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{ć} 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 $\wideparen {\mathcal B} _s$.

arXiv comment50 pages. This is the third version. A thorough reorganisation is performed. Several misprints have been corrected. Some mathematical mistakes are corrected. The old title was "Characterizations of linear operators on spaces related to entire functions by Wick and anti-Wick operators"