Abstract
The adjoint of aC 0-semigroup on a Banach spaceX induces a locally convex σ(X,X ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ℴ)-topology and it is shown that the uniform limit of σ(X,X ℴ)-compact operators is σ(X,X ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Arendt, Resolvent positive operators and integrated semigroups, Semesterbericht Tübingen, Sommersemester 1984.
P.L. Butzer, H. Berens,Semigroups of operators and approximation, Springer Verlag, Berlin-Heidelberg-New York (1967)
J. Diestel, J.J. Uhl,Vector measures, Math. Surveys nr, 15, Amer. Math. Soc., Providence, R.I., (1977).
E. Hille, R.S. Phillips,Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., rev. ed., Providence, R.I. (1957).
J. Lindenstrauss, L. Tzafriri,Classical Banach spaces I Springer Verlag, Berlin-Heidelberg-New York (1977).
J.M.A.M. van Neerven, Hahn-Banach type theorems for adjoint semigroups, to appear in: Math. Ann.
J.M.A.M. van Neerven, Reflexivity, the dual Radon-Nikodym Property, and continuity of adjoint semigroups to appear in: Indag. Math.
B. de Pagter, A characterization of sun-reflexivity, Math. Ann. 283, 511–518 (1989).
H.H. Schaefer,Topological Vector Spaces, 3rd print, Sprigner Verlag, Berlin-Heidelberg-New York (1971).
M. Zippin, A remark on bases and reflexivity in Banach spaces, Isr. J. Math. 6, 74–79 (1968).
Author information
Authors and Affiliations
Additional information
Communicated by Jerome A. Goldstein
Rights and permissions
About this article
Cite this article
van Neerven, J.M.A.M. On the topology induced by the adjoint of a semigroup of operators. Semigroup Forum 43, 378–394 (1991). https://doi.org/10.1007/BF02574280
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02574280