Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

POLYNOMIAL FACTORIZATION THROUGH Lγ(μ) SPACES

  • Cilia, Raffaella (DIPARTIMENTO DI MATEMATICA FACOLTA DI SCIENZE UNICERSITA DI CATANIA VIALE ANDREA DORIA) ;
  • Gutierrez, Joaquin M. (DEPARTAMENTO DE MATEMATICA APLICADA ETS DE INGENIEROS INDUSTRIALES UNIVERSIDAD POLITECNICA DE MADRID)
  • Published : 2009.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

We give conditions so that a polynomial be factorable through an $L_{\gamma}({\mu})$ space. Among them, we prove that, given a Banach space X and an index m, every absolutely summing operator on X is 1-factorable if and only if every 1-dominated m-homogeneous polynomial on X is right 1-factorable, if and only if every 1-dominated m-homogeneous polynomial on X is left 1-factorable. As a consequence, if X has local unconditional structure, then every 1-dominated homogeneous polynomial on X is right and left 1-factorable.

Keywords

References

  1. R. Alencar, On reflexivity and basis for P$(^mE)$, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 131-138.
  2. R. M. Aron and P. D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3-24.
  3. F. Blasco, Complementation in spaces of symmetric tensor products and polynomials, Studia Math. 123 (1997), no. 2, 165-173. https://doi.org/10.4064/sm-123-2-165-173
  4. G. Botelho, Ideals of polynomials generated by weakly compact operators, Note Mat. 25 (2005/06), no. 1, 69-102.
  5. G. Botelho and D. M. Pellegrino, Two new properties of ideals of polynomials and applications, Indag. Math. (N.S.) 16 (2005), no. 2, 157-169. https://doi.org/10.1016/S0019-3577(05)80019-9
  6. D. Carando, Extendible polynomials on Banach spaces, J. Math. Anal. Appl. 233 (1999), no. 1, 359-372. https://doi.org/10.1006/jmaa.1999.6319
  7. R. Cilia, M. D'Anna, and J. M. Gutierrez, Polynomials on Banach spaces whose duals are isomorphic to $\iota_1(\Gamma)$, Bull. Austral. Math. Soc. 70 (2004), no. 1, 117-124. https://doi.org/10.1017/S0004972700035863
  8. R. Cilia and J. M. Gutierrez, Polynomial characterization of Asplund spaces, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 3, 393-400.
  9. R. Cilia and J. M. Guti´errez, Dominated, diagonal polynomials on lp spaces, Arch. Math. (Basel) 84 (2005), no. 5, 421-431. https://doi.org/10.1007/s00013-005-1194-4
  10. R. Cilia and J. M. Gutierrez, Ideals of integral and r-factorable polynomials, to appear in Bol. Soc. Mat. Mexicana.
  11. A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, 176. North-Holland Publishing Co., Amsterdam, 1993.
  12. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, 43. Cambridge University Press, Cambridge, 1995.
  13. J. Diestel and J. J. Uhl, Jr., Vector Measures, Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977.
  14. S. Dineen, Complex Analysis on Infinite-Dimensional Spaces, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1999.
  15. K. Floret, Natural norms on symmetric tensor products of normed spaces, Note Mat. 17 (1997), 153-188.
  16. J. M. Gutierrez and I. Villanueva, Extensions of multilinear operators and Banach space properties, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 3, 549-566. https://doi.org/10.1017/S0308210500002535
  17. J. Lindenstrauss and H. P. Rosenthal, The Lp spaces, Israel J. Math. 7 (1969), 325-349. https://doi.org/10.1007/BF02788865
  18. M. C. Matos, Absolutely summing holomorphic mappings, An. Acad. Brasil. Cienc. 68 (1996), no. 1, 1-13.
  19. Y. Melendez and A. Tonge, Polynomials and the Pietsch domination theorem, Math. Proc. R. Ir. Acad. 99A (1999), no. 2, 195-212.
  20. J. Mujica, Complex Analysis in Banach Spaces, North-Holland Publishing Co., Amsterdam, 1986.
  21. R. A. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, Ph. D. Thesis, Trinity College, Dublin 1980.
  22. R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer-Verlag London, Ltd., London, 2002.