Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators

We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter. 1 Introduction and Statement of Results 1.1 Quadratic forms and singular spaces Recently, there has been a renewed interest in the analysis of spectra and resolvents of non-selfadjoint operators with double characteristics. The study of pseudodifferen-tial operators with double characteristics has a long and distinguished tradition in th