Integral representation of linear operators on Orlicz-Bochner spaces

• Autores: Krzystof Feledziak, Marian Nowak
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 61, Fasc. 3, 2010, págs. 277-290
• Idioma: español
• DOI: 10.1007/bf03191233
• Enlaces
  • Texto completo
• Resumen

  • Let (Ω, Σ, μ) be a σ-finite measure space and let \mathcal{L}(X,Y) stand for the space of all bounded linear operators between Banach spaces (X; ‖ • ‖ ^X ) and (Y; ‖ • ‖ ^Y ). We study the problem of integral representation of linear operators from an Orlicz-Bochner spaceL ^ϕ(μ,X) toY with respect to operator measures m : \sum \to \mathcal{L}(X,Y). It is shown that a linear operatorT:L^ϕ (μ,X) →Y has the integral representationT(f = ∫^Ω f(ω)dm with respect to a ϕ*-variationally μ-continuous operator measurem if and only ifT is (γ_ϕ ‖ • ‖ Y )-continuous, where γ_ϕ stands for a natural mixed topology onL ^ϕ (μ,X). As an application, we derive Vitali-Hahn-Saks type theorems for families of operator measures.

• Referencias bibliográficas
  • C.D. Aliprantis and O. Burkinshaw, Positive Operators, Pure and Applied Mathematics 119, Academic Press Inc., Orlando, FL, 1985.
  • C.D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs 105, American...
  • R.A. Alò, A. De Korwin, and L. Kunes, Topological aspects of q-regular measures, Studia Math. 48 (1973), 49–60.
  • J. Batt, Applications of the Orlicz-Pettis theorem to operator-valued measures and compact and weakly compact transformations on the spaces...
  • A.V. Bukhvalov, On an analytic representation of operators with abstract norm, Izv. Vyss. Uceb. Zaved. Math. 11 (1975), 21–32.
  • A.V. Bukhvalov, On an analytic representation of linear operators by means of measurable vector-valued functions, Izv. Vyss. Uceb. Zaved....
  • J.B. Cooper, Saks Spaces and Applications to Functional Analysis, North-Holland Publishing Co., Amsterdam-New York, 1978.
  • J. Diestel, On the representation of bounded linear operators from Orlicz spaces of Lebesgue-Bochner measurable functions to any Banach space,...
  • J. Diestel and J.J. Uhl, Vector Measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI, 1977.
  • N. Dinculeanu, Espaces d’Orlicz de champs de vecteurs III, Opérations linéaires, Studia Math. 17 (1958), 285–293.
  • N. Dinculeanu, Espaces d’Orlicz de champs de vecteurs IV, Opérations linéaires, Studia Math. 19 (1960), 321–331.
  • N. Dinculeanu, Vector Measures, International Series of Monographs in Pure and Applied Mathematics 95, Pergamon Press, Oxford-New York-Toronto,...
  • N. Dinculeanu, Linear operators on Lp-spaces, Vector and Operator-Valued Measures and Applications, (Proc. Sympos. Alta, Utah, 1972), 109–124,...
  • N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces, Wiley-Interscience, New York, 2000.
  • K. Feledziak and M. Nowak, Locally solid topologies on vector valued function spaces, Collect. Math. 48 (1997), 487–511.
  • K. Feledziak, Uniformly Lebesgue topologies on Köthe-Bochner spaces, Comment. Math. Prace Mat. 37 (1997), 487–511.
  • K. Feledziak, Weakly compact operators on Köthe-Bochner spaces with the mixed topology, Function Spaces VIII, 71–77, Warsaw, 2008.
  • M.A. Krasnosel’skii and Ja.B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.
  • P.W. Lewis, Some regularity conditions on vector measures with finite semi-variation, Rev. Roumaine Math. Pures Appl. 15 (1970), 375–384.
  • P.W. Lewis, Vector measures and topology, Rev. Roumaine Math. Pures Appl. 16 (1971), 1201–1209.
  • M. Nowak, Lebesgue topologies on vector-valued function spaces, Math. Japon. 52 (2000), 171–182.
  • M. Nowak, Linear operators on vector-valued function spaces with Mackey topologies, J. Convex Anal. 15 (2008), 165–178.
  • M. Nowak, Operator-valued measures and linear operators, J. Math. Anal. Appl. 337 (2008), 695–701.
  • M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc., New York,...
  • A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47–68.