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f1f7953 | 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 61 62 63 64 | 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
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