Exponential dichotomy for dynamically defined matrix-valued Jacobi operators

Abstract

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 $\omega \in \Omega$ by the law $[H_{\omega }\text{u}{]}_n:=D(T^{n-1}\omega ){\text{u}}_{n-1}+D(T^n\omega ){\text{u}}_{n+1}+V(T^n\omega ){\text{u}}_n$, where $\Omega$ is a compact metric space, $T:\Omega \to \Omega$ is a minimal homeomorphism and $D,V:\Omega \to M(l,\mathbb{R})$ are continuous maps with $D(\omega )$ invertible for each $\omega \in \Omega$. Namely, we show that for each $\omega \in \Omega$, \[ρ(H_ω)=\{z \in \mathbb{C}\mid (T, A_z)\;\mathrm{is\; uniformly\; hyperbolic}\}, \] 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 $z\in \mathbb{C}$.