Spectral permanence in a space with two norms

Abstract

A generalization of a classical argument of Mark G. Krein leads us to the conclusion that the Neumann–Poincar´e operator associated to the Lamé system of linear elastostatics equations in two dimensions has the same spectrum on the Lebesgue space of the boundary as the more natural energy space. A similar result for the Neumann–Poincaré operator associated to the Laplace equation was stated by Poincaré and was proved rigorously a century ago by means of a symmetrization principle for non-selfadjoint operators. We develop the necessary theoretical framework underlying the spectral analysis of the Neumann–Poincaré operator, including also a discussion of spectral asymptotics of a Galerkin type approximation. Several examples from function theory of a complex variable and harmonic analysis are included.