theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators

open in the book · appendices/A-long-proofs.tex:28073 · p. 3073

Rests on

depends_on definition A.585 The Gaussian average of the Weyl operators ¶
- depends_on definition A.580 Weyl operator ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
    - depends_on definition 12.41 The operator classes ¶
      - depends_on definition 6.9 Compact set ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
  - depends_on definition 12.109 Weyl system ¶
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - proves proof ch:10-hilbert-spaces@proof-37 ¶
    - depends_on theorem 12.66 Stone ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - proves proof ch:10-hilbert-spaces@prooflink-3 ¶
  - depends_on equation 12.71 eq:hilbert-weyl-relation ¶
- depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
  - depends_on lemma A.254 Riemann integral of a continuous curve ¶
    - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶ ↺
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - proves proof app:A-long-proofs@proof-157 ¶
  - depends_on lemma A.582 Joint strong continuity ¶
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    - depends_on lemma A.581 Composition law ¶
      - depends_on definition A.580 Weyl operator ¶ ↺
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on equation 12.71 eq:hilbert-weyl-relation ¶ ↺
      - proves proof app:A-long-proofs@proof-347 ¶
    - proves proof app:A-long-proofs@proof-348 ¶
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
  - proves proof app:A-long-proofs@proof-349 ¶
depends_on lemma A.584 Fourier injectivity in $n$ variables ¶
- depends_on definition 17.29 Fourier transform; the treatise convention ¶
- depends_on equation 17.42 eq:ft-gaussian ¶
- depends_on theorem 17.32 Fourier inversion ¶
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
  - depends_on equation 17.42 eq:ft-gaussian ¶ ↺
  - proves proof ch:15-fourier-integral-transforms@proof-20 ¶
- proves proof app:A-long-proofs@proof-350 ¶
depends_on lemma A.586 Gaussian integral with a complex linear term ¶
- depends_on equation 7.151 eq:ana-iterated ¶
- depends_on equation 17.42 eq:ft-gaussian ¶ ↺
- depends_on equation 17.35 eq:ft-translation ¶
- proves proof app:A-long-proofs@proof-351 ¶
proves proof app:A-long-proofs@proof-352 ¶

Supports

depends_on lemma A.590 The Gram matrix is universal ¶
- depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
- depends_on lemma A.590 The Gram matrix is universal ¶ ↺
- depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
depends_on proposition A.592 The vacuum of the Schrödinger system ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The Gaussian average of the Weyl operators	declared	appendices/A-long-proofs.tex:28082
`depends_on`	→	Fourier injectivity in $n$ variables	declared	appendices/A-long-proofs.tex:28082
`depends_on`	→	Gaussian integral with a complex linear term	declared	appendices/A-long-proofs.tex:28082
`depends_on`	←	The Gram matrix is universal	declared	appendices/A-long-proofs.tex:28296
`depends_on`	←	Cyclic subspaces and the rank of the average	declared	appendices/A-long-proofs.tex:28237
`depends_on`	←	The vacuum of the Schrödinger system	declared	appendices/A-long-proofs.tex:28407
`proves`	←	app:A-long-proofs@proof-352	declared	appendices/A-long-proofs.tex:28086