proposition A.262 Spectral theorem for an unbounded self-adjoint operator

open in the book · appendices/A-long-proofs.tex:13022 · p. 2919

Rests on

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 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 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - 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 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 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 ¶
      - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - proves proof app:A-long-proofs@proof-160 ¶
    - 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 definition 12.41 The operator classes ¶
      - depends_on definition 6.6 Continuous map ¶
    - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on equation A.470 eq:app-stone-domain ¶ ↺
      - depends_on lemma A.254 Riemann integral of a continuous curve ¶ ↺
      - proves proof app:A-long-proofs@proof-158 ¶
    - proves proof app:A-long-proofs@proof-159 ¶
  - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
  - 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 3.35 Union, intersection, difference ¶
      - depends_on definition 6.2 Open 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.241 The polynomial calculus is isometric ¶
    - depends_on lemma A.240 Spectral mapping for polynomials ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      - proves proof app:A-long-proofs@proof-146 ¶
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
      - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - proves proof app:A-long-proofs@proof-145 ¶
    - 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 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 definition 12.58 Projection-valued measure ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - 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 ¶
      - depends_on definition 6.5 Open cover ¶
  - proves proof app:A-long-proofs@proof-148 ¶
- 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 ¶ ↺
  - 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 theorem A.238 Spectral theorem, both forms ¶
  - depends_on definition 12.58 Projection-valued measure ¶ ↺
  - depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
  - 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 ¶
      - 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 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 ¶

Supports

depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
- depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Cayley transform of a self-adjoint operator	declared	appendices/A-long-proofs.tex:13036
`depends_on`	→	Bounded Borel functional calculus	declared	appendices/A-long-proofs.tex:13036
`depends_on`	→	Spectral theorem for a unitary operator	declared	appendices/A-long-proofs.tex:13036
`depends_on`	←	Direct-integral form of the spectral theorem	declared	appendices/A-long-proofs.tex:13962
`proves`	←	app:A-long-proofs@proof-165	declared	appendices/A-long-proofs.tex:13040