Lipschitz extensions of linear operators

Let E, F, E₀ be Banach spaces, with E₀ a subspace of E. For a maximal Banach operator ideal A, we show that a linear operator from E₀ to F can be extended to a linear operator from E to F that belongs to A if and only if it can be extended to a Lipschitz map from E to F belonging to a wide class of Lipschitz Banach operator ideals related with A. As a consequence, we show that linear operators with special Lipschitz factorization through ∞(Γ) has analogous linear factorization through ∞(Γ).