The strong Gelfand-Phillips property in Banach lattices
Let \(X,Y\) be Banach spaces and \(E,F\) be Banach lattices. A subset \(A\subset X\) is called limited if every weak* null sequence \((x_n^*)\) in the dual \(X^*\) converges uniformly on \(A\). If every limited subset of \(X\) is relatively compact, then \(X\) is said to have Gelfand-Phillips property (GPP). \(c_0,l_1\), every separable Banach space, every Schur space has the GP property. A subset \(A\) of \(E\) is almost limited if every disjoint weak* null sequence \((x_n^*)\) in \(E^*\) converges uniformly on \(A\). \(E\) is said to have the strong Gelfand-Phillips property (SGP property) if every almost limited subset in \(E\) is relatively compact. The strong Gelfand-Phillips property implies the Gelfand-Phillips property. Discrete Banach lattices with order continuous norms have the SGP property and if, in addition, \(E\) is \(\sigma\)-Dedekind complete, then \(E\) has the SGP property if and only if \(E\) is discrete and has order continuous norm. If \(E\) is a nondiscrete Banach lattice with order continuous norm, then \(E\) does not have the SGP property. \(X\) has the GP property if and only if every weakly null and limited sequence in \(X\) is norm null. Similarly, if \(E\) has the SGP property, then every weakly null and almost limited sequence in \(E\) is norm null. The converse is false, in general. If \(E\) has order continuous norm and \(A\subset E\) is almost limited in \(E\), then not only is \(A\) conditionally weakly compact, but \(A\) is relatively weakly compact. Weakly null and almost limited sequences in a Banach lattice are not norm null, in general. The Rademacher functions in \(L^1[0,1]\) are weakly null and almost limited but not norm null. If \(E^*\) has a weak order unit or \(E\) has order continuous norm, then \(E\) has the SGP property if and only if every weakly null and almost limited sequence in \(E\) is norm null. A bounded linear operator \(T:E\rightarrow X\) is called almost limited completely continuous (alcc) if it maps weakly null and almost limited sequences in \(E\) to norm null sequences. They are denoted by \(L^a_{cc}(E,X)\). Let \(L(E,X)\) be the space of all bounded operators. The authors give a characterization of the SGP property in terms of almost limited completely continuous operators. Theorem. Suppose that \(E\) has order continuous norm or \(E^*\) has a weak order unit. Then the following are equivalent. {\parindent=6mm \begin{itemize} \item[(1)] \(E\) has the SGP property. \item [(2)] For each \(Y\), \(L^a_{cc}(E,Y) = L(E,Y)\). \item [(3)] \(L^a_{cc}(E,l_\infty) = L(E,l_\infty)\). \end{itemize}} Let \(\phi_x\), and \(\varphi_{y^*}\) be evaluation operators, where \(\phi_x(T) = Tx\), \(\varphi_{y^*}(T) = T^*y^*\) for each \(x\in E\), \(y^*\in Y^*\), \(T\in L(X,Y)\). If \(M\subset L(E,F)\) is a Banach lattice with the SGP property, then all the evaluation operators \(\phi_x\), and \(\varphi_{y^*}\) are almost limited completely continuous operators. The authors show that, if \(Y\) has the Schur property and \(M\subset L(X,Y)\) is a Banach lattice for which \(M^*\) has a weak order unit or \(M\) has order continuous norm and if, for every \(y^*\in Y^*\), the operator \(\varphi_{y^*}\) on \(M\) is almost limited completely continuous, then \(M\) has the SGP property. Continuous linear images of limited sets or sequences are limited. This may not be the case for almost limited sets or sequences, in general. In the final part of the paper, the authors give necessary conditions for continuous linear images of almost limited sets to be almost limited.
scientific article; zbMATH DE number 6495770