Limiting set of second order spectra

Let $M$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. A complex number $z$ is in the second order spectrum of $M$ relative to a finite-dimensional subspace $\mathcal{L}\subset \mathrm{Dom}\, M^2$ iff the truncation to $\mathcal{L}$ of $(M-z)^2$ is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets 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 sets, as $\mathcal{L}$ increases towards $\mathcal{H}$, contain the isolated eigenvalues of $M$ of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.