Extension of invariant linear functionals
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- by Ky Fan PDF
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Abstract:
In a recent paper [4] we considered a semigroup of linear contractions or a group of linear isometries in a normed vector space, and obtained sufficient conditions for the orbits to lie on parallel closed hyperplanes. In the present note, we take a more general viewpoint and shall prove some theorems on extension of continuous linear functionals which are invariant under a group or a semigroup of continuous linear maps on a locally convex topological vector space. These more general results include those in [4] as direct consequences.References
- N. Bourbaki, Eléments de mathématique. XVIII. Première partie: Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. Chapitre III: Espaces d’applications linéaires continues. Chapitre IV: La dualité dans les espaces vectoriels topologiques. Chapitre V: Espaces hilbertiens, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1229, Hermann & Cie, Paris, 1955 (French). MR 0077882
- Jacques Dixmier, Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. (Szeged) 12 (1950), 213–227 (French). MR 37470
- 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
- Ky Fan, Orbits of semi-groups of contractions and groups of isometries, Abh. Math. Sem. Univ. Hamburg 45 (1976), 245–250. MR 410470, DOI 10.1007/BF02992919
- Shizuo Kakutani, Two fixed-point theorems concerning bicompact convex sets, Proc. Imp. Acad. Tokyo 14 (1938), no. 7, 242–245. MR 1568507
- Gottfried Köthe, Topologische lineare Räume. I, Die Grundlehren der mathematischen Wissenschaften, Band 107, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0130551
- M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.) 3 (1948), no. 1(23), 3–95 (Russian). MR 0027128
- I. Namioka and E. Asplund, A geometric proof of Ryll-Nardzewski’s fixed point theorem, Bull. Amer. Math. Soc. 73 (1967), 443–445. MR 209904, DOI 10.1090/S0002-9904-1967-11779-8
- Czesław Ryll-Nardzewski, On fixed points of semigroups of endomorphisms of linear spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 55–61. MR 0215134
- Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 66 (1977), 23-29
- MSC: Primary 46A30; Secondary 47D05
- DOI: https://doi.org/10.1090/S0002-9939-1977-0458111-8
- MathSciNet review: 0458111