Characterization of the order weak compactness of semi-compact operators
Recall that an operator \(T\) from a Banach lattice \(E\) to \(F\) is called semi-compact if for every \(\varepsilon>0\), there is a positive element \(u\) in \(F\) so that the image of the unit ball is contained in the order interval \([-u,u]\) plus \(\varepsilon\) times the unit ball in \(F\), and \(T\) is order weakly compact if for each positive \(x\) in \(E\), the image of the order interval \( [0,x]\) is relatively weakly compact. Relationships between these two types of operators are established. It is shown that for operators from \(E\) to \(F\) with \(F\) order \(\sigma\)-complete, the following are equivalent: (1) each order bounded operator is order weakly compact, (2) each semi-compact operator is order weakly compact, (3) each positive semi-compact operator is order weakly compact, (4) one of \(E\) and \(F\) has order continuous norm. For the same condition on \(F\), it is also established that if each positive semi-compact operator has has a semi-compact adjoint, then either \(E\) is a KB-space or \(F\) has an order continuous norm. These ideas are used to show that, without restriction on \(F\), if every positive operator is semi-compact if and only if its adjoint is semi-compact, then \(E'\) or \(F'\) has an order continuous norm. Examples and further results are provided.
scientific article; zbMATH DE number 5563813