File size: 8,356 Bytes
ddc84c5 | 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | import Mathlib
import OAI.Analysis.Crouzeix.NumericalRange
namespace OAI
/-! Exact polynomial compression on spans of finitely many operator iterates. -/
noncomputable section
open scoped TensorProduct Matrix.Norms.L2Operator InnerProductSpace
open Set
namespace CrouzeixHilbert
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
def compression (A : Operator H) (E : Submodule ℂ H) [E.HasOrthogonalProjection] :
Operator E := E.orthogonalProjectionOnto.comp (A.comp E.subtypeL)
theorem compression_inner (A : Operator H) (E : Submodule ℂ H)
[E.HasOrthogonalProjection] (x : E) :
⟪x, compression A E x⟫_ℂ = ⟪(x : H), A (x : H)⟫_ℂ :=
Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left x (A x)
theorem numericalRange_compression_subset (A : Operator H) (E : Submodule ℂ H)
[E.HasOrthogonalProjection] : numericalRange (compression A E) ⊆ numericalRange A := by
rintro z ⟨x, hx, hz⟩
exact ⟨(x : H), hx, (compression_inner A E x).symm.trans hz⟩
theorem coe_compression_apply (A : Operator H) (E : Submodule ℂ H)
[E.HasOrthogonalProjection] (x : E) (hx : A (x : H) ∈ E) :
(compression A E x : H) = A (x : H) := by
have hh := Submodule.orthogonalProjectionOnto_mem_subspace_eq_self (⟨A x, hx⟩ : E)
exact congrArg (fun v : E => (v : H)) hh
theorem coe_compression_pow (A : Operator H) (E : Submodule ℂ H)
[E.HasOrthogonalProjection] (x : E) (k : ℕ)
(hx : ∀ j : ℕ, j ≤ k → (A ^ j) (x : H) ∈ E) :
(((compression A E) ^ k) x : H) = (A ^ k) (x : H) := by
induction k with
| zero => simp
| succ k ih =>
have hk := ih (fun j hj => hx j (hj.trans (Nat.le_succ k)))
have hmem : A (((compression A E) ^ k) x : H) ∈ E := by
rw [hk]
simpa only [pow_succ', mul_apply_eq_comp] using hx (k + 1) le_rfl
simpa only [pow_succ', mul_apply_eq_comp] using
(coe_compression_apply A E (((compression A E) ^ k) x) hmem).trans (congrArg A hk)
def iterateSpan (A : Operator H) (d : ℕ) {l : ℕ} (ξ : Fin l → H) : Submodule ℂ H :=
Submodule.span ℂ (Set.range (fun p : Fin (d + 1) × Fin l => (A ^ (p.1 : ℕ)) (ξ p.2)))
instance iterateSpanFiniteDimensional (A : Operator H) (d : ℕ) {l : ℕ} (ξ : Fin l → H) :
FiniteDimensional ℂ (iterateSpan A d ξ) :=
FiniteDimensional.span_of_finite ℂ (Set.finite_range _)
theorem iterate_mem_iterateSpan (A : Operator H) (d : ℕ) {l : ℕ} (ξ : Fin l → H)
(j : Fin l) (k : ℕ) (hk : k ≤ d) : (A ^ k) (ξ j) ∈ iterateSpan A d ξ := by
apply Submodule.subset_span
exact ⟨(⟨k, Nat.lt_succ_of_le hk⟩, j), rfl⟩
theorem mem_iterateSpan (A : Operator H) (d : ℕ) {l : ℕ} (ξ : Fin l → H)
(j : Fin l) : ξ j ∈ iterateSpan A d ξ := by
simpa using iterate_mem_iterateSpan A d ξ j 0 (Nat.zero_le _)
def amplificationInclusion (E : Submodule ℂ H) (m : ℕ) :
Amplification E m →L[ℂ] Amplification H m :=
(TensorProduct.mapL E.subtypeL (ContinuousLinearMap.id ℂ (EuclideanSpace ℂ (Fin m)))).completion
theorem norm_amplificationInclusion (E : Submodule ℂ H) (m : ℕ)
(x : Amplification E m) : ‖amplificationInclusion E m x‖ = ‖x‖ := by
refine UniformSpace.Completion.induction_on x
(isClosed_eq (amplificationInclusion E m).continuous.norm continuous_norm) ?_
intro a
simp only [amplificationInclusion, ContinuousLinearMap.completion_apply_coe,
UniformSpace.Completion.norm_coe]
rw [TensorProduct.mapL_apply]
change ‖TensorProduct.map E.subtypeₗᵢ.toLinearMap
(LinearIsometry.id : EuclideanSpace ℂ (Fin m) →ₗᵢ[ℂ] EuclideanSpace ℂ (Fin m)).toLinearMap a‖ = ‖a‖
exact (TensorProduct.mapIsometry E.subtypeₗᵢ (LinearIsometry.id :
EuclideanSpace ℂ (Fin m) →ₗᵢ[ℂ] EuclideanSpace ℂ (Fin m))).norm_map a
@[simp]
theorem amplificationInclusion_tmul (E : Submodule ℂ H) {m : ℕ}
(x : E) (y : EuclideanSpace ℂ (Fin m)) :
amplificationInclusion E m (↑(x ⊗ₜ[ℂ] y)) =
(↑((x : H) ⊗ₜ[ℂ] y) : Amplification H m) := by
simp [amplificationInclusion, ContinuousLinearMap.completion_apply_coe]
theorem polynomialEval_tmul (A : Operator H) {m d : ℕ} (B : Fin (d + 1) → Coeff m)
(x : H) (y : EuclideanSpace ℂ (Fin m)) :
polynomialEval A B (↑(x ⊗ₜ[ℂ] y)) =
∑ k : Fin (d + 1), (↑(((A ^ (k : ℕ)) x) ⊗ₜ[ℂ]
(Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin m) (B k) y)) : Amplification H m) := by
simp only [polynomialEval, sum_apply, tensorOperator,
ContinuousLinearMap.completion_apply_coe, TensorProduct.mapL_tmul]
theorem polynomial_compression_tmul (A : Operator H) (E : Submodule ℂ H)
[E.HasOrthogonalProjection] {m d : ℕ} (B : Fin (d + 1) → Coeff m)
(x : E) (y : EuclideanSpace ℂ (Fin m))
(hx : ∀ j : ℕ, j ≤ d → (A ^ j) (x : H) ∈ E) :
polynomialEval A B (amplificationInclusion E m (↑(x ⊗ₜ[ℂ] y))) =
amplificationInclusion E m (polynomialEval (compression A E) B (↑(x ⊗ₜ[ℂ] y))) := by
simp only [amplificationInclusion_tmul, polynomialEval_tmul, map_sum]
apply Finset.sum_congr rfl
intro k _
rw [coe_compression_pow A E x k (fun j hj => hx j (hj.trans (Nat.le_of_lt_succ k.isLt)))]
theorem completion_coe_finsetSum {V : Type*} [NormedAddCommGroup V]
{ι : Type*} (s : Finset ι) (f : ι → V) :
(↑(∑ i ∈ s, f i) : UniformSpace.Completion V) =
∑ i ∈ s, (↑(f i) : UniformSpace.Completion V) :=
map_sum (UniformSpace.Completion.toCompl : V →+ UniformSpace.Completion V) f s
theorem polynomial_bound_of_finite_compressions (A : Operator H)
{m d : ℕ} (B : Fin (d + 1) → Coeff m) {C : ℝ} (hC : 0 ≤ C)
(hfin : ∀ (E : Submodule ℂ H) [FiniteDimensional ℂ E],
‖polynomialEval (compression A E) B‖ ≤ C) :
‖polynomialEval A B‖ ≤ C := by
apply (polynomialEval A B).opNorm_le_bound hC
intro x
refine UniformSpace.Completion.induction_on x
(isClosed_le (polynomialEval A B).continuous.norm (continuous_const.mul continuous_norm)) ?_
intro a
obtain ⟨l, ξ, η, ha⟩ := TensorProduct.exists_sum_tmul_eq a
let E := iterateSpan A d ξ
let xE (j : Fin l) : E := ⟨ξ j, mem_iterateSpan A d ξ j⟩
let v : Amplification E m := ∑ j : Fin l, (↑(xE j ⊗ₜ[ℂ] η j) : Amplification E m)
have hav : (↑a : Amplification H m) = amplificationInclusion E m v := by
rw [ha, completion_coe_finsetSum]
simp only [v, map_sum, amplificationInclusion_tmul]
rfl
have hinter : polynomialEval A B (amplificationInclusion E m v) =
amplificationInclusion E m (polynomialEval (compression A E) B v) := by
simp only [v, map_sum]
apply Finset.sum_congr rfl
intro j _
apply polynomial_compression_tmul
intro k hk
exact iterate_mem_iterateSpan A d ξ j k hk
calc
‖polynomialEval A B (↑a)‖ =
‖amplificationInclusion E m (polynomialEval (compression A E) B v)‖ := by
rw [hav, hinter]
_ = ‖polynomialEval (compression A E) B v‖ := norm_amplificationInclusion E m _
_ ≤ ‖polynomialEval (compression A E) B‖ * ‖v‖ :=
(polynomialEval (compression A E) B).le_opNorm v
_ ≤ C * ‖v‖ := mul_le_mul_of_nonneg_right (hfin E) (norm_nonneg _)
_ = C * ‖(↑a : Amplification H m)‖ := by rw [hav, norm_amplificationInclusion]
theorem polynomial_inequality_of_finite_compressions (A : Operator H)
{m d : ℕ} (B : Fin (d + 1) → Coeff m)
(hfin : ∀ (E : Submodule ℂ H) [FiniteDimensional ℂ E],
‖polynomialEval (compression A E) B‖ ≤
2 * supNorm (numericalRange (compression A E)) (matrixPolynomial B)) :
‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B) := by
have hbK := supNorm_bddAbove (isCompact_numericalClosure A)
(continuous_matrixPolynomial B).continuousOn
have hbW : BddAbove (insert 0 ((fun z => ‖matrixPolynomial B z‖) '' numericalRange A)) :=
hbK.mono (Set.insert_subset_insert (Set.image_mono subset_closure))
apply polynomial_bound_of_finite_compressions A B
(mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) (supNorm_nonneg hbW))
intro E _
exact (hfin E).trans (mul_le_mul_of_nonneg_left
(supNorm_mono (numericalRange_compression_subset A E) hbW) (by norm_num))
end CrouzeixHilbert
end
end OAI
|