  For a class of non-selfadjoint $h$--pseudodifferential operators with double
characteristics, we give a precise description of the spectrum and establish
accurate semiclassical resolvent estimates in a neighborhood of the origin.
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 give a precise description of the spectrum of the operator
in an ${\cal O}(h)$--neighborhood of the origin. Moreover, when all the
singular spaces are reduced to zero, we establish accurate semiclassical
resolvent estimates of subelliptic type, which depend directly on algebraic
properties of the Hamilton maps associated to the quadratic approximations of
the principal symbol.
