Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T16:11:43.778Z Has data issue: false hasContentIssue false
  • English
  • Français

The Growth Bound for Strongly Continuous Semigroups on Fréchet Spaces

Part of: Topological linear spaces and related structures Groups and semigroups of linear operators, their generalizations and applications Differential equations in abstract spaces

Published online by Cambridge University Press:  23 November 2015

Sven-Ake Wegner
Sven-Ake Wegner*
Affiliation:
Sobolev Institute of Mathematics, Pr. Akad. Koptyuga 4, 630090, Novosibirsk, Russia (wegner@math.uni-wuppertal.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fréchet spaces and show that the Banach space inequality s(A) ⩽ ω 0(T) extends to the new setting. Via a concrete example of an even uniformly continuous semigroup, we illustrate that for Fréchet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.

Keywords

semigroupgrowth boundspectral boundpower-bounded operatorFréchet space

MSC classification

Secondary: 47D06: One-parameter semigroups and linear evolution equations 46A04: Locally convex Fréchet spaces and (DF)-spaces 34G10: Linear equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 801 - 810
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Albanese, A. A., Bonet, J. and Ricker, W. J., Montel resolvents and uniformly mean ergodic semigroups of linear operators, Quaest. Math. 36(2) (2013), 253–290.CrossRefGoogle Scholar
2. Allan, G. R., A spectral theory for locally convex alebras, Proc. Lond. Math. Soc. 15 (1965), 399–421.Google Scholar
3. Arikan, H., Runov, L. and Zahariuta, V., Holomorphic functional calculus for operators on a locally convex space, Results Math. 43(1–2) (2003), 23–36.CrossRefGoogle Scholar
4. Engel, K.-J. and Nagel, R., One-parameter semigroups for linear evolution equations (Springer, 2000).Google Scholar
5. Frerick, L., Jordá, E., T. Kalmes, and J. Wengenroth, , Strongly continuous semigroups on some Fréchet spaces, J. Math. Analysis Applic. 412(1) (2014), 121–124.Google Scholar
6. Jarchow, H., Locally convex spaces (Teubner, Leipzig, Stuttgart, 1981).Google Scholar
7. Joseph, G. A., Boundedness and completeness in locally convex spaces and algebras, J. Austral. Math. Soc. A 24(1) (1977), 50–63.CrossRefGoogle Scholar
8. Kōmura, T., Semigroups of operators in locally convex spaces, J. Funct. Analysis 2 (1968), 258–296.CrossRefGoogle Scholar
9. Köthe, G., Topological vector spaces II, Grundlehren der mathematischen Wissenschaften, Volume 237 (Springer, 1979).CrossRefGoogle Scholar
10. Moore, R. T., Banach algebras of operators on locally convex spaces, Bull. Am. Math. Soc. 75 (1969), 68–73.Google Scholar
11. Yosida, K., Functional analysis (Springer, 1995).Google Scholar