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When every multilinear mapping is multiple summing. (English) Zbl 1191.46041
Multiple summing multilinear mappings, also called fully summing multilinear mappings, were introduced independently by M. C. Matos [Collect. Math. 54, No. 2, 111–136 (2003; Zbl 1078.46031)] and F. Bombal, D. Pérez-García and I. Villanueva [Q. J. Math. 55, No. 4, 441–450 (2004; Zbl 1078.46030)]. The latter used the well-known Grothendieck coincidence results for absolutely summing operators in order to prove, via an induction, Grothendieck type results for multiple summing multilinear mappings. Inspired by their proofs, in this paper the authors identify general linear conditions that yield the following unified and general coincidence result: Every continuous $$n$$-linear mapping $$A:E_1\times \cdots \times E_n \to F$$ between Banach spaces is multiple $$(q;r,\dots,r,p)$$-summing whenever $$E_1,\dots,E_n$$ belong to $$B(p,q,r,F)$$, the collection of all Banach spaces $$E$$ such that every continuous linear operator from $$E$$ to $$F$$ is $$(q,p)$$-summing and every continuous linear operator from $$E$$ to $$\ell_q(F)$$ is $$( q,r)$$-summing. This result not only recovers known coincidence situations, but also gives several new interesting ones.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47L22 Ideals of polynomials and of multilinear mappings in operator theory
##### Keywords:
multiple summing; absolutely summing; multilinear mappings
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##### References:
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