theorem 12.59 Spectral theorem for a bounded self-adjoint operator

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1598 · p. 430

Rests on

depends_on definition 12.58 Projection-valued measure ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - 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 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 5.37 Linear transformation ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
      - 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 theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
- depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
depends_on proposition 12.43 Norm of a self-adjoint operator ¶
- depends_on proposition 12.42 Elementary consequences ¶
  - depends_on definition 12.41 The operator classes ¶ ↺
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-21 ¶
  - proves proof ch:10-hilbert-spaces@proof-22 ¶
- depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-23 ¶
depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶
  - 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 5.24 Orthogonal vectors ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - proves proof ch:10-hilbert-spaces@proof-2 ¶
    - depends_on definition 12.16 Orthogonal complement ¶
      - depends_on definition 5.24 Orthogonal vectors ¶ ↺
      - depends_on equation 5.52 eq:lin-orthogonal-complement ¶
    - proves proof ch:10-hilbert-spaces@proof-8 ¶
  - depends_on theorem 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-10 ¶
- depends_on definition 12.41 The operator classes ¶ ↺
- depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
    - depends_on definition 12.49 Resolvent set; spectrum ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.47 Inverse of a linear transformation ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-27 ¶
  - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
    - depends_on definition 8.5 Complex derivative ¶
    - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-28 ¶
  - depends_on theorem 8.18 Liouville ¶
    - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:06-complex-analysis@proof-5 ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on lemma 8.9 ML estimate ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - proves proof ch:06-complex-analysis@proof-11 ¶
    - proves proof ch:06-complex-analysis@proof-12 ¶
  - proves proof ch:10-hilbert-spaces@proof-29 ¶
- proves proof ch:10-hilbert-spaces@proof-30 ¶
proves proof ch:10-hilbert-spaces@prooflink-2 ¶

Supports

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 lemma A.249 The cyclic case ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    - 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.261 Spectral theorem for a unitary 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 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 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 theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
  - 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.259 The generator is self-adjoint ¶
  - 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 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 definition 25.42 Ordering rule ¶
    - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
    - depends_on remark 24.55 What polarization costs ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Projection-valued measure	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1620
`depends_on`	→	Norm of a self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1620
`depends_on`	→	The spectrum of a self-adjoint operator is real	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1620
`depends_on`	←	Functional calculus	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1658
`depends_on`	←	Nuclear spectral theorem	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2901
`depends_on`	←	Schur's lemma, commutant form	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2490
`depends_on`	←	Stone	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1825
`proves`	←	ch:10-hilbert-spaces@prooflink-2	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1642