On the coincidence of Köthe spaces from the family \((l_p(R):0<p\leq\infty)\) (Q1398559)

Let \(R\) be a set of sequences of elements from \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\). The authors study the cases when the spaces \(l_p(R):=\{(\xi_k)^\infty_{k=1}\in\mathbb{K}^\mathbb{N}:(\xi_kr_k)^\infty_{k=1}\in\ell_p\) for all \((r_k)^\infty_{k=1}\in R\}\) of multipliers from \(R\) to \(l_p\), \(0<p\leq\infty\), coincide. In particular, it is shown that the condition \(l_p(R)=l_q(R)\) for some \(0<p<q<\infty\) implies \(l_p(R)=l_{\infty}(R)\).