Boundedness of Gevrey and Gelfand–Shilov kernels of positive semi-definite operators

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 ∈ D ′ s to positive operators, are completely determined by the behaviour of K alone the diagonal. We also prove that positive elements a in S ′ with respect to twisted convolutions, having Gevrey class property of order s ≥ 1/2 at the origin, then a belongs to the Gelfand-Shilov space S s .

Strong ultra-regularity properties for positive elements in the twisted convolutions

“…for some h > 0 (for every h > 0), then a ∈ S s (a ∈ Σ s ) (cf. [1,Theorem 4.1]). We note that if (0.1) holds true with s < 1/2 in (5), then a is trivially equal to 0, since the Gelfand-Shilov spaces S s and Σ s are trivial for such choices of s.…”