  Let $\left( W,\sigma \right) $ be a symplectic vector space and let $%
T:W\rightarrow W$ be a linear map that satisfies a certain condition of
non-degeneracy. We define the Schur multiplier $\omega _{\sigma ,T}$ on $W$. To
this multiplier we associate a $\omega _{\sigma ,T}$-representation and and we
build the $T$-Weyl calculus, $\mathrm{Op}_{\sigma ,T}$, whose properties are
are systematically studied further.
