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, and many of the counterexamples included are the authors 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.

Intro -- Preface -- Contents -- Part I Bounded Linear Operators -- 1 Some Basic Properties -- 1.1 Basics -- 1.2 Questions -- 1.2.1 Does the ``Banachness'' of B(X,Y) Yield That of Y? -- 1.2.2 An Operator A`" with A2=0 and So ""026B30D A2""026B30D `""026B30D A""026B30D 2 -- 1.2.3 A,BB(H) with ABAB=0 but BABA`" -- 1.2.4 An Operator Commuting with Both A+B and AB, But It Does Not Commute with Any of A and B -- 1.2.5 The Non-transitivity of the Relation of Commutativity -- 1.2.6 Two Operators A,B with ""026B30D AB-BA""026B30D =2""026B30D A""026B30D ""026B30D B""026B30D

1.2.7 Two Nilpotent Operators Such That Their Sum and Their Product Are Not Nilpotent -- 1.2.8 Two Non-nilpotent Operators Such That Their Sum and Their Product Are Nilpotent -- 1.2.9 An Invertible Operator A with ""026B30D A-1""026B30D `"/""026B30D A""026B30D -- 1.2.10 An AB(H) Such That I-A Is Invertible and Yet ""026B30D A""026B30D e" -- 1.2.11 Two Non-invertible A,BB(H) Such That AB Is Invertible -- 1.2.12 Two A,B Such That A+B=AB but AB`"A -- 1.2.13 Left (Resp. Right) Invertible Operators with Many Left (Resp. Right) Inverses -- 1.2.14 An Injective Operator That Is Not Left Invertible

1.2.15 An A`" Such That ""426…