  We prove that the elastic Neumann--Poincar\'e operator defined on the smooth
boundary of a bounded domain in three dimensions, which is known to be
non-compact, is in fact polynomially compact. As a consequence, we prove that
the spectrum of the elastic Neumann-Poincar\'e operator consists of three
non-empty sequences of eigenvalues accumulating to certain numbers determined
by Lam\'e parameters. These results are proved using the surface Riesz
transform, calculus of pseudo-differential operators and the spectral mapping
theorem.
