On the Kleinecke--Shirokov Theorem for families of derivations (Q2773418)

A derivation of an algebra \(A\) is a linear operator \(d: A\to A\) satisfing the Leibniz rule. Given an element \(a\in A\), let \(L_a\) and \(R_a\) be the corresponding multiplication operators on \(A\) defined by \(L_a x=ax\) and \(R_ax=xa\) for all \(x\in A\). If \(A\) is a Banach algebra, an element \(a\in A\) is then called a Riesz element if \(L_aR_a\) is a Riesz operator on \(A\). The authors prove that Riesz elements in the intersection of the kernel and the closure of the image of a family of (not necessarily bounded) derivations on a Banach algebra are quasinilpotent. Some related results are also obtained.