theorem 12.38 Existence and uniqueness of the adjoint

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:950 · p. 423

Rests on

depends_on definition 5.41 Adjoint ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
- depends_on definition 12.35 Bounded operator; operator norm ¶
  - depends_on definition 5.37 Linear transformation ¶ ↺
  - depends_on definition 5.19 Norm ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on proposition 12.8 Absolutely convergent series test ¶
  - depends_on definition 5.19 Norm ¶ ↺
  - 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 ¶
  - proves proof ch:10-hilbert-spaces@proof-4 ¶
- 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 definition 12.35 Bounded operator; operator norm ¶ ↺
  - depends_on definition 5.46 Functional ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
- 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 5.24 Orthogonal vectors ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - proves proof ch:10-hilbert-spaces@proof-2 ¶
    - depends_on definition 12.16 Orthogonal complement ¶
      - depends_on definition 5.24 Orthogonal vectors ¶ ↺
      - depends_on equation 5.52 eq:lin-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 5.7 Vector subspace ¶
    - 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 12.6 Parallelogram law and polarization ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on equation 5.44 eq:lin-norm-assoc ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-3 ¶
    - proves proof ch:10-hilbert-spaces@proof-7 ¶
  - proves proof ch:10-hilbert-spaces@proof-9 ¶
- proves proof ch:10-hilbert-spaces@proof-24 ¶
proves proof ch:10-hilbert-spaces@proof-20 ¶

Supports

depends_on definition 12.41 The operator classes ¶
- depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
  - 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.591 Any two irreducible Weyl systems are equivalent ¶ ↺
  - depends_on theorem A.579 Stone–von Neumann ¶
  - depends_on theorem 12.91 Schur's lemma, commutant form ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
    - depends_on theorem 12.114 Stone–von Neumann ¶
  - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
- depends_on definition 12.58 Projection-valued measure ¶
  - depends_on definition 12.60 Functional calculus ¶
    - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on lemma A.247 Integration against a projection-valued measure ¶
  - depends_on theorem A.238 Spectral theorem, both forms ¶
    - 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.261 Spectral theorem for a unitary operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on example A.283 Momentum on the line ¶
    - depends_on theorem A.253 Stone ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on definition 12.60 Functional calculus ¶ ↺
    - depends_on theorem 12.107 Nuclear spectral theorem ¶
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
    - depends_on theorem 12.66 Stone ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
  - depends_on definition A.580 Weyl operator ¶
    - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
  - depends_on definition 12.109 Weyl system ¶ ↺
  - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
    - depends_on proposition A.256 Density ¶
      - depends_on proposition A.259 The generator is self-adjoint ¶
  - depends_on lemma A.581 Composition law ¶ ↺
  - depends_on lemma A.582 Joint strong continuity ¶ ↺
  - depends_on proposition A.256 Density ¶ ↺
  - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
    - depends_on theorem 12.66 Stone ¶ ↺
  - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
  - depends_on theorem A.253 Stone ¶ ↺
  - depends_on theorem 12.66 Stone ¶ ↺
  - depends_on theorem 25.34 Stone–von Neumann ¶ ↺
- depends_on lemma A.231 Restriction to an invariant closed subspace ¶
  - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
- depends_on lemma A.230 Sequential characterisation ¶
  - depends_on lemma A.232 Attainment ¶
    - depends_on lemma A.233 Construction of the system ¶ ↺
  - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
  - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶ ↺
- depends_on proposition 12.42 Elementary consequences ¶
  - depends_on lemma A.232 Attainment ¶ ↺
  - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    - depends_on lemma A.232 Attainment ¶ ↺
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on proposition A.241 The polynomial calculus is isometric ¶
    - depends_on theorem A.229 Hilbert–Schmidt ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
    - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
- depends_on theorem A.229 Hilbert–Schmidt ¶ ↺
- depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶ ↺
- depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
  - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
  - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
depends_on definition 12.71 Adjoint of a densely defined operator ¶
- depends_on definition 12.79 Deficiency subspaces and indices ¶
  - depends_on definition A.269 Cayley transform ¶
    - depends_on proposition A.270 Properties of the transform ¶
      - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
      - depends_on lemma A.272 The operator attached to an isometry ¶
      - depends_on proposition A.273 Self-adjoint means unitary ¶
  - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
    - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
  - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
    - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
    - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
  - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
    - depends_on definition A.269 Cayley transform ¶ ↺
    - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
    - depends_on proposition A.270 Properties of the transform ¶ ↺
    - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
  - depends_on lemma A.268 The indices do not depend on $\mu$ ¶ ↺
  - depends_on theorem A.266 von Neumann ¶
    - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
    - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
  - depends_on theorem 12.80 von Neumann's criterion ¶
    - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
    - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶ ↺
- depends_on definition 12.72 Symmetric; self-adjoint ¶
  - depends_on definition 12.78 Essential self-adjointness ¶
    - depends_on theorem A.266 von Neumann ¶ ↺
    - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
  - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
    - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺
  - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
  - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶ ↺
  - depends_on lemma A.272 The operator attached to an isometry ¶ ↺
  - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
  - depends_on theorem A.266 von Neumann ¶ ↺
  - depends_on theorem 12.74 Hellinger–Toeplitz ¶
    - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
  - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
- depends_on proposition 12.73 The adjoint is always closed ¶
  - depends_on definition 12.78 Essential self-adjointness ¶ ↺
  - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶ ↺
depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
- depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
- depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
- depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
- depends_on proposition 12.42 Elementary consequences ¶ ↺
- depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
- depends_on proposition 12.96 Operators on a tensor product ¶

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`	→	Adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:959
`depends_on`	→	$\mathcal{B}(\mathcal{H})$ is a Banach algebra	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:959
`depends_on`	→	Riesz representation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:959
`depends_on`	←	The operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1073
`depends_on`	←	Adjoint of a densely defined operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1988
`depends_on`	←	Algebra of the adjoint; the $C^{\ast}$ identity	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1013
`proves`	←	ch:10-hilbert-spaces@proof-20	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:962