Barrier nonsubordinacy and absolutely continuous spectrum of block Jacobi matrices

TL;DR

This paper introduces the concept of barrier nonsubordinacy for block Jacobi matrices and demonstrates its role in ensuring the absolute continuity of their spectrum, extending scalar results to higher dimensions.

Contribution

It defines barrier nonsubordinacy and proves its implication for absolute continuity in block Jacobi operators, extending known scalar conditions to higher-dimensional blocks.

Findings

Barrier nonsubordinacy implies absolute continuity for block Jacobi matrices.

Extended scalar spectral conditions to block matrices with dimension d ≥ 1.

Applied results to specific classes of block Jacobi matrices.

Abstract

We explore to what extent the relation between the absolute continuous spectrum and non-existence of subordinate generalized eigenvectors, known for scalar Jacobi operators, can be formulated also for block Jacobi operators with $d$-dimensional blocks. The main object here allowing to make some progress in that direction is the new notion of the barrier nonsubordinacy. We prove that the barrier nonsubordinacy implies the absolute continuity for block Jacobi operators. Finally, we extend some well-known $d=1$ conditions guaranteeing the absolute continuity to $d \geq 1$ and we give applications of our results to some concrete classes of block Jacobi matrices.