`BookProof.ChapterCayleySpectralModel.unbounded_spectral_multiplication_model` (hxi : ‖xi‖ = 1) (hcyc : DenseRange (cfcVec (cayleyCLM T) (isStarNormal_cayleyCLM T) xi)) : ∃ (mu : M

Proved

BookProof.ChapterCayleySpectralModel.unbounded_spectral_multiplication_model

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCayleySpectralModel.

BookProof.ChapterCayleySpectralModel.unbounded_spectral_multiplication_model (hxi : ‖xi‖ = 1) (hcyc : DenseRange (cfcVec (cayleyCLM T) (isStarNormal_cayleyCLM T) xi)) : ∃ (mu : Measure (spectrum ℂ (cayleyCLM T))) (_ : IsProbabilityMeasure mu) (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H), (∀ u : Lp ℂ 2 mu, ∃ hmem : U (mulRep mu (resSymbol T) u) ∈ T.domain, T.op ⟨U (mulRep mu (resSymbol T) u), hmem⟩ = U (mulRep mu (opSymbol T) u)) ∧ (∀ x : T.domain, ∃ u : Lp ℂ 2 mu, U (mulRep mu (resSymbol T) u) = (x : H))

Formalization note: Lean 4 identifier BookProof.ChapterCayleySpectralModel.unbounded_spectral_multiplication_model.

Preamble

-- Generated from ChapterCayleySpectralModel.lean — theorem BookProof.ChapterCayleySpectralModel.unbounded_spectral_multiplication_model
import Definitions.Def_ChapterUnitaryTransport
import Definitions.Def_ChapterSirkTrotterKato
import Definitions.Def_ChapterCayleyTransform
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralMultiplication
import Mathlib
import Definitions.Def_ChapterCayleySpectralModel
import Definitions.Def_ChapterStoneResolvent
open BookProof.ChapterCayleySpectralModel


open scoped InnerProductSpace
open MeasureTheory


open BookProof.ChapterUnitaryTransport BookProof.ChapterStoneResolvent
open BookProof.ChapterCayleyTransform BookProof.ChapterAbelianGelfandModel
open BookProof.ChapterSpectralMultiplication

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : UnboundedSelfAdjoint H)

variable (xi : H)

Formal statement

theorem BookProof.ChapterCayleySpectralModel.unbounded_spectral_multiplication_model (hxi : ‖xi‖ = 1)
    (hcyc : DenseRange (cfcVec (cayleyCLM T) (isStarNormal_cayleyCLM T) xi)) :
    ∃ (mu : Measure (spectrum ℂ (cayleyCLM T))) (_ : IsProbabilityMeasure mu)
      (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H),
      (∀ u : Lp ℂ 2 mu, ∃ hmem : U (mulRep mu (resSymbol T) u) ∈ T.domain,
        T.op ⟨U (mulRep mu (resSymbol T) u), hmem⟩ = U (mulRep mu (opSymbol T) u)) ∧
      (∀ x : T.domain, ∃ u : Lp ℂ 2 mu, U (mulRep mu (resSymbol T) u) = (x : H)) := by sorry

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