Corollary 5

Let A be a Banach algebra and $x\in A$.

a) Sp$'(x)$ is a compact subset of $\mathbf{C}$;

b) The spectral radius $\varrho (x)$ is the radius of the smallest closed disk with center 0 in $\mathbf{C}$ that contains Sp$'(x)$;

c) In order that $x$ be quasi-nilpotent, it is necessary and sufficient that one have Sp$'(x) =\{0\}$.

Assertions a) and b) result from th. 1 by considering the Banach algebra deduced from A by adjunction of a unit element. Assertion c) results from b).