In this paper, we investigate properties of with closed range satisfying the operator equations , , 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 S satisfying some given conditions. The relation between the ranges and null spaces of these operators is also shown.

Abstract
In this paper, we study the notions of W4-flat para-Kenmotsu, semi-symmetric para-Kenmotsu and symmetric and recurrent relationships with respect to the W4-curvature tensor.