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Traced tensor norms and multiple summing multilinear operators. (English) Zbl 06707935
Given \(1 \leq p \leq \infty\), \(n \in \mathbb{N}\) and \(E_1,\dots,E_n,F\) Banach spaces, we say that an \(n\)-linear operator \(T:E_1 \times \dots \times E_n \rightarrow F\) is absolutely \(p\)-summing if there exists a constant \(C>0\) such that \[ \left( \sum_{j=1}^{m} \|T(x_j^1,\dots,x_j^n)\|^p\right)^{1/p} \leq C \prod_{i=1}^{n}\|(x_j^i)_j\|_{w,p}, \] for every \(m \in \mathbb{N}\) and for every choice of \(x_1^i,\dots,x_m^i \in E_i, i=1,\dots,n\). Using a tensor product point of view the above definition is equivalent to: an \(n\)-linear operator \(T:E_1 \times \dots \times E_n \rightarrow F\) is absolutely \(p\)-summing if the \(n\)-linear map \(\widehat{T}: (\ell_p^0 \otimes_\varepsilon E_1) \times \dots \times (\ell_p^0 \otimes_\varepsilon E_n) \rightarrow \ell_p \otimes_{\Delta_p} F\) given by \[ \widehat{T}\left( \left(\sum_{j=1}^{m} e_j \otimes x_j^1, \dots,\sum_{j=1}^{m} e_j \otimes x_j^n \right)\right) := \sum_{j=1}^{m} e_j \otimes T(x_j^1,\dots,x_j^n) \] is continuous, where \(\ell_p^0\) is the subspace of all sequences of \(\ell_p\) with only a finite nonzero coordinates.
Constructing a unified approach based in tensor product operators as above, the authors define some abstract classes of multilinear operators that encompass well-known classes of summing and multiple summing \(n\)-linear operators. So, they prove an interesting result that characterize these abstract classes by means of the continuity of tensor operators obtained by an order reduction procedure. Applications to the bilinear case are presented for some classes and they use these ideas to provide results even in the linear context of the theory, namely for the classes of \(p\)-summing and \(\tau(p)\)-summing linear operators.
It is worth mentioning that the multiple \(\tau(p)\)-summing operators introduced in this paper are particular cases of the more general notion of weakly mid \((p,q)\)-summing operators recently introduced in [G. Botelho et al., Pac. J. Math. 287, No. 1, 1–17 (2017; Zbl 1373.47058)] (see Theorem 2.3). Having this in mind, Corollary 4.6 of the paper under review is a straightforward consequence of a result due to Sinha and Karn (see Theorem 1.2(ii) in the aforementioned work).

47L22 Ideals of polynomials and of multilinear mappings in operator theory
46A32 Spaces of linear operators; topological tensor products; approximation properties
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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