Perturbation analysis of the Moore–Penrose metric generalized inverse with applications

Abstract

In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called $(\alpha ,\beta )$-USU (uniformly strong uniqueness) property. This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case found to date. Closely connected to the main perturbation results, one application, the error estimate for projecting a point onto a linear manifold problem, is also investigated.