We investigate the far-field operator for the scattering of time-harmonic elastic plane waves by either a rigid body, a cavity, or an absorbing obstacle. Extending results of Colton & Kress for acoustic obstacle scattering, for the spectrum of the far-field operator we show that there exist an infinite number of eigenvalues and determine disks in the complex plane where these eigenvalues lie. In addition, as counterpart of an identity in acoustic scattering due to Kress & Paivarinta, we will establish a factorization for the difference of the far-field operators for two different scatterers. Finally, extending a sampling method for the approximate solution of the acoustic inverse obstacle scattering problem suggested by Kirsch to elasticity, this factorization is used for a characterization of a rigid scatterer in terms of the eigenvalues and eigenelements of the far-field operator.