definition 12.60 Functional calculus

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1650 · p. 431

ground object -- no derivation owed

Rests on

depends_on definition 12.58 Projection-valued measure ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-11 ¶
  - 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 12.21 Characterization of orthogonal projections ¶ ↺
depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
- depends_on definition 12.58 Projection-valued measure ¶ ↺
- depends_on proposition 12.43 Norm of a self-adjoint operator ¶
  - depends_on proposition 12.42 Elementary consequences ¶
    - depends_on definition 12.41 The operator classes ¶ ↺
    - 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 ¶
    - proves proof ch:10-hilbert-spaces@proof-22 ¶
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-23 ¶
- depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
  - 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 ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-10 ¶
  - depends_on definition 12.41 The operator classes ¶ ↺
  - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
    - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-27 ¶
    - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      - depends_on definition 8.5 Complex derivative ¶
      - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-28 ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - proves proof ch:06-complex-analysis@proof-12 ¶
    - proves proof ch:10-hilbert-spaces@proof-29 ¶
  - proves proof ch:10-hilbert-spaces@proof-30 ¶
- proves proof ch:10-hilbert-spaces@prooflink-2 ¶

Supports

depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
- depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
    - depends_on lemma A.249 The cyclic case ¶
    - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
  - depends_on proposition A.246 Bounded Borel functional calculus ¶
    - 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.261 Spectral theorem for a unitary operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺
  - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
    - depends_on lemma A.249 The cyclic case ¶ ↺
    - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺

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`	→	Projection-valued measure	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1658
`depends_on`	→	Spectral theorem for a bounded self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1658
`depends_on`	←	Uniqueness of the continuous functional calculus	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1670