Almost Limited Sets in Banach Lattices

By Jin Xi Chen, Zi-li Chen, Guoxing Ji. 2013

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We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak$^{*}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and $L$-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into $c_{0}$, we give an operator characterization of those $\sigma$-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a $\sigma$-Dedekind Banach lattice $E$, every relatively weakly compact set in $E$ is almost limited if and only if every continuous linear operator $T:E\rightarrow c_{0}$ is an almost Dunford-Pettis operator.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/375ea722eef61bec39e066ccc3a5d17a47c32b75