## Tauberian theorems and stability of one-parameter semigroups

HTML articles powered by AMS MathViewer

- by W. Arendt and C. J. K. Batty PDF
- Trans. Amer. Math. Soc.
**306**(1988), 837-852 Request permission

## Abstract:

The main result is the following stability theorem: Let $\mathcal {T} = {(T(t))_{t \geqslant 0}}$ be a bounded ${C_0}$-semigroup on a reflexive space $X$. Denote by $A$ the generator of $\mathcal {T}$ and by $\sigma (A)$ the spectrum of $A$. If $\sigma (A) \cap i{\mathbf {R}}$ is countable and no eigenvalue of $A$ lies on the imaginary axis, then ${\lim _{t \to \infty }}T(t)x = 0$ for all $x \in X$.## References

- 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. MR**915430**, DOI 10.1112/blms/19.6.537 - Wolfgang Arendt and Günther Greiner,
*The spectral mapping theorem for one-parameter groups of positive operators on $C_0(X)$*, Semigroup Forum**30**(1984), no. 3, 297–330. MR**765499**, DOI 10.1007/BF02573461 - 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. MR**1009162**
A. E. Ingham, - Y. Katznelson and L. Tzafriri,
*On power bounded operators*, J. Funct. Anal.**68**(1986), no. 3, 313–328. MR**859138**, DOI 10.1016/0022-1236(86)90101-1 - J. Korevaar,
*On Newman’s quick way to the prime number theorem*, Math. Intelligencer**4**(1982), no. 3, 108–115. MR**684025**, DOI 10.1007/BF03024240 - Ulrich Krengel,
*Ergodic theorems*, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR**797411**, DOI 10.1515/9783110844641 - 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. MR**839450**, DOI 10.1007/BFb0074922 - D. J. Newman,
*Simple analytic proof of the prime number theorem*, Amer. Math. Monthly**87**(1980), no. 9, 693–696. MR**602825**, DOI 10.2307/2321853 - Béla Sz.-Nagy and Ciprian Foiaş,
*Harmonic analysis of operators on Hilbert space*, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR**0275190**
D. V. Widder, - 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. MR**650263**
D. Zagier,

*On Wiener’a method in Tauberian theorems*, Proc. London Math. Soc. (2)

**38**(1935), 458-480.

*An introduction to transform theory*, Academic Press, New York, 1971.

*Short proof of the prime number theorem*, unpublished manuscript.

## Additional Information

- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**306**(1988), 837-852 - MSC: Primary 47D05; Secondary 34G10
- DOI: https://doi.org/10.1090/S0002-9947-1988-0933321-3
- MathSciNet review: 933321