Some Tauberian theorems related to operator theory

C. Batty

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 21-34
  • ISSN: 0137-6934

Abstract

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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, Puss, 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.

How to cite

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Batty, C.. "Some Tauberian theorems related to operator theory." Banach Center Publications 30.1 (1994): 21-34. <http://eudml.org/doc/262766>.

@article{Batty1994,
abstract = { 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, Puss, 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. },
author = {Batty, C.},
journal = {Banach Center Publications},
keywords = {Tauberian theorems; semigroups of operators; Laplace transform; power series; Dirichlet series},
language = {eng},
number = {1},
pages = {21-34},
title = {Some Tauberian theorems related to operator theory},
url = {http://eudml.org/doc/262766},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Batty, C.
TI - Some Tauberian theorems related to operator theory
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 21
EP - 34
AB - 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, Puss, 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.
LA - eng
KW - Tauberian theorems; semigroups of operators; Laplace transform; power series; Dirichlet series
UR - http://eudml.org/doc/262766
ER -

References

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