Yamamoto type theorems in Banach algebras

Abstract

The aim of this thesis is to study the asymptotic relation between the approximation numbers and isolated spectral points with finite multiplicity in a general Banach algebra setting. In 1967 T. Yamamoto was the first to show that such asymptotic results hold for the algebra of n by n matrices with entries in the complex field. About twenty years later Edmunds and Evans found a meaningful extension of Yamamoto' s Theorem for bounded operators on a Banach space. After an extensive study of the notion of finite rank elements, we extend Yamamoto's Theorem to a general Banach algebra setting. Recently, Nylen and Rodman proved a special case of the result by showing that Yamamoto's Theorem holds for Banach algebras with the spectral radius property and conjectured that any Banach algebra possesses this property. In this thesis we prove their conjecture in the affirmative.