Derivations decreasing the spectral radius
The following theorem is proved: Let \(A\) be a unital Banach algebra and let \(d\) be an inner derivation of \(A\). Suppose that for some positive number \(M\) and for all \(x\in A\) the inequality \(r(d(x))\leq Mr(x)\) holds. Then \(d\) maps \(A\) into \(\text{rad}(A)\).
scientific article; zbMATH DE number 446280