Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Tauberian theorems and stability of one-parameter semigroups


Authors: W. Arendt and C. J. K. Batty
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [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. MR 915430 (89c:47003)
  • [2] W. Arendt and G. Greiner, The spectral mapping theorem for one-parameter groups of positive operators on $ {C_0}(X)$, Semigroup Forum 30 (1984), 297-330. MR 765499 (86h:47068)
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Part I, Wiley, New York, 1958. MR 1009162 (90g:47001a)
  • [4] A. E. Ingham, On Wiener'a method in Tauberian theorems, Proc. London Math. Soc. (2) 38 (1935), 458-480.
  • [5] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Functional Anal. 68 (1986), 313-328. MR 859138 (88e:47006)
  • [6] J. Korevaar, On Newman's quick way to the prime number theorem, Math. Intelligencer 4 (1982), 108-115. MR 684025 (84b:10063)
  • [7] U. Krengel, Ergodic theorems, De Gruyter, Berlin, 1985. MR 797411 (87i:28001)
  • [8] R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin and New York, 1986. MR 839450 (88i:47022)
  • [9] D. J. Newman, Simple analytic proof of the prime number theorem, Amer. Math. Monthly 87 (1980), 693-696. MR 602825 (82h:10056)
  • [10] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. MR 0275190 (43:947)
  • [11] D. V. Widder, An introduction to transform theory, Academic Press, New York, 1971.
  • [12] M. Wolff, A remark on the spectral bound of the generator of a semigroup of positive operators with applications to stability theory, Functional Analysis and Approximation (Proc. Conf., Oberwolfach, 1980) (P. L. Butzer, B. Sz.-Nagy, E. Görlich, eds.), Birkhäuser, Basel, 1981, pp. 39-50. MR 650263 (83i:47054)
  • [13] D. Zagier, Short proof of the prime number theorem, unpublished manuscript.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D05, 34G10

Retrieve articles in all journals with MSC: 47D05, 34G10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0933321-3
Keywords: Tauberian theorems, $ {C_0}$-semigroup, stability, power bounded, Laplace transform, residual spectrum
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society