Advertisement

Completability and optimal factorization norms in tensor products of Banach function spaces

  • J. M. CalabuigEmail author
  • M. Fernández-Unzueta
  • F. Galaz-Fontes
  • E. A. Sánchez-Pérez
Original Paper

Abstract

Given \(\sigma \)-finite measure spaces \((\Omega _1,\Sigma _1, \mu _1)\) and \((\Omega _2,\Sigma _2,\mu _2)\), we consider Banach spaces \(X_1(\mu _1)\) and \(X_2(\mu _2)\), consisting of \(L^0 (\mu _1)\) and \(L^0 (\mu _2)\) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product \(X_1(\mu _1) \otimes _\pi X_2(\mu _2)\) is continuously included in the metric space of measurable functions \(L^0(\mu _1 \otimes \mu _2)\). In particular, we prove that the elements of the completion of the projective tensor product of \(L^p\)-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally show that given a bounded linear operator \(T:X_1(\mu _1) \otimes _\pi X_2(\mu _2) \rightarrow E\) (where E is a Banach space), a norm can be found for T to be bounded, which is ‘minimal’ with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.

Keywords

Product measure Banach function space Bilinear operator Tensor product Factorization 

Mathematics Subject Classification

46E30 28A35 47H60 

Notes

References

  1. 1.
    Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics, Vol 50, AMS (2002)Google Scholar
  2. 2.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)zbMATHGoogle Scholar
  3. 3.
    Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity-Trends in Mathematics. Birkhäser Verlag AG, Basel, pp. 97–26 (2007)Google Scholar
  4. 4.
    Buskes, G., Van Rooij, A.: Bounded variation and tensor products of Banach lattices. Positivity 7, 47–59 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108(2), 353–367 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Equivalent norms in a Banach function space and the subsequence property. J. Korean Math. Soc.  https://doi.org/10.4134/JKMS.j180682
  7. 7.
    Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Curbera, G.P., Ricker, W.J.: Vector measures, integration and applications. In: Positivity. Birkhäuser Basel, pp. 127–160 (2007)Google Scholar
  9. 9.
    Gil de Lamadrid, J.: Uniform cross norms and tensor products. J. Duke Math. 32, 797–803 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dunford, N., Schwartz, J.: Linear Operators, Part I: General Theory. Interscience Publishers Inc., New York (1958)zbMATHGoogle Scholar
  11. 11.
    Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94(3), 777–798 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211(2), 87–106 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yew, K.L.: Completely \(p\)-summing maps on the operator Hilbert space OH. J. Funct. Anal. 255, 1362–1402 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kwapien, S., Pelczynski, A.: The main triangle projection in matrix spaces and its applications. Stud. Math. 34(1), 43–68 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)CrossRefzbMATHGoogle Scholar
  16. 16.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)zbMATHGoogle Scholar
  17. 17.
    Milman, M.: Some new function spaces and their tensor products. Depto. de Matemática, Facultad de Ciencias, U. de los Andes, Mérida, Venezuela (1978)Google Scholar
  18. 18.
    Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. Oper. Theory Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Google Scholar
  19. 19.
    Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 579–591 (1984)Google Scholar
  20. 20.
    Zaanen, A.C.: Integration. North-Holland Publishing Company, Amsterdam-New York (1967)zbMATHGoogle Scholar
  21. 21.
    Zaanen, A.C.: Riesz Spaces II. North-Holland Publishing Company, Amsterdam (1983)zbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Centro de Investigación em Matemáticas, A.C.GuanajuatoMexico

Personalised recommendations