Functions of orthogonal projectors involving the Moore Penrose inverse

Several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors. The results were established in various settings, including finite and infinite dimensional vector spaces over either real or complex numbers, but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors. In the present paper, these results are unified and reestablished in a generalized form in a complex Euclidean vector space, with the generalization obtained by relaxing the nonsingularity assumption and use of the Moore Penrose inverse instead of the ordinary inverse. Additionally, several new formulae of the type are provided.