Counterexamples in Operator Theory
Springer International Publishing (Verlag)
978-3-030-97813-6 (ISBN)
The first half of the book focuses on bounded linear operators, including counterexamples in the areas of operator topologies, matrices of bounded operators, square roots, the spectrum, operator exponentials, and non-normal operators. The second part of the book is devoted to unbounded linear operators in areas such as closedness and closability, self-adjointness, normality, commutativity, and the spectrum, concluding with a chapter that features some open problems. Chapters begin with a brief "Basics" section for the readers' reference, andmany of the counterexamples included are the author's original work.
Counterexamples in Operator Theory can be used by students in graduate courses on operator theory and advanced matrix theory. Previous coursework in advanced linear algebra, operator theory, and functional analysis is assumed. Researchers, quantum physicists, and undergraduate students studying functional analysis and operator theory will also find this book to be a useful reference.
Mohammed Hichem Mortad's primary research interest is operator theory. He received his Ph.D. in mathematical analysis from the University of Edinburgh in 2003.
Preface.- Part 1. Bounded Linear Operators.- Some Basic Properties.- Basic Classes of Bounded Linear Operators.- Operator Topologies.- Positive Operators.- Matrices of Bounded Operators.- (Square) Roots of Bounded Operators.- Absolute Value. Polar Decomposition.- Spectrum.- Spectral Radius. Numerical Range.- Compact Operators.- Functional Calculi.- Fuglede-Putnam Theorems and Intertwining Relations.- Operator Exponentials.- Nonnormal Operators.- Similarity and Unitary Equivalence.- The Sylvester Equation.- More Questions and Some Open Problems.- Part 2. Unbounded Linear Operators.- Basic Notions.- Closedness.- Adjoints. Symmetric Operators.- Self-adjointness.- (Arbitrary) Square Roots.- Normality.- Absolute Value. Polar Decomposition.- Unbounded Nonnormal Operators.- Commutativity.- The Fuglede-Putnam Theorems and Intertwining Relations.- Commutators.- Spectrum.- Matrices of Unbounded Operators.- Relative Boundedness.- More Questions and Some Open Problems II.- Appendix A: A Quick Review of the Fourier Transform.- Appendix B: A Word on Distributions and Sobolev Spaces.- Bibliography.- Index
"The book ... offers a wide range of examples and counterexamples in the theory of linear operators acting on Hilbert spaces. ... The books thus covers a broad range of largely classical topics in operator theory ... . The book is most likely ... a reference work for its intended audience of advanced students and researchers in operator theory, as well as, no doubt, those setting exam questions on the subject." (David Seifert, zbMATH 1512.47002, 2023)
“The book … offers a wide range of examples and counterexamples in the theory of linear operators acting on Hilbert spaces. … The books thus covers a broad range of largely classical topics in operator theory … . The book is most likely … a reference work for its intended audience of advanced students and researchers in operator theory, as well as, no doubt, those setting exam questions on the subject.” (David Seifert, zbMATH 1512.47002, 2023)
| Erscheinungsdatum | 06.05.2022 |
|---|---|
| Zusatzinfo | XXXVI, 598 p. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 1161 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Schlagworte | bounded and unbounded operators • bounded linear operators on Hilbert space • counterexamples in operator theory • Fuglede-Putnam theorem • matrices of operators • unbounded linear operators on Hilbert space |
| ISBN-10 | 3-030-97813-3 / 3030978133 |
| ISBN-13 | 978-3-030-97813-6 / 9783030978136 |
| Zustand | Neuware |
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