Derivations mapping into the radical. III (Q1908093)

The authors obtain commutativity-free characterizations of those derivations \(d\) on a unital complex Banach algebra \(A\) that map \(A\) into its radical: \(dA\subseteq \text{rad}(A)\) if and only if there exists a constant \(M\geq 0\) such that \(r(dx)\leq Mr(x)\) for all \(x\in A\), which in turn is equivalent to \(\sup\{r(z^{-1}dz)\mid z\in A\text{ invertible}\}<\infty\) (where \(r(\cdot)\) is denoting the spectral radius). The second characterization answers positively a question by J. Zemánek. The authors also prove the following result. Let \(\delta= L_a+ d\) with \(a= \delta(1)\) be a generalized derivation on a unital Banach algebra \(A\). The following conditions are equivalent. (a) \(\delta\) is spectrally bounded. (b) Both \(L_a\) and \(d\) are spectrally bounded. [For part II see the second author and \textit{V. Runde}, Bull. Lond. Math. Soc. 24, No. 5, 485-487 (1992; Zbl 0760.46042)].