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We prove that the elastic Neumann--Poincar\'e operator defined on the smoothboundary of a bounded domain in three dimensions, which is known to benon-compact, is in fact polynomially compact. As a consequence, we prove thatthe spectrum of the elastic Neumann-Poincar\'e operator consists of threenon-empty sequences of eigenvalues accumulating to certain numbers determinedby Lam\'e parameters. These results are proved using the surface Riesztransform, calculus of pseudo-differential operators and the spectral mappingtheorem.

Elastic Neumann-Poincar\'e operators on three dimensional smooth domains: Polynomial compactness and spectral structure