Explicit Expression Of Generalized Inverses And Its Application

In this paper, under certain restrictive conditions, which is applied to the study ofthe explicit expression for Drazin inverse and the Moore-Penrose inverse, respectively.The paper is arranged as follows:At first, we give explicit expressions of (P±Q)_d of two matrices P and Q, interms of P, Q, P_d and Qd,(P±I)_d, under the condition that P Q=P, and apply theresult to finding an explicit representation for the Drazin inverse of some2×2blockmatrix. Similarly, analogous discussions are also mentioned on Banach spaces under theconditions P~2Q=P~2+Q, Q~2P=Q~2+P.In the reminder of this paper, under certain restrictive conditions, we investigate theMoore-Penrose inverse of a perturbed bounded linear operator between Hilbert spacesand present its explicit expressions. Moreover, we discuss their relevant perturbationbounds.