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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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