definition 12.35 Bounded operator; operator norm

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ground object -- no derivation owed

Rests on

depends_on definition 5.37 Linear transformation ¶
- 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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.28 Distributivity ¶ ↺
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.30 Ring ¶ ↺
depends_on definition 5.19 Norm ¶
- depends_on definition 4.33 Vector space ¶ ↺

Supports

depends_on definition 12.45 Continuous linear functional; the dual ¶
- depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
  - depends_on definition 12.103 Gelfand triple ¶
    - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
    - depends_on definition 12.105 Generalized eigenvector ¶
      - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
    - depends_on example 12.104 The Schwartz triple ¶
      - depends_on example A.283 Momentum on the line ¶
      - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶ ↺
    - depends_on theorem A.279 Gelfand–Maurin ¶ ↺
    - depends_on theorem 12.107 Nuclear spectral theorem ¶ ↺
- depends_on definition 12.103 Gelfand triple ¶ ↺
- depends_on theorem 12.46 Riesz representation ¶
  - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶ ↺
  - depends_on lemma A.247 Integration against a projection-valued measure ¶
  - 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 ¶ ↺
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 12.41 The operator classes ¶
      - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
      - depends_on definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on proposition 12.42 Elementary consequences ¶
      - 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 definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on definition 12.79 Deficiency subspaces and indices ¶
      - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on proposition 12.73 The adjoint is always closed ¶
    - 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 ¶
depends_on definition 12.88 Reducing subspace ¶
- depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶ ↺
- depends_on proposition 12.89 Reduction is commutation ¶
  - 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 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 definition 12.49 Resolvent set; spectrum ¶
- depends_on definition 12.50 Point, continuous and residual spectrum ¶
  - depends_on definition 12.105 Generalized eigenvector ¶ ↺
  - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
  - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
- depends_on lemma A.240 Spectral mapping for polynomials ¶
  - depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
- 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 proposition 12.52 Neumann series; the spectrum is bounded ¶
  - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶ ↺
  - depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
  - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶ ↺
  - depends_on theorem 12.54 The spectrum is compact and non-empty ¶ ↺
depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
- depends_on lemma A.240 Spectral mapping for polynomials ¶ ↺
- depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
- depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
  - 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.246 Bounded Borel functional calculus ¶ ↺
    - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶ ↺
- depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
- depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
- depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
- depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
  - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
  - depends_on corollary 25.30 The relation cannot be realized by matrices ¶
  - depends_on definition 12.109 Weyl system ¶
    - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
    - depends_on definition A.580 Weyl operator ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on lemma A.581 Composition law ¶
    - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
    - depends_on theorem A.579 Stone–von Neumann ¶ ↺
    - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
depends_on proposition 12.36 Boundedness is continuity ¶
- 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 12.75 The canonical commutation relation admits no bounded solution ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:894
`depends_on`	→	Norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:894
`depends_on`	←	Continuous linear functional; the dual	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1208
`depends_on`	←	Reducing subspace	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2441
`depends_on`	←	Resolvent set; spectrum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1309
`depends_on`	←	$\mathcal{B}(\mathcal{H})$ is a Banach algebra	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:926
`depends_on`	←	Boundedness is continuity	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:902
`depends_on`	←	The canonical commutation relation admits no bounded solution	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2055