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Some Tauberian theorems related to operator theory

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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.
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  • St., John's College, Oxford OX1 3JP, England
  • [1] G. R. Allan, A. G. O'Farrell and T. J. Ransford, A Tauberian theorem arising in operator theory, Bull. London Math. Soc. 19 (1987), 537-545.
  • [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852.
  • [3] W. Arendt and C. J. K. Batty, A complex Tauberian theorem and mean ergodic semigroups, preprint.
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  • [6] C. J. K. Batty, Asymptotic behaviour of semigroups of operators, this volume, 35-52.
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  • [10] G. H. Hardy, Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc. 8 (1910), 301-320.
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  • [12] A. E. Ingham, The Distribution of Prime Numbers, Cambridge Univ. Press, Cambridge, 1932 (reprinted 1990).
  • [13] A. E. Ingham, On Wiener's method in Tauberian theorems, Proc. London Math. Soc. 38 (1935), 458-480.
  • [14] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328.
  • [15] J. Korevaar, On Newman's quick way to the prime number theorem, Math. Intelligencer 4 (1982), 108-115.
  • [16] J. E. Littlewood, On the converse of Abel's theorem on power series, Proc. London Math. Soc. 9 (1911), 434-448.
  • [17] D. J. Newman, Simple analytic proof of the prime number theorem, Amer. Math. Monthly 87 (1980), 693-696.
  • [18] H. R. Pitt, General Tauberian theorems, Proc. London Math. Soc. 44 (1938), 243-288.
  • [19] T. J. Ransford, Some quantitative Tauberian theorems for power series, Bull. London Math. Soc. 20 (1988), 37-44.
  • [20] M. Riesz, Über eine Satz des Herrn Fatou, J. Reine Angew. Math. 140 (1911), 89-99.
  • [21] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
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  • [23] E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, Oxford, 1932.
  • [24] J. van de Lune, An Introduction to Tauberian Theory: from Tauber to Wiener, Stichting Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1986.
  • [25] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, 1941.
  • [26] D. V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971.
  • [27] N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1-100.
  • [28] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge Univ. Press, Cambridge, 1933 (reprinted 1988).
  • [29] N. Wiener, I am a Mathematician, Doubleday, New York, 1956.
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