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Table of Contents
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 ""426830A Ax,x""526930B =0 for All xH
1.2.16 The Open Mapping Theorem Fails to Hold True for Bilinear Mappings
Answers
2 Basic Classes of Bounded Linear Operators
2.1 Basics
2.2 Questions
2.2.1 A Non-unitary Isometry
2.2.2 A Nonnormal A Such That kerA=kerA*
2.2.3 Do Normal Operators A and B Satisfy ""026B30D ABx""026B30D =""026B30D BAx""026B30D for All x?
2.2.4 Do Normal Operators A and B Satisfy ""026B30D ABx""026B30D =""026B30D AB*x""026B30D for All x?
2.2.5 Two Operators B and V Such That ""026B30D BV""026B30D `""026B30D B""026B30D Where V Is an Isometry
2.2.6 An Invertible Normal Operator That Is not Unitary
2.2.7 Two Self-Adjoint Operators Whose Product Is Not Even Normal
2.2.8 Two Normal Operators A,B Such That AB Is Normal, but AB`"A
2.2.9 Two Normal Operators Whose Sum Is Not Normal
2.2.10 Two Unitary U,V for Which U+V Is Not Unitary
2.2.11 Two Anti-commuting Normal Operators Whose Sum Is Not Normal
2.2.12 Two Unitary Operators A and B Such That AB, BA, and A+B Are All Normal yet AB`"A
2.2.13 A Non-self-adjoint A Such That A2 Is Self-Adjoint
2.2.14 Three Self-Adjoint Operators A, B, and C Such That ABC Is Self-Adjoint, Yet No Two of A, B, and C Need to Commute
2.2.15 An Orthogonal Projection P and a Normal A Such That PAP Is Not Normal
2.2.16 A Partial Isometry That Is Not an Isometry
2.2.17 A Non-partial Isometry V Such That V2 Is a Partial Isometry
2.2.18 A Partial Isometry V Such That V2 Is a Partial Isometry, but Neither V3 Nor V4 Is One
2.2.19 No Condition of U=U*, U2=I and U*U=I Needs to Imply Any of the Other Two
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 ""426830A Ax,x""526930B =0 for All xH
1.2.16 The Open Mapping Theorem Fails to Hold True for Bilinear Mappings
Answers
2 Basic Classes of Bounded Linear Operators
2.1 Basics
2.2 Questions
2.2.1 A Non-unitary Isometry
2.2.2 A Nonnormal A Such That kerA=kerA*
2.2.3 Do Normal Operators A and B Satisfy ""026B30D ABx""026B30D =""026B30D BAx""026B30D for All x?
2.2.4 Do Normal Operators A and B Satisfy ""026B30D ABx""026B30D =""026B30D AB*x""026B30D for All x?
2.2.5 Two Operators B and V Such That ""026B30D BV""026B30D `""026B30D B""026B30D Where V Is an Isometry
2.2.6 An Invertible Normal Operator That Is not Unitary
2.2.7 Two Self-Adjoint Operators Whose Product Is Not Even Normal
2.2.8 Two Normal Operators A,B Such That AB Is Normal, but AB`"A
2.2.9 Two Normal Operators Whose Sum Is Not Normal
2.2.10 Two Unitary U,V for Which U+V Is Not Unitary
2.2.11 Two Anti-commuting Normal Operators Whose Sum Is Not Normal
2.2.12 Two Unitary Operators A and B Such That AB, BA, and A+B Are All Normal yet AB`"A
2.2.13 A Non-self-adjoint A Such That A2 Is Self-Adjoint
2.2.14 Three Self-Adjoint Operators A, B, and C Such That ABC Is Self-Adjoint, Yet No Two of A, B, and C Need to Commute
2.2.15 An Orthogonal Projection P and a Normal A Such That PAP Is Not Normal
2.2.16 A Partial Isometry That Is Not an Isometry
2.2.17 A Non-partial Isometry V Such That V2 Is a Partial Isometry
2.2.18 A Partial Isometry V Such That V2 Is a Partial Isometry, but Neither V3 Nor V4 Is One
2.2.19 No Condition of U=U*, U2=I and U*U=I Needs to Imply Any of the Other Two