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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.MatrixBounds.SchattenTriangle
import OAI.Probability.BinarySweep.FiniteLaws.HilbertMainCases
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator Classical
namespace BinaryCoordinateSweeps
open TraceHolder Irrep
lemma norm_le_momentRoot {ι : Type*} [Fintype ι] [DecidableEq ι]
{q : ℕ} (hq : 0<q) (A : M ι) :
‖A‖ ≤ matrixMoment q A ^ (((2*q:ℕ):ℝ)⁻¹) := by
exact norm_le_schatten hq A
lemma norm_le_traceMomentRoot {D q : ℕ} (hq : 0<q)
(K : RepSpace D →L[ℂ] RepSpace D) :
‖K‖≤traceMoment q K ^ (((2*q:ℕ):ℝ)⁻¹) := by
let v := EuclideanSpace.basisFun (Fin D) ℂ
let A := LinearMap.toMatrixOrthonormal v K.toLinearMap
have he : (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin D)) A=K := by
apply ContinuousLinearMap.coe_injective
change (LinearMap.toMatrixOrthonormal v).symm
(LinearMap.toMatrixOrthonormal v K.toLinearMap) = K.toLinearMap
exact StarAlgEquiv.symm_apply_apply _ _
have hm : traceMoment q K=matrixMoment q A := by
have hh := evenMoment_orthonormalMatrix v q K.toLinearMap
simpa only [traceMoment,evenMoment,ContinuousLinearMap.toLinearMap_pow,
ContinuousLinearMap.toLinearMap_mul,ContinuousLinearMap.adjoint_toLinearMap] using hh
rw [hm,←he,Matrix.l2_opNorm_toEuclideanCLM]
exact norm_le_momentRoot hq A
lemma opNorm_of_logMoment {D q : ℕ} (hq : 0<q) (K : RepSpace D →L[ℂ] RepSpace D)
{a : ℝ} (h : logMoment (traceMoment q K)≤(a:EReal)) :
‖K‖≤Real.exp (a/(2*q)) := by
have ht := (logMoment_le_iff_le_exp _ _).mp h
have hm := (norm_le_traceMomentRoot hq K).trans
(Real.rpow_le_rpow (by
have hh := evenMoment_nonneg q K.toLinearMap
simpa only [traceMoment,evenMoment,ContinuousLinearMap.toLinearMap_pow,
ContinuousLinearMap.toLinearMap_mul,ContinuousLinearMap.adjoint_toLinearMap] using hh) ht (by positivity))
rw [← Real.exp_mul] at hm
simpa only [Nat.cast_mul,Nat.cast_ofNat,div_eq_mul_inv] using hm
end BinaryCoordinateSweeps
end
end OAI
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