Counterexamples in Operator Theory

Book excerpt: 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, and many 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.