Representations For Moore-Penrose Inverses Of Multiplication Perturbations Of Adjointable Operators

A multiplication perturbation of T has the form of ETF, where T is fixed, while E and F can be changeable. Moore-Penrose inverses of multiplication perturbations have various applications, and much effort has been made by many mathematicians. Let S+ denote the Moore-Penrose inverse of an operator S. One important topic concerning representations for Moore-Penrose inverses is the study of the relationship between (ETF)+ and T+. When E and F are both invertible, much progress has been made on this topic. Yet, less has been done in the case that either E or F is not invertible, which is the concern of this paper.Let T, E and F be three adjointable operators on a Hilbert C*-module, where T is fixed, while E and F can be changeable which need not to be invertible. Firstly, we provide a formula for (ETE*)+ in terms of T and T+ under the conditions that T= T(?) and ETT++I-TT+ is invertible. Secondly, by using a self-adjoint method we have managed to give a formula for (ETF*)+ under the conditions that both ETT++I-TT+ and T+TF+I-T+T are invertible. The results obtained in this paper have generalized some related works in the literature.