proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:999 · p. 424

Rests on

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 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 5.19 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 ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - proves proof ch:10-hilbert-spaces@proof-4 ¶
- proves proof ch:10-hilbert-spaces@proof-19 ¶
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 5.37 Linear transformation ¶ ↺
- depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
- 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 ¶
    - 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 ¶
  - proves proof ch:10-hilbert-spaces@proof-24 ¶
- proves proof ch:10-hilbert-spaces@proof-20 ¶
proves proof ch:10-hilbert-spaces@proof-21 ¶

Supports

depends_on definition 12.79 Deficiency subspaces and indices ¶
- depends_on definition A.269 Cayley transform ¶
  - depends_on proposition A.270 Properties of the transform ¶
    - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
      - depends_on lemma A.272 The operator attached to an isometry ¶
    - depends_on lemma A.272 The operator attached to an isometry ¶ ↺
    - depends_on proposition A.273 Self-adjoint means unitary ¶
- depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
  - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
- depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
  - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
  - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
- depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
  - depends_on definition A.269 Cayley transform ¶ ↺
  - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
  - depends_on proposition A.270 Properties of the transform ¶ ↺
  - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
- depends_on lemma A.268 The indices do not depend on $\mu$ ¶ ↺
- depends_on theorem A.266 von Neumann ¶
  - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
  - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
- depends_on theorem 12.80 von Neumann's criterion ¶
  - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
  - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶ ↺
depends_on proposition A.241 The polynomial calculus is isometric ¶
- depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
    - depends_on lemma A.249 The cyclic case ¶
    - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
  - depends_on proposition A.246 Bounded Borel functional calculus ¶
    - 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 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.245 The measures $\mu_{x,y}$ ¶
    - depends_on lemma A.249 The cyclic case ¶ ↺
    - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
- 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 definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on lemma A.581 Composition law ¶
    - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
    - 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.258 The generator is closed ¶
    - depends_on proposition A.263 The two constructions are inverse ¶
    - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.263 The two constructions are inverse ¶ ↺
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
    - depends_on theorem A.253 Stone ¶
      - depends_on proposition A.263 The two constructions are inverse ¶ ↺
  - 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 example 25.45 Weyl ordering collapses ¶
    - depends_on theorem 25.38 Groenewold–van Hove ¶
      - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
      - depends_on remark 24.55 What polarization costs ¶
depends_on proposition 12.42 Elementary consequences ¶
- depends_on lemma A.232 Attainment ¶
  - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
- 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 theorem A.229 Hilbert–Schmidt ¶
  - 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.261 Spectral theorem for a unitary operator ¶ ↺
    - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on example A.283 Momentum on the line ¶
    - depends_on theorem A.253 Stone ¶ ↺
  - 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 definition 12.60 Functional calculus ¶
      - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on theorem 12.107 Nuclear spectral theorem ¶
    - 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 12.66 Stone ¶ ↺
depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
depends_on proposition 12.96 Operators on a tensor product ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	$\mathcal{B}(\mathcal{H})$ is a Banach algebra	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1013
`depends_on`	→	Existence and uniqueness of the adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1013
`depends_on`	←	Deficiency subspaces and indices	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2154
`depends_on`	←	The polynomial calculus is isometric	declared	appendices/A-long-proofs.tex:12068
`depends_on`	←	Exponential of a bounded self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1775
`depends_on`	←	Elementary consequences	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1086
`depends_on`	←	Norm of a self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1118
`depends_on`	←	Operators on a tensor product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2653
`proves`	←	ch:10-hilbert-spaces@proof-21	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1016