`BookProof.UnboundedSpectralModel.cayley_pointwise_inc` (w : ℂ) : w - (starRingEnd ℂ) w - (2 * Complex.I) * ((starRingEnd ℂ) w * w) = (2 * Complex.I) * (((w.im - ‖w‖ ^ 2 : ℝ) : ℂ))

Proved

BookProof.UnboundedSpectralModel.cayley_pointwise_inc

by leonardopedro · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterUnboundedSpectralModel.

BookProof.UnboundedSpectralModel.cayley_pointwise_inc (w : ℂ) : w - (starRingEnd ℂ) w - (2 * Complex.I) * ((starRingEnd ℂ) w * w) = (2 * Complex.I) * (((w.im - ‖w‖ ^ 2 : ℝ) : ℂ))

Formalization note: Lean 4 identifier BookProof.UnboundedSpectralModel.cayley_pointwise_inc.

Replaces the unuploadable node BookProof_UnboundedSpectralModel_cayley_pointwise, whose statement name is already declared inside its published definition bundle; the _inc suffix publishes the same statement as its own node.

Preamble

-- Generated from ChapterUnboundedSpectralModel.lean — theorem BookProof.UnboundedSpectralModel.cayley_pointwise_inc
import Definitions.Def_ChapterUnitaryTransport
import Definitions.Def_ChapterSirkTrotterKato
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralMultiplication
import Definitions.Def_ChapterSpectralDirectSum
import Mathlib
import Definitions.Def_ChapterUnboundedSpectralModel
open BookProof.UnboundedSpectralModel


noncomputable section

open MeasureTheory Complex
open scoped InnerProductSpace


open BookProof.ChapterUnitaryTransport BookProof.ChapterStoneResolvent
open BookProof.ChapterAbelianGelfandModel BookProof.ChapterSpectralMultiplication
open BookProof.ChapterSpectralDirectSum

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

Formal statement

theorem BookProof.UnboundedSpectralModel.cayley_pointwise_inc (w : ℂ) :
    w - (starRingEnd ℂ) w - (2 * Complex.I) * ((starRingEnd ℂ) w * w)
      = (2 * Complex.I) * (((w.im - ‖w‖ ^ 2 : ℝ) : ℂ)) := by sorry

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