Banded matrices and their orthogonality

TL;DR

This paper explores the spectral properties of banded matrices representing non-normal oscillatory operators, linking them to orthogonal polynomials and providing a spectral theorem for such matrices.

Contribution

It introduces a spectral theorem for oscillatory banded matrices and connects their properties to multiple orthogonal polynomials, extending classical results.

Findings

Spectral theorem for oscillatory banded matrices

Connection between banded matrices and multiple orthogonal polynomials

Reinterpretation of Favard theorem for shifted Jacobi matrices

Abstract

Banded bounded matrices, which represent non normal operators, of oscillatory type that admit a positive bidiagonal factorization are considered. To motivate the relevance of the oscillatory character the Favard theorem for Jacobi matrices is revisited and it is shown that after an adequate shift of the Jacobi matrix one gets an oscillatory matrix. In this work we present a spectral theorem for this type of operators and show how the theory of multiple orthogonal polynomials apply.