Compactness of L-weakly and M-weakly compact operators on Banach lattices
Let \(E\) be a Banach lattice and \(X\) a Banach space. An operator \(T:E\rightarrow X\) is called M-weakly compact if \(\lim_n|| Tx_n|| = 0\) for each norm bounded disjoint sequence \((x_n)\) in \(E\). \(T\) is called L-weakly compact if \(\lim_n ||y_n|| = 0\) for each disjoint sequence \((y_n)\) in the solid hull of \(T(B_X)\), where \(B_X\) is the closed unit ball of \(X\). L- or M-weakly compact operators need not be compact. Similarly, compact operators need not be L- or M-weakly compact. \textit{Z. L. Chen} and \textit{A. W. Wickstead} [Indag. Math., New Ser. 10, No. 3, 321--336 (1999; Zbl 1028.47028)] have shown that M-weakly compact operators from an AL-space into a Banach space are compact and each L-weakly compact operator from a Banach space into an AM-space is compact. In this paper, these two results are generalized and the converse problem is studied. One of the main result is the following. Theorem. Assume that the dual \(E'\) of \(E\) does not have order continuous norm. Then the following are equivalent: (1) Every L-weakly compact operator \(T:E\rightarrow F\) is compact. (2) Each positive L-weakly compact operator \(T:E\rightarrow F\) is compact. (3) The largest lattice ideal in \(E\) for which the norm is order continuous is discrete. It is shown that, if \(E'\) is discrete with order continuous norm, then each regular L-weakly compact operator \(T:E\rightarrow F\) is compact and, in Theorem 3.6, this property is shown to give a characterization of Banach lattices whose duals are discrete and have order continuous norm. If the range \(Y\) contains an isomorphic copy of \(c_0\), then every M-weakly compact operator \(T:E\rightarrow Y\) is compact if and only if the largest lattice ideal \((E')^a\) in \(E'\) is discrete result. As with L-weakly compact operators, the authors prove the following on M-weakly compact operators. Theorem. The following are equivalent: (1) \(F\) is discrete and has order continuous norm. (2) For each \(E\), every regular M-weakly compact \(T:E\rightarrow F\) is compact. (3) For each \(E\), every positive M-weakly compact \(T:E\rightarrow F\) is compact.
scientific article; zbMATH DE number 6003013