| import OAI.Analysis.FiniteFactor.BoundedMatrixOperator |
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| namespace OAI |
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| namespace FiniteFactor |
| open Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| /-- Coordinate translation on the original l2 fiber. -/ |
| def fiberShiftVector (n : ℤ) (v : SourceFiberHilbert) : SourceFiberHilbert := |
| ⟨fun k => v (k-n), memℓp_gen |
| ((Equiv.subRight n).summable_iff.mpr ((lp.memℓp v).summable (by norm_num)))⟩ |
|
|
| theorem fiberShiftVector_norm (n : ℤ) (v : SourceFiberHilbert) : |
| ‖fiberShiftVector n v‖ = ‖v‖ := by |
| have he : ‖fiberShiftVector n v‖ ^ 2 = ‖v‖ ^ 2 := by |
| rw [show ‖fiberShiftVector n v‖ ^ 2 = |
| ∑' k, ‖fiberShiftVector n v k‖ ^ 2 by |
| simpa using lp.norm_rpow_eq_tsum (p := (2 : ℝ≥0∞)) (by norm_num) (fiberShiftVector n v)] |
| rw [show ‖v‖ ^ 2 = ∑' k, ‖v k‖ ^ 2 by |
| simpa using lp.norm_rpow_eq_tsum (p := (2 : ℝ≥0∞)) (by norm_num) v] |
| exact (Equiv.subRight n).tsum_eq (fun k => ‖v k‖ ^ 2) |
| nlinarith [norm_nonneg (fiberShiftVector n v), norm_nonneg v] |
|
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| def fiberShift (n : ℤ) : SourceFiberHilbert →L[ℂ] SourceFiberHilbert := |
| LinearMap.mkContinuous |
| { toFun := fiberShiftVector n |
| map_add' := fun u v => rfl |
| map_smul' := fun c v => rfl } |
| 1 (fun v => by |
| change ‖fiberShiftVector n v‖ ≤ 1 * ‖v‖ |
| rw [one_mul, fiberShiftVector_norm]) |
|
|
| @[simp] theorem fiberShift_apply (n : ℤ) (v : SourceFiberHilbert) (k : ℤ) : |
| fiberShift n v k = v (k-n) := rfl |
|
|
| theorem fiberShift_single (n m : ℤ) (c : ℂ) : |
| fiberShift n (lp.single 2 m c) = lp.single 2 (m+n) c := by |
| classical |
| apply Subtype.ext |
| funext k |
| change (lp.single 2 m c : SourceFiberHilbert) (k-n) = |
| (lp.single 2 (m+n) c : SourceFiberHilbert) k |
| by_cases h : m = k-n |
| · simp [lp.single_apply, h] |
| · have hh : m+n ≠ k := by omega |
| simp [lp.single_apply, h, hh] |
|
|
| @[simp] theorem fiberShift_mul (n m : ℤ) : fiberShift n * fiberShift m = fiberShift (n+m) := by |
| apply ContinuousLinearMap.ext |
| intro v |
| apply Subtype.ext |
| funext k |
| change v (k-n-m) = v (k-(n+m)) |
| exact congrArg v (by omega) |
|
|
| @[simp] theorem fiberShift_zero : fiberShift 0 = 1 := by |
| apply ContinuousLinearMap.ext |
| intro v |
| apply Subtype.ext |
| funext k |
| change v (k-0) = v k |
| rw [sub_zero] |
|
|
| theorem fiber_covariance_operator |
| (A B : SourceFiberHilbert →L[ℂ] SourceFiberHilbert) (n : ℤ) |
| (h : ∀ k m : ℤ, (B (lp.single 2 m (1 : ℂ))) k = |
| (A (lp.single 2 (m+n) (1 : ℂ))) (k+n)) : |
| fiberShift (-n) * A = B * fiberShift (-n) := by |
| apply fiberOperator_ext |
| intro k m |
| change (A (lp.single 2 m (1 : ℂ))) (k-(-n)) = |
| (B (fiberShift (-n) (lp.single 2 m (1 : ℂ)))) k |
| rw [fiberShift_single, h] |
| simp |
|
|
| /-- Invariance of a projection range persists under every nonnegative power. -/ |
| theorem invariant_projection_pow {R : Type*} [Ring R] (p T : R) |
| (hp : p*p = p) (hinv : (1-p)*T*p = 0) : ∀ n : ℕ, (1-p)*(T^n)*p = 0 := by |
| have hTp : T*p = p*T*p := by |
| have h := hinv |
| rw [sub_mul, one_mul, sub_mul] at h |
| exact sub_eq_zero.mp h |
| have hfix : ∀ n : ℕ, T^n*p = p*(T^n)*p := by |
| intro n |
| induction n with |
| | zero => simpa using hp.symm |
| | succ n ih => |
| calc |
| T^(n+1)*p = T*(T^n*p) := by rw [pow_succ', mul_assoc] |
| _ = T*(p*T^n*p) := by rw [ih] |
| _ = (T*p)*T^n*p := by simp only [mul_assoc] |
| _ = (p*T*p)*T^n*p := by rw [hTp] |
| _ = p*T*(p*T^n*p) := by simp only [mul_assoc] |
| _ = p*T*(T^n*p) := by rw [← ih] |
| _ = p*T^(n+1)*p := by rw [pow_succ']; simp only [mul_assoc] |
| intro n |
| rw [sub_mul, one_mul, sub_mul, hfix n] |
| simp only [mul_assoc, sub_self] |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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