Some resolvent set properties of band operators with matrix elements

TL;DR

This paper characterizes the resolvent set of certain infinite band matrix operators with matrix elements using polynomial solutions of difference equations, and explores the asymptotic behavior of related vector orthogonal polynomials.

Contribution

It provides a new characterization of the resolvent set for band operators with matrix elements via polynomial solutions, linking operator theory and orthogonal polynomial asymptotics.

Findings

Characterization of the resolvent set in terms of polynomial solutions

Description of asymptotic behavior of vector orthogonal polynomials

Connection between difference equations and operator resolvent properties

Abstract

For operators generated by a certain class of infinite band matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order finite difference equations. This enables us to describe some asymptotic behaviour of the corresponding systems of vector orthogonal polynomials on the resolvent set.