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Unbounded banded matrices with positive bidiagonal factorization and mixed-type multiple orthogonal polynomials

Amílcar Branquinho, Ana Foulquié-Moreno, Manuel Mañas

TL;DR

The paper addresses extending the Favard spectral theorem to semi-infinite unbounded banded matrices $T$ that admit a positive bidiagonal factorization and proves a Favard-type spectral theorem in this setting. It develops a constructive truncation scheme where finite truncations have simple positive spectra and positive Christoffel numbers, and uses Gaussian quadrature together with a Helly-type compactness argument to pass to a limit and obtain a matrix-valued spectral measure. The main result is the existence of a mixed-type biorthogonality relation for polynomials $A^(a)$ and $B^(b)$ with respect to a matrix-valued measure $dpsi$, namely sum over a and b of the integral B^(b)_l(x) dpsi_ba(x) A^(a)_k(x) equals delta_{k,l}, along with explicit degree bounds for A and B. The work also shows normality and degree attainment via total positivity, unifying bounded and unbounded cases. Overall, the paper advances matrix-valued spectral theory for non-self-adjoint banded operators and links to mixed-type orthogonality and quadrature theory.

Abstract

A spectral Favard theorem is proved for semi-infinite banded matrices admitting a positive bidiagonal factorization, without assuming boundedness of the associated operator, thus covering both the bounded and unbounded settings. The result yields a matrix-valued spectral measure and an explicit spectral representation of the matrix powers in terms of the associated mixed-type multiple orthogonal polynomials. The argument follows the constructive truncation scheme: principal truncations are oscillatory, hence have simple positive spectra, and a suitable choice of initial conditions ensures positivity of the Christoffel coefficients and of the resulting discrete matrix-valued measures supported at the truncation eigenvalues. The main difficulty is the passage to the limit of these discrete measures beyond the bounded case. This is resolved by combining the available Gaussian quadrature structure with a Helly-type compactness argument, leading to a limiting matrix-valued measure and completing the spectral theorem. The role of normality (maximal degree pattern) for the mixed-type families is also addressed.

Unbounded banded matrices with positive bidiagonal factorization and mixed-type multiple orthogonal polynomials

TL;DR

The paper addresses extending the Favard spectral theorem to semi-infinite unbounded banded matrices that admit a positive bidiagonal factorization and proves a Favard-type spectral theorem in this setting. It develops a constructive truncation scheme where finite truncations have simple positive spectra and positive Christoffel numbers, and uses Gaussian quadrature together with a Helly-type compactness argument to pass to a limit and obtain a matrix-valued spectral measure. The main result is the existence of a mixed-type biorthogonality relation for polynomials and with respect to a matrix-valued measure , namely sum over a and b of the integral B^(b)_l(x) dpsi_ba(x) A^(a)_k(x) equals delta_{k,l}, along with explicit degree bounds for A and B. The work also shows normality and degree attainment via total positivity, unifying bounded and unbounded cases. Overall, the paper advances matrix-valued spectral theory for non-self-adjoint banded operators and links to mixed-type orthogonality and quadrature theory.

Abstract

A spectral Favard theorem is proved for semi-infinite banded matrices admitting a positive bidiagonal factorization, without assuming boundedness of the associated operator, thus covering both the bounded and unbounded settings. The result yields a matrix-valued spectral measure and an explicit spectral representation of the matrix powers in terms of the associated mixed-type multiple orthogonal polynomials. The argument follows the constructive truncation scheme: principal truncations are oscillatory, hence have simple positive spectra, and a suitable choice of initial conditions ensures positivity of the Christoffel coefficients and of the resulting discrete matrix-valued measures supported at the truncation eigenvalues. The main difficulty is the passage to the limit of these discrete measures beyond the bounded case. This is resolved by combining the available Gaussian quadrature structure with a Helly-type compactness argument, leading to a limiting matrix-valued measure and completing the spectral theorem. The role of normality (maximal degree pattern) for the mixed-type families is also addressed.
Paper Structure (4 sections, 2 theorems, 38 equations)

This paper contains 4 sections, 2 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Banded matrices and mixed-type multiple orthogonal polynomials
  3. The Favard spectral theorem for unbounded banded matrices
  4. Conclusions and outlook

Key Result

Theorem 2.1

Let us assume that he matrix $T$ is bounded and admits a positive bidiagonal factorization. Then the following versions of the Favard spectral theorem hold:

Theorems & Definitions (3)

  • Theorem 2.1: Favard spectral representation for bounded banded matrices with PBF
  • Theorem 2.2: Favard spectral representation for unbounded banded matrices with PBF
  • proof : Proof of the Favard spectral representation for unbounded matrices