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Revista Matemática Iberoamericana

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Volume 34, Issue 2, 2018, pp. 621–635
DOI: 10.4171/RMI/998

Published online: 2018-05-28

Spectral permanence in a space with two norms

Hyeonbae Kang[1] and Mihai Putinar[2]

(1) Inha University, Incheon, Republic of Korea
(2) University of California, Santa Barbara, USA and Newcastle University, UK

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.

Keywords: Neumann–Poincaré operator, Lamé system, spectrum, finite section method

Kang Hyeonbae, Putinar Mihai: Spectral permanence in a space with two norms. Rev. Mat. Iberoam. 34 (2018), 621-635. doi: 10.4171/RMI/998