Completion problems for operator matrices are concerned with the question of whether a partially specified operator matrix can be completed to form an operator of a desired type. The research devoted to this topic provides an excellent means to investigate the structure of operators. This book provides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory. It will be useful for graduate students and researchers interested in operator theory and the problem of matrix completions.
Author(s): Dragana S. Cvetković Ilić
Series: Mathematical Surveys and Monographs, 267
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 185
City: Providence
Cover
Title page
Contents
Preface
Notation
Chapter 1. Completion problems for upper triangular operator matrices
1.1. Completions of upper triangular operator matrices to injective and left invertible operator
1.2. Completions of upper triangular operator matrices to invertible and right invertible operator
1.3. Completions of upper triangular operator matrices to regular operator
1.4. Completions of upper triangular operator matrices to Drazin and generalized Drazin invertible operator
1.5. Completions of upper triangular operator matrices to Fredholm operator
1.6. Completions of upper triangular operator matrices to generalized left and right Weyl operator
1.7. Completions of upper triangular operator matrices to Browder operator
1.8. Completions of upper triangular matrices to Kato nonsingular operator
1.9. Fredholm consistency of upper triangular operator matrices
Chapter 2. Invertibility of an operator ?+??
2.1. Auxiliary results
2.2. Invertibility of an operator ?+??
2.3. Injectivity of an operator ?+??
2.4. Surjectivity of an operator ?+??
2.5. Right and left invertibility of an operator ?+??
2.6. Fredholmness of an operator ?+??
2.7. Dense range of an operator ?+??
2.8. Closed range of an operator ?+??
Chapter 3. Completions of operator matrices ?_{?}
3.1. Completions of operator matrices ?_{?} to invertible operator
3.2. Completions of ?_{?} to injective and nondense range operator
3.3. Completions of operator matrices ?_{?} to right and left invertible operator
3.4. Completions of operator matrices ?_{?} to semi-Fredholm operator
3.5. Completions of operator matrices ?_{?} to Fredholm operator
3.6. Completions of operator matrices ?_{?} to closed range operator
Chapter 4. Completions of operator matrices ?_{(?,?)}
4.1. Completions of operator matrices ?_{(?,?)} to invertibility, right and left invertibility
4.2. Consistency of operator matrices ?_{(?,?)} in invertibility
4.3. Consistency of operator matrices ?_{(?,?)} in Fredholmness
Chapter 5. Applications of completion results
5.1. Invertibility of linear combinations of operators
5.2. Injectivity, right and left invertibility of the sum of two operators
5.3. Fredholmness of the sum of two operators
Bibliography
Index
Back Cover
