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 $ω\in Ω$ by the law $[H_ω \textbf{u}]_{n} := D(T^{n - 1}ω) \textbf{u}_{n - 1} + D(T^{n}ω) \textbf{u}_{n + 1} + V(T^{n}ω) \textbf{u}_{n}$, where $Ω$ is a compact metric space, $T: Ω\rightarrow Ω$ is a minimal homeomorphism and $D, V: Ω\rightarrow M(l, \mathbb{R})$ are continuous maps with $D(ω)$ invertible for each $ω\inΩ$. Namely, we show that for each $ω\inΩ$, \[ρ(H_ω)=\{z \in \mathbb{C}\mid (T, A_z)\;\mathrm{is\; uniformly\; hyperbolic}\}, \] where $ρ(H_ω)$ is the resolvent set of $H_ω$ and $(T, A_z)$ is the $SL(2l,\mathbb{C})$-cocycle induced by the eigenvalue equation $[H_ωu]_n=zu_n$ at $z\in\mathbb{C}$.