Boundedness for Gevrey and Gelfand-Shilov kernels to positive operators

TL;DR

This paper investigates the properties of positive operators on Gelfand-Shilov spaces and distributions, establishing that kernel boundedness is determined by diagonal behavior and characterizing positive elements with Gevrey class properties.

Contribution

It provides a new characterization of kernels for positive operators and links Gevrey class properties to Gelfand-Shilov spaces for positive elements.

Findings

Boundedness of kernels determined by diagonal behavior.

Positive elements with Gevrey properties belong to Gelfand-Shilov spaces.

Characterization of positivity with respect to non-commutative convolutions.

Abstract

We study properties of positive operators on Gelfand-Shilov spaces, and distributions which are positive with respect to non-commutative convolutions. We prove that boundedness of kernels $K \in \maclD_s^{\prime}$ to positive operators, are completely determined by the behaviour of $K$ alone the diagonal. We also prove that positive elements $a$ in $\mascS^{\prime}$ with respect to twisted convolutions, having Gevrey class property of order $s\geq 1/2$ at the origin, then $a$ belongs to the Gelfand-Shilov space $\maclS_s$.