# Green matrix estimates of block Jacobi matrices II: Bounded gap in the essential spectrum

## TL;DR

This paper derives decay bounds for Green matrices and eigenvectors of block Jacobi operators within spectral gaps, with refinements for commuting entries and illustrative examples.

## Contribution

It introduces new decay estimates for block Jacobi matrices in spectral gaps, including cases with commuting entries and eigenvalues.

## Key findings

- Decay bounds for Green matrices in spectral gaps
- Refined estimates for commuting matrix entries
- Examples illustrating the decay bounds

## Abstract

This paper provides decay bounds for Green matrices and generalized eigenvectors of block Jacobi operators when the real part of the spectral parameter lies in a bounded gap of the operator's essential spectrum. The case of the spectral parameter being an eigenvalue is also considered. It is also shown that if the matrix entries commute, then the estimates can be refined. Finally, various examples of block Jacobi operators are given to illustrate the results.