Opuscula Math. 45, no. 3 (2025), 351-402
https://doi.org/10.7494/OpMath.2025.45.3.351

 
Opuscula Mathematica

Block Jacobi matrices and Titchmarsh-Weyl function

Marcin Moszyński
Grzegorz Świderski

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.

Keywords: block Jacobi matrix, matrix measures, Titchmarsh-Weyl function, Liouville-Ostrogradsky formulae.

Mathematics Subject Classification: 47B36.

Full text (pdf)

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  • Marcin Moszyński (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-9268-1303
  • University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, ul. Stefana Banacha 2, 02-097 Warsaw, Poland
  • Communicated by P.A. Cojuhari.
  • Received: 2025-03-17.
  • Accepted: 2025-04-16.
  • Published online: 2025-05-30.
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Cite this article as:
Marcin Moszyński, Grzegorz Świderski, Block Jacobi matrices and Titchmarsh-Weyl function, Opuscula Math. 45, no. 3 (2025), 351-402, https://doi.org/10.7494/OpMath.2025.45.3.351

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