Exponential dichotomy for dynamically defined matrix-valued Jacobi operators

We present in this work a proof of the exponential dichotomy for dynamically defined matrix-valued Jacobi operators in $({\mathbb{C}}^l{)}^{\mathbb{Z}}$, given for each $\omega \in \mathrm{\Omega }$ by the law $[H_{\omega }\mathbf{\text{u}}{]}_n:=D(T^{n-1}\omega ){\mathbf{\text{u}}}_{n-1}+D(T^n\omega ){\mathbf{\text{u}}}_{n+1}+V(T^n\omega ){\mathbf{\text{u}}}_n$, where $\mathrm{\Omega }$ is a compact metric space, $T:\mathrm{\Omega }\to \mathrm{\Omega }$ is a minimal homeomorphism and $D,V:\mathrm{\Omega }\to M(l,\mathbb{R})$ are continuous maps with $D(\omega )$ invertible for each $\omega \in \mathrm{\Omega }$. Namely, we show that for each $\omega \in \mathrm{\Omega }$, $$\rho (H_{\omega })=\{z\in \mathbb{C}\mid (T,A_z)\mathrm{i}\mathrm{s}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\},$$ where $\rho (H_{\omega })$ is the resolvent set of $H_{\omega }$ and $(T,A_z)$ is the $SL(2l,\mathbb{C})$-cocycle induced by the eigenvalue equation $[H_{\omega }u{]}_n=z{u}_n$ at $z\in \mathbb{C}$.