Boundedness for Gevrey and Gelfand-Shilov kernels to positive operators

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 $K\in {\text{maclD}}_s^{\mathrm{\prime }}$ to positive operators, are completely determined by the behaviour of $K$ alone the diagonal. We also prove that positive elements $a$ in ${\text{mascS}}^{\mathrm{\prime }}$ with respect to twisted convolutions, having Gevrey class property of order $s\geq 1/2$attheorigin,then$a$belongstotheGelfand-Shilovspace$\maclS_s$.