Exponential dichotomy for dynamically defined matrix-valued Jacobi operators

TL;DR

This paper proves the exponential dichotomy for a class of matrix-valued Jacobi operators defined dynamically over a compact space, linking spectral properties to hyperbolic cocycles.

Contribution

It establishes the exponential dichotomy for these operators and characterizes the resolvent set via hyperbolic cocycles, extending spectral theory in a dynamical setting.

Findings

Spectral set characterized by hyperbolic cocycles

Exponential dichotomy proven for the operators

Connection between resolvent set and hyperbolic dynamics

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