Is Closed spectrum eq
LinearPMap.IsClosed.spectrum_eq
Project documentation
The continuous spectrum, σᶜ, of a partial linear map. A complex number z is in σᶜ T iff the range of T - z • 1 is not closed. -/ def continuousSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ := {z : ℂ | ¬root.IsClosed ((T - z • 1).toFun.range : Set H)} @[inherit_doc continuousSpectrum] scoped notation "σᶜ" => continuousSpectrum lemma continuousSpectrum_eq (T...
Exact Lean statement
lemma IsClosed.spectrum_eq [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) :
σ T = σᵖ T ∪ σʳ T ∪ σᶜ TFormal artifact
Lean source
lemma IsClosed.spectrum_eq [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) : σ T = σᵖ T ∪ σʳ T ∪ σᶜ T := by refine Subset.antisymm ?_ ?_ · intro z hσ apply mem_spectrum_iff.mp at hσ rcases eq_or_ne (T - z • 1).toFun.ker ⊥ with h_ker | h_ker · by_cases h_cont : Continuous (𝑅 T z) · left; right; exact ⟨h_ker, (hσ.neg_resolve_left h_ker).neg_resolve_right h_cont, h_cont⟩ · right rw [mem_continuousSpectrum_iff, ← inverse_domain] refine fun h ↦ h_cont ?_ refine continuous_of_isClosed_domain ?_ h apply (inverse_closed_iff h_ker).mpr exact hT.sub_continuous (Continuous.const_smul (by fun_prop) _) le_top · left; left; exact h_ker · refine union_subset ?_ T.continuousSpectrum_subset_spectrum exact union_subset T.pointSpectrum_subset_spectrum T.residualSpectrum_subset_spectrum- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/Basic.lean:795-811
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