Banach Actions Preserving Unconditional Convergence

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$ be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence ${\left(a_n\right)}_{n\in \omega }$ in A and unconditionally convergent series ${\sum }_{n\in \omega }{x}_n$ in X, the series ${\sum }_{n\in \omega }{a}_n{x}_n$ is unconditionally convergent in Y. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^{*}\in Y^{*}$ the operator $D_{y^{*}}:X\to A^{*}$, $D_{y^{*}}\left(x\right)\left(a\right)=y^{*}\left(ax\right)$ is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ${\ell }_1$ to ${\ell }_2$, we prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis ${\left(e_n\right)}_{n\in \omega }$ such that for every $x\in X$, the series ${\sum }_{n\in \omega }{e}_{n}x$ is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers $p,q,r\in [1,\infty ]$ with $\frac{1}{r}\le \frac{1}{p}+\frac{1}{q}$, the coordinatewise multiplication ${\ell }_p\times {\ell }_q\to {\ell }_r$ preserves unconditional convergence if and only if one of the following conditions holds: (i) $p\le 2$ and $q\le r$, (ii) $2<p<q\le r$, (iii) $2<p=q<r$, (iv) $r=\infty$, (v) $2\le q<p\le r$, (vi) $q<2<p$ and $\frac{1}{p}+\frac{1}{q}\ge \frac{1}{r}+\frac{1}{2}$.