Spectra of pseudo-differential operators in the Hörmander class

Abstract

We characterize the spectra of L^p-bounded translation invariant pseudo-differential operators with symbols in the Hörmander class $S_{\rho }^{m}$. In particular, we obtain for these operators a precise version of their L^p-spectral invariance. We also prove a partial result on the spectra of L^p-bounded pseudodifferential operators with symbols in the Hörmander class $S_{{\rho ,\delta }}^{m}$. We use these results to study the holomorphic functional calculus of translation invariant pseudo-differentia1 operators.