Some Tauberian Theorems Related to Operator Theory

This article is a survey of some Tauberian theorems obtained recently in connection with work on asymptotic behaviour of semigroups of operators on Banach spaces. The results in operator theory are described in [6], where we made little attempt to show the Tauberian aspects. At the end of this article, we will give a sketch of the connections between the results in this article and in [6]; for details, the reader can turn to the original papers. In this article, we make no attempt to describe applications of Tauberian theorems in other areas such as number theory and probability theory, apart from a few historical remarks concerning proofs of the Prime Number Theorem. We begin with a summary of some of the classical Tauberian theorems, which will serve to put the recent results in perspective. Fuller accounts of the classical theory may be found in standard texts such as [9], [26], and in the historical account of van de Lune [24]. In Section 2, we introduce some of the tricks of the trade by applying them to refine the classical theorems. In Section 3, we give the recent results, due to Allan, Arendt, Katznelson, O’Farrell, Prüss, Ransford, Tzafriri and the author [1]–[5], [14], [19], all of which can be obtained from contour integral methods originating in an idea of Newman [17], adapted by Korevaar [15]. Throughout this article, we will state results for the complex-valued case. However, all the results have Banach space-valued versions, and it is those which are required for the applications to operator theory.