proposition A.280 Direct-integral form of the spectral theorem

open in the book · appendices/A-long-proofs.tex:13945 · p. 2929

Rests on

depends_on lemma A.250 Decomposition into cyclic subspaces ¶
- depends_on definition A.248 Cyclic vector and cyclic subspace ¶
  - depends_on proposition A.243 Continuous functional calculus ¶
    - depends_on proposition A.241 The polynomial calculus is isometric ¶
      - depends_on lemma A.240 Spectral mapping for polynomials ¶
      - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
      - proves proof app:A-long-proofs@proof-147 ¶
    - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - proves proof ch:10-hilbert-spaces@proof-32 ¶
    - depends_on theorem A.242 Stone–Weierstrass; quoted ¶
      - depends_on definition 6.9 Compact set ¶
    - proves proof app:A-long-proofs@proof-148 ¶
  - depends_on theorem 12.18 Projection theorem ¶
    - 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 12.14 Closest point in a closed convex set ¶
      - depends_on definition 12.13 Convex set ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - proves proof ch:10-hilbert-spaces@proof-7 ¶
    - proves proof ch:10-hilbert-spaces@proof-9 ¶
- depends_on definition 12.32 Separable Hilbert space ¶
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
  - depends_on definition 12.86 Internal orthogonal decomposition ¶
    - depends_on definition 12.29 Orthonormal basis ¶
      - depends_on definition 12.16 Orthogonal complement ¶ ↺
      - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
    - depends_on definition 12.16 Orthogonal complement ¶ ↺
  - depends_on proposition 12.85 The direct sum is a Hilbert space ¶
    - depends_on definition 12.84 External direct sum ¶
      - depends_on definition 5.101 External direct sum ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
    - depends_on proposition 12.10 $\ell^{2}$ is complete ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-5 ¶
    - proves proof ch:10-hilbert-spaces@proof-41 ¶
  - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - 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 proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶ ↺
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-14 ¶
    - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶ ↺
      - depends_on definition 12.2 Hilbert space ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-15 ¶
    - proves proof ch:10-hilbert-spaces@proof-16 ¶
  - proves proof ch:10-hilbert-spaces@proof-42 ¶
- proves proof app:A-long-proofs@proof-155 ¶
depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
- depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
  - depends_on definition 12.72 Symmetric; self-adjoint ¶
    - depends_on definition 12.69 Operator with a domain ¶
      - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
      - depends_on definition 12.69 Operator with a domain ¶ ↺
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
  - depends_on proposition A.259 The generator is self-adjoint ¶
    - depends_on proposition A.258 The generator is closed ¶
      - depends_on equation A.470 eq:app-stone-domain ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - proves proof app:A-long-proofs@proof-161 ¶
    - depends_on proposition A.256 Density ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - proves proof app:A-long-proofs@proof-159 ¶
    - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on theorem 12.66 Stone ¶
      - proves proof ch:10-hilbert-spaces@proof-34 ¶
    - proves proof app:A-long-proofs@proof-162 ¶
  - depends_on proposition 12.73 The adjoint is always closed ¶
    - depends_on definition 12.70 Graph; closed and closable operators ¶
      - depends_on definition 12.69 Operator with a domain ¶ ↺
      - depends_on definition 6.3 Closed set ¶
    - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-35 ¶
  - proves proof app:A-long-proofs@proof-163 ¶
- depends_on proposition A.246 Bounded Borel functional calculus ¶
  - depends_on proposition A.243 Continuous functional calculus ¶ ↺
  - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
    - depends_on proposition A.243 Continuous functional calculus ¶ ↺
    - depends_on theorem A.244 Riesz–Markov; quoted ¶
      - depends_on definition 6.9 Compact set ¶ ↺
    - 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 app:A-long-proofs@proof-149 ¶
  - depends_on theorem 12.46 Riesz representation ¶ ↺
  - proves proof app:A-long-proofs@proof-150 ¶
- depends_on proposition A.261 Spectral theorem for a unitary operator ¶
  - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
  - 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 theorem A.238 Spectral theorem, both forms ¶
    - 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 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 definition 12.41 The operator classes ¶ ↺
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - proves proof ch:10-hilbert-spaces@proof-30 ¶
    - proves proof app:A-long-proofs@proof-151 ¶
    - proves proof app:A-long-proofs@proof-153 ¶
    - proves proof app:A-long-proofs@proof-156 ¶
  - proves proof app:A-long-proofs@proof-164 ¶
- proves proof app:A-long-proofs@proof-165 ¶
depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
proves proof app:A-long-proofs@proof-177 ¶

Supports

depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Decomposition into cyclic subspaces	declared	appendices/A-long-proofs.tex:13962
`depends_on`	→	Spectral theorem for an unbounded self-adjoint operator	declared	appendices/A-long-proofs.tex:13962
`depends_on`	→	Spectral theorem, both forms	declared	appendices/A-long-proofs.tex:13962
`depends_on`	←	The fibre maps are continuous on $\Phi$	declared	appendices/A-long-proofs.tex:14062
`proves`	←	app:A-long-proofs@proof-177	declared	appendices/A-long-proofs.tex:13966