An index of summability for pairs of Banach spaces (Q276822)

An \(m\)-linear operator \(u \in \mathcal{L}(E_1,\dots, E_m;F)\) is called multiple \((p,q)\)-summing if there is a constant \(C>0\) such that NEWLINE\[NEWLINE\left(\sum_{k_1,\dots, k_m=1}^n\left\| u\left(x_{k_1}^{(1)},\dots,x_{k_m}^{(m)}\right)\right\|^p\right)^{1/p} \leq C \prod_{i=1}^{m}\left\| \left(x_{k_i}^{(i)}\right)_{k_i=1}^n\right\|_{w,q},NEWLINE\]NEWLINE for all \(n \in \mathbb{N}\) and for all \(x_{k_i}^{(i)} \in E_i\), with \(1 \leq k_i \leq n\) and \(1 \leq i \leq m\). When an \(m\)-linear operator is not multiple \((p,q)\)-summing the above constant \(C\) does not exist.NEWLINENEWLINEIn this interesting paper the authors show that when the above inequality fails, choosing any \(n \in \mathbb{N}\), there exists a constant \(C_n = C_1n^s\) that makes the inequality true, for all choices of \(x_{k_i}^{(i)} \in E_i\), with \(1 \leq k_i \leq n\) and \(1 \leq i \leq m\), where the number \(s\) depends on \(m,p\) and \(q\). More specifically, they define an index \(\eta_{(p,q)}^{m\mathrm{-mult}}(E_1,\dots, E_m;F)\) of (non) \((p,q)\)-summability of a pair \((E_1 \times \cdots \times E_m,F)\) as the infimum of numbers \(s_{m,p,q} \geq 0\) that satisfy the following property:NEWLINENEWLINE\noindent There is a constant \(C \geq 0\) such that, for every \(u \in \mathcal{L}(E_1,\dots, E_m;F)\), for all \(n \in \mathbb{N}\) and for all \(x_{k_i}^{(i)} \in E_i\), with \(1 \leq k_i \leq n\) and \(1 \leq i \leq m\), NEWLINE\[NEWLINE\left(\sum_{k_1,\dots, k_m=1}^n\left\| u\left(x_{k_1}^{(1)},\dots,x_{k_m}^{(m)}\right)\right\|^p\right)^{1/p} \leq Cn^{s_{m,p,q}} \prod_{i=1}^{m}\left\| \left(x_{k_i}^{(i)}\right)_{k_i=1}^n\right\|_{w,q}.NEWLINE\]NEWLINE They also define an index for polynomials in the same fashion.NEWLINENEWLINESome of the main presented results are upper bounds, that is the index of summability is always finite, and also upper and lower estimates for the index of summability in some special cases.