Theorems · Theorem · functional analysis

CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts

∀ {A : Type u_1} [inst : NonUnitalRing A] [inst_1 : StarRing A] [inst_2 : TopologicalSpace A] [inst_3 : Module ℝ A]
  [inst_4 : IsScalarTower ℝ A A] [inst_5 : SMulCommClass ℝ A A]
  [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] {a : A},
  IsSelfAdjoint a →
    QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal →
      ∃ x, IsSelfAdjoint x ∧ QuasispectrumRestricts x ⇑ContinuousMap.realToNNReal ∧ x * x = a

Defined in: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
Cited by: 3 results in Mathlib
Foundations: Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NonUnitalRing StarRing TopologicalSpace Module IsScalarTower SMulCommClass NonUnitalContinuousFunctionalCalculus

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Cites28

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Realstatement and proof · cited by 25,697
TopologicalSpacestatement and proof · cited by 24,529
Modulestatement and proof · cited by 20,661
NNRealstatement · cited by 4,310
IsScalarTowerstatement and proof · cited by 3,896
ContinuousMapstatement · cited by 2,491
SMulCommClassstatement and proof · cited by 1,927
StarRingstatement and proof · cited by 1,686
IsSelfAdjointstatement and proof · cited by 545
Real.sqrtproof · cited by 545
NonUnitalRingstatement and proof · cited by 422

Cited by3

Results whose statement or proof uses this declaration.

nonneg_iff_isSelfAdjoint_and_quasispectrumRestrictsproof · cited by 4
Commute.mul_nonnegproof · cited by 3
CStarAlgebra.spectralOrderedRingproof · cited by 1