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Spectral properties of operators obtained by localization methods. (English) Zbl 1356.47052

Summary: We prove that, given two elliptic operators \(A_{1}\) and \(A_{2}\) in \(L^p(\Omega_{1})\) and \(L^p(\Omega_{2})\) respectively whose spectral properties are known, we can deduce those of the operator \(A\) coinciding with \(A_{1}\) on \(\Omega_{1}\) and with \(A_{2}\) on \(\Omega_{2}\). Conversely, if the spectral properties of the operator \(A\) are known in \(L^p(\Omega)\), we deduce those of the restriction of \(A\) to a smaller subset of \(\Omega\).

MSC:

47F05 General theory of partial differential operators
47A10 Spectrum, resolvent
47D06 One-parameter semigroups and linear evolution equations
35J99 Elliptic equations and elliptic systems
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