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
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≠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
4.2.1 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
6.1 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+αI Always Has One (αC*)
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 |`3́9`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|
7.2.4 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
8 Spectrum
8.1 Basics
8.2 Questions
8.2.1 An AB(H) Such That σ(A)=
8.2.2 An A Such That σp(A*)≠{λ: λσp(A)}
8.2.3 An AB(H) with p[σ(A)]≠σ[p(A)] Where p Is a Polynomial
8.2.4 A Non-self-adjoint Operator Having a Real Spectrum
8.2.5 A Non-positive Operator with a Positive Spectrum
8.2.6 A Non-orthogonal Projection A with σ(A)={0,1}
8.2.7 A Self-Adjoint Operator Without Any Eigenvalue
8.2.8 A Self-Adjoint A Such That σ(A)σ(-A)=, Yet A Is Neither Positive Nor Negative
8.2.9 A Self-Adjoint A Such That σ(A)σ(-A)={0}, Yet A Is Neither Positive Nor Negative
8.2.10 A Self-Adjoint Matrix A Such That σ(A)σ(-A)={0} and tr A=0, Yet A Is Neither Positive Nor Negative
8.2.11 A Nilpotent T=A+iB Such That σ(A)σ(-A)={0}
8.2.12 A Nilpotent T=A+iB Such That σ(A)σ(-A)=
8.2.13 Two Operators A,B Such That σ(AB)≠σ(BA)
8.2.14 Two Self-Adjoints A,B with σ(A)+σ(B)σ(A+B)
8.2.15 Two Commuting A,B with σ(A)+σ(B)σ(A+B)
8.2.16 Two Self-Adjoints A and B with σ(A)σ(B)σ(AB)
8.2.17 Two Commuting A,B Such That σ(A)σ(B)σ(AB)
8.2.18 Two Commuting Self-Adjoint Operators A and B Such That σp(AB)σp(A)σp(B) and σp(A+B)σp(A)+σp(B)
8.2.19 A Unitary Operator A Such That |σ(A)|2≠σ(A*)σ(A) and σ(A*)σ(A)R
8.2.20 Two Operators A and B Such That σ(A)σ(-B)=, but A+B Is Not Invertible
8.2.21 ``Bigger'' Operators Defined on Larger Hilbert Spaces Do Not Necessarily Have ``Bigger'' Spectra
8.2.22 An Operator A Such That A≠0 and σ(A)={0}
8.2.23 A Self-Adjoint Operator A Such That σp(A)={0,1}, σc(A)=(0,1), and σ(A)=[0,1]
8.2.24 An Operator A with σ(A)={1}, But A Is Neither Self-Adjoint Nor Unitary
8.2.25 Two Self-Adjoint Operators A,B Such That σ(AB) Is Purely Imaginary
8.2.26 A≤B Does Not Yield σ(A)σ(B)
8.2.27 An Operator A with σp(A)σa(A)σ(A)
8.2.28 A TB(H) Such That |λ|σ(|T|), but λσ(T)
8.2.29 A TB(H) Such That λσ(T), yet |λ|σ(|T|)
8.2.30 A Sequence of Normal Operators Am Converging Strongly to A, butσ(A)≠limm→∞ σ(Am)
8.2.31 T=nN Tn, but nNσ(Tn)≠σ(T)
8.2.32 A=nN An Where All (An) Are Nilpotent, but A Is Not Nilpotent
Answers
9 Spectral Radius, Numerical Range
9.1 Basics
9.2 Questions
9.2.1 An AB(H) Such That r(A)<"026B30D A"026B30D
9.2.2 A Quasinilpotent Operator Which Is Not Nilpotent
9.2.3 A Non-normaloid Operator T Such That Re T≥0
9.2.4 An Operator A with A≠I, σ(A)={1} and "026B30D A"026B30D =1
9.2.5 No Inequality Between r(AB) and r(A)r(B) Need to Hold
9.2.6 The Discontinuity of A→r(A) with Respect to the S.O.T.
9.2.7 Two Commuting Normaloid Operators with a Non-normaloid Product
9.2.8 Two Commuting Self-Adjoint Operators A and B Such That W(A+B)W(A)+W(B)
9.2.9 A Bounded Operator Whose Numerical Range Is Not a Closed Set in C
9.2.10 Neither σ(A)W(A) Nor W(A)σ(A) Need to Hold, Even for Self-Adjoint Operators
9.2.11 Two Self-Adjoint A,BB(H) with Equal Spectra but Different Numerical Ranges
9.2.12 On the Failure of σ(AB)W(A)·W(B)
9.2.13 On the Failure of σ(A-1B)W(B)/W(A) for An Invertible A
9.2.14 A Nonnormal Operator A with W(A)=conv(σ(A))
Answers
10 Compact Operators
10.1 Basics
10.2 Questions
10.2.1 A Non-compact A Such That A2 Is Compact
10.2.2 A Non-compact Operator A Such That "026B30D Aen"026B30D →0 as n→∞ Where (en) Is An Orthonormal Basis
10.2.3 Does the Shift Operator Commute with a Nonzero Compact Operator?
10.2.4 A Compact Operator Which Is Not Hilbert-Schmidt
10.2.5 A Compact Operator Without Eigenvalues
10.2.6 A Non-positive Integral Operator Having a Positive Kernel
10.2.7 On λ-Commutativity Related to Compact and Finite Rank Operators
10.2.8 The Infinite Direct Sum of Compact Operators Need Not Be Compact
10.2.9 A Compact Quasinilpotent Operator A with 0W(A)
Answers
11 Functional Calculi
11.1 Basics
11.2 Questions
11.2.1 A Bounded Borel Measurable Function f and a Self-Adjoint AB(H) Such That σ(f(A))≠f(σ(A))
11.2.2 A Normal Operator A Having Only Real Eigenvalues But A Is Not Self-Adjoint
11.2.3 A Non-self-adjoint Operator Whose Square Root Is Its Adjoint
11.2.4 A,B,CB(H), A Being Self-Adjoint, AB=CA but f(A)B≠Cf(A) for Some Continuous Function f
11.2.5 A Positive A Such That BA=AB* and BA≠AB*
11.2.6 An AB(H) Such That "026B30D A"026B30D ≠sup"026B30D x"026B30D ≤1|"426830A Ax,x"526930B |
11.2.7 A Normal AB(H) (Where H Is Over the Field R) Such That "026B30D A"026B30D ≠sup"026B30D x"026B30D ≤1|"426830A Ax,x"526930B |
11.2.8 An A Such That p(A)≠A* for Any Polynomial p
Answers
12 Fuglede-Putnam Theorems and Intertwining Relations
12.1 Basics
12.2 Questions
12.2.1 Does TA=AT Give TA*=A*T When kerA=kerA*?
12.2.2 If T,N,MB(H) Are Such That N and M Are Normal, Do We Have "026B30D TN-MT"026B30D ="026B30D TN*-M*T"026B30D ? How About |TN-MT|=|TN*-M*T|?
12.2.3 A Self-Adjoint A and a Normal B Such That AB=BA but B*A≠AB
12.2.4 A Unitary UB(H) and a Self-Adjoint AB(H) Such That AU*=UA and AU≠UA
12.2.5 Two Nonnormal Double Commuting Matrices
12.2.6 AN=MBAN*=M*B Even When All of M, N, A and B Are Unitary
12.2.7 AN=MBBN*=M*A Where M Is an Isometry or N Is a Co-isometry
12.2.8 Positive Invertible Operators A,B,N,M Such That AN=MB and AB=BA Yet BN*≠M*A
12.2.9 On Some Generalization of the Fuglede-Putnam Theorem Involving Contractions
12.2.10 A,B,CB(H) with A Being Self-Adjoint Such That AB=λCA and λ Is Arbitrary
12.2.11 Are There Two Normal Operators A and B Such That AB=2BA?
12.2.12 A Normal A and a Self-Adjoint B Such That AB Is Normal but BA Is Not Normal
12.2.13 About Some Embry's Theorem
Answers
13 Operator Exponentials
13.1 Basics
13.2 Questions
13.2.1 An Invertible Matrix Which is Not the Exponential of Any Other Matrix
13.2.2 A Compact AB(H) for Which eA is Not Compact
13.2.3 eA=eB A=B
13.2.4 A Non-Self-Adjoint AB(H) Such That eiA Is Unitary
13.2.5 A Normal A Such That eA=eB But AB≠BA
13.2.6 Two Self-Adjoint A,B Such That A≥B and eA≥eB
13.2.7 A Self-Adjoint AB(H) Such That "026B30D eA"026B30D ≠e"026B30D A"026B30D
13.2.8 A Nonnormal Operator A Such That eA Is Normal
13.2.9 A,BB(H), AB≠BA and eA+B=eAeB =eBeA
13.2.10 A,BB(H), AB≠BA and eA+B=eAeB =eBeA
13.2.11 A,BB(H), AB≠BA and eA+B ≠eAeB=eBeA
13.2.12 A,BB(H), AB≠BA and eA+B=eAeB =eBeA
13.2.13 Real Matrices A and B Such That AB≠BA and eA+B=eAeB=eBeA
13.2.14 An Operator A with eAeA*=eA*eA≠eA+A*
13.2.15 An Operator T Such That |eT|≠eRe T and |eT|≤e|T|
Answers
14 Nonnormal Operators
14.1 Basics
14.2 Questions
14.2.1 A Hyponormal Operator Which Is Not Normal
14.2.2 An Invertible Hyponormal Operator Which Is Not Normal
14.2.3 A Hyponormal Operator with a Non-Hyponormal Square
14.2.4 A Hyponormal Operator AB(H) Such That A+λA* is Not Hyponormal for Some λC
14.2.5 On the Failure of Some Property on the Spectrum for Hyponormal Operators
14.2.6 On the Failure of Some Friedland's Conjecture
14.2.7 On the Failure of the (``Unitary'') Polar Decomposition for Quasinormal Operators
14.2.8 The Inclusions Among the Classes of Quasinormals, Subnormals, Hyponormals, Paranormals, and Normaloids are Proper
14.2.9 A Subnormal (Or Quasinormal) Operator Whose Adjoint Is Not Subnormal
14.2.10 A Paranormal Operator Whose Adjoint Is Not Paranormal
14.2.11 A Semi-Hyponormal Operator Which Is Not Hyponormal
14.2.12 Embry's Criterion for Subnormality: A=A*A2 Does Not Even Yield the Hyponormality of AB(H)
14.2.13 An Invertible Hyponormal Operator Which Is Not Subnormal
14.2.14 Two Commuting Hyponormal Operators A and B Such That A+B Is Not Hyponormal
14.2.15 Two Quasinormal Operators A,BB(H) Such That AB=BA=0 Yet A+B Is Not Even Hyponormal
14.2.16 Two Commuting Subnormal Operators S and T Such That Neither ST nor S+T Is Subnormal
14.2.17 The Failure of the Fuglede Theorem for Quasinormals et al.
14.2.18 A Unitary A and a Quasinormal B with TB=AT but TB*≠A*T
14.2.19 Two Double Commuting Nonnormal Hyponormal Operators
14.2.20 Two Hyponormal (Or Quasinormal) Operators A and B Such That AB*=B*A but AB≠BA
14.2.21 Two Commuting (Resp. Double Commuting) Paranormal Operators Whose Tensor (Resp. Usual) Product Fails to Remain Paranormal
14.2.22 A Subnormal Operator Such That |A2|≠|A|2
14.2.23 A Subnormal A Such That A|A|≠|A|A
14.2.24 A Non-Quasinormal T Such That T*2T2=(T*T)2
14.2.25 A Non-Subnormal Operator Whose Powers Are All Hyponormal
14.2.26 A Hyponormal Operator TB(H) Such That All Tn, n≥2 Are Subnormal but T Is Not
14.2.27 Two Quasinormal Operators A and B Such That AB=BA and |AB|≠|A||B|
14.2.28 Two Quasinormal Operators A and B Such That AB=BA Yet A|B|≠|B|A
14.2.29 Two Quasinormal Operators A and B Such That AB*=B*A and B|A|=|A|B Yet A|B|≠|B|A
14.2.30 The Failure of Some Kaplansky's Theorem for Hyponormal Operators
14.2.31 The Inequality |"426830A Ax,x"526930B |≤"426830A |A|x,x"526930B for All x Does Not Yield Hyponormality
14.2.32 The Failure of Reid's Inequality for Hyponormal Operators
14.2.33 The Weak Limit of Sequences of Hyponormal Operators
14.2.34 Strong (and Weak) Limit of Sequences of Quasinormal Operators
14.2.35 A Quasinormal Operator AB(H) Such That eA Is Not Quasinormal
14.2.36 An Invertible Subnormal Operator A Without Any Bounded Square Root
14.2.37 An Invertible Operator Which Is Not the Exponential of Any Operator
14.2.38 A Positive Answer to the Curto-Lee-Yoon Conjecture About Subnormal and Quasinormal Operators, and a Related Question
14.2.39 A Subnormal Operator T Such That p(T) Is Quasinormal for Some Non-Constant Polynomial p yet T Is Not Quasinormal
14.2.40 A Binormal Operator Which Is Not Normal
14.2.41 An Invertible Binormal Operator T Such That T2 Is Not Binormal
14.2.42 Two Double Commuting Binormal Operators Whose Sum Is Not Binormal
14.2.43 A Non-Binormal Operator T Such That Tn=0 for Some Integer n≥3
14.2.44 A Subnormal Operator Which Is Not Binormal
14.2.45 A Hyponormal Binormal Operator Whose Square Is Not Binormal
14.2.46 Binormal Operators and the Polar Decomposition
14.2.47 Does the θ-Class Contain Subnormals? Hyponormals?
14.2.48 An Operator in the θ-Class Which Is Not Even Paranormal
14.2.49 Is the θ-Class Closed Under Addition?
14.2.50 Is the θ-Class Closed Under the Usual Product of Operators?
14.2.51 Posinormal Operators
14.2.52 (α,β)-Normal Operators
Answers
15 Similarity and Unitary Equivalence
15.1 Questions
15.1.1 Two Similar Operators Which Are Not Unitarily Equivalent
15.1.2 Two Operators Having Equal Spectra but They Are Not Metrically Equivalent
15.1.3 Two Metrically Equivalent Operators yet They Have Unequal Spectra
15.1.4 A Non-Self-Adjoint Normal Operator Which Is Unitarily Equivalent to Its Adjoint
15.1.5 Two Commuting Self-Adjoint Invertible Operators Having the Same Spectra yet They Are Not Unitarily Equivalent
15.1.6 A Matrix Which Is Not Unitarily Equivalent to Its Transpose
15.1.7 On Some Similarity Result by J. P. Williams
15.1.8 Two Self-Adjoint Operators A and B Such That AB Is Not Similar to BA
15.1.9 Two Self-Adjoint Matrices A and B Such That AB Is Not Unitarily Equivalent to BA
15.1.10 Two Normal Matrices A and B Such That AB Is Not Similar to BA
15.1.11 Two Self-Adjoint A,B with dimkerAB=dimkerBA yet AB Is Not Similar to BA
15.1.12 Two Similar Congruent Operators A and B Which Are Not Unitarily Equivalent
15.1.13 Two Quasi-Similar Operators A,B with σ(A)≠σ(B)
15.1.14 Two Quasi-Similar Operators A and B Such That One Is Compact and the Other Is Not
15.1.15 Two Similar Subnormal Operators Which Are Not Unitarily Equivalent
15.1.16 Two Quasi-Similar Hyponormal Operators Which Are Not Similar
15.1.17 A Quasinilpotent Operator Not Similar to Its Multiple
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16 The Sylvester Equation
16.1 Basics
16.2 Questions
16.2.1 The Condition σ(A)σ(B)= Is Not Necessary for the Existence of a Solution to Sylvester's Equation
16.2.2 An Equation AX-XB=C Without a Solution X for Some CB(H) Where σ(A)σ(B)≠
16.2.3 Unitary Equivalence and Sylvester's Equation
16.2.4 Schweinsberg's Theorem and the Sylvester Equation for Quasinormal Operators
Answers
17 More Questions and Some Open Problems
17.1 More Questions
17.2 Some Open Problems
Part II Unbounded Linear Operators
18 Basic Notions
18.1 Basics
18.2 Questions
18.2.1 An Unbounded Linear Functional That Is Everywhere Defined
18.2.2 An Unbounded Linear Operator That Is Everywhere Defined from H into H
18.2.3 A Non-densely Defined T≠0 with "426830A Tx,x"526930B =0 for All xD(T)
18.2.4 A Densely Defined Linear Operator A Satisfying D(A)=D(A2)=@汥瑀瑯步渠=D(An)≠H
18.2.5 A Densely Defined Operator A Such That D(A2)={0}
18.2.6 A Bounded B (Not Everywhere Defined) and a Densely Defined A Such That D(BA)≠D(A)
18.2.7 A Densely Defined T with ran TD(T)={0} yet D(T2)≠{0}
18.2.8 Two Densely Defined Operators A and B Such That D(A+B)=D(A)D(B)={0}
18.2.9 The Same Symbol T with Two Nontrivial Domains D and D' Such That DD'={0}
18.2.10 The Sum of a Bounded Operator and an Unbounded Operator Can Be Bounded
18.2.11 Two Densely Defined T and S Such That T-S0 But TS
18.2.12 Three Densely Defined Operators A, B, and C Satisfying A(B+C)AB+AC
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19 Closedness
19.1 Basics
19.2 Questions
19.2.1 A Closed Operator Having Any Other Operator as an Extension
19.2.2 A Bounded Operator That Is Not Closed, and an Unbounded Operator That Is Closed
19.2.3 A Left Invertible Operator That Is Not Closed
19.2.4 A Right Invertible Operator That Is Not Closed
19.2.5 A Closable A That Is Injective but A Is Not Injective
19.2.6 A Closed Densely Defined Unbounded Operator A and a VB(H) Such That AV Is Not Densely Defined
19.2.7 Two Closed Operators A and B Such That BA Is Not Closed
19.2.8 A Compact BB(H) and a Closed Operator A Such That BA Is Not Closed
19.2.9 The Closedness of AB with BB(H) Need Not Yield the Closedness of A
19.2.10 The Closedness of BA with BB(H) Need Not Yield the Closability of A
19.2.11 A Left Invertible Operator A and a Closed Operator B Such That AB Is Unclosed
19.2.12 A Right Invertible Operator A and a Closed Operator B Such That AB Is Unclosed
19.2.13 Two Closable A and B Such That AB≠A B
19.2.14 Three Densely Defined Closed Operators A, B, and C Satisfying ABC≠ABC
19.2.15 A Densely Defined Unbounded Closed (Nilpotent) Operator A Such That A2 Is Bounded and Unclosed
19.2.16 Another Densely Defined Unbounded Closed Operator T Such That T2 Is Bounded and Unclosed
19.2.17 A Densely Defined Unbounded and Closed Operator A Such That D(A)=D(A2)=@汥瑀瑯步渠=D(An)≠H
19.2.18 A Non-closable Unbounded Nilpotent Operator
19.2.19 Unbounded Closed (Resp., Unclosable) Idempotent Operators
19.2.20 An Unclosed (but Closable) A Such That A2 Is Closed
19.2.21 An Unclosed and Closable T Such That T2 Is Closed (Such That D(T2)≠{0})
19.2.22 A Densely Defined Unclosed Closable A Such That A2 Is Closed and Densely Defined
19.2.23 An Unbounded Densely Defined Closed Operator A Such That A2≠0 and A3=0
19.2.24 A Non-closable Operator A Such That A2=0 Everywhere on H
19.2.25 An Unclosable Operator T Such That T2=I Everywhere on H
19.2.26 An Everywhere Defined Unbounded Operator That Is Neither Injective Nor Surjective
19.2.27 An Everywhere Defined Unbounded Operator That Is Injective but Non-surjective
19.2.28 An Everywhere Defined Unbounded Operator That Is Surjective But Non-injective
19.2.29 A Non-closable Unbounded Operator T Such That Tn=I Everywhere on H and Tn-1≠I
19.2.30 An Unclosable Unbounded Operator T Such That Tn=0 Everywhere on H While Tn-1≠0
19.2.31 Three Densely Defined Closed Operators A, B, and C Satisfying A(B+C)AB+AC
19.2.32 A Closed A and a Bounded (Not Everywhere Defined) B Such That A+B Is Unclosed
19.2.33 Two Closed Operators with an Unclosed Sum
19.2.34 Closable Operators A and B with A+B≠A+B
19.2.35 Three Densely Defined Closed Operators A, B, and C Satisfying: A+B+C≠A+B+C
19.2.36 Two Non-closable Operators Whose Sum Is Bounded and Self-Adjoint
19.2.37 Two Closed Operators Whose Sum Is Not Even Closable
19.2.38 Two Densely Defined Closed Operators S and T Such That D(S)D(T)={0}
19.2.39 An Unbounded Nilpotent N Such That I+N Is Not Boundedly Invertible
Answers
20 Adjoints, Symmetric Operators
20.1 Basics
20.2 Questions
20.2.1 Is There Some Densely Defined Non-closable Operator A Such That A*A=I?
20.2.2 A Densely Defined T Such That H≠D(T) +ran(T*)
20.2.3 A Densely Defined Symmetric Operator A Such That ker(A)≠ker(A*)
20.2.4 The Failure of Some Maximality Relations
20.2.5 A Non-closable Symmetric Operator
20.2.6 A Densely Defined Operator T Such That D(T*)={0}
20.2.7 A Densely Defined Operator T on a Hilbert Space Such That T*=0 on D(T*)≠{0}
20.2.8 An Everywhere Defined Linear Operator T Such That D(T*)={0}
20.2.9 A Densely Defined T with D(TT*)=D(T*T)=D(T)
20.2.10 Two Densely Defined Operators A and B Such That (A+B)*≠A*+B*
20.2.11 Two Densely Defined A and B (One of Them Is Bounded but Non-everywhere Defined) with (A+B)*≠A*+B*
20.2.12 Two Densely Defined Operators A and B Such That (BA)*≠A*B*
20.2.13 A Densely Defined Left Invertible Operator T and a Densely Defined Operator S Such That (ST)*≠T*S*
20.2.14 A Densely Defined Operator A with (A2)*≠(A*)2
20.2.15 A Densely Defined Operator T with (T2)*=(T*)2
20.2.16 Two Bounded (Non-everywhere Defined) or Unbounded Operators A and B Such That AB and A*=B*
20.2.17 A Non-closed Densely Defined Operator T Such That ranTD(T) and ranT*D(T*)
20.2.18 A Closed Densely Defined Operator T Such That ranTD(T) and ranT*D(T*)
20.2.19 A Densely Defined Unbounded Closed Operator A Such That ranAD(A) But ranA*D(A*)
20.2.20 A Densely Defined Closed Operator A with A+A* Being Densely Defined But Unclosed
20.2.21 A Closed Operator A Satisfying AA*=A*A+I
20.2.22 A Closed T with D(T)=D(T*) but D(TT*)≠D(T*T)
20.2.23 An Unbounded Closed Symmetric and Positive Operator A Such That D(A2)={0}
Answers
21 Self-Adjointness
21.1 Basics
21.2 Questions
21.2.1 Self-Adjoint Operators A with D(A)D(A2)
21.2.2 Symmetric (Closed) Operators Are Not Maximally Self-Adjoint
21.2.3 A*AAA*A*A= AA*
21.2.4 Two Self-Adjoint Operators (T,D) and (T,D') Such That T Is Not Self-Adjoint on DD'
21.2.5 A Dense Domain D(A)L2(R) for Af(x)=xf(x) on which A Is Closed and Symmetric but Non-self-adjoint
21.2.6 A Dense Domain D(B)L2(R) for Bf(x)=if'(x) on which B Is Only Closed and Symmetric
21.2.7 A Closed Densely Defined Non-self-Adjoint A Such That A2 Is Self-Adjoint
21.2.8 A Bounded Densely Defined Essentially Self-Adjoint Operator A Such That D(A2)={0}
21.2.9 An Unbounded Densely Defined Essentially Self-Adjoint Operator A Such That D(A2)={0}
21.2.10 A Densely Defined Closed Symmetric Operator Which Has Many Self-Adjoint Extensions
21.2.11 A Closed and Symmetric Operator Without Self-Adjoint Extensions
21.2.12 An Unbounded Operator A with Domain D(A) Such That "426830A Ax,x"526930B ≥0 for All xD(A), yet A Is Not Self-Adjoint
21.2.13 A Densely Defined Operator A Such That Neither A*A nor AA* Is Self-Adjoint
21.2.14 A Non-closed A Such That A*A Is Self-Adjoint
21.2.15 Two Closed Operators A and B Such That A+B Is Unclosed and Unbounded
21.2.16 Two Unbounded Self-Adjoint Operators A and B Such That D(A)D(B)={0}
21.2.17 Two Unbounded Self-Adjoint, Positive, and Boundedly Invertible Operators A and B with D(A)D(B)={0}
21.2.18 An Unbounded Self-Adjoint Operator A Defined on D(A)H Such That D(A)D(U*AU)≠{0} for any unitary UB(H)
21.2.19 D(A)D(B)={0} Is Equivalent to D(S)D(T)={0} where A and B are Positive Boundedly Invertible Self-Adjoint Operators, and S and T Are Closed
21.2.20 Three Unbounded Self-Adjoint Operators R, S, and T with D(R)D(S)≠{0}, D(R)D(T)≠{0}, and D(S)D(T)≠{0}, yet D(R)D(S)D(T)={0}
21.2.21 Self-Adjoint Positive Operators C and B Such That D(C)D(B)={0}, yet D(Cα)D(Bα) Is Dense for All α(0,1)
21.2.22 Invertible Unbounded Self-Adjoint Operators A and B Such That D(A)D(B)=D(A-1)D(B-1)={0}
21.2.23 Positive Operators T and S with Dense Ranges Such That ran(T12)=ran(S12) and ran(T)ran(S)={0}
21.2.24 An Unbounded Self-Adjoint Positive Boundedly Invertible A and an Everywhere Defined Bounded Self-Adjoint B Such That D(AB)={0} and D(BA)≠{0}
21.2.25 An Unbounded Self-Adjoint Positive C and a Positive SB(H) Such That D(CS)={0}, D(CSα)≠{0}, and D(CαSα)≠{0} for Each 0<α<1
21.2.26 An Unbounded Self-Adjoint Positive Operator A on L2(R) Such That D(A)D(F*AαF) for Any 0<α<1 Where F Is the L2(R)-Fourier Transform
21.2.27 Another Densely Defined Linear Operator T Such That D(T*)={0}
21.2.28 A T with D(T2)=D(T*)=D(TT*)=D(T*T)={0}
21.2.29 An Unclosed Operator T Such That TT* and T*T Are Closed, but Neither TT* Nor T*T Is Self-Adjoint
21.2.30 Yet Another Densely Defined Operator T Such That D(T*) Is Not Dense
21.2.31 On the Operator Equation A*A=A2
21.2.32 A Closed Operator A with A*AA2 (or A2A*A), yet A Is Not Self-Adjoint
21.2.33 A Rank One Self-Adjoint Operator B and an Unbounded, Self-Adjoint, Positive, and Invertible A Such That BA Is Not Even Closable
21.2.34 Two Unbounded Self-Adjoint Operators A and B Such That A+B Is Not Self-Adjoint
21.2.35 On the Failure of the Essential Self-Adjointness of A+B for Some Closed and Symmetric A and B
21.2.36 A Densely Defined Closed A Such That A+A* Is Closed, Densely Defined, Symmetric but Non-self-adjoint
21.2.37 A Densely Defined and Closed but Non-symmetric Operator A Such That A+A* Is Self-Adjoint
21.2.38 Values of λ for which A+λ|A| Is (Not) Self-Adjoint or (Not) Closed, Where A Is Closed and Symmetric
21.2.39 An Unbounded Self-Adjoint Operator A Such That |A|A Are Not Even Closed
21.2.40 Two Unbounded Self-Adjoint and Positive Operators A and B Such That AB+BA Is Not Self-Adjoint
21.2.41 A Closed and Densely Defined Operator T Such That D(T+T*)=D(TT*)D(T*T)={0}
21.2.42 Closed S and T with D(ST)=D(TS)=D(S+T)={0}
21.2.43 Two Densely Defined Closed Operators A and B Such That D(A*)D(B)=H and D(A)D(B*)={0}
21.2.44 Two Unbounded Self-Adjoint Positive Invertible Operators A and B Such That D(A-1B)=D(BA-1)={0}
21.2.45 A Densely Defined Unbounded Closed Operator B Such That B2 and |B|B Are Bounded, Whereas B|B| Is Unbounded and Closed
21.2.46 A Closed Operator T with D(T2)=D(T*2)={0}
21.2.47 A Densely Defined Closed T with D(T2)≠{0} and D(T*2)≠{0} but D(T3)=D(T*3)={0}
21.2.48 A Densely Defined Closed T Such That D(T3)≠{0} and D(T*3)≠{0} yet D(T4)=D(T*4)={0}
21.2.49 A Densely Defined Closed T Such That D(T5)≠{0} and D(T*5)≠{0} While D(T6)=D(T*6)={0}
21.2.50 For Each nN, There Is a Closed Operator T with D(T2n-1)≠{0} and D(T*2n-1)≠{0} but D(T2n)=D(T*2n)={0}
21.2.51 Two Unbounded Self-Adjoint Operators A and B with D(A)=D(B) While D(A2)≠D(B2)
21.2.52 Self-Adjoint Operators A and B with D(A)=D(B), but Neither A2-B2 nor AB+BA Is Even Densely Defined
21.2.53 On the Impossibility of the Self-Adjointness of AB and BA Simultaneously When B Is Closable (Unclosed) and A Is Self-Adjoint
21.2.54 The Non-self-adjointness of PAP Where P Is an Orthogonal Projection and A Is Self-Adjoint
21.2.55 The Unclosedness of PAP Where P Is an Orthogonal Projection and A Is Self-Adjoint
21.2.56 Two Self-Adjoint Operators A and B Such That A Is Positive, BB(H), and D(A1/2BA1/2)={0}
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22 (Arbitrary) Square Roots
22.1 Basics
22.2 Questions
22.2.1 Compact Operators Having Unbounded Square Roots
22.2.2 A Bounded Invertible Operator Without Any Closed Square Root
22.2.3 A Non-closable Operator Without Any Closable Square Root
22.2.4 An Operator S with a Square Root T but T* Is Not a Square Root of S*
22.2.5 An Operator S with a Square Root T but T Is Not a Square Root of S
22.2.6 A Densely Defined Closed Operator T Such That T2 Is Densely Defined and Non-closable
22.2.7 Square Roots of a Self-Adjoint Operator Need Not Have Equal Domains
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23 Normality
23.1 Basics
23.2 Questions
23.2.1 A Normal T* Such That T Is Not Normal
23.2.2 A T Such That TT*=T*T yet T Is Not Normal
23.2.3 An Unbounded Densely Defined Closed Nonnormal Operator T Such That D(T)=D(T*) and D(TT*)=D(T*T)
23.2.4 A Nonnormal Densely Defined Closed S that Satisfies TS*S and TSS*
23.2.5 Two Unbounded Self-Adjoint Operators A and B, B Is Positive, Such That AB Is Normal Without Being Self-Adjoint
23.2.6 Two Unbounded Self-Adjoint Operators A and B Such That A Is Positive, AB is Normal But Non-self-adjoint
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24 Absolute Value. Polar Decomposition
24.1 Basics
24.2 Questions
24.2.1 Closed Densely Defined Operators S and T Such That ST but |S||T|
24.2.2 A Closed Operator T and an Unclosed Operator A Such That |T*|=|T|=|A|
24.2.3 A Non-closed A with D(A)≠D(|A|)
24.2.4 A Non-closed Densely Defined Operator A Without Any Polar Decomposition
24.2.5 A Non-closed Densely Defined Operator A and a Unitary UB(H) Such That AU|A|
24.2.6 A Non-closed Densely Defined Operator A and a Unitary UB(H) Such That AU|A| and UA|A|
24.2.7 A Closed and Symmetric A Such That |An|≠|A|n
24.2.8 A Self-Adjoint BB(H) and a Self-Adjoint Positive A Such That BA Is Closed Without Being Self-Adjoint
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25 Unbounded Nonnormal Operators
25.1 Questions
25.1.1 An SB(H) Such That 0W(S), and an Unbounded Closed Hyponormal T Such That STT*S but T≠T*
25.1.2 A Closed A with A*AAAA*A Does Not Yield the Quasinormality of A
25.1.3 An Unbounded Paranormal Operator T Such That D(T*)={0}
25.1.4 A Closable Paranormal Operator Whose Closure Is Not Paranormal
25.1.5 A Densely Defined Closed Operator T Such That Both T and T* Are Paranormal and kerT=kerT*={0} but T Is Not Normal
25.1.6 Q-Normal Operators
25.1.7 On the Operator Equations A*A=An, n≥3
Answers
26 Commutativity
26.1 Basics
26.2 Questions
26.2.1 BAABBAAB, BB(H)
26.2.2 A Self-Adjoint BB(H) and an Unclosed A with BAAB but f(B)AAf(B) for Some Continuous Function f
26.2.3 A Self-Adjoint SB(H) and a Non-closable T with STTS but f(S)TTf(S) for Some Continuous Function f
26.2.4 An Unbounded Self-Adjoint Operator Commuting with the L2-Fourier Transform
26.2.5 A Closed Symmetric A and a Unitary U Such That AU*AU
26.2.6 A Self-Adjoint A and B Where B Is a Bounded Multiplication Operator and A Is a Differential Operator which Commute Strongly
26.2.7 Is the Product of Two ``Commuting'' Unbounded Self-Adjoint (Respectively, Normal) Operators Always Self-Adjoint (Respectively, Normal)?
26.2.8 Two Unbounded Self-Adjoint Operators A and B which Commute Pointwise on Some Common Core but A and B Do Not Commute Strongly: Nelson-Like Counterexample
26.2.9 A Densely Defined Closed T Such That T2=T*2 yet T2 Is Not Self-Adjoint
26.2.10 Is There a Normal T Such That (T2)*≠T*2? What About When T2 Is Normal?
26.2.11 BATBA=T Even if BB(H), A and T Are All Self-Adjoint
26.2.12 TAB T=AB Where T, A, and B Are Normal
26.2.13 Two Unbounded Self-Adjoint Operators A and B which Commute Strongly Where B Is a Multiplication Operator and A Is a Differential Operator
26.2.14 On the Failure of the Ôta–Schmüdgen Criterion of Strong Commutativity for Normal Operators
26.2.15 Anti-Commutativity and Exponentials
26.2.16 A Densely Defined Unbounded Operator Without a Cartesian Decomposition
26.2.17 Are There Unbounded Self-Adjoint Operators A and B with A+iB0?
26.2.18 Two Unbounded Self-Adjoint Operators A and B Such That A+iB≠A+iB
26.2.19 A Closed Operator T Such That D(T)=D(T*) but (T+T*)/2 and (T-T*)/2i Are Not Essentially Self-Adjoint
26.2.20 An Unbounded Normal T Such That Both (T+T*)/2 and (T-T*)/2i Are Unclosed
26.2.21 Closed Nilpotent Operators Having a Cartesian Decomposition Are Always Everywhere Defined and Bounded
26.2.22 An Unbounded ``Nilpotent'' Closed Operator Having Positive Real and Imaginary Parts
26.2.23 Two Commuting Unbounded Normal Operators Whose Sum Fails to Remain Normal
26.2.24 A Formally Normal Operator Without Any Normal Extension
Answers
27 The Fuglede–Putnam Theorems and Intertwining Relations
27.1 Basics
27.2 Questions
27.2.1 A Boundedly Invertible Positive Self-Adjoint Unbounded Operator A and an Unbounded Normal Operator N Such That AN*=NA but ANN*A
27.2.2 A Closed T and a Normal M Such That TMMT but TM*M*T and M*TTM*
27.2.3 A Normal BB(H) and a Densely Defined Closed A Such That BAAB Yet B*AAB*
27.2.4 A Self-Adjoint T and a Unitary B with BTTB* but B*TTB
27.2.5 A Closed Operator Which Does Not Commute with Any (Nontrivial) Everywhere Defined Bounded Operator
27.2.6 A Self-Adjoint and a Closed Operators Which Are Not Intertwined by Any Bounded Operator Apart from the Zero Operator
27.2.7 A Self-Adjoint Operator A and a Closed Symmetric Restriction of A Not Intertwined by Any Bounded Operator Except the Zero Operator
27.2.8 Two Densely Defined Closed Operators A and B Not Intertwined by Any Densely Defined Closed (Nonzero) Operator
27.2.9 On the Failure of a Generalization to Unbounded Operators of Some Similarity Result by M.R. Embry
27.2.10 Are There Two Normal Operators A and BB(H) Such That BA2AB with AB Is Normal?
27.2.11 On a Result About Commutativity by C. R. Putnam
Answers
28 Commutators
28.1 Basics
28.2 Questions
28.2.1 Two Operators A,B, One of Them Is Unbounded, with BA-ABI
28.2.2 On Some Theorem of F. E. Browder About Commutators of Unbounded Operators
28.2.3 Are There Two Closable Unbounded Operators A and B Such That AB-BA is Everywhere Defined?
28.2.4 Two Self-Adjoint Operators A and B Such That AB+BA Is Bounded, While AB-BA Is Unbounded
28.2.5 Two Self-Adjoint Positive Operators (One of Them Is Unbounded) A and B Such That AB-BA Is Unbounded
28.2.6 On the Positivity of Some Commutator
28.2.7 Two Unbounded and Self-Adjoint Operators A and B Such That D(A)=D(B) and A2-B2 Is Bounded but AB-BA Is Unbounded
28.2.8 Two Densely Defined Unbounded and Closed Operators C and B Such That CB-BC Is Bounded and Unclosed, While |C|B-B|C| Is Unbounded and Closed
28.2.9 Two Unbounded Injective Self-Adjoint Operators A and B Such That AB-BA Is Bounded, While |A|B-B|A| Is Unbounded
28.2.10 Two Unbounded (Injective) Self-Adjoint Operators A and B Such That AB-BA Is Bounded and Commutes with B (and A) Yet |A|B-B|A| Is Unbounded
28.2.11 Two Unbounded Self-Adjoint, Positive, and Boundedly Invertible Operators A and B Such That AB-BA Is Bounded While AB-BA Is Unbounded
28.2.12 A Densely Defined Closed Operator T Such That |T*||T|-|T||T*| Is Unbounded
28.2.13 A Densely Defined and Closed Operator T Such That TT*-T*T Is Unbounded and Self-Adjoint but |T||T*|-|T*||T| Is Bounded and Unclosed
28.2.14 A Densely Defined and Closed Operator T Such That TT*-T*T Is Bounded Whereas |T||T*|-|T*||T| Is Unbounded
28.2.15 Self-Adjoint Operators A and B Such That 12|"426830A [A,B]f,f"526930B |≤"026B30D Af"026B30D "026B30D Bf"026B30D
28.2.16 Unbounded Skew-Adjoint Operators Cannot Be Universally Commutable
Answers
29 Spectrum
29.1 Basics
29.2 Questions
29.2.1 A Densely Defined Closed Operator with an Empty Spectrum
29.2.2 A Densely Defined Operator A with σ(A)=C
29.2.3 A Densely Defined Closed Operator A with σ(A)=C
29.2.4 Two Unbounded Strongly Commuting Self-Adjoint Operators A and B Such That σ(A+B)σ(A)+σ(B)
29.2.5 Two Densely Closed Unbounded Operators A and B Such That σ(A)+σ(B) Is Not Closed
29.2.6 Two Commuting Self-Adjoint Unbounded Operators A and B Such That σ(BA)σ(B)σ(A)
29.2.7 A Densely Defined Operator A with σ(A2) ≠[σ(A)]2
29.2.8 A Normal Operator T with p[σ(T)]≠σ[p(T,T*)] for Some Polynomial p
29.2.9 Two Closed (Unbounded) Operators A and B Such That σ(AB)-{0}≠σ(BA)-{0}
29.2.10 A (Non-closed) Densely Defined Operator A Such That σ(AA*)-{0}≠σ(A*A)-{0}
29.2.11 Two Self-Adjoint Operators A and B with BB(H) Such That B Commutes with AB-BA yet σ(AB-BA)≠{0}
29.2.12 An Unbounded, Closed, and Nilpotent Operator N with σ(N)≠{0}
29.2.13 An AB(H) and an Unbounded Closed Nilpotent Operator N with AN=NA but σ(A+N)≠σ(A)
29.2.14 A Closed, Symmetric, and Positive Operator T with σ(T)=C
29.2.15 The (Only) Four Possible Cases for the Spectrum of Closed Symmetric Operators
29.2.16 Unbounded Densely Defined Closable Operators A and B Such That AB*, BA Is Essentially Self-Adjoint, and σ(BA)=σ(A B)≠σ(AB)
Answers
30 Matrices of Unbounded Operators
30.1 Questions
30.1.1 Equal Matrices and Pairwise Different Entries
30.1.2 A Bounded Matrix with All Unbounded Entries
30.1.3 The Failure of the Product Formula for Some Matrices of Operators
30.1.4 A Closed Matrix Yet All Entries Are Unclosed
30.1.5 A Matrix Whose Closure Is Not Equal to the Matrix of Closures
30.1.6 A Non-closed (Essentially Self-Adjoint) Matrix Whose Entries Are All Self-Adjoint
30.1.7 A Non-closable Matrix Yet All Entries Are Closed
30.1.8 A Densely Defined Matrix Whose Formal Adjoint Is Not Densely Defined
30.1.9 A Closed Matrix Whose Adjoint Does Not Admit a Matrix Representation
30.1.10 A Matrix Whose Adjoint Differs from Its Formal Adjoint
30.1.11 A Non-boundedly Invertible Matrix Yet All Its Entries Pairwise Commute and Its Formal Determinant Is Boundedly Invertible
30.1.12 A Self-Adjoint Matrix Yet None of Its Entries Is Even Closed
30.1.13 A Matrix of Operators A of Size nn, Where All of Its Entries Are Unclosable and Everywhere Defined, Each Ap, 1≤p≤n-1, Does Not Contain Any Zero Entry but An=0
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31 Relative Boundedness
31.1 Questions
31.1.1 A Function φL2(R2) Such That ∂2φ∂x∂yL2(R2) but φ L∞(R2)
31.1.2 A uL2(Rn+1) with (-i∂/∂t-x)uL2(Rn+1) yet uLqt(Lrx) for Given Values of q and r
31.1.3 A Function uL2(R2) with (∂/∂t+∂3/∂x3)uL2(R2) yet uLp(R2) for Any p>8
31.1.4 A Negative V in L2loc Such That -Δ+V Is Not Essentially Self-Adjoint on C0∞
31.1.5 A Positive V in L2loc Such That ∂2∂t2-∂2∂x2+V Is Not Essentially Self-Adjoint on C0∞
31.1.6 If B Is A-Bounded, Is B2 A2-Bounded? Is B* A*-Bounded?
31.1.7 The Kato–Rellich Theorem for Three Operators?
31.1.8 The Kato–Rellich Theorem for Normal Operators?
Answers
32 More Questions and Some Open Problems II
32.1 More Questions
32.2 Some Open Problems
Appendix A: A Quick Review of the Fourier Transform
Appendix B: A Word on Distributions and Sobolev Spaces
Bibliography
Index
