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

TL;DR

This paper explores the relationships between Wick, anti-Wick, and kernel operators on spaces of power series, establishing conditions under which these operators coincide and analyzing their algebraic properties.

Contribution

It introduces classes of kernels linking Wick, anti-Wick, and analytic kernel operators, and characterizes their equivalences and algebraic structures on spaces of power series.

Findings

Wick, anti-Wick, and kernel operators coincide under certain kernel classes.

Established algebraic properties like ring and module structures for these operator classes.

Identified conditions for operator equivalence on spaces of power series.

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 $\wideparen {\mathcal B} _s$.