Derivations in Banach algebras (Q1607888)

The authors prove the following results. 1. Let \(d\) and \(g\) be derivations on a Banach algebra \(A\). If there exists a positive integer \(n\) and \(\alpha\in \mathbb{C}\) such that \(\alpha d^2+g\) is \(n\)-commuting on \(A\), then both \(d\) and \(g\) map \(A\) into \(\text{rad} (A)\). 2. Let \(d\) and \(g\) be continuous derivations on a Banach algebra \(A\). If there exist a positive integer \(n\) and \(\alpha\in \mathbb{C}\) such that the mapping \(\alpha d^2+g\) is \(n\)-centralizing on \(A\), then both \(d\) and \(g\) map \(A\) into \(\text{rad} (A)\).