Abstract and Applied Analysis

Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces

Angela A. Albanese, José Bonet, and Werner J. Ricker

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Abstract

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operator T to the operator norm convergence of certain sequences of operators generated by T , are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 179027, 16 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858209

Digital Object Identifier
doi:10.1155/2014/179027

Mathematical Reviews number (MathSciNet)
MR3166576

Zentralblatt MATH identifier
07021878

Citation

Albanese, Angela A.; Bonet, José; Ricker, Werner J. Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 179027, 16 pages. doi:10.1155/2014/179027. https://projecteuclid.org/euclid.aaa/1395858209


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References

  • N. Dunford and J. T. Schwartz, Linear Operators I: General Theory, John Wiley & Sons, New York, NY, USA, 1964.
  • J. J. Koliha, “Power convergence and pseudoinverses of operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 48, pp. 446–469, 1974.
  • M. Mbekhta and J. Zemánek, “Sur le théorème ergodique uniforme et le spectre,” Comptes Rendus de l'Académie des Sciences Paris, vol. 317, no. 12, pp. 1155–1158, 1993.
  • U. Krengel, Ergodic Theorems, vol. 6, Walter de Gruyter, Berlin, Germany, 1985.
  • M. Lin, “On the uniform ergodic theorem,” Proceedings of the American Mathematical Society, vol. 43, pp. 337–340, 1974.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “Convergence of arith-metic means of operators in Fréchet spaces,” Journal of Mathematical Analysis and Applications, vol. 401, no. 1, pp. 160–173, 2013.
  • Y. Katznelson and L. Tzafriri, “On power bounded operators,” Journal of Functional Analysis, vol. 68, no. 3, pp. 313–328, 1986.
  • W. F. Eberlein, “Abstract ergodic theorems and weak almost peri odic functions,” Transactions of the American Mathematical Society, vol. 67, pp. 217–240, 1949.
  • V. A. Pietsch, “Quasi-präkompakte Endomorphismen und ein Ergodensatz in lokalkonvexen Vektorräumen,” Journal für die Reine und Angewandte Mathematik, vol. 207, pp. 16–30, 1961.
  • R. Meise and D. Vogt, Introduction to Functional Analysis, vol. 2, Clarendon Press, Oxford, UK, 1997.
  • G. Köthe, Topological Vector Spaces II, Springer, New York, NY, USA, 1979.
  • S. F. Bellenot and E. Dubinsky, “Fréchet spaces with nuclear Köthe quotients,” Transactions of the American Mathematical Society, vol. 273, no. 2, pp. 579–594, 1982.
  • J. Bonet, M. Maestre, G. Metafune, V. B. Moscatelli, and D. Vogt, “Every quojection is the quotient of a countable product of Banach spaces,” in Advances in the Theory of Fréchet Spaces, T. Terzioğlu, Ed., vol. 287 of NATO ASI, pp. 355–356, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
  • V. B. Moscatelli, “Fréchet spaces without continuous norms and without bases,” The Bulletin of the London Mathematical Society, vol. 12, no. 1, pp. 63–66, 1980.
  • P. Domański, “Twisted Fréchet spaces of continuous functions,” Results in Mathematics, vol. 23, no. 1-2, pp. 45–48, 1993.
  • H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, Germany, 1981.
  • S. Dierolf and D. N. Zarnadze, “A note on strictly regular Fréchet spaces,” Archiv der Mathematik, vol. 42, no. 6, pp. 549–556, 1984.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “On the continuous Cesàro operatorčommentComment on ref. [18?]: Please provide more information for this reference. in certain function spaces,” 2013, submitted.
  • S. Dierolf and V. B. Moscatelli, “A note on quojections,” Functiones et Approximatio Commentarii Mathematici, vol. 17, pp. 131–138, 1987.
  • S. Önal and T. Terzioğlu, “Unbounded linear operators and nuclear Köthe quotients,” Archiv der Mathematik, vol. 54, no. 6, pp. 576–581, 1990.
  • D. Vogt, “On two problems of Mityagin,” Mathematische Nachrichten, vol. 141, pp. 13–25, 1989.
  • E. Behrends, S. Dierolf, and P. Harmand, “On a problem of Bellenot and Dubinsky,” Mathematische Annalen, vol. 275, no. 3, pp. 337–339, 1986.
  • V. B. Moscatelli, “Strongly nonnorming subspaces and prequojections,” Studia Mathematica, vol. 95, no. 3, pp. 249–254, 1990.
  • G. Metafune and V. B. Moscatelli, “Quojection and prequojections,” in Advances in the Theory of Fréchet Spaces, T. Terzioğlu, Ed., vol. 287 of NATO ASI, pp. 235–254, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
  • G. Metafune and V. B. Moscatelli, “Prequojections and their duals,” in Progress in Functional Analysis, K. D. Bierstedt, J. Bonet, J. Horváth, and M. Maestre, Eds., vol. 170, pp. 215–232, North-Holland, Amsterdam, The Netherlands, 1992.
  • A. A. Albanese, “A Fréchet space of continuous functions which is a prequojection,” Bulletin de la Société Royale des Sciences de Liège, vol. 60, no. 6, pp. 409–417, 1991.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “Grothendieck spaces with the Dunford-Pettis property,” Positivity, vol. 14, no. 1, pp. 145–164, 2010.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “On mean ergodic operators,” in Vector Measures, Integration and Related Topics, G. P. Curbera, G. Mockenhaupt, and W. J. Ricker, Eds., vol. 201 of Operator Theory: Advances and Applications, pp. 1–20, Birkhäuser, Basel, Switzerland, 2010.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “${C}_{0}$-semigroups and mean ergodic operators in a class of Fréchet spaces,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 142–157, 2010.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “Uniform mean ergo-dicity of ${C}_{0}$-semigroups in a class of Fréchet spaces,” Functiones et Approximatio Commentarii Mathematici. In press.
  • H. R. Dowson, Spectral Theory of Linear Operators, vol. 12, Academic Press, London, UK, 1978.
  • J. Zemánek, “On the Gelfand-Hille theorems,” in Functional Analysis and Operator Theory, vol. 30 of Banach Center Publications, pp. 369–385, Polish Academy of Sciences, Warsaw, Poland, 1994.
  • S. Dierolf and P. Domański, “Factorization of Montel operators,” Studia Mathematica, vol. 107, no. 1, pp. 15–32, 1993.
  • J. C. Díaz and P. Domański, “Reflexive operators with domain in Köthe spaces,” Manuscripta Mathematica, vol. 97, no. 2, pp. 189–204, 1998.
  • A. Grothendieck, Topological Vector Spaces, Gordon and Breach, London, UK, 1975.
  • R. E. Edwards, Functional Analysis, Reinhart and Winston, New York, NY, USA, 1965.
  • K. Floret, Weakly Compact Sets, vol. 801 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1980.
  • B. Cascales and J. Orihuela, “On compactness in locally convex spaces,” Mathematische Zeitschrift, vol. 195, no. 3, pp. 365–381, 1987.
  • A. A. Albanese, J. Bonet, and W. J. Ricker, “Mean ergodic operators in Fréchet spaces,” Annales Academiae Scientiarum Fennicae, vol. 34, no. 2, pp. 401–436, 2009.
  • M. Altman, “Mean ergodic theorem in locally convex linear top-ological spaces,” Studia Mathematica, vol. 13, pp. 190–193, 1953.
  • M. V. Deshpande and S. M. Padhye, “An ergodic theorem for quasi-compact operators in locally convex spaces,” Journal of Mathematical and Physical Sciences, vol. 11, no. 2, pp. 95–104, 1977. \endinput