theorem 12.80 von Neumann's criterion

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2167 · p. 436

Rests on

depends_on definition 12.79 Deficiency subspaces and indices ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶
  - 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 12.18 Projection theorem ¶
    - depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
    - 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 ¶
  - proves proof ch:10-hilbert-spaces@proof-10 ¶
- 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 corollary 12.19 Double complement; the density criterion ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
      - 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 theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
- 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 ¶
depends_on definition 12.78 Essential self-adjointness ¶
- depends_on definition 12.72 Symmetric; self-adjoint ¶
  - depends_on definition 12.69 Operator with a domain ¶ ↺
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
- 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 ¶
depends_on definition 12.72 Symmetric; self-adjoint ¶ ↺
proves proof ch:10-hilbert-spaces@prooflink-4 ¶

Supports

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 ¶ ↺

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`	→	Deficiency subspaces and indices	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2179
`depends_on`	→	Essential self-adjointness	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2179
`depends_on`	→	Symmetric; self-adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2179
`depends_on`	←	Momentum on the half-line: no self-adjoint extension	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2291
`depends_on`	←	Momentum on a finite interval: a circle of self-adjoint momenta	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2225
`proves`	←	ch:10-hilbert-spaces@prooflink-4	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2193