  This work is concerned with extending the results of Calder\' on and
Vaillancourt proving the boundedness of Weyl pseudo differential operators
Op_h^{weyl} (F) in L^2(\R^n). We state conditions under which the norm of such
operators has an upper bound independent of n. To this aim, we apply a
decomposition of the identity to the symbol F, thus obtaining a sum of
operators of a hybrid type, each of them behaving as a Weyl operator with
respect to some of the variables and as an anti-Wick operator with respect to
the other ones. Then we establish upper bounds for these auxiliary operators,
using suitably adapted classical methods like coherent states.
