Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2015, Article ID 796753, 13 pages
http://dx.doi.org/10.1155/2015/796753
Research Article

Some Classes of Continuous Operators on Spaces of Bounded Vector-Valued Continuous Functions with the Strict Topology

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Ulica Szafrana 4A, 65–516 Zielona Góra, Poland

Received 28 October 2014; Accepted 2 January 2015

Academic Editor: Luisa Di Piazza

Copyright © 2015 Marian Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. N. Dinculeanu, Vector Measures, Pergamon Press, New York, NY, USA, 1967. View at MathSciNet
  2. N. Dinculeanu, Integration and Stochastic Integration in Banach Spaces, John Wiley & Sons, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  3. F. D. Sentilles, “Bounded continuous functions on a completely regular space,” Transactions of the American Mathematical Society, vol. 168, pp. 311–336, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  4. R. F. Wheeler, “A survey of Baire measures and strict topologies,” Expositiones Mathematicae, vol. 1, no. 2, pp. 97–190, 1983. View at Google Scholar · View at MathSciNet
  5. R. A. Fontenot, “Strict topologies for vector-valued functions,” Canadian Journal of Mathematics, vol. 26, no. 4, pp. 841–853, 1974. View at Google Scholar · View at MathSciNet
  6. S. S. Khurana, “Topologies on spaces of vector-valued continuous functions,” Transactions of the American Mathematical Society, vol. 241, pp. 195–211, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. S. Khurana and S. I. Othman, “Convex compactness property in certain spaces of measures,” Mathematische Annalen, vol. 279, no. 2, pp. 345–348, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. S. Khurana and S. I. Othman, “Completeness and sequential completeness in certain spaces of measures,” Mathematica Slovaca, vol. 45, no. 2, pp. 163–170, 1995. View at Google Scholar · View at MathSciNet
  9. S. S. Khurana and J. Vielma, “Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions,” Journal of Mathematical Analysis and Applications, vol. 195, no. 1, pp. 251–260, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. E. E. Granirer, “On Baire measures on D-topological spaces,” Fundamenta Mathematicae, vol. 60, pp. 1–22, 1967. View at Google Scholar
  11. A. Katsaras, “Continuous linear functionals on spaces of vector-valued functions,” Bulletin Société Mathématique de Grèce, vol. 15, pp. 13–19, 1974. View at Google Scholar · View at MathSciNet
  12. M. Nowak, “Operators on spaces of bounded vector-valued continuous functions with strict topologies,” Journal of Function Spaces, vol. 2014, Article ID 407521, 12 pages, 2014. View at Google Scholar
  13. J. Diestel and J. J. Uhl, Vector Measures, vol. 15 of Mathematical Surveys, American Mathematical Society, Providence, RI, USA, 1977. View at MathSciNet
  14. P. W. Lewis, “Strongly bounded operators,” Pacific Journal of Mathematics, vol. 53, no. 1, pp. 207–209, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  15. P. W. Lewis, “Variational semi-regularity and norm convergence,” Journal für die Reine und Angewandte Mathematik, vol. 260, pp. 21–30, 1973. View at Google Scholar · View at MathSciNet
  16. J. K. Brooks and P. W. Lewis, “Operators on continuous function spaces and convergence in the spaces of operators,” Advances in Mathematics, vol. 29, no. 2, pp. 157–177, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. C. A. Abbott, E. M. Bator, R. G. Bilyeu, and P. W. Lewis, “Weak precompactness, strong boundedness, and weak complete continuity,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 108, no. 2, pp. 325–335, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  18. C. Abbott, E. Bator, and P. Lewis, “Strictly singular and strictly cosingular operators on spaces of continuous functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 110, no. 3, pp. 505–521, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Batt, “Applications of the Orlicz-Pettis theorem to operator-valued measures and compact and weakly compact transformations on the spaces of continuous functions,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 14, pp. 907–935, 1969. View at Google Scholar
  20. J. Batt and E. J. Berg, “Linear bounded transformations on the space of continuous functions,” Journal of Functional Analysis, vol. 4, no. 2, pp. 215–239, 1969. View at Google Scholar · View at MathSciNet
  21. R. Bilyeu and P. Lewis, “Some mapping properties of representing measures,” Annali di Matematica Pura ed Applicata, vol. 109, pp. 273–287, 1976. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. F. Bombal and P. Cembranos, “Characterization of some classes of operators on spaces of vector-valued continuous functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 97, no. 1, pp. 137–146, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  23. F. Bombal and P. Cembranos, “Dieudonné operators on C(K,E),” Bulletin of the Polish Academy of Sciences Mathematics, vol. 34, pp. 301–305, 1986. View at Google Scholar
  24. F. Bombal, “On weakly compact operators on spaces of vector valued continuous functions,” Proceedings of the American Mathematical Society, vol. 97, no. 1, pp. 93–96, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  25. F. Bombal and B. Porras, “Strictly singular and strictly cosingular operators on C(K, E),” Mathematische Nachrichten, vol. 143, pp. 355–364, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  26. F. Bombal and B. Rodriguez-Salinas, “Some classes of operators on C(K,E). Extension and applications,” Archiv der Mathematik, vol. 47, no. 1, pp. 55–65, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. J. K. Brooks and P. W. Lewis, “Linear operators and vector measures,” Transactions of the American Mathematical Society, vol. 192, pp. 139–162, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  28. I. Dobrakov, “On representation of linear operators on Co(T, X),” Czechoslovak Mathematical Journal, vol. 21, pp. 13–30, 1971. View at Google Scholar · View at MathSciNet
  29. P. Saab, “Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95, no. 1, pp. 101–108, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  30. E. Saab and P. Saab, “On unconditionally converging and weakly precompact operators,” Illinois Journal of Mathematics, vol. 35, no. 3, pp. 522–531, 1991. View at Google Scholar · View at MathSciNet
  31. P. Saab and B. Smith, “Nuclear operators on spaces of continuous vector-valued functions,” Glasgow Mathematical Journal, vol. 33, no. 2, pp. 223–230, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  32. P. Saab and B. Smith, “Spaces on which unconditionally converging operators are weakly completely continuous,” The Rocky Mountain Journal of Mathematics, vol. 22, no. 3, pp. 1001–1009, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. C. Swartz, “Unconditionally converging operators on the space of continuous functions,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 17, pp. 1695–1702, 1972. View at Google Scholar · View at MathSciNet
  34. A. Katsaras, “Spaces of vector measures,” Transactions of the American Mathematical Society, vol. 206, pp. 313–328, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
  35. F. Topsoe, “Compactness in spaces of measures,” Studia Mathematica, vol. 36, pp. 195–212, 1970. View at Google Scholar
  36. A. Grothendieck, “Sur les applications lineaires faiblement compactnes d’espaces de type C(K),” Canadian Journal of Mathematics, vol. 5, pp. 129–173, 1953. View at Google Scholar · View at MathSciNet
  37. J. Diestel, Sequences and Series in Banach Spaces, vol. 92 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  38. W. H. Graves and W. Ruess, “Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded maesurable functions,” Contemporary Mathematics, vol. 2, pp. 180–203, 1980. View at Google Scholar
  39. I. Dobrakov, “On integration in Banach spaces, I,” Czechoslovak Mathematical Journal, vol. 20, no. 3, pp. 511–536, 1970. View at Google Scholar · View at MathSciNet
  40. M. Nowak, “Operators on the space of bounded strongly measurable functions,” Journal of Mathematical Analysis and Applications, vol. 388, no. 1, pp. 393–403, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. P. Cembranos and J. Mendoza, Banach Spaces of Vector-Valued Functions, vol. 1676 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1997.
  42. J. Bourgain, “An averaging result for l1-sequences and applications to weakly conditionally compact sets in LX1,” Israel Journal of Mathematics, vol. 32, no. 4, pp. 289–298, 1979. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. C. D. Aliprantis and O. Burkinshaw, Positive Operators, vol. 119 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1985. View at MathSciNet
  44. C. Bessaga and A. Pełczyński, “On bases and unconditional convergence of series in Banach spaces,” Studia Mathematica, vol. 17, no. 2, pp. 151–164, 1958. View at Google Scholar · View at MathSciNet
  45. R. E. Edwards, Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York, NY, USA, 1965. View at MathSciNet
  46. H. H. Schaefer, Topological Vector Spaces, Springer, New York, NY, USA, 1971. View at MathSciNet
  47. J. Howard, “The comparison of an unconditionally converging operator,” Studia Mathematica, vol. 33, pp. 295–298, 1969. View at Google Scholar · View at MathSciNet