The Sherman–Morrison–Woodbury formula for the Moore–Penrose metric generalized inverse

In this paper, we focus on the Moore–Penrose metric generalized inverse of the modified operator B=A+UGV, where A,U,G,V are bounded linear operators between some Banach spaces. We establish conditions that guarantee the existence of the Moore–Penrose metric generalized inverse of B. As a consequence, we present an extension of the so-called classical Sherman–Morrison–Woodbury formula for the Moore–Penrose metric generalized inverse. Some particular cases and applications will be also considered.