{F : Type} [NormedAddCommGroup F] [NormedSpace ℂ F] (T : F →L[ℂ] F) (X : F →L[ℂ] F) (γ z : ℂ) (v : F) (hTv : T v = z • v) (hz : γ - z ≠ 0) (_hXr : (γ • (1 : F →L[ℂ] F) - T) X = 1) (hXl : X *...

Proved

BookProof.ChapterH1.resolvent_eigenvector

by leonardopedro · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

timepiece

Lean 4 theorem BookProof.ChapterH1.resolvent_eigenvector (module BookProof.ChapterH1), source chapter BookProof/ChapterChapterH1.lean.

Preamble

-- Generated from ChapterH1.lean — theorem BookProof.ChapterH1.resolvent_eigenvector
import Mathlib
import Definitions.Def_ChapterH1
open BookProof.ChapterH1






open scoped BigOperators
open intervalIntegral


noncomputable section














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

Formal statement

theorem BookProof.ChapterH1.resolvent_eigenvector {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
    (T : F →L[ℂ] F) (X : F →L[ℂ] F) (γ z : ℂ) (v : F)
    (hTv : T v = z • v) (hz : γ - z ≠ 0)
    (_hXr : (γ • (1 : F →L[ℂ] F) - T) * X = 1)
    (hXl : X * (γ • (1 : F →L[ℂ] F) - T) = 1) :
    X v = (γ - z)⁻¹ • v := by sorry

Source

https://github.com/leonardopedrio/timepiece/blob/61595bc/BookProof/ChapterChapterH1.lean

View graph

{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] (T : F →L[ℂ] F) (X : F →L[ℂ] F) (γ z : ℂ) (v : F) (hTv : T v = z • v) (hz : γ - z ≠ 0) (_hXr : (γ • (1 : F →L[ℂ] F) - T) * X = 1) (hXl : X *...

{F : Type} [NormedAddCommGroup F] [NormedSpace ℂ F] (T : F →L[ℂ] F) (X : F →L[ℂ] F) (γ z : ℂ) (v : F) (hTv : T v = z • v) (hz : γ - z ≠ 0) (_hXr : (γ • (1 : F →L[ℂ] F) - T) X = 1) (hXl : X *...