2-summing multiplication operators (Q2839271)

Let \(1\leq p<\infty\) and \(\mathcal{X}=(X_n)_{n\in\mathbb{N}}\) be a sequence of Banach spaces. Let \(\ell_p(\mathcal{X})\) denote the Banach space of all sequences \((x_n)_{n\in\mathbb{N}}\) with \(x_n\in X_n\) for all \(n\in\mathbb{N}\), \(\sum_{n=1}^\infty\|x_n\|_{X_n}^p<\infty\), endowed with the norm \(\|(x_n)_{n\in\mathbb{N}}\|_{\ell_p(\mathcal{X})}:=(\sum_{n=1}^\infty\|x_n\|_{X_n}^p)^{1/p}\).NEWLINENEWLINELet \(\mathcal{X}=(X_n)_{n\in\mathbb{N}}\) and \(\mathcal{Y}=(Y_n)_{n\in\mathbb{N}}\) be two sequences of Banach spaces and let \(\mathcal{V}=(V_n)_{n\in\mathbb{N}}\), \(V_n: X_n\to Y_n\), be a sequence of bounded linear operators, and \(1\leq p, q<\infty\). The multiplication operator \(M_{\mathcal{V}}: \ell_p(\mathcal{X})\to \ell_p(\mathcal{Y})\) is defined by \(M_{\mathcal{V}}((x_n)_{n\in\mathbb{N}}):=(V_n(x_n))_{n\in\mathbb{N}}\). As in the scalar case, the operator \(M_{\mathcal{V}}\) is well defined if and only if it is bounded linear if and only if \((\|V_n\|)_{n\in\mathbb{N}}\in \ell_\infty\), for \(p\leq q\), respectively, \((\|V_n\|)_{n\in\mathbb N}\in \ell_s\) for \(q<p\), where \(1/q=1/p+1/s\).NEWLINENEWLINEIn the paper under review, the author gives necessary and sufficient conditions for the multiplication operator \(M_{\mathcal{V}}\) to be 2-summing when \((p,q)\) is one of the couples \((1,2)\), \((2,1)\), \((2,2)\), \((1,1)\), \((p,1)\), \((p,2)\), \((2,p)\), \((1,p)\), \((p,q)\) with \(1<p<2\), \(1<q<\infty\). For example, in the case \((1,2)\), the author shows that \(M_{\mathcal{V}}\) is 2-summing if and only all \(V_n\) are 2-summing and \((\pi_2(V_n))_{n\in \mathbb{N}}\in \ell_\infty\). In the other cases, similar results are presented.