Non‐harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary

Abstract

AbstractWe use Operator Ideals Theory and Gershgorin theory to obtain explicit information in terms of the symbol concerning the spectrum of pseudo‐differential operators, on a smooth manifold Ω with boundary , in the context of the non‐harmonic analysis of boundary value problems introduced in [29] in terms of a model operator . For symbols in the Hörmander class , we provide a non‐harmonic version of Gohberg's lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo‐differential operator is a compact operator in , or a Riesz operator in in the case of Riemannian manifolds with smooth boundary. We extend to the context of the non‐harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations.