Boundedness of certain linear operators on twisted Fock spaces

TL;DR

This paper characterizes when convolution operators on twisted Fock spaces are bounded, establishing necessary and sufficient conditions, and shows that for non-constant symbols, at least one operator is unbounded.

Contribution

It provides a complete characterization of boundedness for convolution operators on twisted Fock spaces and reveals that non-constant symbols lead to unboundedness of at least one operator.

Findings

Necessary and sufficient conditions for boundedness of $S_$ and $ ilde{S}_$

At least one of the operators is unbounded for non-constant $$

Boundedness depends on the properties of the symbol $$

Abstract

On the twisted Fock spaces $ \mathcal{F}^\lambda(\mathbb{C}^{2n}) $ we consider two types of convolution operators $ S_\varphi $ and $ \widetilde{S}_\varphi $ associated to an element $ \varphi \in \mathcal{F}^\lambda(\mathbb{C}^{2n}) .$ We find a necessary and sufficient condition on $ \varphi $ so that $ S_\varphi $ (resp. $ \widetilde{S}_\varphi $ ) is bounded on $ \mathcal{F}^\lambda(\mathbb{C}^{2n}).$ We show that for any given non constant $ \varphi $ at least one of these two operators is unbounded.