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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
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