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STRICT TOPOLOGIES AND OPERATORS ON SPACES OF VECTOR-VALUED CONTINUOUS FUNCTIONS

  • Nowak, Marian
  • Received : 2014.05.18
  • Published : 2015.01.01

Abstract

Let X be a completely regular Hausdorff space, and E and F be Banach spaces. Let $C_{rc}(X,E)$ be the Banach space of all continuous functions $f:X{\rightarrow}E$ such that f(X) is a relatively compact set in E. We establish an integral representation theorem for bounded linear operators $T:C_{rc}(X,E){\rightarrow}F$. We characterize continuous operators from $C_{rc}(X,E)$, provided with the strict topologies ${\beta}_z(X,E)$ ($z={\sigma},{\tau}$) to F, in terms of their representing operator-valued measures.

Keywords

spaces of vector-valued continuous functions;strict topologies;vector measures;integration operators

References

  1. C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York, 1985.
  2. J. Aguayo and J. Sanchez, The Dunford-Pettis property on vector-valued continuous and bounded functions, Bull. Austr. Math. Soc. 48 (1993), no. 2, 303-311. https://doi.org/10.1017/S0004972700015719
  3. J. Aguayo-Garrido and M. Nova-Yanez, Weakly compact operators and u-additive measures, Ann. Math. Blaise Pascal 7 (2000), no. 2, 1-11.
  4. J. Batt, Applications of the Orlicz-Pettis theorem to operator-valued measures and compact and weakly compact linear transformations on the space of continuous functions, Rev. Roumaine Math. Pures Appl. 14 (1969), 907-935.
  5. R. Cristescu, Topological Vector Spaces, Ed. Acad. Bucaresti, Noordhoff Inter. Publ., Leyden 1977.
  6. J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., Math. Surveys 15, Providence, RI, 1977.
  7. N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967.
  8. N. Dinculeanu, Vector Measures, Vector Integration and Stochastic Integration in Banach Spaces, John Wiley and Sons Inc., 2000.
  9. A. Katsaras, Spaces of vector measures, Trans. Amer. Math. Soc. 206 (1975), 313-328. https://doi.org/10.1090/S0002-9947-1975-0365111-8
  10. R. E. Edwards, Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York, 1965.
  11. R. Fontenot, Strict topologies for vector-valued function spaces, Canad. J. Math. 26 (1974), no. 4, 841-853. https://doi.org/10.4153/CJM-1974-079-1
  12. A. Katsaras, Continuous linear functionals on spaces of vector-valued functions, Bull. Soc. Math. Greece 15 (1974), 13-19.
  13. A. Katsaras, Locally convex topologies on spaces of continuous vector functions, Math. Nachr. 71 (1976), 211-226. https://doi.org/10.1002/mana.19760710117
  14. A. Katsaras, Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals, Trans. Amer. Math. Soc. 216 (1976), 367-387. https://doi.org/10.1090/S0002-9947-1976-0390733-9
  15. A. Katsaras and D. B. Liu, Integral representation of weakly compact operators, Pacific J. Math. 56 (1975), no. 2, 547-556. https://doi.org/10.2140/pjm.1975.56.547
  16. S. S. Khurana, Topologies on spaces of vector-valued continuous functions, Trans. Amer. Math. Soc. 241 (1978), 195-211. https://doi.org/10.1090/S0002-9947-1978-0492297-X
  17. S. S. Khurana, Integral representation of a class of operators, J. Math. Anal. Appl. 350 (2009), no. 1, 290-293. https://doi.org/10.1016/j.jmaa.2008.09.059
  18. S. S. Khurana and S. I. Othman, Grothendieck measures, J. London Math. Soc. 39 (1989), no. 3, 481-486.
  19. S. S. Khurana and J. Vielma, Strict topology and perfect measures, Czechoslovak Math. J. 40(115) (1990), no. 1, 1-7. https://doi.org/10.1007/BF01598348
  20. M. Nowak and A. Rzepka, Locally solid topologies on spaces of vector-valued continuous functions, Comment. Math. Univ. Carolinae 43 (2002), no. 3, 473-483.
  21. R. Wheeler, A survey of Baire measures and strict topologies, Expo. Math. 2 (1983), no. 2, 97-190.

Cited by

  1. Extension of operators on spaces of vector-valued continuous functions vol.27, pp.1, 2016, https://doi.org/10.1016/j.indag.2015.07.003
  2. Strongly bounded operators on Crc(X,E) with the strict topology βσ vol.27, pp.4, 2016, https://doi.org/10.1016/j.indag.2016.06.008