Resolvent Set eq empty
LinearPMap.resolventSet_eq_empty
Plain-language statement
If an operator is not closed then its resolvent set is empty.
Exact Lean statement
lemma resolventSet_eq_empty [CompleteSpace H] {T : H →ₗ.[ℂ] H} (h : ¬T.IsClosed) : ρ T = ∅Formal artifact
Lean source
lemma resolventSet_eq_empty [CompleteSpace H] {T : H →ₗ.[ℂ] H} (h : ¬T.IsClosed) : ρ T = ∅ := by refine eq_empty_iff_forall_notMem.mpr fun z ⟨h_ker, h_range, h_cont⟩ ↦ ?_ suffices (T - z • 1).IsClosed by have hTz : T - z • 1 + z • 1 = T := eq_of_le_of_domain_eq (sub_add_le_cancel _ _) (by simp [add_domain, sub_domain]) exact h <| hTz ▸ this.add_continuous (Continuous.const_smul (by fun_prop) _) (by simp) apply (inverse_closed_iff h_ker).mp apply (isClosed_iff_isClosed_domain_of_continuous h_cont).mpr simp [inverse_domain, h_range]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/Basic.lean:644-652
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Person-level attribution pending.
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