Wick and anti-Wick characterizations of linear operators on spaces of power series expansions

Abstract

We study the link between Wick, anti-Wick and analytic kernel operators on the Bargmann transform side. We find classes of kernels, e.g. $\wideparen {\mathcal B} _s$, whose corresponding operators agree with the sets of linear and continuous operators on $\mathcal A _s$, the images of Pilipovi{\'c} under the Bargmann transform. We show that in several situations, the sets of Wick, anti-Wick and kernel operators with symbols and kernels in $\wideparen {\mathcal B} _s$ agree. We also show some ring, module and composition properties for $\wideparen {\mathcal B} _s$, and similarly for other spaces related to $\wideparen {\mathcal B} _s$.