Translation of the spectrum by a scalar ($\sigma (x+\alpha e)=\alpha +\sigma (x)$)

The identity $(x+\alpha e)-\lambda e=x-(\lambda -\alpha )e$ proves the formula directly from invertibility. In a nonzero complex unital Banach algebra, the nonemptiness of the Banach-algebra spectrum implies that if $x$ is a quasinilpotent element, then $\rho (x+e)=1$. Consequently every nonscalar element cannot simultaneously be quasinilpotent: adding the identity preserves nonscalarity while translating its spectrum.