Theorems · Theorem · functional analysis
IsSelfAdjoint.mem_spectrum_eq_re
∀ {A : Type u_1} [inst : CStarAlgebra A] {a : A}, IsSelfAdjoint a → ∀ {z : ℂ}, z ∈ spectrum ℂ a → z = ↑z.reAny element of the spectrum of a selfadjoint is real.
- Defined in
- Mathlib.Analysis.CStarAlgebra.Spectrum
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CStarAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Complexstatement and proof · cited by 5,565
- Unitsproof · cited by 2,804
- Complex.ofRealstatement · cited by 1,654
- NormedAlgebraproof · cited by 1,165
- Complex.restatement and proof · cited by 882
- Complex.Iproof · cited by 866
- Complex.improof · cited by 591
- IsSelfAdjointstatement and proof · cited by 545
- spectrumstatement and proof · cited by 510
- Metric.sphereproof · cited by 371
- unitaryproof · cited by 207
Cited by3
Results whose statement or proof uses this declaration.
- IsSelfAdjoint.val_re_map_spectrumproof · cited by 1
- IsSelfAdjoint.im_eq_zero_of_mem_spectrumproof · cited by 1
- selfAdjoint.mem_spectrum_eq_reproof · cited by 0