Boundedness of certain linear operators on twisted Fock spaces

On the twisted Fock spaces \( {\mathcal {F}}^\lambda ({{\mathbb {C}}}^{2n}) \) we consider two types of convolution operators \( S_\varphi ^\lambda \) and \( {\widetilde{S}}_\varphi ^\lambda \) associated to an element \( \varphi \in {\mathcal {F}}^\lambda ({{\mathbb {C}}}^{2n}).\) We find a necessary and sufficient condition on \( \varphi \) so that \( S_\varphi ^\lambda \) (resp. \( {\widetilde{S}}_\varphi ^\lambda \) ) 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.