lemma A.260 Cayley transform of a self-adjoint operator

open in the book · appendices/A-long-proofs.tex:12914 · p. 2918

Rests on

depends_on definition 12.72 Symmetric; self-adjoint ¶
- depends_on definition 12.69 Operator with a domain ¶
  - 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 12.16 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 ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
    - proves proof ch:10-hilbert-spaces@proof-10 ¶
  - 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 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 A.259 The generator is self-adjoint ¶
- depends_on proposition A.258 The generator is closed ¶
  - depends_on equation A.470 eq:app-stone-domain ¶
  - depends_on lemma A.257 Integrated form of the equation of motion ¶
    - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - proves proof app:A-long-proofs@proof-157 ¶
    - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on theorem 12.66 Stone ¶
      - proves proof ch:10-hilbert-spaces@proof-34 ¶
    - proves proof app:A-long-proofs@proof-160 ¶
  - proves proof app:A-long-proofs@proof-161 ¶
- depends_on proposition A.256 Density ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
  - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    - depends_on equation A.470 eq:app-stone-domain ¶ ↺
    - depends_on lemma A.254 Riemann integral of a continuous curve ¶ ↺
    - proves proof app:A-long-proofs@proof-158 ¶
  - proves proof app:A-long-proofs@proof-159 ¶
- depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
- proves proof app:A-long-proofs@proof-162 ¶
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 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-35 ¶
proves proof app:A-long-proofs@proof-163 ¶

Supports

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 $\Phi$ ¶

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`	→	Symmetric; self-adjoint	declared	appendices/A-long-proofs.tex:12927
`depends_on`	→	The generator is self-adjoint	declared	appendices/A-long-proofs.tex:12927
`depends_on`	→	The adjoint is always closed	declared	appendices/A-long-proofs.tex:12927
`depends_on`	←	Spectral theorem for an unbounded self-adjoint operator	declared	appendices/A-long-proofs.tex:13036
`proves`	←	app:A-long-proofs@proof-163	declared	appendices/A-long-proofs.tex:12931