lean/OAI/Analysis/FiniteFactor/BoundedMatrixOperator.lean

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