Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality

TL;DR

This paper investigates the spectral properties of Jacobi matrices on trees with coefficients derived from multiple orthogonal polynomials, providing a detailed spectral analysis and decomposition into cyclic subspaces.

Contribution

It introduces a novel spectral theory framework for Jacobi matrices on trees with coefficients from multiple orthogonality, including decomposition and eigenfunction characterization.

Findings

Hilbert space decomposes into orthogonal cyclic subspaces

Generators and eigenfunctions are expressed via orthogonal polynomials

Spectrum and spectral types are characterized for various measures

Abstract

We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.