`BookProof.ChapterSchurIrreducible.selfAdjoint_commutant_scalar` (M : System ℂ V) (hirr : M.IsIrreducible) {T : V →L[ℂ] V} (hT : IsSelfAdjoint T) (hcomm : M.Commutes T) : ∃ c : ℝ,

Proved

BookProof.ChapterSchurIrreducible.selfAdjoint_commutant_scalar

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSchurIrreducible.

BookProof.ChapterSchurIrreducible.selfAdjoint_commutant_scalar (M : System ℂ V) (hirr : M.IsIrreducible) {T : V →L[ℂ] V} (hT : IsSelfAdjoint T) (hcomm : M.Commutes T) : ∃ c : ℝ, T = (c : ℂ) • (1 : V →L[ℂ] V)

Formalization note: Lean 4 identifier BookProof.ChapterSchurIrreducible.selfAdjoint_commutant_scalar.

Preamble

-- Generated from ChapterSchurIrreducible.lean — theorem BookProof.ChapterSchurIrreducible.selfAdjoint_commutant_scalar
import Mathlib
import Definitions.Def_ChapterSchurIrreducible
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterSchurIrreducible


open scoped ComplexConjugate InnerProductSpace


open BookProof.ChapterA BookProof.ChapterA

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

Formal statement

theorem BookProof.ChapterSchurIrreducible.selfAdjoint_commutant_scalar (M : System ℂ V) (hirr : M.IsIrreducible)
    {T : V →L[ℂ] V} (hT : IsSelfAdjoint T) (hcomm : M.Commutes T) :
    ∃ c : ℝ, T = (c : ℂ) • (1 : V →L[ℂ] V) := by sorry

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