  For a class of non-selfadjoint semiclassical pseudodifferential operators
with double characteristics, we study bounds for resolvents and estimates for
low lying eigenvalues. Specifically, assuming that the quadratic approximations
of the principal symbol of the operator along the double characteristics enjoy
a partial ellipticity property along a suitable subspace of the phase space,
namely their singular space, we establish semiclassical hypoelliptic a priori
estimates with a loss of the full power of the semiclassical parameter, giving
a localization for the low lying spectral values of the operator.
