Theorems · Theorem · functional analysis
spectrum.subset_circle_of_unitary
∀ {𝕜 : Type u_1} [inst : NormedField 𝕜] {E : Type u_2} [inst_1 : NormedRing E] [inst_2 : StarRing E] [CStarRing E]
[inst_4 : NormedAlgebra 𝕜 E] [CompleteSpace E] {u : E}, u ∈ unitary E → spectrum 𝕜 u ⊆ Metric.sphere 0 1- Defined in
- Mathlib.Analysis.CStarAlgebra.Spectrum
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement · cited by 25,697
- Submonoidstatement · cited by 3,086
- CompleteSpacestatement and proof · cited by 2,532
- StarRingstatement and proof · cited by 1,686
- NormedAlgebrastatement and proof · cited by 1,165
- NormedFieldstatement and proof · cited by 1,084
- NormedRingstatement and proof · cited by 924
- spectrumstatement · cited by 510
- Metric.spherestatement · cited by 371
- unitarystatement and proof · cited by 207
- CStarRingstatement and proof · cited by 61
Cited by5
Results whose statement or proof uses this declaration.
- IsSelfAdjoint.mem_spectrum_eq_reproof · cited by 3
- spectrum.norm_eq_one_of_unitaryproof · cited by 2
- Unitary.spectrum_subset_slitPlane_iff_norm_lt_twoproof · cited by 2
- Unitary.norm_sub_one_sq_eqproof · cited by 2
- Unitary.two_mul_one_sub_le_norm_sub_one_sqproof · cited by 2