Stable perturbations of operators and related topics / Yifeng Xue.

Includes bibliographical references and index.

1. Basics in functional analysis. 1.1. Banach spaces and Hilbert spaces. 1.2. Reduced minimum moduli of densely-defined operators. 1.3. The gap function of subspaces. 1.4. Banach algebras. 1.5. C[symbol]-algebras. 1.6. Notes -- 2. Stable perturbation of densely-defined closed operators. 2.1. Generalized inverses of linear operators on Banach spaces. 2.2. Stable perturbation of operators. 2.3. Perturbation analysis of the operator equation Ax=b. 2.4. Generalized inverses in Banach algebras. 2.5. Some applications of stable perturbations. 2.6. Notes -- 3. Moore-Penrose inverse and its stable perturbation. 3.1. The Moore-Penrose inverse. 3.2. The reverse order law of the Moore-Penrose inverses. 3.3. Stable perturbation of closed operators in Hilbert spaces. 3.4. The representations of certain Moore-Penrose inverses. 3.5. Moore-Penrose inverses in C[symbol]-algebras. 3.6. Notes -- 4. Drazin inverses, T-S generalized inverses and their perturbations. 4.1. Drazin inverses on Banach spaces. 4.2. The outer generalized inverse with prescribed range and kernel. 4.3. The stable perturbation of Drazin inverses. 4.4. The perturbation analysis of the outer generalized inverse with prescribed range and kernel. 4.5. Drazin inverses and generalized Drazin inverses in Banach algebras. 4.6. Stable perturbation of Drazin inverses. 4.7. Representations for the Drazin inverses of sums, products and block matrices. 4.8. Notes -- 5. Miscellaneous applications. 5.1. The generalized Bott-Duffin inverse of an operator on a Hilbert Space. 5.2. Some applications of Moore-Penrose inverses in frame theory. 5.3. Generalized Cowen-Douglas mappings. 5.4. Condition numbers related to Moore-Penrose inverses and Drazin inverses. 5.5. The local conjugacy of a C[symbol]-mapping -- 6. Some related topics. 6.1. Perturbation of reduced minimum modulus. 6.2. Perturbation analysis for the consistent operator equation on Banach spaces. 6.3. Fredholm elements relative to a homomorphism and their indices. 6.4. Lifting a Moore-Penrose inverse from quotient C[symbol]-algebras. 6.5. Approximate polar decomposition in C[symbol]-algebras. 6.6. The reduced minimum modulus relative to a quotient C[symbol]-algebra. 6.7. Notes.

This book provides a broad introduction to the generalized inverses, Moore-Penrose inverses, Drazin inverses and T-S outer generalized inverses and their perturbation analyses in the spaces of infinite-dimensional. This subject has many applications in operator theory, operator algebras, global analysis and approximation theory and so on. Stable Perturbations of Operators and Related Topics is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students. It is also suitable for researchers as a reference. The proofs of statements and explanations in the book are detailed enough that interested readers can study it by themselves.

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