Compact-friendly multiplication operators on Banach function spaces
It was proved by \textit{Y. A. Abramovich} et al. [Indag. Math., New Ser. 10, 161--171 (1999; Zbl 1043.47504)], that a multiplication operator on each \(L_p(\mu)\)-space or \(C(K)\) space is compact-friendly if and only if the multiplier is constant on a set of positive measure. The proof depended heavily on the metric properties of the \(L_p\) and \(C(K)\) spaces. Answering a question posed in that article, the author shows in the present paper that the metric properties of the spaces involved are not essential and that the result remains true for each Banach function space. The main result of the paper under review is as follows: Let \((\Omega,\Sigma,\mu)\) be an arbitrary measure space without infinite atoms. Consider a Banach function space \(X\) over \(\Omega\). Then for any \(0\leq h\in L_\infty(\Omega)\) the multiplication operator \(M_h\in L(X)\) is compact-friendly if and only if \(h\) is constant on a set of positive measure.
scientific article; zbMATH DE number 1836666