`BookProof.UnboundedSpectralModel.cayley_symbol_identity_inc` (T : UnboundedSelfAdjoint H) : coordFn (resOp T) - star (coordFn (resOp T)) - (2 * Complex.I) • (star (coordFn (resOp T))

Proved

BookProof.UnboundedSpectralModel.cayley_symbol_identity_inc

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterUnboundedSpectralModel.

BookProof.UnboundedSpectralModel.cayley_symbol_identity_inc (T : UnboundedSelfAdjoint H) : coordFn (resOp T) - star (coordFn (resOp T)) - (2 * Complex.I) • (star (coordFn (resOp T)) * coordFn (resOp T)) = (2 * Complex.I) • cayleyFn T

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

Replaces the unuploadable node BookProof_UnboundedSpectralModel_cayley_symbol_identity, 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_symbol_identity_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
import Definitions.Def_ChapterStoneResolvent
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_symbol_identity_inc (T : UnboundedSelfAdjoint H) :
    coordFn (resOp T) - star (coordFn (resOp T))
        - (2 * Complex.I) • (star (coordFn (resOp T)) * coordFn (resOp T))
      = (2 * Complex.I) • cayleyFn T := by sorry

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