Reverse Order Law for Moore-Penrose Inverse in Hilbert and Indefinite inner Product Spaces

Abstract

The reverse order law (AB)−1 = B−1A−1 is true for invertible matrices but it does not hold good for various generalized inverses, in general. In 1966, Greville found some necessary and sufficient conditions for reverse order law for the Moore-Penrose inverse to hold in matrix setting. The concept of the Moore-Penrose inverse of a matrix in an indefinite inner product space was introduced by Kamaraj and Sivakumar in 2005. In this thesis, we derive some necessary and sufficient conditions for the Moore-Penrose inverse of a product of two matrices to be the product of the Moore-Penrose inverse of the matrices between indefinite inner product spaces. We also find some equiva lencies with rank conditions for the reverse order law and obtain some estimates of norms on Krein spaces. Using the elementary properties of bounded operators with closed range, we present 61 results to characterize the reverse order law for the Moore Penrose inverse of such operators between Hilbert spaces. The reverse order law for the Moore-Penrose inverse of closed operators with a closed range was studied by Fa-peng and Yi-feng using the extension of orthogonal projections in 2013. We obtain some characterizations of the reverse order law for the Moore-Penrose inverse of such oper ators between Hilbert spaces by using basic properties of unbounded operators. Using the polar decomposition, we also find a method to compute the Moore-Penrose inverse of such operator as an application of the reverse order law.