# On the commutation properties of finite convolution and differential operators II: sesquicommutation

## TL;DR

This paper introduces and analyzes a new sesquicommutation relation between finite convolution operators and differential operators, expanding understanding of their spectral properties and kernel classifications.

## Contribution

It presents a comprehensive analysis of sesquicommutation relations, complementing previous work on commutation, and characterizes kernels that admit such differential operators.

## Key findings

- New sesquicommutation relation introduced
- Complete characterization of kernels with sesquicommuting operators
- Implications for spectral analysis of convolution operators

## Abstract

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ and differential operators $L_1$ and $L_{2}$, that has implications for spectral properties of $K$. This work complements our explicit characterization of commuting pairs $KL=LK$ and provides an exhaustive list of kernels admitting commuting or sesquicommuting differential operators.