| 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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|
