A semi-invertible operator Oseledets theorem (Q2925264)

Let \(X\) be a separable Banach space and \(L(X)\) be a space of bounded linear operator on \(X\) equipped with the strong topology \(\tau\) and the Borel \(\sigma\)-algebra \(\mathcal{S}\) generated by \(\tau\). We remind that the topology \(\tau\) is generated by the sub-base consisting of sets of the form \(\Big\{T : ||Tx-y||<\epsilon\Big\}\).NEWLINENEWLINEA map \(\mathcal{L} : \Omega \longrightarrow X\) is \(\mathcal{S}\) measurable if, for each \(x \in X\), the map \(\omega\longmapsto\mathcal{L}(\omega)(x)\) is measurable.NEWLINENEWLINEIn this paper, the authors extended the well-known Oseledets multiplicative ergodic theorem as follows.NEWLINENEWLINE {Theorem. } Let \(\sigma\) be an invertible ergodic measure-preserving transformation of the Lebesgue space \((\Omega,\mathcal{F},\mathbb{P})\). Let \(X\) be a separable Banach space. Let \(\mathcal{L} : \Omega \longrightarrow L(X)\) be a strongly measurable family of mappings such that \(\log^{+}\|\mathcal{L}(\omega)\| \in L^1(\mathbb{P})\) and suppose that the random linear system \(\mathcal{R}=(\Omega,\mathcal{F},\mathbb{P},\sigma,X,\mathcal{L})\) is quasi-compact (that is, the analogue of the spectral radius, \(\lambda^*\), is larger than the analogue of the essential spectral radius, \(\kappa^*\)). Then there exist \(1 \leq l \leq \infty\) and a sequence of exceptional Lyapunov exponents \(\lambda^*=\lambda_1>\lambda_2>\lambda_3>\cdots>\lambda_l>\kappa^*\) (or in the case \(l=\infty,\lambda^*=\lambda_1>\lambda_2>\cdots>\cdots;\) \(\lim_{n \longrightarrow +\infty}\lambda_n=\kappa^*)\). For a \(\mathbb{P}\)-almost every \(\omega\) there exists a unique measurable equivariant splitting of \(X\) into closed subspaces \(X=V(\omega)\oplus \bigoplus_{j=1}^{l}Y_j(\omega)\) where the \(Y_j(\omega),\) are finite-dimensional. For each \(y \in Y_j(\omega)\setminus\{0\},\) NEWLINE\[NEWLINE\lim_{ n \longrightarrow +\infty} \frac1{n}\log \|\mathcal{L}_{\omega}^n\|=\lambda_j.NEWLINE\]NEWLINE For \(y \in V(\omega)\), NEWLINE\[NEWLINE\lim_{ n \longrightarrow +\infty} \frac1{n}\log \|\mathcal{L}_{\omega}^n\| \leq \kappa^*.NEWLINE\]