lemma A.272 The operator attached to an isometry

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 lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
  • depends_on definition 12.72 Symmetric; self-adjoint ¶ ↺
  • 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 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 ¶
• depends_on proposition A.270 Properties of the transform ¶ ↺
• proves proof app:A-long-proofs@proof-172 ¶

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