Theorems · Theorem · functional analysis

CFC.posPart_negPart_unique

∀ {A : Type u_1} [inst : NonUnitalRing A] [inst_1 : Module ℝ A] [inst_2 : SMulCommClass ℝ A A]
  [inst_3 : IsScalarTower ℝ A A] [inst_4 : StarRing A] [inst_5 : TopologicalSpace A]
  [inst_6 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [inst_7 : PartialOrder A] [StarOrderedRing A]
  [NonnegSpectrumClass ℝ A] [IsSemitopologicalRing A] [T2Space A] {a b c : A},
  a = b - c →
    b * c = 0 →
      autoParam (0 ≤ b) CFC.posPart_negPart_unique._auto_1 →
        autoParam (0 ≤ c) CFC.posPart_negPart_unique._auto_3 → a⁺ = b ∧ a⁻ = c

The positive and negative parts of a selfadjoint element a are unique. That is, if a = b - c is the difference of nonnegative elements whose product is zero, then these are precisely a⁺ and a⁻.

Defined in

Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Cited by

0 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonUnitalRingModuleSMulCommClassIsScalarTowerStarRingTopologicalSpaceNonUnitalContinuousFunctionalCalculusPartialOrderStarOrderedRingNonnegSpectrumClassIsSemitopologicalRingT2Space