Theorems · Theorem · functional analysis
isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts
∀ {A : Type u_1} [inst : TopologicalSpace A] [inst_1 : NonUnitalRing A] [inst_2 : StarRing A] [inst_3 : Module ℂ A]
[inst_4 : IsScalarTower ℂ A A] [inst_5 : SMulCommClass ℂ A A] [NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal]
{a : A}, IsSelfAdjoint a ↔ IsStarNormal a ∧ QuasispectrumRestricts a ⇑Complex.reCLMAn element in a non-unital C⋆-algebra is selfadjoint if and only if it is normal and its
quasispectrum is contained in ℝ.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
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
- RingHom.idstatement · cited by 18,349
- Complexstatement and proof · cited by 5,565
- ContinuousLinearMapstatement · cited by 5,352
- Algebra.algebraMapproof · cited by 4,706
- IsScalarTowerstatement and proof · cited by 3,896
- SMulCommClassstatement and proof · cited by 1,927
- StarRingstatement and proof · cited by 1,686
- Star.starproof · cited by 1,082
Cited by2
Results whose statement or proof uses this declaration.
- IsSelfAdjoint.quasispectrumRestrictsproof · cited by 6
- QuasispectrumRestricts.isSelfAdjointproof · cited by 0