Kotani Theory for ergodic matrix-like Jacobi operators

We extend the so-called Kotani Theory for a particular class of ergodic matrix-like Jacobi operators defined in $l^2(\mathbb{Z};{\mathbb{C}}^l)$ 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}$,where$T: \Omega \rightarrow \Omega$isanergodicautomorphisminthemeasurespace$(\Omega, \nu)$,themap$D: \Omega \rightarrow GL(l, $\mathbb{R}$)$ is bounded, and for each $\omega \in \mathrm{\Omega }$, $D(\omega )$ is symmetric. Namely, it is shown that for each $r\in \{1,\dots ,l\}$, the essential closure of $$\mathcal{Z}$_{r} := {x \in $\mathbb{R}$\mid$exactly$2r$Lyapunovexponentsof$A_z$ are zero$\}$ coincides with ${\sigma }_{ac,2r}(H_{\omega })$, the absolutely continuous spectrum of multiplicity $2r$, where $A_z$ is a Schr"odinger-like cocycle induced by $H_{\omega }$. Moreover, if $k\in \{1,\dots ,2l\}$ is odd, then ${\sigma }_{ac,k}(H_{\omega })=\mathrm{\varnothing }$ for $\nu$-a.e. $\omega \in \mathrm{\Omega }$. We also provide a Thouless Formula for such class of operators.