Prove2Me

`BookProof.ChapterCyclicDirectSum.starProjection_commutes_cfcHom` {M : Submodule ℂ H} [M.HasOrthogonalProjection] (hM : Invariant T hT M) (g : C(spectrum ℂ T, ℂ)) (v : H) : M.starP

Proved
BookProof.ChapterCyclicDirectSum.starProjection_commutes_cfcHom

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCyclicDirectSum.

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

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

Preamble
-- Generated from ChapterCyclicDirectSum.lean — theorem BookProof.ChapterCyclicDirectSum.starProjection_commutes_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.starProjection_commutes_cfcHom {M : Submodule ℂ H} [M.HasOrthogonalProjection]
    (hM : Invariant T hT M) (g : C(spectrum ℂ T, ℂ)) (v : H) :
    M.starProjection (cfcHom hT g v) = cfcHom hT g (M.starProjection v) := by sorry

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