Let $\mathcal{L}$ be a cocycle on a Banach space $X$. An Oseledets decomposition of $\mathcal{L}$ consists of a number $L \in \{1, 2, \ldots, \infty\}$, exponents $\lambda_i$, multiplicities $m_i$, fast spaces $E_i(\omega)$, slow spaces $V_l(\omega)$ and projections $\Pi_l(\omega)$ with exactly the properties asserted by the semi-invertible Oseledets decomposition theorem:

• the $\lambda_i$ ($i \le L$) are almost surely the distinct Lyapunov exponents of $\mathcal{L}$, $i < L$ holds exactly when $m_i \ne 0$ almost surely, and $L \ge 2$ exactly when $\mathcal{L}$ is quasicompact;
• the $E_i$ are measurable and the $\Pi_l$ strongly measurable and tempered;
• for $i < L$, almost surely $E_i(\omega)$ has dimension $m_i$, $\mathcal{L}_\omega E_i(\omega) = E_i(\sigma\omega)$, and both $\|\mathcal{L}^{(n)}_\omega|_{E_i(\omega)}\|$ and $g(\mathcal{L}^{(n)}_\omega, E_i(\omega))$ grow at exponential rate $\lambda_i$;
• for $2 \le l \le L$, almost surely $X = E_1(\omega) \oplus \cdots \oplus E_{l-1}(\omega) \oplus V_l(\omega)$, $\mathcal{L}_\omega V_l(\omega) \subseteq V_l(\sigma\omega)$, $\Pi_l(\omega)$ is the projection onto $V_l(\omega)$ along $E_1(\omega) \oplus \cdots \oplus E_{l-1}(\omega)$, and $V_l(\omega) = \{x \in X : \lambda_\omega(x) \le \lambda_l\}$.

Indices are zero-based as in the decomposition theorem: $lam i$ is $\lambda_{i+1}$, $E i$ is $E_{i+1}$, $mdim i + 1$ is $m_{i+1}$, $Lval$ is $L$, and $V (l + 1)$, $P (l + 1)$ are $V_{l+2}$, $\Pi_{l+2}$. The fast spaces $E i hi$ are given only at the indices with $hi : i + 2 ≤ Lval$, so on the zero space, where $L = 1$, there are none. The values of $lam$, $mdim$, $V$ and $P$ at indices beyond $L$ are unconstrained.