Unbounded banded matrices, shifted positive bidiagonal factorizations, and mixed-type multiple orthogonality

TL;DR

This paper extends Favard-type spectral representations to unbounded banded matrices by using shift-dependent positive bidiagonal factorizations, leading to a generalized spectral theory and recovering classical results for Jacobi matrices.

Contribution

It introduces a novel approach to spectral representation for unbounded banded matrices via shift-dependent factorizations and mixed-type multiple orthogonality, broadening classical Favard theory.

Findings

Established a limiting matrix-valued measure for unbounded matrices.

Derived Favard-type spectral representation for shifted unbounded banded matrices.

Reproduced classical spectral results for Jacobi matrices as a special case.

Abstract

This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N I_{N+1}$ admits a positive bidiagonal factorization (PBF). Allowing $s_N$ to depend on $N$ leads to a natural recentering step: the discrete Gauss-type quadrature measures associated with $A_N$ are translated by $x\mapsto x-s_N$, producing a uniformly bounded family of distribution functions. Combining moment stabilization for banded truncations with Helly-type compactness theorems yields a limiting matrix-valued measure, together with a Favard-type spectral representation and the corresponding mixed-type multiple biorthogonality relations. As a consequence, the classical Favard theorem for (possibly unbounded) Jacobi matrices is recovered as a special case. Indeed, for a tridiagonal $J$ with positive sub- and superdiagonals, each truncation $J^{[N]}$ admits a shift $s_N\ge 0$ such that $J^{[N]}+s_N I_{N+1}$ is oscillatory and therefore admits a PBF. The preceding construction then produces the usual spectral measure for $J$.