theorem 12.55 The spectrum of a self-adjoint operator is real

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1446 · p. 428

Rests on

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 definition 5.18 Inner product ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - 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 ¶
  - depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
  - depends_on theorem 12.14 Closest point in a closed convex set ¶
    - depends_on definition 12.13 Convex set ¶
      - depends_on definition 5.7 Vector subspace ¶
    - depends_on definition 12.2 Hilbert space ¶ ↺
    - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
      - proves proof ch:10-hilbert-spaces@proof-3 ¶
    - proves proof ch:10-hilbert-spaces@proof-7 ¶
  - proves proof ch:10-hilbert-spaces@proof-9 ¶
- proves proof ch:10-hilbert-spaces@proof-10 ¶
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 definition 6.1 Topological space ¶
    - 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 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 proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - 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 theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - proves proof ch:10-hilbert-spaces@proof-20 ¶
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 definition 5.37 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 ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 equation 8.6 eq:cpx-contour-integral ¶ ↺
      - proves proof ch:06-complex-analysis@proof-4 ¶
    - depends_on theorem 8.16 Cauchy integral formula ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on lemma 8.9 ML estimate ¶ ↺
      - proves proof ch:06-complex-analysis@proof-10 ¶
    - 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 ¶

Supports

depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
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 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 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 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 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 ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Double complement; the density criterion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1451
`depends_on`	→	The operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1451
`depends_on`	→	The spectrum is compact and non-empty	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1451
`depends_on`	←	Multiplication by the coordinate: spectrum without eigenvectors	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1504
`depends_on`	←	Spectral theorem, both forms	declared	appendices/A-long-proofs.tex:11966
`depends_on`	←	Spectral theorem for a bounded self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1620
`proves`	←	ch:10-hilbert-spaces@proof-30	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1455