  It is well known that the standard projection methods allow one to recover
the whole spectrum of a bounded self-adjoint operator but they often lead to
spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the
essential spectrum. Methods using second order relative spectra are free from
this problem, but they have not been proven to approximate the whole spectrum.
L. Boulton (2006, 2007) has shown that second order relative spectra
approximate all isolated eigenvalues of finite multiplicity. The main result of
the present paper is that second order relative spectra do not in general
approximate the whole of the essential spectrum of a bounded self-adjoint
operator.
