024), with the index of compactness in place of $\nu$. Such a $\kappa$ always exists, by the concept $IndexOfCompactness$ ($exists_compactnessIndex$).