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The inclusion theorem for multiple summing operators. (English) Zbl 1064.47057
Let $$X$$ be a Banach space, $$X^*$$ be its dual and $$B_X$$ its unit ball. For a finite sequence $$(x_i)_{i=1}^m\subset X$$ and $$1\leq p<\infty,$$ write $$\big\| (x_i)_{i=1}^m\big\| _p^\omega$$ for $\sup\Biggl\{\Biggl(\displaystyle\sum_{i=1}^m| x^*(x_i)| ^p\Biggr)^{1/p}:\;x^*\in B_{X^*}\Biggr\}.$ Let $$1\leq p<\infty$$. A multilinear operator $$T:\,X_1\times\cdots\times X_n\rightarrow Y$$ is multiple $$p$$-summing if there exists a constant $$K>0$$ such that for every choice sequence $$(x^j_{i_j})_{i_j=1}^{m_j}\subset X_j$$, $\Biggl(\sum_{i_1,\cdots,i_n=1}^{m_1,\cdots,m_n} \big\| T(x_{i_1}^1,\dots,x_{i_n}^n)\big\| ^p\Biggr)^{1/p}\leq K\prod_{j=1}^n\Big\| (x^j_{i_j})_{i_j=1}^{m_j}\Big\| ^\omega_p.$ In that case, the multiple $$p$$-summing norm $$\Pi_p(T)$$ of $$T$$ is defined the minimum $$K$$ such that the above inequality holds. Denote by $$\Pi_p^n(X_1,\dots,X_n; Y)$$ the class of multiple $$p$$-summing $$n$$-linear operators, which is a Banach space with the norm $$\Pi_p$$. A Banach space $$X$$ is called to be a GT space, i.e., $$X$$ satisfies Grothendieck’s theorem, if there exists $$K>0$$ such that each linear operator $$u: X\to\ell_2$$ is $$1$$-summing and satisfies $$\Pi_1(u)\leq K\| u\|$$. In this paper, the author proves that for $$1\leq p\leq q<2$$, each multiple $$p$$-summing multilinear operator between Banach spaces is also $$q$$-summing. The author also gives an improvement of this result for the case of an image space of cotype 2.
A multilinear operator $$T: H_1\times\cdots\times H_n\to H$$ between Hilbert spaces is said to be Hilbert-Schmidt if there exists $$K>0$$ such that $\biggl(\displaystyle\sum_{i_1\in I_1,\dots,i_n\in I_n} \| T(e_{i_1}^1,\dots,e_{i_n}^n)\| ^2\biggr)^{1/2}<K,$ where $$(e^i_{i_j})_{i_j\in I_j}\subset H_j$$ is an orthonormal basis, $$1\leq j\leq n$$. In this case, the above least constant $$K$$ is called the Hilbert-Schmidt norm of $$T$$. Denote by $$S_2^n(H_1,\dots, H_n; H)$$ the class of Hilbert-Schmidt multilinear operators. In this paper, the author proves that if $$H_1,\dots, H_n$$ and $$H$$ are Hilbert spaces and $$T: H_1\times\cdots\times H_n\to H$$ is a multilinear operator, then $$T\in S_2^n(H_1,\dots, H_n; H)$$ if and only if $$T\in \Pi_p^n(H_1,\dots,H_n; H)$$ for every $$p\in[1,\infty)$$ if and only if $$T\in \Pi_p^n(H_1,\dots,H_n; H)$$ for some $$p\in[1,\infty)$$, which is a multilinear version of the classical characterization of Hilbert-Schmidt linear operators given by A. Pełczyński in [Stud. Math. 28, 355–360 (1967; Zbl 0156.38001)]. Moreover, the author proves that for GT spaces, every multilinear operator into a Hilbert space is $$1$$-summing with an optimal constant, which is a multilinear generalization of Grothendieck’s theorem for GT spaces.

##### MSC:
 47H60 Multilinear and polynomial operators 46B25 Classical Banach spaces in the general theory 46C99 Inner product spaces and their generalizations, Hilbert spaces
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