Abstract

Let  E \,E\, be an ideal of  L0 \,L^0\, over a  σ\,\sigma-finite measure space  (Ω,Σ,μ) \,(\Om,\Si,\mu)\, and let E′E' be the Köthe dual of  E\,E. Let  (X,∥⋅∥X) \,(X,\|\cdot\|_X)\, be a real Banach space, and  X∗ \,X^*\, the Banach dual of  X\,X. Let  E(X) \,E(X)\, be a subspace of the space  L0(X) \,L^0(X)\, of μ\mu-equivalence classes of all strongly Σ\Si-measurable function  f:Ω→X\,f:\Om\ps X, and consisting of all those  f∈L0(X) \,f\in L^0(X)\, for which the scalar function f~\wf, defined by  f~(ω)=∥f(ω)∥X \,\wf(\om)=\|f(\om)\|_X\, for  ω∈Ω\,\om\in\Om, belongs to EE. Assume that a Banach space  X \,X\, is an Asplund space. It is shown that a subset CC of  E′(X∗) \,E'(X^*)\, is relatively  σ(E′(X∗),E(X))\,\si(E'(X^*),E(X))-compact iff the set  {g~:g∈E′(X∗)} \,\{\wg:g\in E'(X^*)\}\, in E′E' is relatively  σ(E′,E)\,\si(E',E)-compact. We consider the topology  τ(E,E′)‾ \,\overline{\tau(E,E')}\, on E(X)E(X) associated with the Mackey topology  τ(E,E′) \,\tau(E,E')\, on EE. It is shown that  τ(E,E′)‾ \,\overline{\tau(E,E')}\, is strongly Mackey topology; hence  τ(E,E′)‾ \,\overline{\tau(E,E')}\, coincides with the Mackey topology  τ(E(X),E′(X∗))\,\tau(E(X),E'(X^*)). Moreover,  E′(X∗) \,E'(X^*)\, is  σ(E′(X∗),E(X))\,\si(E'(X^*), E(X))-sequentially complete whenever E′E' is perfect. We examine the space Lτ(E(X),Y)\cl_\tau(E(X),Y) of all  (τ(E(X),E′(X∗)),∥⋅∥Y)\,(\tau(E(X),E'(X^*)),\|\cdot\|_Y)-continuous linear operators from  E(X) \,E(X)\, to a Banach space  (Y,∥⋅∥Y)\,(Y,\|\cdot\|_Y), equipped with the weak operator topology (briefly WOT) and the strong operator topology (briefly SOT). It is shown that if EE is perfect, then Lτ(E(X),Y)\cl_\tau(E(X),Y) is WOT-sequentially complete, and every SOT-compact subset of Lτ(E(X),Y)\cl_\tau(E(X),Y) is  (τ(E(X),E′(X∗)),∥⋅∥Y)\,(\tau(E(X),E'(X^*)),\|\cdot\|_Y)-equicontinuous. Moreover, a Vitali-Hahn-Saks type theorem for Lτ(E(X),Y)\cl_\tau(E(X),Y) is obtained.

Author information

Contact details are reproduced from the original publication and may be historical.

Marian Nowak

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

M.Nowak@wmie.uz.zgora.pl

Keywords

  • Vector-valued function spaces
  • Mackey topologies
  • strongly Mackey topologies
  • weak compactness
  • Radon-Nikodym property
  • Asplund spaces
  • sequential completeness
  • convex compactness property
  • weak operator topology
  • strong operator topology
  • linear operator

Mathematics Subject Classification

46E40, 46E30, 46A20, 46A70

References

  1. C. D. Aliprantis, O. Burkinshaw: Locally Solid Riesz Spaces, Academic Press, New York (1978).Google Scholar
  2. A. V. Bukhvalov: On an analytic representation of operators with abstract norm, Izv. Vyssh. Uchebn. Zaved. Mat 11 (1975) 21–32.Google Scholar
  3. A. V. Bukhvalov: On an analytic representation of linear operators by means of measurable vector-valued functions, Izv. Vyssh. Uchebn. Zaved. Mat 7 (1977) 21–32.Google Scholar
  4. A. V. Bukhvalov, G. Ya. Lozanowskii: On sets closed in measure in spaces of measurable function, Trans. Mosc. Math. Soc. 2 (1978) 127–148.Google Scholar
  5. P. Cembranos, J. Mendoza: Banach Spaces of Vector-Valued Functions, Lect. Notes in Math. 1676, Springer, Berlin (1997).https://doi.org/10.1007/BFb0096765Google Scholar
  6. M. Duhoux: Mackey topologies on Riesz spaces and weak compactness, Math. Z. 158 (1978) 199–209.https://doi.org/10.1007/BF01214793Google Scholar
  7. N. Dunford, J. Schwartz: Linear Operators, Part. I, Interscience, New York (1958).Google Scholar
  8. J. Diestel, J. J. Uhl: Vector Measures, Math. Surveys 15, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  9. K. Feledziak, M. Nowak: Locally solid topologies on vector-valued function spaces, Collect. Math. 48(4–6) (1997) 487–511.Google Scholar
  10. L. V. Kantorovitch, A. V. Akilov: Functional Analysis, Third Edition, Nauka, Moskow (1984) (in Russian).Google Scholar
  11. P.-K. Lin: Köthe-Bochner Function Spaces, Birkhäuser, Boston (2003).https://doi.org/10.1007/978-0-8176-8188-3Google Scholar
  12. I. Namioka, R. R. Phelps: Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975) 735–750.https://doi.org/10.1215/S0012-7094-75-04261-1Google Scholar
  13. M. Nowak: A characterization of the Mackey topology τ (Lϕ, Lϕ) ∗ on Orlicz spaces, Bull. Pol. Acad. Sci., Math. 34 (1986) 577–583.Google Scholar
  14. M. Nowak: Duality of vector-valued function spaces I, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 37 (1997) 195–215.Google Scholar
  15. M. Nowak: Strong topologies on vector-valued function spaces, Czechoslovak Math. J. 50 (125) (2000) 401–414.https://doi.org/10.1023/A:1022487305977Google Scholar
  16. M. Nowak: Lebesgue topologies on vector-valued function spaces, Math. Jap. 52(2) (2000) 171–182.https://doi.org/10.1023/A:1022487305977Google Scholar
  17. M. Nowak: Conditional weak compactness in vector-valued function spaces, Proc. Amer. Math. Soc. 129(10) (2001) 2947–2953.https://doi.org/10.1090/S0002-9939-01-06064-6Google Scholar
  18. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math. 1364, Springer, Berlin (1989).https://doi.org/10.1007/978-3-662-21569-2Google Scholar
  19. M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Marcel Dekker, New York (1991).Google Scholar
  20. W. Ruess: (Weakly) compact operators and DF spaces, Pac. J. Math. 98(2) (1982) 419–441.https://doi.org/10.2140/pjm.1982.98.419Google Scholar
  21. H. Schaefer, Xiao-Dong Zhang: On the Vitali-Hahn-Saks theorem, Oper. Theory Adv. Appl. 75 (1995) 289–297.https://doi.org/10.1007/978-3-0348-9076-2_15Google Scholar
  22. R. Wheeler: A survey of Baire measures and strict topologies, Expo. Math. 2 (1983) 97–190.Google Scholar
  23. A. Wilansky: Modern Methods in Topological Vector-Spaces, McGraw-Hill, Düsseldorf (1978).Google Scholar
  24. A. Wiweger: Linear spaces with mixed topology, Stud. Math. 20 (1961) 47–68.https://doi.org/10.4064/sm-20-1-47-68Google Scholar
  25. A. C. Zaanen: Riesz Spaces II, North-Holland, Amsterdam (1983).Google Scholar

Suggested citation

M. Nowak. “Linear Operators on Vector-Valued Function Spaces with Mackey Topologies.” Journal of Convex Analysis 15 (2008), No. 1, 165–178. https://doi.org/10.68381/jca15011

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.