Green matrix estimates of block Jacobi matrices I: Unbounded gap in the essential spectrum

TL;DR

This paper establishes decay bounds for Green matrices and eigenvectors of block Jacobi matrices within spectral gaps, considering both commutative and noncommutative cases, with an illustrative example.

Contribution

It provides new decay estimates for Green matrices in spectral gaps of block Jacobi operators, including noncommutative cases, expanding understanding of spectral properties.

Findings

Decay bounds for Green matrices in spectral gaps

Results for both commutative and noncommutative matrix entries

An example illustrating the theoretical results

Abstract

This work deals with decay bounds for Green matrices and generalized eigenvectors of block Jacobi matrices when the real part of the spectral parameter lies in an infinite gap of the operator's essential spectrum. We consider the cases of commutative and noncommutative matrix entries separately. An example of a block Jacobi operator with noncommutative entries and nonnegative essential spectrum is given to illustrate the results.