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Barrier nonsubordinacy and absolutely continuous spectrum of block Jacobi matrices

Marcin Moszyński, Grzegorz Świderski

TL;DR

The paper extends the scalar subordinacy framework to block Jacobi operators with $d$-dimensional blocks by introducing barrier nonsubordinacy, which provides a sufficient route to absolute continuity of the spectral matrix measure $M$ via the matrix Weyl function $W$. It defines and analyzes barrier functions that uniformly control generalized eigenvector growth, derives key inequalities linking $W$ to spectral ac parts, and proves that barrier nonsubordinacy on a set $G$ yields absolute continuity on $G$ with invertible densities on $L(W)$. It then adapts and extends prominent sufficiency criteria (GLS, GBS, and $H$-class) to the block setting, showing how these conditions imply barrier nonsubordinacy and ac spectrum. The paper also presents concrete examples and applications demonstrating both the reach and the limitations of the approach, including a counterexample where density invertibility does not guarantee nonsubordinacy and applications to classes of block Jacobi operators studied previously.

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.

Barrier nonsubordinacy and absolutely continuous spectrum of block Jacobi matrices

TL;DR

The paper extends the scalar subordinacy framework to block Jacobi operators with -dimensional blocks by introducing barrier nonsubordinacy, which provides a sufficient route to absolute continuity of the spectral matrix measure via the matrix Weyl function . It defines and analyzes barrier functions that uniformly control generalized eigenvector growth, derives key inequalities linking to spectral ac parts, and proves that barrier nonsubordinacy on a set yields absolute continuity on with invertible densities on . It then adapts and extends prominent sufficiency criteria (GLS, GBS, and -class) to the block setting, showing how these conditions imply barrier nonsubordinacy and ac spectrum. The paper also presents concrete examples and applications demonstrating both the reach and the limitations of the approach, including a counterexample where density invertibility does not guarantee nonsubordinacy and applications to classes of block Jacobi operators studied previously.

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 -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 conditions guaranteeing the absolute continuity to and we give applications of our results to some concrete classes of block Jacobi matrices.
Paper Structure (22 sections, 22 theorems, 156 equations)

This paper contains 22 sections, 22 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some abbreviations and common symbols
  4. Introductory notation and notions
  5. Block Jacobi operator. Finite cyclicity and the spectral matrix measure
  6. The two associated difference equations and "the solution extensions to -1"
  7. Transfer matrices
  8. $\ell^2$ matrix solutions and the matrix Weyl function $W$
  9. The Cauchy transform of the spectral matrix measure and the matrix Weyl function
  10. The boundary limits of the matrix Weyl function and spectral consequences for $J$
  11. Some analogs of "J--L continuous interpolation" of a discrete family of semi-norms
  12. A barrier approach to uniform estimates of matrix Weyl function
  13. The nonsubordinacy
  14. Barriers and barrier nonsubordinacy
  15. The main result and its consequences
  16. ...and 7 more sections

Key Result

Theorem 1.3

(see Theorem thm:10) Assume that $J$ is self-adjoint, $G \in \mathrm{Bor}(\mathbb{R})$ and $\mathfrak{b}$ is a barrier for $J$ on $G$. If $J$ is $\mathfrak{b}$-nonsubordinate on $G$, then If, moreover, $J$ is uniformly $\mathfrak{b}$-nonsubordinate on $G$, then there exist $c_1, c_2>0$ such that the above density satisfies for a.e. $\lambda \in G$.

Theorems & Definitions (45)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.7
  • Proposition 2.8
  • ...and 35 more