Perturbation of the Moore-Penrose Metric generalized inverse with applications to the best approximate solution problem in Lp(Ω, μ)

Let X = L-p(Omega, mu) (1 < p < infinity), let T is an element of B(X) with closed range. In this paper, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we present some new perturbation results of the Moore-Penrose metric generalized inverse. As applications of our results, we also investigate the best approximate solution problem for the ill-posed operator equation Tx = y under some conditions. The main results have three parts, part one covers the null space preserving case, part two covers the range preserving case, and part three covers the general case. Examples in connection with the theoretical results will be also presented.