Prove2Me

`BookProof.ChapterCyclicDirectSum.commute_starProjection_cfcHom` {M : Submodule ℂ H} [M.HasOrthogonalProjection] (hM : Invariant T hT M) (g : C(spectrum ℂ T, ℂ)) : Commute M.starPr

Proved
BookProof.ChapterCyclicDirectSum.commute_starProjection_cfcHom

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCyclicDirectSum.

BookProof.ChapterCyclicDirectSum.commute_starProjection_cfcHom {M : Submodule ℂ H} [M.HasOrthogonalProjection] (hM : Invariant T hT M) (g : C(spectrum ℂ T, ℂ)) : Commute M.starProjection (cfcHom hT g)

Formalization note: Lean 4 identifier BookProof.ChapterCyclicDirectSum.commute_starProjection_cfcHom.

Preamble
-- Generated from ChapterCyclicDirectSum.lean — theorem BookProof.ChapterCyclicDirectSum.commute_starProjection_cfcHom
import Definitions.Def_ChapterCyclicDecomposition
import Mathlib
import Definitions.Def_ChapterCyclicDirectSum
open BookProof.ChapterCyclicDirectSum


noncomputable section

open MeasureTheory Complex


open BookProof.ChapterCyclicDecomposition

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : H →L[ℂ] H) (hT : IsStarNormal T)
Formal statement
theorem BookProof.ChapterCyclicDirectSum.commute_starProjection_cfcHom {M : Submodule ℂ H} [M.HasOrthogonalProjection]
    (hM : Invariant T hT M) (g : C(spectrum ℂ T, ℂ)) :
    Commute M.starProjection (cfcHom hT g) := by sorry

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