Remark 2

Let A be a unital Banach algebra and let $x\in A$. By Theorem 1 of I, p. 24$,\mathbf{C}-$ Sp$_A(x)$ is an open subset of $\mathbf{C}$, hence is locally connected. Therefore the connected components of $\mathbf{C}-$ Sp$_A(x)$ are open. By Theorem 1, one of these connected components contains the set of $\lambda \in \mathbf{C}$ such that $|\lambda |> \varrho (x)$; all the other connected components are therefore bounded.