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)

 

 

Optimal individual stability estimates for $C_0$-semigroups in Banach spaces


Author: Volker Wrobel
Journal: Trans. Amer. Math. Soc. 351 (1999), 4981-4994
MSC (1991): Primary 47D06
Published electronically: July 22, 1999
MathSciNet review: 1473458
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In a previous paper we proved that the asymptotic behavior of a $C_0$-semigroup is completely determined by growth properties of the resolvent of its generator and geometric properties of the underlying Banach space as described by its Fourier type. The given estimates turned out to be optimal. The method of proof uses complex interpolation theory and reflects the full semigroup structure. In the present paper we show that these uniform estimates have to be replaced by weaker ones, if individual initial value problems and local resolvents are considered because the full semigroup structure is lacking. In a different approach this problem has also been studied by Huang and van Neerven, and a part of our straightforward estimates can be inferred from their results. We mainly stress upon the surprising fact that these estimates turn out to be optimal. Therefore it is not possible to obtain the optimal uniform estimates mentioned above from individual ones. Concerning Hardy-abscissas, individual orbits and their local resolvents behave as badly as general vector valued functions and their Laplace-transforms. This is in strict contrast to the uniform situation of a $C_0$-semigroup itself and the resolvent of its generator where a simple dichotomy holds true.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47D06

Retrieve articles in all journals with MSC (1991): 47D06


Additional Information

Volker Wrobel
Affiliation: Mathematisches Seminar, Universität Kiel, D-24098 Kiel, Germany

DOI: https://doi.org/10.1090/S0002-9947-99-02200-X
Received by editor(s): April 1, 1997
Published electronically: July 22, 1999
Article copyright: © Copyright 1999 American Mathematical Society