Abstract

We study multilinear operators from quasi-Banach lattices to quasi-Banach spaces. We prove that certain vector valued norm inequalities for these operators are equivalent to domination theorems. As an application we show that under some mild assumptions these domination theorems can be expressed in terms of factorization through Orlicz spaces. In the case of the multilinear functionals on C(K)-spaces we recover a multilinear variant of the Grothendieck factorization theorem.

Author information

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Mieczysław Mastyło

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
and: Institute of Mathematics, Polish Academy of Sciences (Poznań branch), Umultowska 87, 61-614 Poznań, Poland

mastylo@amu.edu.pl

Enrique A. Sánchez Pérez

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

easancpe@mat.upv.es

Keywords

  • Factorization
  • Banach function lattice
  • Banach envelope
  • Orlicz space

Mathematics Subject Classification

46E30; 47B38, 46B42

References

  1. R. C. Blei: Multi-linear measure theory and the Grothendieck factorization theorem, Proc. Lond. Math. Soc., III. Ser. 56 (3) (1988) 529–546.https://doi.org/10.1112/plms/s3-56.3.529Google Scholar
  2. 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
  3. A. Defant, M. Mastylo: Interpolation of Fremlin tensor products and Schur factorization of matrices, J. Funct. Anal. 262 (2012) 3981–3999.https://doi.org/10.1016/j.jfa.2012.02.007Google Scholar
  4. 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
  5. 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
  6. J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators, Cambridge Univ. Press, Cambridge (1995).https://doi.org/10.1017/CBO9780511526138Google Scholar
  7. N. J. Kalton, S. J. Montgomery-Smith: Set-functions and factorization, Arch. Math. 61(2) (1993) 183–200.https://doi.org/10.1007/BF01207468Google Scholar
  8. A. Kamińska, M. Mastylo: Abstract duality Sawyer formula and its applications, Monatsh. Math. 151(3) (2007) 223–245.https://doi.org/10.1007/s00605-007-0445-9Google Scholar
  9. L. V. Kantorovich, G. P. Akilov: Functional Analysis, 2nd Ed., Pergamon Press, Oxford (1982).Google Scholar
  10. S. G. Krein, Ju. I. Petunin, E. M. Semenov: Interpolation of Linear Operators, Translations of Mathematical Monographs 54, AMS, Providence (1982).Google Scholar
  11. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, Springer, Berlin (1979).https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  12. M. Mastylo, E. A. Sánchez Pérez: Maurey-Rosenthal factorization of operators through Orlicz spaces, preprint.Google Scholar
  13. S. Okada, W. J. Ricker, E. A. Sánchez Pérez: Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory: Advances and Applications 180, Birkhäuser, Basel (2008).https://doi.org/10.1007/978-3-7643-8648-1Google Scholar
  14. A. Pietsch: Operator Ideals, North-Holland, Amsterdam (1980).Google Scholar
  15. G. Pisier: Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Reg. Conf. Ser. Math. 60, Amer. Math. Soc., Providence (1985).https://doi.org/10.1090/cbms/060Google Scholar
  16. P. Wojtaszczyk: Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge Univ. Press, Cambridge (1991).https://doi.org/10.1017/CBO9780511608735Google Scholar

Suggested citation

M. Mastyło, E. A. Sánchez Pérez. “Domination and Factorization of Multilinear Operators.” Journal of Convex Analysis 20 (2013), No. 4, 999–1012. https://doi.org/10.68381/jca20054

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