lemma A.271 Injectivity of $\identity-V$ for any isometric extension

open in the book · appendices/A-long-proofs.tex:13469 · p. 2924

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.270 Properties of the transform ¶
- depends_on definition A.269 Cayley transform ¶
  - depends_on definition 12.79 Deficiency subspaces and indices ¶
    - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
    - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
    - 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 lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
    - depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
    - depends_on definition 12.72 Symmetric; self-adjoint ¶ ↺
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof app:A-long-proofs@proof-168 ¶
- depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
- proves proof app:A-long-proofs@proof-170 ¶
proves proof app:A-long-proofs@proof-171 ¶

Supports

depends_on lemma A.272 The operator attached to an isometry ¶

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:13473
`depends_on`	→	Properties of the transform	declared	appendices/A-long-proofs.tex:13473
`depends_on`	←	The operator attached to an isometry	declared	appendices/A-long-proofs.tex:13503
`proves`	←	app:A-long-proofs@proof-171	declared	appendices/A-long-proofs.tex:13476