Spectral properties of operators obtained by localization methods

Abstract

We prove that, given two elliptic operators A₁ and A₂ in Lᵖ(Ω₁) and Lᵖ(Ω₂) respectively whose spectral properties are known, we can deduce those of the operator A coinciding with A₁ on Ω₁ and with A₂ on Ω₂. Conversely, if the spectral properties of the operator A are known in Lᵖ(Ω), we deduce those of the restriction of A to a smaller subset of Ω.