On the orthogonality of solutions for higher-order non-Hermitian difference equations

TL;DR

This paper investigates higher-order non-Hermitian difference equations, establishing orthogonality relations for polynomial solutions under specific matrix conditions, and explores related matrix moment problems and perturbations.

Contribution

It introduces orthogonality properties for solutions of higher-order difference equations with non-Hermitian matrices, expanding the understanding of their spectral and structural characteristics.

Findings

Existence of positive matrix measure for polynomial solutions

Orthogonality relations in complex symmetric and specific matrix cases

Analysis of a related matrix moment problem and matrix perturbations

Abstract

In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix ($\mathbf{J}=(g_{m,n})_{m,n=0}^\infty$, $| g_{m,n} |< C$, $C>0$; and $g_{k,l}=0$ if $|k-l|>N$), $y_j$s are unknowns, $\lambda$ is a complex parameter, $N\in\mathbb{N}$. It is assumed that all $g_{k,k+N}$ and $g_{l-N,l}$ are nonzero. Two special cases are considered: \noindent \textit{Case A}: The matrix $\mathbf{J}$ is complex symmetric, i.e. $\mathbf{J} = \mathbf{J}^T$. \noindent \textit{Case B}: The matrix $\mathbf{J}$ is such that $g_{k,k+N}=1$, $k=0,1,2,...$. Notice that this condition can be attained by changing $y_j$s by their multiples. In both cases there exists a \textit{positive} matrix measure $M$ on a circle in the complex plane such that polynomial solutions satisfy some orthogonality relations. Namely, in case~A this is related to a $J$-orthogonality in the Hilbert space $L^2(M)$ ($J$ is a complex conjugation). In case~B we have a left $J$-orthogonality in $L^2(M)$. As a tool, a related matrix moment problem is studied. A complex rank-one perturbation of a free Jacobi matrix is discussed.