lemma A.267 Isometry of $A\pm\ii\mu$, and closed range

open in the book · appendices/A-long-proofs.tex:13328 · p. 2922

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 12.72 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 theorem 12.18 Projection theorem ¶ ↺
proves proof app:A-long-proofs@proof-168 ¶

Supports

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 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 ¶ ↺

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`	→	Deficiency subspaces and indices	declared	appendices/A-long-proofs.tex:13342
`depends_on`	→	Symmetric; self-adjoint	declared	appendices/A-long-proofs.tex:13342
`depends_on`	→	Projection theorem	declared	appendices/A-long-proofs.tex:13342
`depends_on`	←	Cayley transform	declared	appendices/A-long-proofs.tex:13421
`depends_on`	←	The indices do not depend on $\mu$	declared	appendices/A-long-proofs.tex:13373
`depends_on`	←	Properties of the transform	declared	appendices/A-long-proofs.tex:13441
`depends_on`	←	Self-adjoint means unitary	declared	appendices/A-long-proofs.tex:13551
`proves`	←	app:A-long-proofs@proof-168	declared	appendices/A-long-proofs.tex:13346