On the commutation properties of finite convolution and differential operators I: commutation

TL;DR

This paper classifies all pairs of finite convolution and differential operators that commute or sesquicommute, extending spectral analysis tools to complex and non self-adjoint cases, and introduces a new commutation concept.

Contribution

It provides a complete list of commuting and sesquicommuting operator pairs, expanding spectral analysis methods to broader classes of operators.

Findings

Complete classification of commuting operator pairs

Introduction of sesquicommutation concept

Extension to complex-valued, non self-adjoint operators

Abstract

Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. In this paper we compile a complete list of such commuting pairs, extending previous work to complex-valued and non self-adjoint operators. In addition, we introduce a new kind of commutation relation, which we call sesquicommutation, that also has implications for the spectral properties of the integral operator. In this case we also compute a complete list of sequicommuting pairs of integral and differential operators.