LEADER 10928nam 2200505 450 001 9910568266203321 005 20230718113952.0 010 $a9783030978143$b(electronic bk.) 010 $z9783030978136 035 $a(MiAaPQ)EBC6975908 035 $a(Au-PeEL)EBL6975908 035 $a(CKB)21957562800041 035 $a(PPN)269154965 035 $a(EXLCZ)9921957562800041 100 $a20221126d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCounterexamples in operator theory /$fMohammed Hichem Mortad 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (613 pages) 311 08$aPrint version: Mortad, Mohammed Hichem Counterexamples in Operator Theory Cham : Springer International Publishing AG,c2022 9783030978136 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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?0 with A2=0 and So "026B30D A2"026B30D ?"026B30D A"026B30D 2 -- 1.2.3 A,BB(H) with ABAB=0 but BABA?0 -- 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 ?1/"026B30D A"026B30D -- 1.2.10 An AB(H) Such That I-A Is Invertible and Yet "026B30D A"026B30D ?1 -- 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?BA -- 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?0 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. 327 $a2.2.7 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?BA -- 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?BA -- 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 -- 2.2.20 A B Such That BB*+B*B=I and B2=B*2=0 -- 2.2.21 A Nonnormal Solution of (A*A)2=A*2A2 -- 2.2.22 An AB(H) Such That An=I, While An-1?I, n?2 -- 2.2.23 A Unitary A Such That An?I for All nN, n?2 -- 2.2.24 A Normal Non-self-adjoint Operator AB(H) Such That A*A=An -- 2.2.25 A Nonnormal A Satisfying A*pAq=An -- Answers -- 3 Operator Topologies -- 3.1 Questions -- 3.1.1 Strong Convergence Does Not Imply Convergence in Norm, and Weak Convergence Does Not Entail Strong Convergence -- 3.1.2 s-limn? An=As-limn? A*n=A* -- 3.1.3 (A,B)AB Is Not Weakly Continuous -- 3.1.4 The Uniform Limit of a Sequence of Invertible Operators -- 3.1.5 A Sequence of Self-adjoint Operators Such That None of Its Terms Commutes with the (Uniform) Limit of the Sequence -- 3.1.6 Strong (or Weak) Limit of Sequences of Unitary or Normal Operators -- Answers -- 4 Positive Operators -- 4.1 Basics -- 4.2 Questions. 327 $a4.2.1 Two Positive Operators A,B Such That AB=0 -- 4.2.2 Two A,B Such That A?0, A?0, B?0, B?0, yet AB?0 -- 4.2.3 KAK*?A Where A?0 and K Is a Contraction -- 4.2.4 KAK*?A AK=KA Where A?0 and K Is an Isometry -- 4.2.5 KAK*?A AK*=KA Where A?0 and K Is Unitary -- 4.2.6 The Operator Norm Is Not Strictly Increasing -- 4.2.7 A?B?0A2?B2 -- 4.2.8 A,B?0AB+BA?0 -- 4.2.9 Two Non-self-adjoint A and B Such That An+Bn?0 for All n -- 4.2.10 Two Positive A,B (with A?0 and B?0) and Such That AB?0 but A2+B2 Is Not Invertible -- 4.2.11 Two A, B Satisfying "026B30D AB-BA"026B30D =1/2"026B30D A"026B30D "026B30D B"026B30D -- 4.2.12 Two A, B Satisfying "026B30D AB-BA"026B30D ="026B30D A"026B30D "026B30D B"026B30D -- 4.2.13 On Normal Solutions of the Equations AA*=qA*A, qR -- Answers -- 5 Matrices of Bounded Operators -- 5.1 Basics -- 5.2 Questions -- 5.2.1 A Non-invertible Matrix Whose Formal Determinant Is Invertible -- 5.2.2 An Invertible Matrix Whose Formal Determinant Is Not Invertible -- 5.2.3 Invertible Triangular Matrix vs. Left and Right Invertibility of Its Diagonal Elements -- 5.2.4 Non-invertible Triangular Matrix vs. Left and Right Invertibility of Its Diagonal Elements -- 5.2.5 An Invertible Matrix yet None of Its Entries Is Invertible -- 5.2.6 A Normal Matrix yet None of Its Entries Is Normal -- 5.2.7 A Unitary Matrix yet None of Its Entries Is Unitary -- 5.2.8 Two Non-comparable Self-Adjoint Matrices yet the Corresponding Entries Are Comparable -- 5.2.9 An Isometry S Such That S2 Is Unitarily Equivalent to SS -- 5.2.10 An Infinite Direct Sum of Invertible Operators Need Not Be Invertible -- 5.2.11 The Similarity of AB to CD Does Not Entail the Similarity of A to C or That of B to D -- 5.2.12 A Matrix of Operators T on H2 Such That T3=0 But T2?0 -- 5.2.13 Block Circulant Matrices Are Not Necessarily Circulant -- Answers -- 6 (Square) Roots of Bounded Operators. 327 $a6.1 Basics -- 6.2 Questions -- 6.2.1 A Self-Adjoint Operator with an Infinitude of Self-Adjoint Square Roots -- 6.2.2 An Operator Without Any Square Root -- 6.2.3 A Nilpotent Operator with Infinitely Many Square Roots -- 6.2.4 An Operator Having a Cube Root but Without Any Square Root -- 6.2.5 An Operator Having a Square Root but Without Any Cube Root -- 6.2.6 A Non-invertible Operator with Infinitely Many Square Roots -- 6.2.7 An Operator A Without Any Square Root, but A+?I Always Has One (?C*) -- 6.2.8 A2?0A?0 Even When A Is Normal -- 6.2.9 A3?0A?0 Even When A Is Normal -- 6.2.10 An Operator Having Only Two Square Roots -- 6.2.11 Can an Operator Have Only One Square Root? -- 6.2.12 Can an Operator Have Only Two Cube Roots? -- 6.2.13 A Rootless Operator -- 6.2.14 On Some Result By B. Yood on Rootless Matrices -- 6.2.15 A Non-nilpotent Rootless Matrix -- 6.2.16 Two (Self-Adjoint) Square Roots of a Self-Adjoint Operator Need Not Commute -- 6.2.17 A BB(H) Commuting with A Need Not Commute with an Arbitrary Root of A -- 6.2.18 A Self-Adjoint Operator Without Any Positive Square Root -- 6.2.19 Three Positive Operators A,B,CB(H) Such That A?B?0 and C Is Invertible Yet (CA2C)12?(CB2C)12 -- 6.2.20 Three Positive Operators A,B,CB(H) Such That A?C and B?C Yet (A2+B2)12 ?2 C -- 6.2.21 On Some Result by F. Kittaneh on Normal Square Roots -- 6.2.22 On the Normality of Roots of Normal Operators Having Co-prime Powers -- 6.2.23 An Isometry Without Square or Cube Roots -- 6.2.24 Two Operators A and B Without Square Roots, Yet AB Has a Square Root -- Answers -- 7 Absolute Value, Polar Decomposition -- 7.1 Basics -- 7.2 Questions -- 7.2.1 An A Such That |Re A|?|A| and |`3?9`42`"?613A``45`47`"603AImA|?|A| -- 7.2.2 A Weakly Normal T Such That T2 Is Not Normal -- 7.2.3 Two Self-Adjoints A,B Such That |A+B| ?|A|+|B|. 327 $a7.2.4 Two Self-Adjoint Operators A,B That Do Not Satisfy |A||B|+|B||A|?AB+BA -- 7.2.5 Two Self-Adjoint Operators A and B Such That "026B30D |A|-|B|"026B30D ?"026B30D A-B"026B30D -- 7.2.6 Two Non-commuting Operators A and B That Are Not Normal and Yet |A+B|=|A|+|B| -- 7.2.7 Two Positive Operators A and B with |A-B|?A+B -- 7.2.8 Two Self-adjoint Operators A and B Such That I+|AB-I|?(I+|A-I|)(I+|B-I|) -- 7.2.9 Two Self-Adjoints A,BB(H) Such That |AB|?|A||B| -- 7.2.10 Two Operators A and B Such That AB=BA, However, |A||B|?|B||A| -- 7.2.11 A Pair of Operators A and B Such That A|B|=|B|A and B|A|=|A|B, But AB?BA and AB*?B*A -- 7.2.12 An Operator A Such That A|A|?|A|A -- 7.2.13 An A Such That |A||A*|=|A*||A| But AA*?A*A -- 7.2.14 An Operator A Such That |A2|?|A|2 -- 7.2.15 A Non-surjective A Such That |A| Is Surjective -- 7.2.16 Two Self-Adjoint Operators A,B with B?0 Such That -B?A?B but |A|?B -- 7.2.17 The Failure of the Inequality |"426830A Ax,x"526930B |?"426830A |A|x,x"526930B -- 7.2.18 On the Generalized Cauchy-Schwarz Inequality -- 7.2.19 On the Failure of Some Variants of the Generalized Cauchy-Schwarz Inequality -- 7.2.20 A Sequence of Self-Adjoint Operators (An) Such That "026B30D |An|-|A|"026B30D 0 But "026B30D An-A"026B30D 0 -- 7.2.21 The Non-weakly Continuity of A|A| -- 7.2.22 A Sequence of Operators (An) Converging Strongly to A, but (|An|) Does Not Converge Strongly to |A| -- 7.2.23 An Invertible A=U|A| with U|A|?|A|U, UA?AU, and A|A|?|A|A -- 7.2.24 Left or Right Invertible Operators Do Not Enjoy a (``Unitary'') Polar Decomposition -- 7.2.25 A Normal Operator Whose Polar Decomposition Is Not Unique -- 7.2.26 On a Result of the Uniqueness of the Polar Decomposition By Ichinose-Iwashita -- 7.2.27 An Operator A Expressed as A=V|A| with A3=0 but V3?0 -- 7.2.28 An Invertible Operator A Expressed as A=U|A| with A3=I but U3?I -- Answers. 327 $a8 Spectrum. 606 $aOperator theory 606 $aTeoria d'operadors$2thub 608 $aLlibres electrònics$2thub 615 0$aOperator theory. 615 7$aTeoria d'operadors 676 $a515.724 700 $aMortad$b Mohammed Hichem$f1978-$01264582 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910568266203321 996 $aCounterexamples in operator theory$92965421 997 $aUNINA