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Scalar-valued dominated polynomials on Banach spaces


Authors: Geraldo Botelho and Daniel M. Pellegrino
Journal: Proc. Amer. Math. Soc. 134 (2006), 1743-1751
MSC (2000): Primary 46G25; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-05-08501-1
Published electronically: December 20, 2005
MathSciNet review: 2204287
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Abstract: It is well known that 2-homogeneous polynomials on $ {\mathcal L}_\infty$-spaces are 2-dominated. Motivated by the fact that related coincidence results are possible only for polynomials defined on symmetrically regular spaces, we investigate the situation in several classes of symmetrically regular spaces. We prove a number of non-coincidence results which makes us suspect that there is no infinite dimensional Banach space $ E$ such that every scalar-valued homogeneous polynomial on $ E$ is $ r$-dominated for every $ r \geq 1$.


References [Enhancements On Off] (What's this?)

  • 1. R. M. Aron and S. Dineen. Q-reflexive Banach spaces, Rocky Mountain J. Math. 27 (1997), 1009-1025. MR 1627646 (99h:46010)
  • 2. R. M. Aron, C. Hervés and M. Valdivia. Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), 189-204. MR 0707203 (84g:46066)
  • 3. G. Botelho. Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. Roy. Irish Acad. Sect. A 97 (1997), 145-153. MR 1645283 (99i:46006)
  • 4. G. Botelho. Almost summing polynomials, Math. Nachr. 211 (2000), 25-36. MR 1743489 (2001d:46069)
  • 5. G. Botelho. Ideals of polynomials generated by weakly compact operators, Note Mat. 25 (2005).
  • 6. G. Botelho and D. Pellegrino. Dominated polynomials on $ {\mathcal L}_p$-spaces, Arch. Math. 83 (2004), 364-370. MR 2096810 (2005e:46076)
  • 7. J. Bourgain. New classes of $ {\mathcal L}_p$-spaces, Lecture Notes in Math. 889, 1981. MR 0639014 (83j:46028)
  • 8. J. Bourgain. New Banach space properties of certain spaces of analytic functions, Proc. Internat. Congress of Mathematicians, vol. II. Warzawa, 1983, 945-951. MR 0804748 (86m:46050)
  • 9. J. Bourgain. New Banach space properties of the disc algebra and $ H^\infty$, Acta. Math. 152 (1984), 1-48. MR 0736210 (85j:46091)
  • 10. J. Bourgain. Bilinear forms on $ H^{\infty}$ and bounded bianalytic functions, Trans. Amer. Math. Soc. 286 (1984), 313-337. MR 0756042 (86c:46060)
  • 11. P. G. Casazza and T. J. Shura. Tsirelson's spaces, Lecture Notes in Math. 1363 (1989). MR 0981801 (90b:46030)
  • 12. J. Castillo and F. Sánchez. Remarks on some basic properties of Tsirelson's space, Note Mat. XIII (1993), 117-122. MR 1283523 (95f:46011)
  • 13. A. Defant and K. Floret. Tensor Norms and Operator Ideals, North-Holland Mathematics Studies 176, North-Holland, 1993. MR 1209438 (94e:46130)
  • 14. J. Diestel, H. Jarchow and A. Tonge. Absolutely summing operators, Cambridge University Press, Cambridge, 1995. MR 1342297 (96i:46001)
  • 15. S. Dineen. Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London, 1999. MR 1705327 (2001a:46043)
  • 16. H. Fetter and B. Gamboa. The James forest, London Mathematical Society Lecture Notes Series, 236, 1997. MR 1474498 (98k:46013)
  • 17. M. González and J. Gutiérrez. Injective factorization of holomorphic mappings, Proc. Amer. Math. Soc. 127 (1999), 1715-1721. See also Erratum in vol. 129 (2001), 1255-1256. MR 1610897 (99i:46033) MR 1814156 (2002a:46059)
  • 18. J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces I and II, Springer-Verlag, 1996.
  • 19. M. Matos. Absolutely summing holomorphic mappings, An. Acad. Brasil. Ciênc. 68 (1996), 1-13. MR 1752625 (2001c:46086)
  • 20. A. Pe\lczynski. Banach spaces of analytic functions and absolutely summing operators, CBMS Regional Conf. Series 30, Amer. Math. Soc., 1977. MR 0511811 (58:23526)
  • 21. D. Pellegrino. Cotype and absolutely summing homogeneous polynomials in $ {\mathcal L}_p$ spaces, Studia Math. 157 (2003), 121-131. MR 1980709 (2004f:46019)
  • 22. D. Pellegrino. On scalar-valued nonlinear absolutely summing mappings, Ann. Polon. Math. 83 (2004), 281-288. MR 2111715 (2005h:46022)
  • 23. A. Pietsch. Ideals of multilinear functionals, Proceedings of the Second International Conference on Operator Algebras, Ideals and Their Applications in Theoretical Physics, 185-199, Teubner-Texte, Leipzig, 1983. MR 0763541
  • 24. M. Valdivia. Complemented subspaces and interpolation properties in spaces of polynomials, J. Math. Anal. Appl. 208 (1997), 1-30. MR 1440340 (99d:46026)
  • 25. M. Venkova. Properties of Q-reflexive Banach spaces, J. Math. Anal. Appl. 264 (2001), 96-106. MR 1868330 (2003a:46034)

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Additional Information

Geraldo Botelho
Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uber- lândia, Brazil
Email: botelho@ufu.br

Daniel M. Pellegrino
Affiliation: Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58.109-970, Campina Grande Brazil
Email: pellegrino@dme.ufcg.edu.br

DOI: https://doi.org/10.1090/S0002-9939-05-08501-1
Received by editor(s): January 18, 2005
Published electronically: December 20, 2005
Additional Notes: The authors were partially supported by Instituto do Milênio, IMPA. The second author was also supported by CNPq/FAPESQ
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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