zbMATH — the first resource for mathematics

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.

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