Theorems · Theorem · functional analysis
IsSelfAdjoint.spectrumRestricts
∀ {A : Type u_1} [inst : TopologicalSpace A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : Algebra ℂ A]
[ContinuousFunctionalCalculus ℂ A IsStarNormal] {a : A}, IsSelfAdjoint a → SpectrumRestricts a ⇑Complex.reCLM- Cited by
- 13 results in Mathlib
- Foundations
- Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Realstatement · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- RingHom.idstatement · cited by 18,349
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Complexstatement and proof · cited by 5,565
- ContinuousLinearMapstatement · cited by 5,352
- StarRingstatement and proof · cited by 1,686
- IsSelfAdjointstatement and proof · cited by 545
- ContinuousFunctionalCalculusstatement and proof · cited by 331
- IsStarNormalstatement and proof · cited by 117
Cited by13
Results whose statement or proof uses this declaration.
- cfc_real_eq_complexproof · cited by 4
- selfAdjoint.norm_sq_expUnitary_sub_oneproof · cited by 3
- IsSelfAdjoint.toReal_spectralRadius_eq_normproof · cited by 2
- SpectrumRestricts.nnreal_iff_nnnormproof · cited by 2
- CStarAlgebra.nnnorm_mem_spectrum_of_nonnegproof · cited by 1
- CStarAlgebra.norm_or_neg_norm_mem_spectrumproof · cited by 1
- spectrum_imaginaryPart'proof · cited by 1
- spectrum_realPart'proof · cited by 1
- cfcHom_real_eq_restrictstatement and proof · cited by 0
- cfc_complex_eq_realproof · cited by 0
- IsSelfAdjoint.coe_mem_spectrum_complexproof · cited by 0
- argSelfAdjoint_expUnitaryproof · cited by 0