Theorems · Theorem · functional analysis

cfc_map_spectrum

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
  [inst_7 : StarRing A] [inst_8 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] (f : R → R) (a : A),
  autoParam (p a) cfc_map_spectrum._auto_1 →
    autoParam (ContinuousOn f (spectrum R a)) cfc_map_spectrum._auto_3 → spectrum R (cfc f a) = f '' spectrum R a

The spectral mapping theorem for the continuous functional calculus.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

11 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculus