Uniqueness under spectral variation in the socle of a Banach algebra (Q739550)

Let \(A\) be a semisimple Banach algebra. By \(\sigma(a)\) and \(\rho(a)\) we denote the spectrum and the spectral radius of \(a\), respectively. The paper is centered around conditions (1) \(\sigma(ax)=\sigma(bx)\) and (2) \(\rho(ax)\leq \rho(bx)\) for all \(x\) from some subset of \(A\). We will state only two sample results. If \(a,b\) lie in \(\operatorname{Soc}A\), the socle of \(A\), and (1) holds for every rank one element \(x\in A\), then \(a=b\). If \(A\) is prime, \(\operatorname{Soc}A\neq \{0\}\), and (2) holds for every \(x\in A\), then \(a\) and \(b\) are linearly dependent.