theorem 12.91 Schur's lemma, commutant form

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2483 · p. 440

Rests on

depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
- 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 definition 12.88 Reducing subspace ¶
  - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
  - depends_on definition 12.20 Orthogonal projection operator ¶ ↺
depends_on proposition 12.89 Reduction is commutation ¶
- depends_on definition 12.88 Reducing subspace ¶ ↺
- depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-43 ¶
depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
- depends_on definition 12.58 Projection-valued measure ¶
  - depends_on definition 12.41 The operator classes ¶ ↺
  - 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 12.16 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 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 theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - 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 ¶
proves proof ch:10-hilbert-spaces@proof-44 ¶

Supports

depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
depends_on theorem 12.114 Stone–von Neumann ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Self-adjoint family; commutant; irreducibility	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2490
`depends_on`	→	Reduction is commutation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2490
`depends_on`	→	Spectral theorem for a bounded self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2490
`depends_on`	←	Any two irreducible Weyl systems are equivalent	declared	appendices/A-long-proofs.tex:28332
`depends_on`	←	Stone–von Neumann	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3196
`proves`	←	ch:10-hilbert-spaces@proof-44	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2494