Barrier nonsubordinacy and absolutely continuous spectrum 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.