  In this article we further develop a perturbation approach to the
Rayleigh--Ritz approximations from our earlier work. We both sharpen the
estimates and extend the applicability of the theory to nonnegative definite
operators . The perturbation argument enables us to solve two problems in one
go: We determine which part of the spectrum of the operator is being
approximated by the Ritz values and compute the approximation estimates. We
also present a Temple--Kato like inequality which --unlike the original
Temple--Kato inequality-- applies to any test vectors from the quadratic form
domain of the operator.
