theorem 12.54 The spectrum is compact and non-empty

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1414 · p. 428

Rests on

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 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶
  - depends_on definition 12.35 Bounded operator; operator 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 ¶
      - depends_on definition 6.24 Metric ¶
    - proves proof ch:10-hilbert-spaces@proof-4 ¶
  - proves proof ch:10-hilbert-spaces@proof-19 ¶
- 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 corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-24 ¶
    - 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 definition 6.17 Simply connected space ¶
      - depends_on theorem 8.12 Cauchy ¶
      - proves proof ch:06-complex-analysis@proof-9 ¶
    - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶ ↺
      - depends_on proposition 8.4 Euler's formula ¶
      - proves proof ch:06-complex-analysis@proof-6 ¶
    - 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 ¶

Supports

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 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 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 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 ¶

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`	→	Neumann series; the spectrum is bounded	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1418
`depends_on`	→	The resolvent set is open, the resolvent analytic	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1418
`depends_on`	→	Liouville	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1418
`depends_on`	←	The spectrum of a self-adjoint operator is real	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1451
`proves`	←	ch:10-hilbert-spaces@proof-29	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1422