Some compactness properties of \(L\)-weakly and \(M\)-weakly compact operators (Q1928055)

Recall that an operator \(T\) from a Banach space \(X\) into a Banach lattice \(F\) is called order weakly compact if \(T\) maps order bounded sets to relatively weakly compact sets. The authors first prove that, if \(T: E\rightarrow X\) is an order weakly compact operator and if there exists a positive element \(y\in E\) for which \((I_y)'\) is discrete, where \(I_y\) is the order ideal generated by \(y\), then the restriction of the operator \(T\) to \(I_y\), where \(I_y\) is equipped with the order unit norm \(\| \cdot\| _\infty\), is compact. Next, they show that, if \(T:E\rightarrow X\) is an \(M\)-weakly compact operator, then there exist a positive element \(y\in E\) and positive scalars \(\alpha_n\) such that \[ T(B_E)\subseteq \alpha_n T[0,y] + n^{-1}B_X \] for each \(n\). The main result of the paper is the following Theorem. Let \(E\) be a Banach lattice and \(X\) be a Banach space and \(T: E\rightarrow X\) be \(M\)-weakly compact. If there exists \(0< y\in E\) such that \((I_y)'\) is discrete, then \(T\) is a compact operator. If the operator \(T:E\rightarrow X\) is both order weakly compact and semicompact and if there exists positive vector \(y\in E\) for which \((I_y)'\) is discrete, then \(T^2\) is compact.