definition 12.49 Resolvent set; spectrum

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1300 · p. 427

ground object -- no derivation owed

Rests on

depends_on definition 12.35 Bounded operator; operator norm ¶
- 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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 5.19 Norm ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on definition 5.47 Inverse of a linear transformation ¶
- depends_on definition 5.37 Linear transformation ¶ ↺

Supports

depends_on definition 12.50 Point, continuous and residual spectrum ¶
- depends_on definition 12.105 Generalized eigenvector ¶
  - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
    - depends_on example A.283 Momentum on the line ¶
  - depends_on theorem A.279 Gelfand–Maurin ¶
    - depends_on example A.283 Momentum on the line ¶ ↺
  - depends_on theorem 12.107 Nuclear spectral theorem ¶
- depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
- depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
depends_on lemma A.240 Spectral mapping for polynomials ¶
- depends_on proposition A.241 The polynomial calculus is isometric ¶
  - 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.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.261 Spectral theorem for a unitary 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 ¶ ↺
depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
- depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
- depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶ ↺
- depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
- depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
  - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
    - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
      - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶ ↺
      - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
- depends_on theorem 12.54 The spectrum is compact and non-empty ¶ ↺

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`	→	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1309
`depends_on`	→	Inverse of a linear transformation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1309
`depends_on`	←	Point, continuous and residual spectrum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1325
`depends_on`	←	Spectral mapping for polynomials	declared	appendices/A-long-proofs.tex:12022
`depends_on`	←	The norm of a self-adjoint operator lies in its spectrum	declared	appendices/A-long-proofs.tex:11985
`depends_on`	←	Neumann series; the spectrum is bounded	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1364