Prove2Me

`BookProof.BrstUnboundedLeakage.norm_omega_stoneU_eq` {Om : H →L[ℂ] H} (hcomm : ∀ (s : ℝ) (y : H), Om (T.stoneU s y) = T.stoneU s (Om y)) (t : ℝ) (x : H) : ‖Om (T.stoneU t x)‖ = ‖O

Proved
BookProof.BrstUnboundedLeakage.norm_omega_stoneU_eq

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBrstUnboundedLeakage.

BookProof.BrstUnboundedLeakage.norm_omega_stoneU_eq {Om : H →L[ℂ] H} (hcomm : ∀ (s : ℝ) (y : H), Om (T.stoneU s y) = T.stoneU s (Om y)) (t : ℝ) (x : H) : ‖Om (T.stoneU t x)‖ = ‖Om x‖

Formalization note: Lean 4 identifier BookProof.BrstUnboundedLeakage.norm_omega_stoneU_eq.

Preamble
-- Generated from ChapterBrstUnboundedLeakage.lean — theorem BookProof.BrstUnboundedLeakage.norm_omega_stoneU_eq
import Definitions.Def_ChapterBrstTruncationLeakage
import Definitions.Def_ChapterSirkTrotterKato
import Mathlib
import Definitions.Def_ChapterBrstUnboundedLeakage
import Definitions.Def_ChapterStoneResolvent
import Definitions.Def_ChapterStoneUnitary
open BookProof.BrstUnboundedLeakage


open NormedSpace Filter Topology
open scoped InnerProductSpace


open BookProof.BrstLeakage BookProof.ChapterStoneResolvent

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

variable (T : UnboundedSelfAdjoint H)
Formal statement
theorem BookProof.BrstUnboundedLeakage.norm_omega_stoneU_eq {Om : H →L[ℂ] H}
    (hcomm : ∀ (s : ℝ) (y : H), Om (T.stoneU s y) = T.stoneU s (Om y)) (t : ℝ) (x : H) :
    ‖Om (T.stoneU t x)‖ = ‖Om x‖ := by sorry

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