  Different variants of approximate inverse iteration like the locally optimal
block preconditioned conjugate gradient method became in recent years
increasingly popular for the solution of the large matrix eigenvalue problems
arising from the discretization of selfadjoint elliptic partial differential
equations, in particular for the calculation of the minimum eigenvalue. We
extend in this little note the classical convergence theory of D'yakonov and
Orekhov [Math. Notes 27 (1980)] to the case of operators with an essential
spectrum on infinite dimensional Hilbert spaces and allow for arbitrary,
sufficiently small perturbations of the solutions of the equation that links
the iterates. The note complements the much more elaborate convergence theory
of Neymeyr and Knyazev and Neymeyr for the matrix case (see [Knyazev and
Neymeyr, SIAM J. Matrix Anal. Appl. 31 (2009)] and the references therein) and
is suitable for classroom presentation.
