Estimates for the resolvent and spectral gaps for non self-adjoint operators

In mathematical physics and dynamical systems one encounters non self-adjoint operators with spectrum not intersecting a strip, called spectral gap. The estimates of the norm of the resolvent of P in the strip play an important role in the investigation of the local decay of the energy, the analysis of the scattering resonances and the analytic continuation of the dynamical zeta function. We will discuss results and open problems concerning the estimate of the inverse of a holomorphic function without zeros in a strip and the estimates of the cut-off resolvent of the Dirichlet Laplacian for trapping perturbations.