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Tauberian theorems and stability of one-parameter semigroups. (English) Zbl 0652.47022
Let $$\{T(t):t\geq 0\}$$ be a bounded $$C_ 0$$-semigroup, with generator A, on a Banach space. Assume that the intersection of the spectrum of A with the imaginary axis is at most countable, containing no eigenvalues of the adjoint $$A^*$$. Then, for every vector x, we have $$T(t)x\to 0$$ as $$t\to \infty$$. The authors’ proof is based on Tauberian techniques involving the Laplace transform. A shorter proof of the same result was obtained a year earlier by Yu. I. Lyubich and Vũ Quôc Phóng [Stud. Math. 88, No.1, 37-42 (1988; Zbl 0639.34050)]. On the other hand, the present paper contains some interesting examples and discrete analogues for power bounded operators. A very simple proof of the result of Y.Katznelson and L. Tzafriri, quoted in Theorem 5.6, can be found in a forthcoming paper by G. R. Allan and T. J. Ransford “Power- dominated elements in a Banach algebra” [Stud. Math. 94 (1989), to appear].
Reviewer: J.Zemánek

##### MSC:
 47D03 Groups and semigroups of linear operators 47A10 Spectrum, resolvent 44A10 Laplace transform
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##### References:
 [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), no. 6, 537 – 545. · Zbl 0652.46041 · doi:10.1112/blms/19.6.537 · doi.org [2] Wolfgang Arendt and Günther Greiner, The spectral mapping theorem for one-parameter groups of positive operators on \?$$_{0}$$(\?), Semigroup Forum 30 (1984), no. 3, 297 – 330. · Zbl 0536.47032 · doi:10.1007/BF02573461 · doi.org [3] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. · Zbl 0635.47002 [4] A. E. Ingham, On Wiener’a method in Tauberian theorems, Proc. London Math. Soc. (2) 38 (1935), 458-480. · Zbl 0010.35202 [5] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313 – 328. · Zbl 0611.47005 · doi:10.1016/0022-1236(86)90101-1 · doi.org [6] J. Korevaar, On Newman’s quick way to the prime number theorem, Math. Intelligencer 4 (1982), no. 3, 108 – 115. · Zbl 0496.10027 · doi:10.1007/BF03024240 · doi.org [7] Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. · Zbl 0575.28009 [8] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. [9] D. J. Newman, Simple analytic proof of the prime number theorem, Amer. Math. Monthly 87 (1980), no. 9, 693 – 696. · Zbl 0444.10033 · doi:10.2307/2321853 · doi.org [10] Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. · Zbl 0201.45003 [11] D. V. Widder, An introduction to transform theory, Academic Press, New York, 1971. · Zbl 0219.44001 [12] Manfred Wolff, A remark on the spectral bound of the generator of semigroups of positive operators with applications to stability theory, Functional analysis and approximation (Oberwolfach, 1980) Internat. Ser. Numer. Math., vol. 60, Birkhäuser, Basel-Boston, Mass., 1981, pp. 39 – 50. · Zbl 0468.47024 [13] D. Zagier, Short proof of the prime number theorem, unpublished manuscript. · Zbl 0887.11039
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