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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 | import OAI.Analysis.FiniteFactor.Convolution
import OAI.Analysis.FiniteFactor.Fiber.ProjectionDiagonal
namespace OAI
namespace FiniteFactor
open scoped ComplexConjugate
noncomputable section
/-- Composition of the two genuine range inclusions used in the convolution. -/
theorem invariant_transport_composition {R : Type*} [Ring R] (P Q D W : R)
(hD : (1-Q)*D*P = 0) (hW : (1-Q)*W*Q = 0) :
(1-Q)*W*D*P = 0 := by
have hd : D*P = Q*D*P := by
apply sub_eq_zero.mp
simpa only [sub_mul,one_mul] using hD
calc
_ = (1-Q)*W*(D*P) := by simp only [mul_assoc]
_ = (1-Q)*W*(Q*D*P) := by rw [hd]
_ = ((1-Q)*W*Q)*D*P := by simp only [mul_assoc]
_ = 0 := by rw [hW,zero_mul,zero_mul]
theorem fiber_complement_inner_zero
(Q : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) (hQ : IsStarProjection Q)
(u v : SourceFiberHilbert) (hv : (1-Q) v = 0) :
inner ℂ ((1-Q) u) v = 0 := by
rw [← ContinuousLinearMap.adjoint_inner_right (1-Q) u v,
← ContinuousLinearMap.star_eq_adjoint,hQ.one_sub.isSelfAdjoint.star_eq,hv,inner_zero_right]
/-- The source convolution vanishes whenever the actual transport and weighted
power identities hold. No square summability of rescaled coefficients is assumed. -/
theorem fiber_transport_convolution_zero
(P Q D W : SourceFiberHilbert →L[ℂ] SourceFiberHilbert)
(hQ : IsStarProjection Q) (F G : ℤ → ℝ) (hG : ∀ k, G k ≠ 0) (n : ℤ)
(hD : (1-Q)*D*P = 0) (hW : (1-Q)*W*Q = 0)
(hDa : ∀ (v : SourceFiberHilbert) (k : ℤ), D v k = (G k / F k : ℂ) * v k)
(hWa : ∀ (v : SourceFiberHilbert) (k : ℤ), W v (n+k) = (G (n+k) / G k : ℂ) * v k) :
(∑' k, leftCoefficient F (P (lp.single 2 0 (1 : ℂ))) k *
rightCoefficient G ((1-Q) (lp.single 2 0 (1 : ℂ))) (-n-k)) = 0 := by
let ξ := P (lp.single 2 0 (1 : ℂ))
let γ := (1-Q) (lp.single 2 0 (1 : ℂ))
let η := W (D ξ)
have hη : ∀ k, η (n+k) = ((G (n+k) / F k : ℝ) : ℂ) * ξ k := by
intro k
dsimp [η]
rw [hWa,hDa,Complex.ofReal_div]
have hg : (G k : ℂ) ≠ 0 := by exact_mod_cast hG k
field_simp
have hz : (1-Q) η = 0 := by
exact congrArg (fun A : SourceFiberHilbert →L[ℂ] SourceFiberHilbert => A (lp.single 2 0 (1 : ℂ)))
(invariant_transport_composition P Q D W hD hW)
simp only [cocycle_convolution_term_eq]
rw [weighted_shift_convolution_eq_inner ξ γ η (fun k => G (n+k)/F k) n hη]
exact fiber_complement_inner_zero Q hQ _ η hz
end
end FiniteFactor
end OAI
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