On strongly \(p\)-summing sublinear operators (Q2479969)

In the paper under review, the authors generalize the class of strongly \(p\)-summing linear operators given by \textit{J. S. Cohen} [Math. Ann. 201, 177--200 (1973; Zbl 0233.47019)] to the class of sublinear operators. Precisely, a sublinear operator \(T\) from a Banach space \(X\) into a Banach lattice \(Y\) is said to be strongly \(p\)-summing (\(1<p<\infty\)) if there is a positive constant \(C\) such that for any \(n\in \mathbb N\), \(x_1,\dots,x_n\in X\) and \(y_1^*,\dots,y^*_n\in Y^*\), we have \(\|(\langle T(x_i),y_i^*\rangle)\|_{\ell_1^n}\leq C\|(x_i)\|_{\ell_p^n(X)}\|(y_i^*)\|_{\ell_{p^*}^{n\omega}}\). In particular, the authors give an analogue to Pietsch's domination theorem for this category of operators and study some relation between the strongly \(p\)-summing sublinear operators \(T\) and the linear operators \(u\in \nabla T\), where \(\nabla T=\{u\in {\mathcal L}(X,Y):\;u\geq T\}\) and \({\mathcal L}(X,Y)\) denotes the space of all linear operators from \(X\) into \(Y\).