Scalar-valued dominated polynomials on Banach spaces
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- by Geraldo Botelho and Daniel M. Pellegrino PDF
- Proc. Amer. Math. Soc. 134 (2006), 1743-1751 Request permission
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
- Richard M. Aron and Seán Dineen, $Q$-reflexive Banach spaces, Rocky Mountain J. Math. 27 (1997), no. 4, 1009–1025. MR 1627646, DOI 10.1216/rmjm/1181071856
- R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2
- Geraldo Botelho, Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. Roy. Irish Acad. Sect. A 97 (1997), no. 2, 145–153. MR 1645283
- Geraldo Botelho, Almost summing polynomials, Math. Nachr. 211 (2000), 25–36. MR 1743489, DOI 10.1002/(SICI)1522-2616(200003)211:1<25::AID-MANA25>3.3.CO;2-W
- G. Botelho. Ideals of polynomials generated by weakly compact operators, Note Mat. 25 (2005).
- Geraldo Botelho and Daniel M. Pellegrino, Dominated polynomials on $\scr L_p$-spaces, Arch. Math. (Basel) 83 (2004), no. 4, 364–370. MR 2096810, DOI 10.1007/s00013-004-1035-x
- Jean Bourgain, New classes of ${\cal L}^{p}$-spaces, Lecture Notes in Mathematics, vol. 889, Springer-Verlag, Berlin-New York, 1981. MR 639014
- J. Bourgain, New Banach space properties of certain spaces of analytic functions, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 945–951. MR 804748
- J. Bourgain, New Banach space properties of the disc algebra and $H^{\infty }$, Acta Math. 152 (1984), no. 1-2, 1–48. MR 736210, DOI 10.1007/BF02392189
- J. Bourgain, Bilinear forms on $H^{\infty }$ and bounded bianalytic functions, Trans. Amer. Math. Soc. 286 (1984), no. 1, 313–337. MR 756042, DOI 10.1090/S0002-9947-1984-0756042-5
- Peter G. Casazza and Thaddeus J. Shura, Tsirel′son’s space, Lecture Notes in Mathematics, vol. 1363, Springer-Verlag, Berlin, 1989. With an appendix by J. Baker, O. Slotterbeck and R. Aron. MR 981801, DOI 10.1007/BFb0085267
- Jesús M. F. Castillo and Fernando Sánchez, Remarks on some basic properties of Tsirelson’s space, Note Mat. 13 (1993), no. 1, 117–122. MR 1283523
- Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
- Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
- Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
- Helga Fetter and Berta Gamboa de Buen, The James forest, London Mathematical Society Lecture Note Series, vol. 236, Cambridge University Press, Cambridge, 1997. With a foreword by Robert C. James and a prologue by Bernard Beauzamy. MR 1474498, DOI 10.1017/CBO9780511662379
- Manuel González and Joaquín M. Gutiérrez, Injective factorization of holomorphic mappings, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1715–1721. MR 1610897, DOI 10.1090/S0002-9939-99-04917-5
- J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces I and II, Springer-Verlag, 1996.
- Mário C. Matos, Absolutely summing holomorphic mappings, An. Acad. Brasil. Ciênc. 68 (1996), no. 1, 1–13. MR 1752625
- Aleksander Pełczyński, Banach spaces of analytic functions and absolutely summing operators, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 30, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at Kent State University, Kent, Ohio, July 11–16, 1976. MR 0511811, DOI 10.1090/cbms/030
- Daniel Pellegrino, Cotype and absolutely summing homogeneous polynomials in $\scr L_p$ spaces, Studia Math. 157 (2003), no. 2, 121–131. MR 1980709, DOI 10.4064/sm157-2-2
- Daniel Pellegrino, On scalar-valued nonlinear absolutely summing mappings, Ann. Polon. Math. 83 (2004), no. 3, 281–288. MR 2111715, DOI 10.4064/ap83-3-10
- Albrecht Pietsch, Ideals of multilinear functionals (designs of a theory), Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics (Leipzig, 1983) Teubner-Texte Math., vol. 67, Teubner, Leipzig, 1984, pp. 185–199. MR 763541
- Manuel Valdivia, Complemented subspaces and interpolation properties in spaces of polynomials, J. Math. Anal. Appl. 208 (1997), no. 1, 1–30. MR 1440340, DOI 10.1006/jmaa.1997.5191
- M. Venkova, Properties of $Q$-reflexive Banach spaces, J. Math. Anal. Appl. 264 (2001), no. 1, 96–106. MR 1868330, DOI 10.1006/jmaa.2001.7644
Additional Information
- Geraldo Botelho
- Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uber- lândia, Brazil
- MR Author ID: 638411
- 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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2204287