Every Banach algebra has the spectral radius property

Nylen and Rodman [NR] introduced the notion of spectral radius property in Banach algebras in order to generalize a classical theorem of Yamamoto on the asymptotic behaviour of the singular values of an n x n matrix. In this paper we prove a conjecture of theirs in the affirmative, namely that any unital Banach algebra has the spectral radius property. In fact a slightly more general spectral property holds. Wie show that for every element which has spectral points which are not of finite multiplicity, the essential spectral radius is the supremum of the set of absolute values of the spectral points that are not of finite multiplicity.