| import OAI.Analysis.HilbertCrouzeix.NumericalRange |
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| namespace OAI |
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| noncomputable section |
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| open Complex Set |
| open scoped TensorProduct Matrix.Norms.L2Operator Classical ComplexConjugate |
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| namespace HilbertCrouzeix |
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| universe u |
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| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| section Elementary |
| omit [CompleteSpace H] |
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|
| def compression (A : H →L[ℂ] H) (E : Submodule ℂ H) [E.HasOrthogonalProjection] : |
| E →L[ℂ] E := E.orthogonalProjectionOnto.comp (A.comp E.subtypeL) |
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| theorem numericalRange_compression_subset (A : H →L[ℂ] H) (E : Submodule ℂ H) |
| [E.HasOrthogonalProjection] : numericalRange (compression A E) ⊆ numericalRange A := by |
| rintro z ⟨x, hx, rfl⟩ |
| refine ⟨(x : H), hx, ?_⟩ |
| exact (Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left x (A x)).symm |
|
|
| theorem compression_pow (A : H →L[ℂ] H) (E : Submodule ℂ H) |
| [E.HasOrthogonalProjection] (x : E) {d : ℕ} |
| (hmem : ∀ k ≤ d, (A ^ k) (x : H) ∈ E) {k : ℕ} (hk : k ≤ d) : |
| (((compression A E) ^ k) x : H) = (A ^ k) (x : H) := by |
| induction k with |
| | zero => simp |
| | succ k ih => |
| have hk' : k ≤ d := by omega |
| have hn : A ((A ^ k) (x : H)) ∈ E := by |
| simpa only [pow_succ', mul_apply_eq_comp] using hmem (k + 1) hk |
| rw [pow_succ', pow_succ', mul_apply_eq_comp, mul_apply_eq_comp] |
| change E.starProjection (A (((compression A E) ^ k) x : H)) = _ |
| rw [ih hk'] |
| exact Submodule.starProjection_eq_self_iff.mpr hn |
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| def iterateSpace {q : ℕ} (A : H →L[ℂ] H) (d : ℕ) (x : Fin q → H) : Submodule ℂ H := |
| Submodule.span ℂ (Set.range fun t : Fin (d + 1) × Fin q => (A ^ t.1.val) (x t.2)) |
|
|
| instance iterateSpace_finiteDimensional {q : ℕ} (A : H →L[ℂ] H) (d : ℕ) |
| (x : Fin q → H) : FiniteDimensional ℂ (iterateSpace A d x) := |
| FiniteDimensional.span_of_finite ℂ (Set.finite_range _) |
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| theorem pow_mem_iterateSpace {q : ℕ} (A : H →L[ℂ] H) (d : ℕ) (x : Fin q → H) |
| {k : ℕ} (hk : k ≤ d) (j : Fin q) : (A ^ k) (x j) ∈ iterateSpace A d x := by |
| apply Submodule.subset_span |
| exact ⟨(⟨k, by omega⟩, j), rfl⟩ |
|
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| theorem mem_iterateSpace {q : ℕ} (A : H →L[ℂ] H) (d : ℕ) (x : Fin q → H) |
| (j : Fin q) : x j ∈ iterateSpace A d x := by |
| simpa using pow_mem_iterateSpace A d x (Nat.zero_le d) j |
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| def completedIsometry {V : Type*} {W : Type*} |
| [NormedAddCommGroup V] [NormedAddCommGroup W] [NormedSpace ℂ V] [NormedSpace ℂ W] |
| (f : V →ₗᵢ[ℂ] W) : UniformSpace.Completion V →ₗᵢ[ℂ] UniformSpace.Completion W := |
| { f.toContinuousLinearMap.completion.toLinearMap with |
| norm_map' := f.isometry.completion_map.norm_map_of_map_zero |
| (f.toContinuousLinearMap.completion.map_zero) } |
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| def tensorInclusion (E : Submodule ℂ H) (m : ℕ) : |
| Amplification E m →ₗᵢ[ℂ] Amplification H m := |
| completedIsometry (TensorProduct.mapIsometry E.subtypeₗᵢ (.id : CoeffSpace m →ₗᵢ[ℂ] CoeffSpace m)) |
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| @[simp] theorem tensorInclusion_tmul (E : Submodule ℂ H) {m : ℕ} |
| (x : E) (v : CoeffSpace m) : |
| tensorInclusion E m (↑(x ⊗ₜ[ℂ] v)) = (↑((x : H) ⊗ₜ[ℂ] v) : Amplification H m) := by |
| change (TensorProduct.mapIsometry E.subtypeₗᵢ (.id : CoeffSpace m →ₗᵢ[ℂ] CoeffSpace m) |
| ).toContinuousLinearMap.completion (↑(x ⊗ₜ[ℂ] v)) = _ |
| rw [ContinuousLinearMap.completion_apply_coe] |
| rfl |
|
|
| @[simp] theorem tensorOp_tmul {m : ℕ} (A : H →L[ℂ] H) (B : Coeff m) |
| (x : H) (v : CoeffSpace m) : |
| tensorOp A B (↑(x ⊗ₜ[ℂ] v)) = |
| (↑(A x ⊗ₜ[ℂ] (Matrix.toEuclideanCLM (n := Fin m) (𝕜 := ℂ) B) v) : Amplification H m) := by |
| simp only [tensorOp, ContinuousLinearMap.completion_apply_coe, TensorProduct.mapL_tmul] |
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| theorem polynomialEval_compression_tmul (A : H →L[ℂ] H) (E : Submodule ℂ H) |
| [E.HasOrthogonalProjection] {m d : ℕ} (B : Fin (d + 1) → Coeff m) |
| (x : E) (v : CoeffSpace m) (hmem : ∀ k ≤ d, (A ^ k) (x : H) ∈ E) : |
| tensorInclusion E m (polynomialEval (compression A E) B (↑(x ⊗ₜ[ℂ] v))) = |
| polynomialEval A B (↑((x : H) ⊗ₜ[ℂ] v)) := by |
| simp only [polynomialEval, sum_apply, map_sum, tensorOp_tmul, |
| tensorInclusion_tmul] |
| apply Finset.sum_congr rfl |
| intro k hk |
| rw [compression_pow A E x hmem (Nat.le_of_lt_succ k.isLt)] |
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| @[simp] theorem completion_coe_sum {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 |
|
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| theorem exists_finite_compression {m d : ℕ} (A : H →L[ℂ] H) |
| (B : Fin (d + 1) → Coeff m) (ξ : H ⊗[ℂ] CoeffSpace m) : |
| ∃ (E : Submodule ℂ H) (hE : FiniteDimensional ℂ E), |
| letI : FiniteDimensional ℂ E := hE |
| ∃ η : Amplification E m, |
| tensorInclusion E m η = (↑ξ : Amplification H m) ∧ |
| tensorInclusion E m (polynomialEval (compression A E) B η) = |
| polynomialEval A B (↑ξ) := by |
| obtain ⟨q, x, v, rfl⟩ := TensorProduct.exists_sum_tmul_eq ξ |
| let E := iterateSpace A d x |
| let xE (j : Fin q) : E := ⟨x j, mem_iterateSpace A d x j⟩ |
| let η : Amplification E m := ∑ j : Fin q, (↑(xE j ⊗ₜ[ℂ] v j) : Amplification E m) |
| refine ⟨E, inferInstance, η, ?_, ?_⟩ |
| · simp only [η, map_sum, tensorInclusion_tmul, completion_coe_sum] |
| rfl |
| · simp only [η, map_sum, completion_coe_sum] |
| apply Finset.sum_congr rfl |
| intro j hj |
| exact polynomialEval_compression_tmul A E B (xE j) (v j) |
| (fun k hk => pow_mem_iterateSpace A d x hk j) |
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| theorem exists_finite_compression_norm_bound {m d : ℕ} (A : H →L[ℂ] H) |
| (B : Fin (d + 1) → Coeff m) (ξ : H ⊗[ℂ] CoeffSpace m) : |
| ∃ (E : Submodule ℂ H) (hE : FiniteDimensional ℂ E), |
| letI : FiniteDimensional ℂ E := hE |
| ‖polynomialEval A B (↑ξ)‖ ≤ ‖polynomialEval (compression A E) B‖ * ‖ξ‖ := by |
| obtain ⟨E, hE, η, hη, hp⟩ := exists_finite_compression A B ξ |
| let : FiniteDimensional ℂ E := hE |
| refine ⟨E, hE, ?_⟩ |
| calc |
| ‖polynomialEval A B (↑ξ)‖ = |
| ‖tensorInclusion E m (polynomialEval (compression A E) B η)‖ := congrArg norm hp.symm |
| _ = ‖polynomialEval (compression A E) B η‖ := (tensorInclusion E m).norm_map _ |
| _ ≤ ‖polynomialEval (compression A E) B‖ * ‖η‖ := |
| (polynomialEval (compression A E) B).le_opNorm η |
| _ = ‖polynomialEval (compression A E) B‖ * ‖ξ‖ := by |
| rw [← (tensorInclusion E m).norm_map η, hη, UniformSpace.Completion.norm_coe] |
|
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| end Elementary |
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| end HilbertCrouzeix |
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| end |
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| end OAI |
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