definition 12.41 The operator classes

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1059 · p. 424

ground object -- no derivation owed

Rests on

depends_on definition 6.9 Compact set ¶
- depends_on definition 6.5 Open cover ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on equation 3.51 eq:set-indexed ¶ ↺
depends_on proposition 12.21 Characterization of orthogonal projections ¶
- depends_on definition 12.20 Orthogonal projection operator ¶
  - depends_on theorem 12.18 Projection theorem ¶
    - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on definition 12.16 Orthogonal complement ¶
      - proves proof ch:10-hilbert-spaces@proof-8 ¶
    - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - depends_on definition 12.13 Convex set ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - proves proof ch:10-hilbert-spaces@proof-7 ¶
    - proves proof ch:10-hilbert-spaces@proof-9 ¶
- depends_on theorem 12.18 Projection theorem ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-11 ¶
depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
- depends_on definition 5.41 Adjoint ¶
  - depends_on definition 5.18 Inner product ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
  - depends_on definition 12.35 Bounded operator; operator norm ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on proposition 12.8 Absolutely convergent series test ¶
    - depends_on definition 5.19 Norm ¶ ↺
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - proves proof ch:10-hilbert-spaces@proof-4 ¶
  - proves proof ch:10-hilbert-spaces@proof-19 ¶
- depends_on theorem 12.46 Riesz representation ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
    - depends_on definition 5.46 Functional ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
  - depends_on theorem 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-24 ¶
- proves proof ch:10-hilbert-spaces@proof-20 ¶

Supports

depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
- depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
  - depends_on lemma A.590 The Gram matrix is universal ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
  - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
- depends_on theorem A.579 Stone–von Neumann ¶
- depends_on theorem 12.91 Schur's lemma, commutant form ¶
  - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
  - depends_on theorem 12.114 Stone–von Neumann ¶
- depends_on theorem 12.114 Stone–von Neumann ¶ ↺
depends_on definition 12.58 Projection-valued measure ¶
- depends_on definition 12.60 Functional calculus ¶
  - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on proposition A.243 Continuous functional calculus ¶
      - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
- depends_on lemma A.247 Integration against a projection-valued measure ¶
- depends_on theorem A.238 Spectral theorem, both forms ¶
  - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
  - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
    - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶ ↺
  - depends_on theorem A.279 Gelfand–Maurin ¶
    - depends_on example A.283 Momentum on the line ¶
  - depends_on theorem A.253 Stone ¶
    - depends_on proposition A.263 The two constructions are inverse ¶
- depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
  - depends_on definition 12.60 Functional calculus ¶ ↺
  - depends_on theorem 12.107 Nuclear spectral theorem ¶
  - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
  - depends_on theorem 12.66 Stone ¶
    - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
    - depends_on definition 12.109 Weyl system ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
      - depends_on definition A.580 Weyl operator ¶
      - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem A.579 Stone–von Neumann ¶ ↺
      - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
    - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.263 The two constructions are inverse ¶ ↺
      - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
      - depends_on theorem A.253 Stone ¶ ↺
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
    - depends_on theorem 25.34 Stone–von Neumann ¶
      - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on theorem 25.38 Groenewold–van Hove ¶
depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
- depends_on definition A.580 Weyl operator ¶ ↺
- depends_on definition 12.109 Weyl system ¶ ↺
- depends_on lemma A.255 Smoothed vectors lie in the domain ¶
  - depends_on proposition A.256 Density ¶
    - depends_on proposition A.259 The generator is self-adjoint ¶ ↺
- depends_on lemma A.581 Composition law ¶
  - depends_on lemma A.582 Joint strong continuity ¶
    - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
  - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
- depends_on lemma A.582 Joint strong continuity ¶ ↺
- depends_on proposition A.256 Density ¶ ↺
- depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
  - depends_on theorem 12.66 Stone ¶ ↺
- depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
- depends_on theorem A.253 Stone ¶ ↺
- depends_on theorem 12.66 Stone ¶ ↺
- depends_on theorem 25.34 Stone–von Neumann ¶ ↺
depends_on lemma A.231 Restriction to an invariant closed subspace ¶
- depends_on lemma A.233 Construction of the system ¶
  - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
depends_on lemma A.230 Sequential characterisation ¶
- depends_on lemma A.232 Attainment ¶
  - depends_on lemma A.233 Construction of the system ¶ ↺
- depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
- depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶ ↺
depends_on proposition 12.42 Elementary consequences ¶
- depends_on lemma A.232 Attainment ¶ ↺
- depends_on proposition 12.43 Norm of a self-adjoint operator ¶
  - depends_on lemma A.232 Attainment ¶ ↺
  - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
    - depends_on proposition A.241 The polynomial calculus is isometric ¶
      - depends_on proposition A.243 Continuous functional calculus ¶ ↺
  - depends_on theorem A.229 Hilbert–Schmidt ¶
  - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
  - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
    - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
    - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
      - depends_on lemma A.472 The pairing identity ¶
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
depends_on theorem A.229 Hilbert–Schmidt ¶ ↺
depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶ ↺
depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
- depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
- depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
- depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Compact set	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1073
`depends_on`	→	Characterization of orthogonal projections	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1073
`depends_on`	→	Existence and uniqueness of the adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1073
`depends_on`	←	Self-adjoint family; commutant; irreducibility	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2480
`depends_on`	←	Projection-valued measure	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1588
`depends_on`	←	Strongly continuous one-parameter unitary group	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1749
`depends_on`	←	Restriction to an invariant closed subspace	declared	appendices/A-long-proofs.tex:11601
`depends_on`	←	Sequential characterisation	declared	appendices/A-long-proofs.tex:11573
`depends_on`	←	Elementary consequences	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1086
`depends_on`	←	Hilbert–Schmidt	declared	appendices/A-long-proofs.tex:11554
`depends_on`	←	Hilbert–Schmidt: compact self-adjoint operators	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1163
`depends_on`	←	The spectrum of a self-adjoint operator is real	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1451