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 $S_{\phi }$ and ${\tilde{S}}_{\phi }$ associated to an element $\phi \in {\mathcal{F}}^{\lambda }({\mathbb{C}}^{2n})\mathrm{.}$ We find a necessary and sufficient condition on $\phi$ so that $S_{\phi }$ (resp. ${\tilde{S}}_{\phi }$ ) is bounded on ${\mathcal{F}}^{\lambda }({\mathbb{C}}^{2n})\mathrm{.}$ We show that for any given non constant $\phi$ at least one of these two operators is unbounded.