04380nam 22005655 450 991056826620332120251113210907.09783030978143(electronic bk.)978303097813610.1007/978-3-030-97814-3(MiAaPQ)EBC6975908(Au-PeEL)EBL6975908(CKB)21957562800041(PPN)269154965(OCoLC)1319219251(DE-He213)978-3-030-97814-3(EXLCZ)992195756280004120220503d2022 u| 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierCounterexamples in Operator Theory /by Mohammed Hichem Mortad1st ed. 2022.Cham :Springer International Publishing :Imprint: Birkhäuser,2022.1 online resource (613 pages)Mathematics and Statistics SeriesPrint version: Mortad, Mohammed Hichem Counterexamples in Operator Theory Cham : Springer International Publishing AG,c2022 9783030978136 Includes bibliographical references and index.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.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.Mathematics and Statistics SeriesOperator theoryFunctional analysisOperator TheoryFunctional AnalysisOperator theory.Functional analysis.Operator Theory.Functional Analysis.515.724515.724Mortad Mohammed Hichem1978-1264582MiAaPQMiAaPQMiAaPQ9910568266203321Counterexamples in operator theory2965421UNINA