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Little G.T. for \(l_p\)-lattice summing operators. (English) Zbl 1164.46309

Let \(1 \leq p \leq \infty\) and \(u:E \rightarrow X\) be a linear operator between a Banach space \(E\) and a Banach lattice \(X\). Then \(u\) is called \(p\)-lattice summing (notation: \(u \in \Lambda_p(E,X)\)) whenever there exists a constant \(C>0\) such that \(\| ( \sum_1^n |uv(e_i)|^p)^{1/p}\| \leq C \, \|v\|\) for every linear operator \(v: l_{p'} \rightarrow E\), \(1/p+1/{p'}=1\). For an operator space \(E\), the author now calls the operator \(u\) \(l_p\)-lattice summing (notation: \(u \in \Lambda_{l_p}(E,X)\)) if the above inequality holds with the operator norm \(\|v\|\) replaced by the completely bounded norm \(\|v\|_{cb}\). The author proves some facts about and characterizations of \(l_p\)-lattice summing operators. Eventually, it is shown that the Little G.T. does not hold within this setting of operator spaces. More precisely, whereas in the commutative case \(\Lambda_\infty(l_\infty,l_2)=B(l_\infty,l_2)=\Pi_2(l_\infty,l_2)=\Lambda_2(l_\infty,l_2)\), the noncommutative analogue does not hold: \(\Lambda_{l_\infty}(B(H),OH) \not= \Lambda_{l_2}(B(H),OH)\), where \(\Pi_2(l_\infty,l_2)\) and \(B(l_\infty, l_2)\) denote the spaces of all \(2\)-summing and bounded operators between \(l_\infty\) and \(l_2\), respectively, \(B(H)\) the space of all bounded operators on a Hilbert space \(H\), and \(OH\) Pisier’s operator Hilbert space.
Reviewer: Carsten Michels

MSC:

46B28 Spaces of operators; tensor products; approximation properties
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L25 Operator spaces (= matricially normed spaces)
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