| import Mathlib.Analysis.InnerProductSpace.Adjoint | |
| import Mathlib.Analysis.Normed.Operator.Banach | |
| import Mathlib.Algebra.Algebra.Spectrum.Basic | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.OperatorRange | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| theorem denseRange_of_adjoint_injective (T : H →L[ℂ] H) | |
| (hT : Function.Injective (star T : H →L[ℂ] H)) : DenseRange T := by | |
| change Dense (T.range : Set H) | |
| rw [Submodule.dense_iff_topologicalClosure_eq_top, | |
| Submodule.topologicalClosure_eq_top_iff, ContinuousLinearMap.orthogonal_range] | |
| exact LinearMap.ker_eq_bot.mpr hT | |
| theorem denseRange_of_selfAdjoint_injective (T : H →L[ℂ] H) | |
| (hself : IsSelfAdjoint T) (hinj : Function.Injective T) : DenseRange T := by | |
| apply denseRange_of_adjoint_injective | |
| simpa only [hself.star_eq] using hinj | |
| theorem unitary_injective (U : H →L[ℂ] H) | |
| (hU : U ∈ unitary (H →L[ℂ] H)) : Function.Injective U := by | |
| intro x y hxy | |
| have h := congrArg (fun z => star U z) hxy | |
| change (star U * U) x = (star U * U) y at h | |
| simpa only [(Unitary.mem_iff.mp hU).1, one_apply_eq_self] using h | |
| theorem unitary_surjective (U : H →L[ℂ] H) | |
| (hU : U ∈ unitary (H →L[ℂ] H)) : Function.Surjective U := by | |
| intro x | |
| refine ⟨star U x, ?_⟩ | |
| change (U * star U) x = x | |
| rw [(Unitary.mem_iff.mp hU).2, one_apply_eq_self] | |
| theorem unitary_mul_selfAdjoint_range (U D : H →L[ℂ] H) | |
| (hU : U ∈ unitary (H →L[ℂ] H)) (hD : IsSelfAdjoint D) | |
| (hinj : Function.Injective D) : | |
| Function.Injective (U * D) ∧ DenseRange (U * D) := by | |
| constructor | |
| · exact (unitary_injective U hU).comp hinj | |
| · exact (unitary_surjective U hU).denseRange.comp | |
| (denseRange_of_selfAdjoint_injective D hD hinj) U.continuous | |
| theorem not_surjective_of_zero_mem_spectrum (T : H →L[ℂ] H) | |
| (hinj : Function.Injective T) (hspec : (0 : ℂ) ∈ spectrum ℂ T) : | |
| ¬ Function.Surjective T := by | |
| intro hsurj | |
| exact ((spectrum.zero_mem_iff (R := ℂ)).mp hspec) | |
| (ContinuousLinearMap.isUnit_iff_bijective.mpr ⟨hinj, hsurj⟩) | |
| omit [CompleteSpace H] in | |
| theorem not_isClosed_range_of_dense_not_surjective (T : H →L[ℂ] H) | |
| (hdense : DenseRange T) (hnot : ¬ Function.Surjective T) : | |
| ¬ IsClosed (Set.range T) := by | |
| intro hclosed | |
| apply hnot | |
| have hrange : Set.range T = Set.univ := | |
| hclosed.closure_eq.symm.trans hdense.closure_range | |
| intro x | |
| have : x ∈ Set.range T := by rw [hrange]; trivial | |
| exact this | |
| theorem eq_zero_of_eigenvector_of_spectrum_zero (T : H →L[ℂ] H) | |
| (hinj : Function.Injective T) (hspec : spectrum ℂ T = {0}) | |
| (z : ℂ) (x : H) (heigen : T x = z • x) : x = 0 := by | |
| by_cases hz : z = 0 | |
| · apply hinj | |
| simpa only [hz, zero_smul, map_zero] using heigen | |
| · have hunit : IsUnit (algebraMap ℂ (H →L[ℂ] H) z - T) := by | |
| apply spectrum.notMem_iff.mp | |
| rw [hspec] | |
| simpa only [Set.mem_singleton_iff] using hz | |
| apply (ContinuousLinearMap.isUnit_iff_bijective.mp hunit).1 | |
| simp only [sub_apply, Algebra.algebraMap_eq_smul_one, | |
| smul_apply, one_apply_eq_self, heigen, sub_self, map_zero] | |
| end Invariant233.OperatorRange | |
| end | |
| end OAI | |