openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /PartialResolventIntertwining.lean
| import Mathlib.Analysis.InnerProductSpace.LinearPMap | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.PartialResolvent | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] | |
| variable (H : E →ₗ.[ℂ] E) (R : E →L[ℂ] E) | |
| variable (hD : ∀ x : E, R x ∈ H.domain) | |
| variable (hHR : ∀ x : E, H ⟨R x, hD x⟩ = x - R x) | |
| variable (hRH : ∀ x : H.domain, R ((x : E) + H x) = x) | |
| include hD hHR hRH | |
| theorem normalized_intertwining (A B : E →L[ℂ] E) | |
| (hA : ∀ x : H.domain, A (x : E) ∈ H.domain) | |
| (hI : ∀ x : H.domain, H ⟨A (x : E), hA x⟩ = B (H x)) : | |
| (1 - R) * A * R = R * B * (1 - R) := by | |
| ext x | |
| change A (R x) - R (A (R x)) = R (B (x - R x)) | |
| have hi := hI ⟨R x, hD x⟩ | |
| rw [hHR] at hi | |
| have hr := hRH ⟨A (R x), hA ⟨R x, hD x⟩⟩ | |
| rw [hi, map_add] at hr | |
| exact sub_eq_iff_eq_add.mpr (by simpa only [add_comm] using hr.symm) | |
| theorem domain_intertwining_of_normalized (A B : E →L[ℂ] E) | |
| (hI : (1 - R) * A * R = R * B * (1 - R)) : | |
| ∃ hA : ∀ x : H.domain, A (x : E) ∈ H.domain, | |
| ∀ x : H.domain, H ⟨A (x : E), hA x⟩ = B (H x) := by | |
| have hrep : ∀ x : H.domain, R (A (x : E) + B (H x)) = A (x : E) := by | |
| intro x | |
| have he := congrArg (fun T : E →L[ℂ] E => T ((x : E) + H x)) hI | |
| change A (R ((x : E) + H x)) - R (A (R ((x : E) + H x))) = | |
| R (B ((x : E) + H x - R ((x : E) + H x))) at he | |
| rw [hRH, add_sub_cancel_left] at he | |
| rw [map_add] | |
| exact (by simpa only [add_comm] using (sub_eq_iff_eq_add.mp he).symm) | |
| let hA : ∀ x : H.domain, A (x : E) ∈ H.domain := | |
| fun x => hrep x ▸ hD (A (x : E) + B (H x)) | |
| refine ⟨hA, fun x => ?_⟩ | |
| have hx : (⟨R (A (x : E) + B (H x)), hD _⟩ : H.domain) = | |
| ⟨A (x : E), hA x⟩ := Subtype.ext (hrep x) | |
| have hh := hHR (A (x : E) + B (H x)) | |
| rw [hx, hrep x, add_sub_cancel_left] at hh | |
| exact hh | |
| theorem normalized_intertwining_iff (A B : E →L[ℂ] E) : | |
| (1 - R) * A * R = R * B * (1 - R) ↔ | |
| ∃ hA : ∀ x : H.domain, A (x : E) ∈ H.domain, | |
| ∀ x : H.domain, H ⟨A (x : E), hA x⟩ = B (H x) := by | |
| constructor | |
| · exact domain_intertwining_of_normalized H R hD hHR hRH A B | |
| · rintro ⟨hA, hI⟩ | |
| exact normalized_intertwining H R hD hHR hRH A B hA hI | |
| end Invariant233.PartialResolvent | |
| end | |
| end OAI | |