Kotani Theory for ergodic matrix-like Jacobi operators

Abstract

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 $[H_{\omega }\text{u}{]}_n:=D^{\ast }(T^{n-1}\omega ){\text{u}}_{n-1}+D(T^n\omega ){\text{u}}_{n+1}+V(T^n\omega ){\text{u}}_n$, where $T:\Omega \to \Omega$ is an ergodic automorphism in the measure space $(\Omega ,\nu )$, the map $D:\Omega \to GL(l,\mathbb{R})$ is bounded, and for each $\omega \in \Omega$, $D(\omega )$ 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 $2r$ Lyapunov exponents of $A_z$ are zero$\}$ coincides with ${\sigma }_{ac,2r}(H_{\omega })$, the absolutely continuous spectrum of multiplicity $2r$, where $A_z$ is a Schrödinger-like cocycle induced by $H_{\omega }$. Moreover, if $k\in \{1,\dots ,2l\}$ is odd, then ${\sigma }_{ac,k}(H_{\omega })=\varnothing$ for $\nu$-a.e. $\omega \in \Omega$. We also provide a Thouless Formula for such class of operators.