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Multiple summing operators on Banach spaces. (English) Zbl 1044.46037
There are various methods to generalize properties of linear operators, like compactness, summability, etc., to multilinear operators. In this paper, the authors construct a predual of certain spaces of multiple $$(q; p_1,\ldots,p_n)$$-summing multilinear operators. The deduced results are also used to improve several results due to K. Floret and M. C. Matos [Math. Nachr. 176, 65–72 (1995; Zbl 0839.46040)] and Y. Meléndez and A. Tonge [Math. Proc. R. Ir. Acad. 99A, 195–212 (1999; Zbl 0973.46037)] or to give a much easier approach to a result essentially contained in the work of H. P. Rosenthal and S. J. Szarek [Functional analysis, Proc. Semin., Austin/TX (USA) 1987-89, Lect. Notes Math. 1470, 108–132 (1991; Zbl 0758.47022)].

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47H60 Multilinear and polynomial operators 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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##### References:
 [1] Alencar, R., Multilinear mappings of nuclear and integral type, Proc. amer. math. soc., 94, 33-38, (1985) · Zbl 0537.47011 [2] Aron, R.; Choi, S.Y.; Llavona, J., Estimates by polynomials, Bull. austral. math. soc., 52, 475-486, (1995) · Zbl 0839.46039 [3] F. Bombal, D. Pérez-Garcı́a, I. Villanueva, Multilinear extensions of Grothendieck’s theorem, preprint [4] Cilia, R.; D’Anna, M.; Gutiérrez, J., Polynomial characterization of $$L∞$$-spaces, J. math. anal. appl., 275, 900-912, (2002) · Zbl 1031.46051 [5] Cobos, F.; Kühn, T.; Peetre, J., Schatten – von Neumann classes of multilinear forms, Duke math. J., 65, 121-156, (1992) · Zbl 0779.47016 [6] Defant, K.; Floret, A., Tensor norms and operator ideals, (1993), North-Holland · Zbl 0774.46018 [7] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators, (1995), Cambridge Univ. Press · Zbl 0855.47016 [8] Dwyer, T.A.W., Partial differential equations in fischer – fock spaces for the hilbert – schmidt holomorphy type, Bull. amer. math. soc., 77, 725-730, (1971) · Zbl 0222.46019 [9] Floret, S.; Hunfeld, K., Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces, Proc. amer. math. soc., 130, 1425-1435, (2001) · Zbl 1027.46054 [10] Floret, M.C.; Matos, K., Applications of a Khintchine inequality to holomorphic mappings, Math. nachr., 176, 65-72, (1995) · Zbl 0839.46040 [11] M.C. Matos, Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math., to appear · Zbl 1078.46031 [12] Matos, M.C., On multilinear mappings of nuclear type, Rev. mat. complut., 6, 61-81, (1993) · Zbl 0807.46022 [13] Meléndez, Y.; Tonge, A., Polynomials and the Pietsch domination theorem, Math. proc. roy. irish acad. sect. A, 99, 195-212, (1999) · Zbl 0973.46037 [14] D. Pérez-Garcı́a, I. Villanueva, Multiple summing operators on C(K) spaces, to appear · Zbl 1063.46032 [15] Pietsch, A., Ideals of multilinear functionals (designs of a theory), (), 185-199 [16] Ramanujan, M.S.; Schock, E., Operator ideals and spaces of bilinear operators, Linear and multilinear algebra, 18, 307-318, (1985) · Zbl 0607.47042 [17] Rosenthal, H.P.; Szarek, S.J., On tensor products of operators from Lp to lq, () · Zbl 0758.47022 [18] M.L.V. Souza, Aplicações multilineares completamente absolutamente somantes, Ph.D. thesis, Univ. Estadual de Campinas, 2003 [19] I. Villanueva, Integral mappings between Banach spaces, J. Math. Anal. Appl., to appear · Zbl 1030.47503
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