Semiclassical Hypoelliptic Estimates for Non-Selfadjoint Operators with Double Characteristics

For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, with a leading symbol with a non-negative real part, 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 spaces, 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.