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Multilinear extensions of Grothendieck’s theorem. (English) Zbl 1078.46030
Let $$X_1,\dots ,X_n$$, and $$Y$$ be Banach spaces and $$1\leq q_1,\dots ,q_n \leq p<\infty$$. A multilinear mapping $$T:X_1 \times \dots \times X_n \to Y$$ is multiple $$(p;q_1 ,\dots ,q_n )$$-summing if there exists a constant $$K>0$$ such that $( \sum\limits_{i_1 =1}^{m_1} \dots \sum\limits_{i_n =1}^{m_n}\| T(x_{i_1}^1 ,\dots ,x_{i_n}^n )\| ^p)^{\frac{1}{p}} \leq K \prod\limits_{j=1}^{n} w_{q_j} ( ( x_i^j )_{i=1}^{m_j})$ for every choice of positive integers $$m_j$$ and all $$x_{i_j}^j \in X_j$$ $$(1\leq j\leq n,~1\leq i_j \leq m_j )$$. In this case, the multiple $$(p;q_1 ,\dots ,q_n)$$-summing norm of $$T$$ is defined by $\pi_{(p;q_1 ,\dots ,q_n )}(T)=\min K ,$ where the minimum is taken over all such $$K$$. This is a generalization of a definition due to M. S. Ramanujan and E. Schock [Linear Multilinear Algebra 18, 307–318 (1985; Zbl 0607.47042)]. Using this concept, the authors prove several multilinear versions of Grothendieck’s inequality. More precisely, every multilinear mapping on a product of $${\mathcal L}_{\infty}$$-spaces into an $${\mathcal L}_1$$-space is multiple $$(2;2,\dots ,2)$$-summing or every multilinear mapping on a product of $${\mathcal L}_1$$-spaces into a Hilbert space is multiple $$(1;1,\dots ,1)$$-summing. As an application, a vector-valued version of Littlewood’s inequality is proved.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47H60 Multilinear and polynomial operators
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