English
If an invariant subspace p is disjoint from the μ-eigenspace, then the restricted μ-eigenspace is the bottom subspace ⊥.
Русский
Если инвариантное подпространство p бючится дисjoint от μ-эйгенпространства, то ограниченное μ-эйгенпространство равно нулю.
LaTeX
$$$\\text{Disjoint}(f.eigenspace(μ), p) \\Rightarrow \\mathrm{eigenspace}(f|_p, μ) = ⊥$$$
Lean4
/-- If an invariant subspace `p` of an endomorphism `f` is disjoint from the `μ`-eigenspace of `f`,
then the restriction of `f` to `p` has trivial `μ`-eigenspace. -/
theorem eigenspace_restrict_eq_bot {f : End R M} {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p) {μ : R}
(hμp : Disjoint (f.eigenspace μ) p) : eigenspace (f.restrict hfp) μ = ⊥ :=
by
rw [eq_bot_iff]
intro x hx
simpa using hμp.le_bot ⟨eigenspace_restrict_le_eigenspace f hfp μ ⟨x, hx, rfl⟩, x.prop⟩
