`BookProof.ChapterNumericalRangeSemigroup.shiftEquiv_apply` {A : E →L[ℂ] E} {ω : ℝ} (h : NumReLE A ω) {z : ℂ} (hz : ω < z.re) (x : E) : shiftEquiv h hz x = (z • (1 : E →L[ℂ] E) - A

Proved

BookProof.ChapterNumericalRangeSemigroup.shiftEquiv_apply

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeSemigroup.

BookProof.ChapterNumericalRangeSemigroup.shiftEquiv_apply {A : E →L[ℂ] E} {ω : ℝ} (h : NumReLE A ω) {z : ℂ} (hz : ω < z.re) (x : E) : shiftEquiv h hz x = (z • (1 : E →L[ℂ] E) - A) x

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeSemigroup.shiftEquiv_apply.

Preamble

-- Generated from ChapterNumericalRangeSemigroup.lean — theorem BookProof.ChapterNumericalRangeSemigroup.shiftEquiv_apply
import Mathlib
import Definitions.Def_ChapterNumericalRangeSemigroup
import Definitions.Def_ChapterContinuityUnitaryInfinite
open BookProof.ChapterContinuityUnitaryInfinite
open BookProof.ChapterNumericalRangeSemigroup


open scoped InnerProductSpace


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

Formal statement

theorem BookProof.ChapterNumericalRangeSemigroup.shiftEquiv_apply {A : E →L[ℂ] E} {ω : ℝ} (h : NumReLE A ω) {z : ℂ} (hz : ω < z.re)
    (x : E) : shiftEquiv h hz x = (z • (1 : E →L[ℂ] E) - A) x := by sorry

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