Resolvent Set is Open
LinearPMap.resolventSet_isOpen
Plain-language statement
The resolvent set is an open subset of ℂ.
Exact Lean statement
lemma resolventSet_isOpen [CompleteSpace H] (T : H →ₗ.[ℂ] H) : IsOpen (ρ T)
Formal artifact
Lean source
lemma resolventSet_isOpen [CompleteSpace H] (T : H →ₗ.[ℂ] H) : IsOpen (ρ T) := by by_cases hT : T.IsClosed · rw [hT.resolventSet_eq'] apply isOpen_iff_forall_mem_open.mpr intro z₁ hz₁ refine ⟨connectedComponentIn T.regularityDomain z₁, fun z₂ hz₂ ↦ ⟨?_, ?_⟩, ?_, ?_⟩ · exact connectedComponentIn_subset _ _ hz₂ · simp_all [hT.isClosable.defectNumber_const hz₂] · exact T.regularityDomain_isOpen.connectedComponentIn · exact mem_connectedComponentIn hz₁.1 · simp [resolventSet_eq_empty hT]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/Basic.lean:681-691
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