Abstract
In this paper we introduce a new class of operators acting on a locally convex space. We show that for some Fréchet spaces all these operators are mean ergodic. This leads to the conclusion that the classes of reflexive and non-reflexive Fréchet spaces are, in a sense, close to each other.
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Albanese A.A., Bonet J., Ricker W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)
K. D. Bierstedt, An introduction to locally convex inductive limits, Functional Analysis and Its Applications (Nice, 1986. H. Hogbe-Nlend, ed.), pp. 35–133, ICPAM Lecture Notes, World Sci. Publishing, Singapore, 1988.
Civin P., Yood B.: Quasi-reflexive spaces. Proc. Amer. Math. Soc. 8, 906–911 (1957)
Cuttle Y.: On quasi-reflexive Banach spaces. Proc. Amer. Math. Soc. 12, 936–940 (1961)
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, 2nd ed., Wiley-Interscience, New York, 1964.
Fonf V.P., Lin M., Wojtaszczyk P.: Ergodic characterizations of reflexivity of Banach spaces. J. Funct. Anal. 187, 146–162 (2001)
V. P. Fonf, M. Lin, and P. Wojtaszczyk, A non-reflexive Banach space with all contractions mean ergodic, preprint.
B. V. Godun, Equivalent norms on quasireflexive spaces with a basis (Russian), Dokl. Akad. Nauk Ukrain. SSR Ser. A 1977, 778–781, 863.
James R.C.: Bases and reflexivity of Banach spaces. Ann. of Math. 52, 518–527 (1950)
James R.C.: Banach spaces quasi-reflexive of order one. Studia Math. 60, 157–177 (1977)
Köthe G.: Topological Vector Spaces I. Springer-Verlag New York Inc., New York (1969)
Köthe G.: Topological Vector Spaces II. Springer-Verlag, New York-Berlin (1979)
U. Krengel, Ergodic theorems. With a supplement by Antoine Brunel, de Gruyter Studies in Mathematics 6, Walter de Gruyter Co., Berlin, 1985.
V. K. Maslyuchenko and A. N. Plichko, Quasireflexive locally convex spaces without Banach subspaces (Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen. 44 (1985), 78–84; translation in J. Soviet Math. 48 (1990), 307–312.
Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)
Piszczek K.: Quasi-reflexive Fréchet spaces and mean ergodicity. J. Math. Anal. Appl. 361, 224–233 (2010)
Singer I.: On bases in quasi-reflexive Banach spaces. Rev. Math. Pures Appl. (Bucarest) 8, 309–311 (1963)
Singer I.: Bases and quasi-reflexivity of Banach spaces. Math. Ann. 153, 199–209 (1964)
L. Sucheston, Problems. Probability in Banach Spaces, Lecture Notes in Math. 526, Springer-Verlag, Berlin, 1976.
Valdivia M.: Bases and quasi-reflexivity in Fréchet spaces. Math. Nachr. 278, 712–729 (2005)
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The research of the author has been supported in the years 2007-2010 by the Ministry of Science and Higher Education, Poland, Grant no. N N201 2740 33.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Piszczek, K. Quasi-reflexive Fréchet spaces and contractively power bounded operators. Arch. Math. 96, 49–58 (2011). https://doi.org/10.1007/s00013-010-0197-y
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DOI: https://doi.org/10.1007/s00013-010-0197-y