Abstract

Let T ⁣:X1→Y1T\colon X_1\to Y_1 and S ⁣:X2→Y2S\colon X_2\to Y_2 be two continuous linear operators between Banach function spaces related to a finite measure space. Under some lattice requirements on the spaces involved, we give characterizations by means of inequalities of when TT can be strongly factorized through SS, that is, T=Mg∘S∘MfT=M_g\circ S\circ M_f with Mf ⁣:X1→X2M_f\colon X_1\to X_2 and Mg ⁣:Y2→Y1M_g\colon Y_2\to Y_1 being multiplication operators defined by some measurable functions ff and gg. In particular, we study the cases when SS is a composition operator or a kernel operator.

Author information

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

Olvido Delgado

Dep. de Matemática Aplicada I, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain

olvido@us.es

Enrique A. Sánchez Pérez

Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

easancpe@mat.upv.es

Keywords

  • Banach function spaces
  • factorization of operators
  • multiplication operators
  • product spaces
  • vector measures

Mathematics Subject Classification

46E30, 47B38; 46B42

References

  1. W. Arendt, S. Thomaschewski: Local operators and forms, Positivity 9 (2005) 357–367.https://doi.org/10.1007/s11117-005-3558-1Google Scholar
  2. J. M. Calabuig, O. Delgado, E. A. Sánchez Pérez: Generalized perfect spaces, Indag. Math. 19 (2008) 359–378.https://doi.org/10.1016/S0019-3577(09)00008-1Google Scholar
  3. J. M. Calabuig, O. Delgado, E. A. Sánchez Pérez: Factorizing operators on Banach function spaces through spaces of multiplication operators, J. Math. Anal. Appl. 364 (2010) 88–103.https://doi.org/10.1016/j.jmaa.2009.10.034Google Scholar
  4. G. P. Curbera, W. J. Ricker: Vector measures, integration and applications, in: Positivity (Tunis, 2006), K. Boulabiar et al. (ed.), Trends Math., Birkhäuser, Basel (2007) 127–160.https://doi.org/10.1007/978-3-7643-8478-4_5Google Scholar
  5. A. Defant: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces, Positivity 5 (2001) 153–175.https://doi.org/10.1023/A:1011466509838Google Scholar
  6. A. Defant, E. A. Sánchez Pérez: Maurey-Rosenthal factorization of positive operators and convexity, J. Math. Anal. Appl. 297 (2004) 771–790.https://doi.org/10.1016/j.jmaa.2004.04.047Google Scholar
  7. A. Defant, E. A. Sánchez Pérez: Domination of operators on function spaces, Math. Proc. Camb. Philos. Soc. 146 (2009) 57–66.https://doi.org/10.1017/S0305004108001734Google Scholar
  8. O. Delgado, E. A. Sánchez Pérez: Summability properties for multiplication operators on Banach function spaces, Integral Equations Oper. Theory 66 (2010) 197–214.https://doi.org/10.1007/s00020-010-1741-7Google Scholar
  9. J. Diestel, J. J. Uhl, Jr.: Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  10. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces II, Springer, Berlin (1979).https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  11. G. Ya. Lozanovskii: On some Banach lattices, Sib. Math. J. 10 (1969) 419–430.https://doi.org/10.1007/BF01078332Google Scholar
  12. L. Maligranda, L. E. Persson: Generalized duality of some Banach function spaces, Indag. Math. 51 (1989) 323–338.https://doi.org/10.1016/S1385-7258(89)80007-1Google Scholar
  13. K. Musial: A Radon-Nikodým theorem for the Bartle-Dunford-Schwartz integral, Atti Semin. Mat. Fis. Univ. Modena 41 (1993) 227–233.Google Scholar
  14. S. Okada, W. J. Ricker, E. A. Sánchez Pérez: Optimal Domain and Integral Extension of Operators acting in Function Spaces, Oper. Theory Adv. Appl. 180, Birkhäuser, Basel (2008).https://doi.org/10.1007/978-3-7643-8648-1Google Scholar
  15. A. Pietsch: Operator Ideals, North Holland, Amsterdam (1980).Google Scholar
  16. P. Wojtaszczyk: Banach Spaces for Analysts, Cambridge University Press, Cambridge (1991).https://doi.org/10.1017/CBO9780511608735Google Scholar
  17. A. C. Zaanen: Integration, 2nd Rev. Ed., North Holland, Amsterdam (1967).Google Scholar
  18. A. C. Zaanen: Examples of orthomorphisms, J. Approx. Theory 13 (1975) 192–204.https://doi.org/10.1016/0021-9045(75)90052-0Google Scholar

Suggested citation

O. Delgado, E. A. Sánchez Pérez. “Strong Factorizations between Couples of Operators on Banach Function Spaces.” Journal of Convex Analysis 20 (2013), No. 3, 599–616. https://doi.org/10.68381/jca20035

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.