Algebraic Properties of Generalized Inverses

Description

This book addresses selected topics in the theory of generalized inverses. Following a discussion of the “reverse order law” problem and certain problems involving completions of operator matrices, it subsequently presents a specific approach to solving the problem of the reverse order law for {1} -generalized inverses. Particular emphasis is placed on the existence of Drazin invertible completions of an upper triangular operator matrix; on the invertibility and different types of generalized invertibility of a linear combination of operators on Hilbert spaces and Banach algebra elements; on the problem of finding representations of the Drazin inverse of a 2x2 block matrix; and on selected additive results and algebraic properties for the Drazin inverse. In addition to the clarity of its content, the book discusses the relevant open problems for each topic discussed. Comments on the latest references on generalized inverses are also included. Accordingly, the book will be useful for graduate students, PhD students and researchers, but also for a broader readership interested in these topics. Front Matter ....Pages i-viii Definitions and Motivations (Dragana S. Cvetković Ilić, Yimin Wei)....Pages 1-10 Reverse Order Law (Dragana S. Cvetković Ilić, Yimin Wei)....Pages 11-50 Completions of Operator Matrices and Generalized Inverses (Dragana S. Cvetković Ilić, Yimin Wei)....Pages 51-88 Generalized Inverses and Idempotents (Dragana S. Cvetković Ilić, Yimin Wei)....Pages 89-108 Drazin Inverse of a \(2 \times 2\) Block Matrix (Dragana S. Cvetković Ilić, Yimin Wei)....Pages 109-158 Additive Results for the Drazin Inverse (Dragana S. Cvetković Ilić, Yimin Wei)....Pages 159-192 Back Matter ....Pages 193-194