The Schur property on projective and injective tensor products
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- by Geraldo Botelho and Pilar Rueda PDF
- Proc. Amer. Math. Soc. 137 (2009), 219-225 Request permission
Abstract:
The problem of whether the Schur property is passed from a Banach space to its (symmetric) projective $n$-fold tensor product is reformulated in the language of polynomial ideals. As a result, a very closely related question is solved in the negative. It is also proved that the injective tensor product of infrabarrelled locally convex spaces with the Schur property has the Schur property as well.References
- Raymundo Alencar, Richard M. Aron, and Seán Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), no. 3, 407–411. MR 728358, DOI 10.1090/S0002-9939-1984-0728358-5
- Raymundo Alencar and Klaus Floret, Weak-strong continuity of multilinear mappings and the Pełczyński-Pitt theorem, J. Math. Anal. Appl. 206 (1997), no. 2, 532–546. MR 1433955, DOI 10.1006/jmaa.1997.5253
- Alvaro Arias and Jeff D. Farmer, On the structure of tensor products of $l_p$-spaces, Pacific J. Math. 175 (1996), no. 1, 13–37. MR 1419470
- Fernando Blasco, Complementation in spaces of symmetric tensor products and polynomials, Studia Math. 123 (1997), no. 2, 165–173. MR 1439028, DOI 10.4064/sm-123-2-165-173
- Geraldo Botelho, Weakly compact and absolutely summing polynomials, J. Math. Anal. Appl. 265 (2002), no. 2, 458–462. MR 1876152, DOI 10.1006/jmaa.2001.7674
- Geraldo Botelho, Daniel Pellegrino, and Pilar Rueda, On composition ideals of multilinear mappings and homogeneous polynomials, Publ. Res. Inst. Math. Sci. 43 (2007), no. 4, 1139–1155. MR 2389796
- Jean Bourgain and Gilles Pisier, A construction of ${\scr L}_{\infty }$-spaces and related Banach spaces, Bol. Soc. Brasil. Mat. 14 (1983), no. 2, 109–123. MR 756904, DOI 10.1007/BF02584862
- V. Dimant and I. Zalduendo, Bases in spaces of multilinear forms over Banach spaces, J. Math. Anal. Appl. 200 (1996), no. 3, 548–566. MR 1393101, DOI 10.1006/jmaa.1996.0224
- 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
- Manuel González and Joaquín M. Gutiérrez, Gantmacher type theorems for holomorphic mappings, Math. Nachr. 186 (1997), 131–145. MR 1461217, DOI 10.1002/mana.3211860108
- Manuel González and Joaquín Gutiérrez, The Dunford-Pettis property on tensor products, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 1, 185–192. MR 1833082, DOI 10.1017/S0305004101005175
- Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257
- Kamil John, Tensor product of several spaces and nuclearity, Math. Ann. 269 (1984), no. 3, 333–356. MR 761310, DOI 10.1007/BF01450699
- Françoise Lust, Produits tensoriels injectifs d’espaces de Sidon, Colloq. Math. 32 (1975), no. 2, 285–289 (French). MR 390794, DOI 10.4064/cm-32-2-285-289
- A. Pełczyński, On weakly compact polynomial operators on $B$-spaces with Dunford-Pettis property, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 371–378. MR 161160
- A. P. Robertson and Wendy Robertson, Topological vector spaces, 2nd ed., Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, London-New York, 1973. MR 0350361
- R. Ryan. Applications of topological tensor products to infinite dimensional holomorphy, Thesis, Trinity College, 1980.
- Raymond A. Ryan, The Dunford-Pettis property and projective tensor products, Bull. Polish Acad. Sci. Math. 35 (1987), no. 11-12, 785–792 (English, with Russian summary). MR 961717
Additional Information
- Geraldo Botelho
- Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, 38.400-902, Uberlândia, Brazil
- MR Author ID: 638411
- Email: botelho@ufu.br
- Pilar Rueda
- Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, 46.100 Burjasot, Valencia, Spain
- MR Author ID: 636319
- Email: pilar.rueda@uv.es
- Received by editor(s): September 24, 2007
- Received by editor(s) in revised form: December 27, 2007
- Published electronically: May 22, 2008
- Additional Notes: The first author was supported by CNPq Project 202162/2006-0
The second author was supported by MEC and FEDER Project MTM2005-08210 - Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 219-225
- MSC (2000): Primary 46G20; Secondary 46A04, 46A32
- DOI: https://doi.org/10.1090/S0002-9939-08-09486-0
- MathSciNet review: 2439444