Positive compact operators on Banach lattices: Some loose ends
The following two topics are the main concern of this paper. Let \(0\leq S\leq T\) be two operators on a Banach lattice \(E\) and assume that \(T\) is compact. As the well known theorem of C. D. Aliprantis and O. Burkinshaw states the operator \(S^3\) is also compact, and if \(E\) or \(E^*\) has order continuous norm, then \(S^2\) is compact. Moreover, the well known theorem of P. Dodds and D. Fremlin asserts that if both \(E\) and \(E^*\) have order continuous norm, then \(S\) itself is compact. The author investigates as to what extent the order continuity condition in the above results is actually necessary. The second topic is aimed at studying the closure in the regular-norm topology of finite rank operators. For precise statements the reader is referred to this interesting paper.
scientific article; zbMATH DE number 1538738