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Block Jacobi matrices and Titchmarsh-Weyl function

Marcin Moszyński, Grzegorz Świderski

TL;DR

This paper develops a matrix-generalized subordinacy framework for block Jacobi operators with $d\times d$ blocks, focusing on the matrix Weyl function $W(z)$ and the spectral matrix measure $M$ to connect generalized eigenvectors with the absolutely continuous and singular spectrum. It constructs a spectral representation via $M$ and proves that the Cauchy transform of $M$ coincides with $W$, enabling boundary-limit analysis and a structured description of ac and sing parts of the spectrum through $L(W)$, $S_{\mathrm{ac}}$, and $S_{\mathrm{sing}}$. The work introduces a block-analogue of Jitomirskaya--Last semi-norms, develops transfer-matrix techniques, Liouville–Ostrogradsky identities, and matrix orthogonal polynomials $P$ and $Q$, and establishes finite-cyclicity for the block operator. It further derives spectral consequences of matrix-subordinacy theory, including a concise nonsubordinacy concept and its implications, while signaling a barrier-nonsubordinacy program explored in a parallel paper for stronger absolute-continuity results.

Abstract

We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators.

Block Jacobi matrices and Titchmarsh-Weyl function

TL;DR

This paper develops a matrix-generalized subordinacy framework for block Jacobi operators with blocks, focusing on the matrix Weyl function and the spectral matrix measure to connect generalized eigenvectors with the absolutely continuous and singular spectrum. It constructs a spectral representation via and proves that the Cauchy transform of coincides with , enabling boundary-limit analysis and a structured description of ac and sing parts of the spectrum through , , and . The work introduces a block-analogue of Jitomirskaya--Last semi-norms, develops transfer-matrix techniques, Liouville–Ostrogradsky identities, and matrix orthogonal polynomials and , and establishes finite-cyclicity for the block operator. It further derives spectral consequences of matrix-subordinacy theory, including a concise nonsubordinacy concept and its implications, while signaling a barrier-nonsubordinacy program explored in a parallel paper for stronger absolute-continuity results.

Abstract

We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators.

Paper Structure

This paper contains 32 sections, 20 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Short review on spectral analysis methods for block Jacobi Operators
  3. Perturbations theory for block Toeplitz operators
  4. Combes--Thomas type estimates
  5. Estimates of quadratic forms
  6. Commutator methods
  7. Compactness of the resolvent
  8. Matrix Titchmarsh--Weyl Function and subordinacy methods
  9. The present work
  10. Preliminaries
  11. Abbreviations and symbols
  12. Introductory notation and notions
  13. Some matrix formulae and inequalities
  14. Asymptotic symbols and the affine interpolation
  15. Analogs of "J--L continuous interpolation" of a discrete family of semi-norms
  16. ...and 17 more sections

Key Result

Proposition 2.2

Let $A \in M_{d}(\mathbb{C})$ and $v \in \mathbb{C}^d$. Then:

Theorems & Definitions (51)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.5
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 41 more