×

Hahn-Banach extension of multilinear forms and summability. (English) Zbl 1161.46025

This paper deals with the relations between the validity of Hahn-Banach extension theorems for multilinear forms on Banach spaces and summability properties of sequences from these spaces. As the authors show, this is closely related to the so-called property BEP. A pair \((X, Y )\) of Banach spaces has the BEP if every bilinear form on \(X \times Y\) is extendible to any superspace containing \(X \times Y.\) The notion of \(n\)EP is defined analogously.
Among other things, the authors show
Theorem: If \((X,Y)\) has the BEP, then \(X {\widetilde{\bigotimes}}_\pi Y\) contains uniformly complemented copies of the \(\ell_2^n\)’s.
In addition,
Theorem: Suppose that \(X\) is an infinite-dimensional Banach space with the BEP. Then, for \(n \geq 2,\;{\mathcal L}(^nX) = {\mathcal L}_{(1;2,\dots,2)}(^nX). \)
Summarizing some, but by no means all, of the authors’ interesting results, we have the following.
Theorem: For every Banach space \(X,\;{\mathcal L}^2_{\text{ext}}(X) = {\mathcal L}^2_{(1;2,2)}(X) \subseteq {\mathcal L}^2(X),\) with equality if and only if \(X\) has the BEP.
Theorem: For every \(n \geq 3\) and every Banach space \(X,\) \({\mathcal L}^n_{\text{ext}}(X) \subseteq {\mathcal L}^n_{(1;2,\dots,2)}(X) \subseteq {\mathcal L}^n(X). \) Moreover, we have equality in the first inclusion if and only if \(X\) has the \(n\)EP, and in the second inclusion if and only if \(X\) has the BEP.

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47H60 Multilinear and polynomial operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Alencar, M.C. Matos, Some classes of multilinear mappings between Banach spaces, Publ. Dep. de Análisis Matemático, Sec. 1, no. 12, UCM, 1989; R. Alencar, M.C. Matos, Some classes of multilinear mappings between Banach spaces, Publ. Dep. de Análisis Matemático, Sec. 1, no. 12, UCM, 1989
[2] Aron, R. M.; Berner, P. D., A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France, 106, 1, 3-24 (1978) · Zbl 0378.46043
[3] Blei, R. C., Multidimensional extensions of Grothendieck’s inequality and applications, Ark. Mat., 17, 51-68 (1979) · Zbl 0461.43005
[4] Blei, R. C., Multilinear measure theory and the Grothendieck factorization theorem, Proc. London. Math. Soc., 56, 3, 529-546 (1988) · Zbl 0661.46057
[5] Bombal, F.; Villanueva, I.; Pérez-García, D., Multilinear extensions of Grothendieck’s theorem, Q. J. Math., 55, 441-450 (2004) · Zbl 1078.46030
[6] Bourgain, J., New Banach space properties of the disc algebra and \(H^\infty \), Acta Math., 152, 1-48 (1984) · Zbl 0574.46039
[7] Bourgain, J., Bilinear forms on \(H^\infty\) and bounded bianalytic functions, Trans. Amer. Math. Soc., 286, 313-337 (1984) · Zbl 0572.46048
[8] Cabello, F.; Pérez-García, D.; Villanueva, I., Unexpected subspaces of tensor products, J. London. Math. Soc. (2), 74, 2, 512-526 (2006) · Zbl 1122.46008
[9] Carando, D., Extendible polynomials on Banach spaces, J. Math. Anal. Appl., 233, 359-372 (1999) · Zbl 0942.46029
[10] D. Carando, V. Dimant, Extension of polynomials and John’s theorem for symmetric tensor products, Proc. Amer. Math. Soc., in press; D. Carando, V. Dimant, Extension of polynomials and John’s theorem for symmetric tensor products, Proc. Amer. Math. Soc., in press · Zbl 1118.46042
[11] Carne, T. K., Banach lattices and extensions of Grothendieck’s inequality, J. London Math. Soc., 21, 496-516 (1980) · Zbl 0434.46019
[12] Casazza, P.; Nielsen, N. J., The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures, Studia Math., 155, 1-21 (2003) · Zbl 1019.46008
[13] Castillo, J. M.F.; García, R.; Jaramillo, J., Extension of bilinear forms on Banach spaces, Proc. Amer. Math. Soc., 129, 12, 3647-3656 (2001) · Zbl 0988.46010
[14] Castillo, J. M.F.; García, R.; Jaramillo, J., Extension of bilinear forms from subspaces of \(L_1\)-spaces, Ann. Acad. Sci. Fenn. Math., 27, 91-96 (2002) · Zbl 1021.46008
[15] Choi, Y. S.; Kim, S. G.; Meléndez, Y.; Tonge, A., Estimates for absolutely summing norms of polynomials and multilinear maps, Q. J. Math., 52, 1-12 (2001) · Zbl 0991.46024
[16] Davie, A. M.; Gamelin, T. W., A theorem on polynomial-star approximation, Proc. Amer. Math. Soc., 106, 2, 351-356 (1989) · Zbl 0683.46037
[17] Defant, A.; Floret, K., Tensor Norms and Operator Ideals (1993), North-Holland: North-Holland Amsterdam · Zbl 0774.46018
[18] A. Defant, D. García, M. Maestre, D. Pérez-García, Extension of multilinear forms and polynomials from subspaces of \(\mathcal{L}_1\); A. Defant, D. García, M. Maestre, D. Pérez-García, Extension of multilinear forms and polynomials from subspaces of \(\mathcal{L}_1\)
[19] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43 (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0855.47016
[20] Dineen, S., Holomorphically complete locally convex topological vector spaces, (Lecture Notes in Math., vol. 332 (1973)), 77-111 · Zbl 0278.46005
[21] Galindo, P.; García, D.; Maestre, M.; Mújica, J., Extension of multilinear mappings on Banach spaces, Studia Math., 108, 1, 55-76 (1994) · Zbl 0852.46004
[22] S. Geiss, Ideale multilinearer Abbildungen, Diplomarbeit Universität Jena, 1984; S. Geiss, Ideale multilinearer Abbildungen, Diplomarbeit Universität Jena, 1984
[23] Gordon, Y.; Lewis, D. R., Absolutely summing operators and local unconditional structures, Acta Math., 133, 27-48 (1974) · Zbl 0291.47017
[24] Gordon, Y.; Lewis, D. R.; Retherford, J. R., Banach ideals of operators with applications to the finite dimensional structure of Banach spaces, Israel J. Math., 13, 348-360 (1972)
[25] Jarchow, H., On Hilbert-Schmidt spaces, Rend. Circ. Mat. Palermo (Suppl.), II, 153-160 (1982) · Zbl 0503.46014
[26] Kirwan, P.; Ryan, R. A., Extendibility of homogeneous polynomials on Banach spaces, Proc. Amer. Math. Soc., 126, 4, 1023-1029 (1998) · Zbl 0890.46032
[27] Matos, M. C., Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math., 54, 111-136 (2003) · Zbl 1078.46031
[28] Maurey, B., Un théorème de prolongement, C. R. Acad. Sci. Paris Ser. A, 279, 329-332 (1974) · Zbl 0291.47001
[29] Pelczyński, A., Sur certaines propriétés isomorphiques nouvelles des espaces de Banach de fonctions holomorphes \(A\) et \(H^\infty \), C. R. Acad. Sci. Paris Ser. A, 279, 9-12 (1974) · Zbl 0285.46020
[30] Pérez-García, D., A counterexample using 4-linear forms, Bull. Austral. Math. Soc., 70, 469-473 (2004) · Zbl 1074.46029
[31] Pérez-García, D., Comparing different classes of absolutely summing multilinear operators, Arch. Math. (Basel), 85, 258-267 (2005) · Zbl 1080.47047
[32] Pérez-García, D., The trace class is a \(Q\)-algebra, Ann. Acad. Sci. Fenn. Math., 31, 2, 287-295 (2006) · Zbl 1101.47046
[33] Pietsch, A., Operator Ideals (1980), North-Holland: North-Holland Amsterdam · Zbl 0399.47039
[34] Pietsch, A., Ideals of multilinear functionals (designs of a theory), (Proc. 2nd Int. Conf. on Operator Algebras, Ideals and their Applications in Theoretical Physics. Proc. 2nd Int. Conf. on Operator Algebras, Ideals and their Applications in Theoretical Physics, Leipzig. Proc. 2nd Int. Conf. on Operator Algebras, Ideals and their Applications in Theoretical Physics. Proc. 2nd Int. Conf. on Operator Algebras, Ideals and their Applications in Theoretical Physics, Leipzig, Teubner-Texte (1983)), 185-199
[35] Pisier, G., Factorization of Linear Operators and Geometry of Banach Spaces, CBMS, vol. 60 (1985), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[36] Tonge, A., The Von Neumann inequality for polynomials in several Hilbert-Schmidt operators, J. London Math. Soc., 18, 519-526 (1978) · Zbl 0399.47006
[37] Meléndez, Y.; Tonge, A., Polynomials and the Pietsch domination theorem, Math. Proc. R. Ir. Acad. A, 99, 2, 195-212 (1999) · Zbl 0973.46037
[38] Wojtaszczyk, P., Banach Spaces for Analysts, Cambridge Stud. Adv. Math., vol. 25 (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0724.46012
[39] Zalduendo, I., Extending polynomials on Banach spaces—A survey, Rev. Un. Mat. Argentina, 46, 2, 45-72 (2005) · Zbl 1122.46026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.