Limiting set of second order spectra

Let M be a self-adjoint operator acting on a Hilbert space H. A complex number z is in the second order spectrum of M relative to a finite dimensional subspace L ⊂ DomM, iff the truncation to L of (M − z) is not invertible. This definition was first introduced in [4] and according to the results of [7] and [13], these set provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on M, the uniform limit of these set, as L increases towardsH, contains the isolated eigenvalues of M of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine non-pollution and approximation at a very high level of generality.