Mem resolvent Set of range eq top
LinearPMap.IsSelfAdjoint.mem_resolventSet_of_range_eq_top
Plain-language statement
(T - z • 1).range = ⊤ is a sufficient condition for z ∈ ρ T (and it is a necessary condition by definition of ρ).
Exact Lean statement
lemma mem_resolventSet_of_range_eq_top {z : ℂ} (h : (T - z • 1).toFun.range = ⊤) : z ∈ ρ TFormal artifact
Lean source
lemma mem_resolventSet_of_range_eq_top {z : ℂ} (h : (T - z • 1).toFun.range = ⊤) : z ∈ ρ T := by by_cases hz_im : z.im = 0 · rw [(isClosed hT).resolventSet_eq] refine ⟨?_, h⟩ have h_orthog := (isUnbounded hT).orthogonal_closure_sub_range z rwa [isSelfAdjoint_def.mp hT, (isClosed hT).closure_eq, conj_eq_iff_im.mpr hz_im, h, Submodule.top_orthogonal_eq_bot, Eq.comm, ← LinearMap.le_ker_iff_map, Submodule.ker_subtype, le_bot_iff] at h_orthog · exact mem_resolventSet_of_im_ne_zero hT hz_im- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/SpectralTheory/SelfAdjoint.lean:80-88
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