Moore-Penrose inverse of linear operators in Hilbert space

In this paper, we investigate properties of with closed range satisfying the operator equations In particular, we investigate the invertibility of with closed range where the Moore-Penrose inverse of T turns out to be the usual inverse of T under some classes of operators. We also deduce the Moore-Penrose inverse of a perturbed linear operator with closed range where such that has closed ranges and satisfying some given conditions. The relation between the ranges and null spaces of these operators is also shown.