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Multiple summing operators on \(C(K)\) spaces. (English) Zbl 1063.46032
Let \(X\) be a Banach space and \(B_X\) be its unit ball. The weak \(p\)-norm of a finite family \((x_i)_{i=1}^m \subseteq X\) is denoted by \(w_p(x_i)\). The authors consider the Banach spaces \(\Pi_{(q;p_1,\ldots , p_n)}^n(X_1,\ldots,X_n;Y)\) of all multiple \((q;p_1,\ldots , p_n)\)-summing multilinear operators for Banach spaces \(X_1,\ldots,X_n,Y\) and \(1\leq q,p_1,\ldots , p_n < \infty \). A continuous multilinear operator \(T:X_1 \times \ldots \times X_n \rightarrow Y\) is multiple \((q;p_1,\ldots , p_n)\)-summing if where is a constant \(C\) such that \[ \left(\sum_{i_1=1}^{m_1} \cdots \sum_{i_n=1}^{m_n} \| T(x_{i_1},\ldots,x_{i_n})\| ^q\right)^{1/q} \leq C\cdot w_{p_1}(x_{i_1})\cdot \cdots \cdot w_{p_n}(x_{i_n}) \] for all finite families \((x_{i_1})_{i_1=1}^{m_1} \subseteq X_1,\ldots,(x_{i_n})_{i_n=1}^{m_n} \subseteq X_n\). The associated norm is defined by \(\pi_{(q;p_1,\ldots , p_n)}(T) = \inf\{C:C\) as above\(\}\). Write \((q;p)\) instead of \((q;p_1,\ldots , p_n)\) if \(p_1 = \cdots= p_n =p\).
The operator \(T\) is integral if there is a regular \(Y''\)-valued Borel measure \(G\) of bounded variation on the product \(B=B_{X_1'} \times \cdots \times B_{X_n'}\) such that \[ T(x_1,\ldots, x_n) = \int_B x_1'(x_1)\cdots x_n'(x_n) dG(x_1',\ldots, x_n') \] for all \((x_1,\ldots, x_n) \in X_1 \times \ldots \times X_n\).
A Banach space \(Y\) is a GT space if every bounded linear operator from \(Y\) into \(\ell_2\) is \((1;1)\)-summing.
The main results are the following, inter alia: In [H. P. Rosenthal and S. J. Szarek, Lect. Notes Math. 1470, 108–132 (1991; Zbl 0758.47022)] it was shown that \(u_1\otimes u_2 \in L(X_1 \widehat{\otimes }_\varepsilon X_2;Y_1 \widehat{\otimes }_\pi Y_2)\) if \(u_j \in L(X_j;Y_j)\) for \({\mathcal L}_\infty \) spaces \(X_j\) and \({\mathcal L}_1 \) spaces \(Y_j\), \( j=1,2\). In the present paper, it is shown that this is also true if \(u_j \in L(X_j;Y_j)\) for \({\mathcal L}_\infty \) spaces \(X_j\) and GT spaces with cotype 2 \(Y_j\), \( 1\leq j\leq n\). Next, it is proved that given Banach spaces \(X_1,\ldots,X_n\) are \({\mathcal L}_\infty \) spaces iff for every Banach space \(Y\) all multiple (1,1)-summing operators \(T:X_1 \times \ldots \times X_n \rightarrow Y\) are integral.
In Theorem 3.9., the authors show: Let \(K_j\) be compact Hausdorff spaces, \(Y\) a Banach space, \(T: C(K_1)\times \cdots \times C(K_n) \rightarrow Y\) with representing polymeasure \(\gamma \). Then \(T\) is multiple \((p;1)\)-summing iff the variation \(p\)-norm of \(\gamma \), \(v_p(\gamma )\), is finite. Moreover, in this case, \(v_p(\gamma ) \leq \pi_{(p;1)}(T) \leq 2^{n(1-1/p)}v_p(\gamma )\) in the real case (and \(2^{n(2-1/p)}v_p(\gamma )\) in the complex case).
A question stated in [M. S. Ramanujan and E. Schock, Linear Multilinear Algebra 18, 307–318 (1985; Zbl 0607.47042)] is solved in the present paper by showing the existence of a multiple (2;2) summing operator \(T:X\times Y \rightarrow Z\) such that the linearization \(T_1\) is not in \(\Pi_2(X;\Pi_2(Y;Z))\).

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
46B20 Geometry and structure of normed linear spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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