| import Mathlib.Analysis.InnerProductSpace.l2Space |
| import Mathlib.Analysis.Normed.Operator.Extend |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open Filter |
| open scoped ENNReal Topology |
| noncomputable section |
|
|
| abbrev SourceFiberHilbert := lp (fun _ : ℤ => ℂ) 2 |
|
|
| /-- Canonical inclusion of finite coordinate vectors in l2(Z). -/ |
| def fiberFiniteEmbedding : (ℤ →₀ ℂ) →ₗ[ℂ] SourceFiberHilbert := |
| Finsupp.linearCombination ℂ (fun n => lp.single 2 n (1 : ℂ)) |
|
|
| theorem fiberFiniteEmbedding_single (n : ℤ) (c : ℂ) : |
| fiberFiniteEmbedding (Finsupp.single n c) = lp.single 2 n c := by |
| simp [fiberFiniteEmbedding, ← lp.single_smul] |
|
|
| theorem fiberFiniteEmbedding_sum (a : ℤ →₀ ℂ) : |
| fiberFiniteEmbedding a = ∑ n ∈ a.support, lp.single 2 n (a n) := by |
| simp [fiberFiniteEmbedding, Finsupp.linearCombination_apply, Finsupp.sum, ← lp.single_smul] |
|
|
| theorem fiberFiniteEmbedding_norm_sq (a : ℤ →₀ ℂ) : |
| ‖fiberFiniteEmbedding a‖ ^ 2 = ∑ n ∈ a.support, ‖a n‖ ^ 2 := by |
| rw [fiberFiniteEmbedding_sum] |
| simpa using lp.norm_sum_single (p := (2 : ℝ≥0∞)) (by norm_num) a a.support |
|
|
| theorem fiberFiniteEmbedding_dense : DenseRange fiberFiniteEmbedding := by |
| intro x |
| apply isClosed_closure.mem_of_tendsto (lp.hasSum_single (by norm_num : (2 : ℝ≥0∞) ≠ ∞) x) |
| apply Eventually.of_forall |
| intro s |
| apply subset_closure |
| refine ⟨∑ n ∈ s, Finsupp.single n (x n), ?_⟩ |
| simp [fiberFiniteEmbedding_single] |
|
|
| /-- The algebraic action of a matrix on finite vectors. -/ |
| def fiberMatrixLinear (m : ℤ → ℤ → ℂ) : (ℤ →₀ ℂ) →ₗ[ℂ] (ℤ → ℂ) := |
| Finsupp.linearCombination ℂ (fun n k => m k n) |
|
|
| theorem fiberMatrixLinear_apply (m : ℤ → ℤ → ℂ) (a : ℤ →₀ ℂ) (k : ℤ) : |
| fiberMatrixLinear m a k = ∑ n ∈ a.support, a n * m k n := by |
| simp [fiberMatrixLinear, Finsupp.linearCombination_apply, Finsupp.sum, Finset.sum_apply, |
| Pi.smul_apply, smul_eq_mul] |
|
|
| /-- Every bounded operator is determined by its coordinate matrix. -/ |
| theorem fiberOperator_ext {A B : SourceFiberHilbert →L[ℂ] SourceFiberHilbert} |
| (h : ∀ k n : ℤ, (A (lp.single 2 n (1 : ℂ))) k = (B (lp.single 2 n (1 : ℂ))) k) : A = B := by |
| apply lp.ext_continuousLinearMap (by norm_num : (2 : ℝ≥0∞) ≠ ∞) |
| intro n |
| apply ContinuousLinearMap.ext |
| intro c |
| apply Subtype.ext |
| funext k |
| change (A (lp.single 2 n c)) k = (B (lp.single 2 n c)) k |
| have he : lp.single 2 n c = c • (lp.single 2 n (1 : ℂ) : SourceFiberHilbert) := by |
| simp [← lp.single_smul] |
| rw [he, map_smul, map_smul] |
| change c * (A (lp.single 2 n 1)) k = c * (B (lp.single 2 n 1)) k |
| rw [h] |
|
|
| theorem fiberOperator_hasSum (A : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (v : SourceFiberHilbert) : |
| HasSum (fun n : ℤ => v n • A (lp.single 2 n (1 : ℂ))) (A v) := by |
| convert (lp.hasSum_single (by norm_num : (2 : ℝ≥0∞) ≠ ∞) v).map A A.continuous using 1 |
| funext n |
| have he : lp.single 2 n (v n) = v n • (lp.single 2 n (1 : ℂ) : SourceFiberHilbert) := by |
| simp [← lp.single_smul] |
| change v n • A (lp.single 2 n (1 : ℂ)) = A (lp.single 2 n (v n)) |
| rw [he, map_smul] |
|
|
| theorem fiberOperator_action (A : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) |
| (v : SourceFiberHilbert) (k : ℤ) : |
| (A v) k = ∑' n : ℤ, (A (lp.single 2 n (1 : ℂ))) k * v n := by |
| have h := (fiberOperator_hasSum A v).map |
| (lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 k) (lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 k).continuous |
| have he := h.tsum_eq.symm |
| change (A v) k = ∑' n : ℤ, v n * (A (lp.single 2 n (1 : ℂ))) k at he |
| simpa only [mul_comm] using he |
|
|
| variable (m : ℤ → ℤ → ℂ) (C : ℝ) |
| variable (hb : ∀ (s t : Finset ℤ) (a : ℤ → ℂ), |
| (∑ k ∈ t, ‖∑ n ∈ s, a n * m k n‖ ^ 2) ≤ C ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2) |
|
|
| include C hb in |
| theorem fiberMatrixLinear_mem (a : ℤ →₀ ℂ) : Memℓp (fiberMatrixLinear m a) 2 := by |
| apply memℓp_gen' (C := C ^ 2 * ∑ n ∈ a.support, ‖a n‖ ^ 2) |
| intro t |
| simpa [fiberMatrixLinear_apply] using hb a.support t a |
| |
| /-- The finite matrix action takes values in actual l2, by the finite-sum bound. -/ |
| def fiberMatrixFinite : (ℤ →₀ ℂ) →ₗ[ℂ] SourceFiberHilbert where |
| toFun a := ⟨fiberMatrixLinear m a, fiberMatrixLinear_mem m C hb a⟩ |
| map_add' a b := by apply Subtype.ext; exact map_add (fiberMatrixLinear m) a b |
| map_smul' c a := by apply Subtype.ext; exact map_smul (fiberMatrixLinear m) c a |
| |
| theorem fiberMatrixFinite_norm_le (hC : 0 ≤ C) (a : ℤ →₀ ℂ) : |
| ‖fiberMatrixFinite m C hb a‖ ≤ C * ‖fiberFiniteEmbedding a‖ := by |
| have he : ‖fiberMatrixFinite m C hb a‖ ^ 2 = |
| ∑' k, ‖fiberMatrixLinear m a k‖ ^ 2 := by |
| simpa [fiberMatrixFinite] using lp.norm_rpow_eq_tsum (p := (2 : ℝ≥0∞)) (by norm_num) |
| (fiberMatrixFinite m C hb a) |
| have hsum : (∑' k, ‖fiberMatrixLinear m a k‖ ^ 2) ≤ |
| C ^ 2 * ∑ n ∈ a.support, ‖a n‖ ^ 2 := by |
| apply Real.tsum_le_of_sum_le |
| · intro k; positivity |
| · intro t; simpa [fiberMatrixLinear_apply] using hb a.support t a |
| rw [← he, ← fiberFiniteEmbedding_norm_sq] at hsum |
| have hpos := mul_nonneg hC (norm_nonneg (fiberFiniteEmbedding a)) |
| nlinarith [norm_nonneg (fiberMatrixFinite m C hb a)] |
| |
| /-- Extension of the bounded finite matrix action to every l2 vector. -/ |
| def boundedMatrixOperator : SourceFiberHilbert →L[ℂ] SourceFiberHilbert := |
| (fiberMatrixFinite m C hb).extendOfNorm fiberFiniteEmbedding |
| |
| theorem boundedMatrixOperator_norm (hC : 0 ≤ C) : ‖boundedMatrixOperator m C hb‖ ≤ C := |
| LinearMap.opNorm_extendOfNorm_le fiberFiniteEmbedding_dense hC |
| (fiberMatrixFinite_norm_le m C hb hC) |
| |
| theorem boundedMatrixOperator_entry (hC : 0 ≤ C) (k n : ℤ) : |
| (boundedMatrixOperator m C hb (lp.single 2 n (1 : ℂ))) k = m k n := by |
| rw [← fiberFiniteEmbedding_single n 1] |
| change ((fiberMatrixFinite m C hb).extendOfNorm fiberFiniteEmbedding |
| (fiberFiniteEmbedding (Finsupp.single n 1))) k = _ |
| rw [LinearMap.extendOfNorm_eq fiberFiniteEmbedding_dense |
| ⟨C, fiberMatrixFinite_norm_le m C hb hC⟩] |
| change fiberMatrixLinear m (Finsupp.single n 1) k = _ |
| simp [fiberMatrixLinear] |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
