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.


47H60 Multilinear and polynomial operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
Full Text: DOI


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