Summability properties for multiplication operators on Banach function spaces (Q989941)

A Banach function space \(X\subset L^0(\Omega, \Sigma, \mu)\), where \((\Omega, \Sigma, \mu)\) is \(\sigma\)-finite, is called saturated if, for all \(A\in \Sigma\) with \(\mu(A)>0\), there exists \(B\in \Sigma\) with \(B\subset A\) such that \(\chi_B\in X\). This is known to be a necessary and sufficient condition for the \(Y\)-dual of \(X\), \(X^Y=\{f\in L^0: hf\in Y, \text{ for all } h\in X\}\), to be a Banach function space. Given \(1\leq p\leq \infty\) and three saturated Banach function spaces \(X\), \(Y\) and \(Z\), the spaces \(X\pi Y\) and \(Xd_{p,Z} Y\) are given by those \(z\in L^0\) such that \(|z|\leq \sum_{i\geq 1}|x_i||y_i|\) for some sequences \((x_i)\subset X\) and \((y_i)\subset Y\) satisfying \[ \sum_{i\geq 1}\|x_i\|\|y_i\|<\infty \] and \[ \left(\sum_{i\geq 1} \|y_i\|^{p'}\right)^{1/p'}\left(\sup_{f\in X^Z, \|f\|\leq 1}\sum_{i\geq 1} \|fx_i\|^p\right)^{1/p}<\infty, \] respectively. Connections of these notions with multiplication operators having some \((p,Z)\)-summing properties are provided and some applications to Reisner's theorem and the Maurey-Rosenthal theorem are given.