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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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