`BookProof.SpectralGapStability.notMem_spectrum_of_gapAt` {A : F →L[ℂ] F} (hA : IsSelfAdjoint A) {lam d : ℝ} (hd : 0 < d) (h : GapAt A lam d) : (lam : ℂ) ∉ spectrum ℂ A

Proved

BookProof.SpectralGapStability.notMem_spectrum_of_gapAt

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSpectralGapStability.

BookProof.SpectralGapStability.notMem_spectrum_of_gapAt {A : F →L[ℂ] F} (hA : IsSelfAdjoint A) {lam d : ℝ} (hd : 0 < d) (h : GapAt A lam d) : (lam : ℂ) ∉ spectrum ℂ A

Formalization note: Lean 4 identifier BookProof.SpectralGapStability.notMem_spectrum_of_gapAt.

Preamble

-- Generated from ChapterSpectralGapStability.lean — theorem BookProof.SpectralGapStability.notMem_spectrum_of_gapAt
import Mathlib
import Definitions.Def_ChapterSpectralGapStability
open BookProof.SpectralGapStability


noncomputable section


open scoped InnerProductSpace
open Filter Topology

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

Formal statement

theorem BookProof.SpectralGapStability.notMem_spectrum_of_gapAt {A : F →L[ℂ] F} (hA : IsSelfAdjoint A) {lam d : ℝ}
    (hd : 0 < d) (h : GapAt A lam d) : (lam : ℂ) ∉ spectrum ℂ A := by sorry

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