Counterexamples in Operator Theory

This text is the first of its kind exclusively devoted to counterexamples in operator theory and includes over 500 problems on bounded and unbounded linear operators in Hilbert spaces. This volume is geared towards graduate students studying operator theory, and the author has designated the difficulty level for each counterexample, indicating which ones are also suitable for advanced undergraduate students.
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.

Presents over 500 counterexamples in operator theory at varying levels of difficulty Includes counterexamples on both bounded and unbounded linear operators, many of which are the author's original work Divided into brief subsections, making it easy for readers to navigate the wide variety of topics included

Autorentext

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.

Inhalt
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