Tauberian theorems and stability of one-parameter semigroups

Arendt, W.; Batty, C. J. K.

doi:10.1090/S0002-9947-1988-0933321-3

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

On Wiener’a method in Tauberian theorems

38

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

An introduction to transform theory

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

Short proof of the prime number theorem