Some additive and multiplicative results for generalized inverses

Abstract

In this paper we give necessary and sufficient conditions for $A_1\{1,3\}+A_2\{1,3\}+...+A_k\{1,3\}\subseteq (A_1+A_2+...+A_k)\{1,3\}$ and $A_1\{1,4\}+A_2\{1,4\}+...+A_k\{1,4\}\subseteq (A_1+A_2+...+A_k)\{1,4\}$ for regular operators on Hilbert space. We also consider similar inclusions for $\{1,2,3\}$- and $\{1,2,4\}$-inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.