Is Closed resolvent Set eq
LinearPMap.IsClosed.resolventSet_eq'
Plain-language statement
For a closed operator the resolvent set consists of those regular points for which the defect number is zero.
Exact Lean statement
lemma IsClosed.resolventSet_eq' [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) :
ρ T = T.regularityDomain ∩ T.defectNumber ⁻¹' {0}Formal artifact
Lean source
lemma IsClosed.resolventSet_eq' [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) : ρ T = T.regularityDomain ∩ T.defectNumber ⁻¹' {0} := by ext z refine ⟨fun hρ ↦ ?_, fun ⟨h_reg, h_defect⟩ ↦ ?_⟩ · have hz : z ∈ T.regularityDomain := T.resolventSet_subset_regularityDomain hρ exact ⟨hz, (hT.defectNumber_eq_zero_iff hz).mpr hρ.2.1⟩ · obtain ⟨h_ker, h_cont⟩ := mem_regularityDomain_iff.mp h_reg exact ⟨h_ker, (hT.defectNumber_eq_zero_iff h_reg).mp h_defect, h_cont⟩- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/Basic.lean:671-678
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.