Corollary

Let A be a Banach algebra, and let $x$ and $y$ be permutable elements of A.

a) One has $\varrho (xy)\leqslant \varrho (x)\varrho (y)$ and $\varrho (x+y)\leqslant \varrho (x) +\varrho (y)$;

b) If $y$ is quasi-nilpotent, then Sp$'_A(x) =$ Sp$'_A(x+y)$; if moreover A is unital, then Sp$_A(x) =$ Sp$_A(x+y)$.

By considering the Banach algebra deduced from A by adjoining a unit element, one is first reduced to the case where the algebra A is unital. Then by considering the closed full subalgebra of A generated by $x$ and $y$, one is reduced to the case where A is commutative and unital. Assertion a) is then a consequence of Prop. 7, a), and assertion b) follows from Prop. 6 of I, p. 37 and Cor. 5 of I, p. 28.