definition 12.71 Adjoint of a densely defined operator

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1978 · p. 434

ground object -- no derivation owed

Rests on

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 definition 5.18 Inner product ¶
    - 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 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 5.7 Vector subspace ¶
    - depends_on definition 12.2 Hilbert space ¶ ↺
    - 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-10 ¶
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 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 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 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 ¶
    - 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 ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-24 ¶
- proves proof ch:10-hilbert-spaces@proof-20 ¶

Supports

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

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Double complement; the density criterion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1988
`depends_on`	→	Operator with a domain	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1988
`depends_on`	→	Existence and uniqueness of the adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1988
`depends_on`	←	Deficiency subspaces and indices	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2154
`depends_on`	←	Symmetric; self-adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1998
`depends_on`	←	The adjoint is always closed	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2006