`BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general_inc` : ∃ (S : Set H) (mu : S → Measure (spectrum ℂ T)) (V : ∀ x : S, Lp ℂ 2 (mu x) →ₗᵢ[ℂ] H), (∀ x :...

Proved

BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general_inc

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSpectralDirectSum.

BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general_inc : ∃ (S : Set H) (mu : S → Measure (spectrum ℂ T)) (V : ∀ x : S, Lp ℂ 2 (mu x) →ₗᵢ[ℂ] H), (∀ x : S, IsProbabilityMeasure (mu x)) ∧ IsHilbertSum ℂ (fun x : S => Lp ℂ 2 (mu x)) V ∧ (∀ (x : S) (u : Lp ℂ 2 (mu x)), V x (mulRep (mu x) (coordFn T) u) = T (V x u))

Formalization note: Lean 4 identifier BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general_inc.

Replaces the retired node BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general, whose stored statement omitted the hT hypothesis binder (dropped include).

Preamble

-- Generated from ChapterSpectralDirectSum.lean — theorem BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general_inc
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralMultiplication
import Definitions.Def_ChapterCyclicDecomposition
import Definitions.Def_ChapterCyclicDirectSum
import Mathlib
import Definitions.Def_ChapterSpectralDirectSum
open BookProof.ChapterSpectralDirectSum


noncomputable section

open MeasureTheory Complex


open BookProof.ChapterAbelianGelfandModel BookProof.ChapterSpectralMultiplication
open BookProof.ChapterCyclicDecomposition BookProof.ChapterCyclicDirectSum

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : H →L[ℂ] H) (hT : IsStarNormal T) (xi : H)


include hT in

Formal statement

theorem BookProof.ChapterSpectralDirectSum.spectral_multiplication_model_general_inc :
    ∃ (S : Set H) (mu : S → Measure (spectrum ℂ T))
      (V : ∀ x : S, Lp ℂ 2 (mu x) →ₗᵢ[ℂ] H),
      (∀ x : S, IsProbabilityMeasure (mu x)) ∧
      IsHilbertSum ℂ (fun x : S => Lp ℂ 2 (mu x)) V ∧
      (∀ (x : S) (u : Lp ℂ 2 (mu x)),
        V x (mulRep (mu x) (coordFn T) u) = T (V x u)) := by sorry

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