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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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