On operator valued sequences of multipliers and \(R\)-boundedness
Given Banach spaces \(X\) and \(Y\) as well as two spaces of sequences \(E(X)\) and \(F(Y)\) containing \(c_{00}(X)\) and \(c_{00}(Y)\), respectively, a sequence \((u_n) \subseteq L(X,Y)\) is called a \textit{multiplier sequence from \(E(X)\) to \(F(Y)\)} if the map \((x_i)\mapsto(u_ix_i)\) is bounded from \(E(X)\) into~\(F(Y)\). In the first part (Section~2) of the present paper, the authors discuss and summarize recent results on \((p,q)\)-summing multipliers, that is, multipliers for which \(E(X)=l_q^w(X)\) is the space of weakly \(q\)-summable sequences in \(X\) and \(F(Y)=l_p(Y)\) is the space of strongly \(p\)-summable sequences in \(Y\). In the second part (Section~3), the concept of Rademacher boundedness of a sequence \((u_i)\) of operators is considered, meaning that \((u_i)\) is a multiplier sequence from \(Rad(X)\) to \(Rad(Y)\). Several other related properties are considered and their relations among each other studied. In particular, conditions on the geometry of the spaces \(X\) and \(Y\) yield embedding results of the respective multiplier spaces. The setting of multiplier sequences proves also useful in simplifying proofs of several known results.
scientific article; zbMATH DE number 5124049
scientific article