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Inclusion and coincidence properties for Cohen strongly summing multilinear operators. (English) Zbl 1307.47068
In light of the importance of certain ideals of linear operators, whose definitions involve notions of summability, many multilinear “relatives” have been developed in recent times. One of these classes, which is explored in the article under review, is the ideal of Cohen strongly $$p$$-summing multilinear operators.
Through a property of a tensor product operator, the authors prove that the composition of any $$m$$-linear mapping $$A$$ with linear operators $$u_1, \dots, u_m$$, where $$u_j$$ is Cohen strongly $$p_j$$-summing, produces an $$m$$-linear mapping $$A\circ(u_1, \dots, u_m)$$ which is Cohen strongly $$p$$-summing, with $$\frac{1}{p}=\frac{1}{p_1}+\cdots + \frac{1}{p_m}$$.
One of the statements of a classical theorem of S. Kwapień [Stud. Math. 38, 277–278 (1970; Zbl 0203.43302)] (which is an isomorphic version of an isometric result by J. S. Cohen [Math. Ann. 201, 177–200 (1973; Zbl 0233.47019)]) characterizes Hilbert spaces in terms of inclusions between absolutely and strongly 2-summing linear operators. Here, some multilinear versions of this result are obtained.

##### MSC:
 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] Achour, D., Mezrag, L.: On the Cohen strongly $$p$$ -summing multilinear operators. J. Math. Anal. Appl. 327, 550–563 (2007) · Zbl 1121.47013 [2] Achour, D., Saadi, K.: A polynomial characterization of Hilbert spaces. Collect. Math. 61(3), 291–301 (2010) · Zbl 1226.46042 [3] Berrios, S., Botelho, G.: Approximation properties determined by operator ideals and approximability of homogeneous polynomials and holomorphic functions. Studia Mathematica 208(2), 97–116 (2012) · Zbl 1250.46032 [4] Botelho, G., Pellegrino, D.: Two new properties of ideals of polynomials and applications. Indag. Mathem. N.S. 16(2), 157–169 (2005) · Zbl 1089.46027 [5] Botelho, G., Pellegrino, D., Rueda, P.: On composition ideals of multilinear mappings and homogeneous polynomials. Publ. RIMS Kyoto Univ. 43, 1139–1155 (2007) · Zbl 1169.46023 [6] Botelho, G.: Ideals of polynomials generated by weakly compact operators. Note Math. 25, 69–102 (2005) · Zbl 1223.46047 [7] Braunss, H.A.: Multi-ideals with special properties. Blätter Potsdamer Forschungen1/87, Potsdam (1987) [8] Bu, Q.: Some mapping properties of $$p$$ -summing operators with hilbertian domain. Contemp. Math. 328, 145–149 (2003) · Zbl 1066.47019 [9] Campos, J.R.: Multiple Cohen strongly $$p$$ -summing operators, ideals, coherence and compatibility, arXiv: 1207.6664v2 [10] Cilia, R., D’Anna, M., Gutiérrez, J.M.: Polynomial characterization of $$$$\backslash$$cal L_{$$\backslash$$infty }$$ -spaces. J. Math. Anal. Appl 275, 900–912 (2002) · Zbl 1031.46051 [11] Cobos, F., Kühn, T., Peetre, J.: Schatten-von Neumann classes of multilinear forms. Duke Math. J. 65, 121–156 (1992) · Zbl 0779.47016 [12] Cohen, J.S.: Absolutely $$p$$ -summing, $$p$$ -nuclear operators and their conjugates. Math. Ann. 201, 177–200 (1973) · Zbl 0233.47019 [13] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995) · Zbl 0855.47016 [14] Dimant, V.: Strongly $$p$$ -summing multilinear mappings. J. Math. Anal. Appl. 278, 182–193 (2003) · Zbl 1043.47018 [15] Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999) · Zbl 1034.46504 [16] Dwyer III, T.A.W.: Partial differential equations in Fischer-Fock spaces for the Hilbert–Schmidt holomorphy type. Bull. Am. Math. Soc 77, 725–730 (1971) · Zbl 0222.46019 [17] Kwapień, S.: A linear topological characterization of inner product space. Studia Math. 38, 277–278 (1970) · Zbl 0203.43302 [18] Matos, M.C.: Fully absolutely summing and Hilbert–Schmidt multilinear mappings. Collect. Math. 54(2), 111–136 (2003) · Zbl 1078.46031 [19] Mezrag, L., Saadi, K.: Inclusion theorems for Cohen strongly summing multilinear operators. Bull. Belg. Math. Soc. Simon Stevin 16, 1–11 (2009) · Zbl 1176.47047 [20] Pérez-García, D., Villanueva, I.: Multiple summing operators on $$C(K)$$ spaces. Ark. Mat. 42, 153–171 (2004) · Zbl 1063.46032 [21] Pietsch, A.: Absolut p-summierende Abbildungen in normierten Räumen. Studia Math. 28, 333–353 (1967) · Zbl 0156.37903 [22] Pietsch, A.: Ideals of multilinear functionals (designs of a theory). In: Proceedings of the Second International Conference on Operator Algebras, Ideals and their Applications in Theoretical Physics, pp. 185–199. Teubner-Texte, Leipzig (1983) · Zbl 0561.47037 [23] Ryan, R.A.: Introduction to tensor products of Banach spaces. Springer Monographs in Mathematics (2001) [24] Stegall, C.P., Retherford, J.R.: Fully nuclear and completely nuclear operators with applications to $$$$\backslash$$cal L_{1}$$ - and $$$$\backslash$$cal L$$\backslash$$cal _{$$\backslash$$infty }$$ -spaces. Trans. Am. Math. Soc 163, 457–492 (1972) · Zbl 0205.43003
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