Is Closed resolvent Set eq
LinearPMap.IsClosed.resolventSet_eq
Plain-language statement
For a closed operator the continuity of the resolvent is redundant in the definition of the resolvent set.
Exact Lean statement
lemma IsClosed.resolventSet_eq [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) :
ρ T = {z : ℂ | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤}Formal artifact
Lean source
lemma IsClosed.resolventSet_eq [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) : ρ T = {z : ℂ | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤} := by ext z rw [mem_resolventSet_iff, mem_setOf_eq, and_congr_right_iff, and_iff_left_iff_imp] intro h_ker h_range refine continuous_of_isClosed_domain ?_ ?_ · apply (inverse_closed_iff h_ker).mpr exact hT.sub_continuous (Continuous.const_smul (by fun_prop) _) (by simp) · simp [inverse_domain, h_range]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/Basic.lean:659-667
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