A tensor norm preserving unconditionality for $\mathcal {L}_p$-spaces
HTML articles powered by AMS MathViewer
- by Andreas Defant and David Pérez-García PDF
- Trans. Amer. Math. Soc. 360 (2008), 3287-3306 Request permission
Abstract:
We show that, for each $n\in \mathbb {N}$, there is an $n$-tensor norm $\alpha$ (in the sense of Grothendieck) with the surprising property that the $\alpha$-tensor product $\tilde {\bigotimes }_\alpha (Y_1, \ldots , Y_n)$ has local unconditional structure for each choice of $n$ arbitrary $\mathcal {L}_{p_j}$-spaces $Y_j$. In fact, $\alpha$ is the tensor norm associated to the ideal of multiple $1$-summing $n$-linear forms on Banach spaces.References
- Lev Aizenberg, Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1147–1155. MR 1636918, DOI 10.1090/S0002-9939-99-05084-4
- G. Bennett, Schur multipliers, Duke Math. J. 44 (1977), no. 3, 603–639. MR 493490
- Harold P. Boas, Majorant series, J. Korean Math. Soc. 37 (2000), no. 2, 321–337. Several complex variables (Seoul, 1998). MR 1775963
- Harold P. Boas and Dmitry Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2975–2979. MR 1443371, DOI 10.1090/S0002-9939-97-04270-6
- H. Bohr, A theorem concerning power series, Proc. London Math. Soc. 13 (1914), 1–5.
- Fernando Bombal, David Pérez-García, and Ignacio Villanueva, Multilinear extensions of Grothendieck’s theorem, Q. J. Math. 55 (2004), no. 4, 441–450. MR 2104683, DOI 10.1093/qjmath/55.4.441
- T. K. Carne, Tensor products and Banach algebras, J. London Math. Soc. (2) 17 (1978), no. 3, 480–488. MR 493403, DOI 10.1112/jlms/s2-17.3.480
- Andreas Defant, Juan Carlos Díaz, Domingo García, and Manuel Maestre, Unconditional basis and Gordon-Lewis constants for spaces of polynomials, J. Funct. Anal. 181 (2001), no. 1, 119–145. MR 1818112, DOI 10.1006/jfan.2000.3702
- Andreas Defant, Domingo García, and Manuel Maestre, Bohr’s power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197. MR 1978407, DOI 10.1515/crll.2003.030
- Andreas Defant, Domingo García, Manuel Maestre, and David Pérez-García, Extension of multilinear forms and polynomials from subspaces of $\scr L_1$-spaces, Houston J. Math. 33 (2007), no. 3, 839–860. MR 2335739
- Andreas Defant and Nigel Kalton, Unconditionality in spaces of $m$-homogeneous polynomials, Q. J. Math. 56 (2005), no. 1, 53–64. MR 2124579, DOI 10.1093/qmath/hah022
- A. Defant, M. Maestre, and C. Prengel, Domains of convergence for monomial series expansions of holomorphic functions in infinitely many variables, preprint.
- Andreas Defant, Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces, Positivity 5 (2001), no. 2, 153–175. MR 1825653, DOI 10.1023/A:1011466509838
- 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
- Seán Dineen and Richard M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), no. 3, 227–234. MR 1019790, DOI 10.4064/sm-94-3-227-234
- Klaus Floret and Stephan Hunfeld, Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1425–1435. MR 1879966, DOI 10.1090/S0002-9939-01-06228-1
- B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 (1961), 1281–1286. MR 147881
- Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27–48. MR 410341, DOI 10.1007/BF02392140
- Y. Gordon, D. R. Lewis, and J. R. Retherford, Banach ideals of operators with applications, J. Functional Analysis 14 (1973), 85–129. MR 0380488, DOI 10.1016/0022-1236(73)90031-1
- A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 94682
- S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43–68. MR 270118, DOI 10.4064/sm-34-1-43-67
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Mário C. Matos, Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 54 (2003), no. 2, 111–136. MR 1995136
- David Pérez-García, The inclusion theorem for multiple summing operators, Studia Math. 165 (2004), no. 3, 275–290. MR 2110152, DOI 10.4064/sm165-3-5
- David Pérez-García, Comparing different classes of absolutely summing multilinear operators, Arch. Math. (Basel) 85 (2005), no. 3, 258–267. MR 2172384, DOI 10.1007/s00013-005-1125-4
- David Pérez-García and Ignacio Villanueva, Unconditional bases in tensor products of Hilbert spaces, Math. Scand. 96 (2005), no. 2, 280–288. MR 2153415, DOI 10.7146/math.scand.a-14957
- David Pérez-García and Ignacio Villanueva, Multiple summing operators on Banach spaces, J. Math. Anal. Appl. 285 (2003), no. 1, 86–96. MR 2000141, DOI 10.1016/S0022-247X(03)00352-4
- David Pérez-García and Ignacio Villanueva, Multiple summing operators on $C(K)$ spaces, Ark. Mat. 42 (2004), no. 1, 153–171. MR 2056549, DOI 10.1007/BF02432914
- David Pérez-García and Ignacio Villanueva, A composition theorem for multiple summing operators, Monatsh. Math. 146 (2005), no. 3, 257–261. MR 2184227, DOI 10.1007/s00605-005-0316-1
- David Pérez-García and Ignacio Villanueva, There is no lattice preserving natural tensor norm, Quaest. Math. 27 (2004), no. 3, 267–273. MR 2109666, DOI 10.2989/16073600409486099
- Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
- Gilles Pisier, Some results on Banach spaces without local unconditional structure, Compositio Math. 37 (1978), no. 1, 3–19. MR 501916
- Carsten Schütt, Unconditionality in tensor products, Israel J. Math. 31 (1978), no. 3-4, 209–216. MR 516148, DOI 10.1007/BF02761492
- I. Schütt, Banachräume absolut $p$-summierender Operatoren, Ph.D. Thesis, Kiel, 1986.
- N. Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals, Longman Scientific & Technical, 1989.
Additional Information
- Andreas Defant
- Affiliation: Fachbereich Mathematik, Universitaet Oldenburg, D–26111, Oldenburg, Germany
- Email: defant@mathematik.uni-oldenburg.de
- David Pérez-García
- Affiliation: Área de Matemática Aplicada, Universidad Rey Juan Carlos, C/ Tulipan s/n, 28933 Móstoles (Madrid), Spain
- Address at time of publication: Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Email: david.perez.garcia@urjc.es, dperez@mat.ucm.es
- Received by editor(s): September 15, 2005
- Received by editor(s) in revised form: September 27, 2006
- Published electronically: January 10, 2008
- Additional Notes: This work was partially supported by Spanish projects MTM2005-00082 and MTM2005-08210
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3287-3306
- MSC (2000): Primary 46G25, 46M05, 47L20
- DOI: https://doi.org/10.1090/S0002-9947-08-04428-0
- MathSciNet review: 2379797