On the reproducing kernel of the Segal-Bargmann space

Abstract

This article revolves around the properties on the Lp scale of spaces of the integral kernel operator K whose kernel function is the reproducing kernel of the Segal-Bargmann space. We find sufficient conditions on p and q for K to be a Hille-Tamarkin (and hence compact) operator from Lp to Lq with respect to the standard Gaussian measure as well as with respect to a weighted measure on the codomain space. We also find sufficient conditions for K to be unbounded with respect to the standard Gaussian measure. Finally we give sufficent conditions for a Toeplitz operator to be Hille-Tamarkin on the Lp scale of spaces with respect to both the standard Gaussian measure and a weighted measure on the codomain space.