Banach lattices of \(L\)-weakly and \(M\)-weakly compact operators
Suppose that \(E\) and \(F\) are two Banach lattices. A bounded linear operator \( T:E\to F\) is called \(M\)-weakly compact\ if, whenever \((x_{n})_{n}\) is a norm bounded disjoint sequence in \(E\), we have \(\| T(x_{n})\| \to 0\). A bounded linear operator \(S:E\to F\) is called \(L\)-weakly compact if, for every disjoint sequence \((y_{n})_{n}\) in the solid hull of \(\{ T(x):x\in E, \| x\| \leq 1\}\), we have \(\| y_{n}\| \to 0\). We put \(\mathcal{W}_{M}(E,F)\) (respectively, \(\mathcal{W}_{M}^{r}(E,F)\)) for the space of all \(M\)-weakly compact operators from \(E\) into \(F\) (respectively, the linear span of positive \(M\)-weakly compact operators in \(\mathcal{W}_{M}(E,F))\). The spaces \(\mathcal{W}_{L}(E,F)\) and \(\mathcal{W}_{L}^{r}(E,F)\) are defined in a similar manner. The paper under review is devoted to the order properties of the spaces \(\mathcal{W}_{L}^{r}(E,F)\) and \(\mathcal{W}_{M}^{r}(E,F)\) endowed with the regular norm. For example, it is proved that \(\mathcal{W}_{L}^{r}(E,F)\) is always a Dedekind complete Banach lattice (and the same is true for \(\mathcal{W}_{M}^{r}(E,F)\) provided that \(F\) is Dedekind complete). Also, necessary and sufficient conditions are indicated for these spaces to have an order continuous norm, to be an \(AL\)-space, or an \(AM\)-space.
scientific article; zbMATH DE number 6696930