Derivations Mapping into the Radical III

We obtain commutativity-free characterizations of those derivations d on a unital complex Banach algebra A that map A into its radical: dA ⊆ rad(A) if and only if there exists a constant M ≥ 0 such that r(dx) ≤ Mr(x) for all x ∈ A, which in turn is equivalent to sup{r(z−1dz) | z ∈ A invertible} < ∈ (where r(·) is denoting the spectral radius). The second characterization answers positively a question raised by J. Zemánek.