proposition 12.61 Uniqueness of the continuous functional calculus

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1661 · p. 431

Rests on

depends_on definition 12.60 Functional calculus ¶
- 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 proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on definition 12.20 Orthogonal projection operator ¶
      - 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 proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on theorem 12.46 Riesz representation ¶
      - proves proof ch:10-hilbert-spaces@proof-20 ¶
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
- depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
  - depends_on definition 12.58 Projection-valued measure ¶ ↺
  - 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 ¶
      - 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 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 proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      - depends_on theorem 8.18 Liouville ¶
      - proves proof ch:10-hilbert-spaces@proof-29 ¶
    - proves proof ch:10-hilbert-spaces@proof-30 ¶
  - proves proof ch:10-hilbert-spaces@prooflink-2 ¶
depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
proves proof ch:10-hilbert-spaces@proof-32 ¶

Supports

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.282 The fibre maps are continuous on $\Phi$ ¶
- 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 ¶ ↺

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`	→	Functional calculus	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1670
`depends_on`	→	$\mathcal{B}(\mathcal{H})$ is a Banach algebra	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1670
`depends_on`	←	Continuous functional calculus	declared	appendices/A-long-proofs.tex:12117
`proves`	←	ch:10-hilbert-spaces@proof-32	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1673