×

b-weak compactness of weak Dunford-Pettis operators. (English) Zbl 1279.46011

The aim of this paper is to study when weak Dunford-Pettis operators are b-weakly compact. Recall that an operator from a Banach lattice \(E\) into a Banach space \(X\) is called weakly Dunford-Pettis if for each weakly null sequence \((x_n)\) in \(E\) and weakly null sequence \((y_n')\) in \(X'\), we have \(y'_n(Tx_n)\rightarrow 0\). An operator \(T:E\rightarrow X\) is called b-weakly compact if for each disjoint sequence in \(E\) which is order bounded in \(E''\), the bidual of \(E\), we have that \(||T(x_n)||\rightarrow 0\).
The main results of the paper are as follows:
Theorem. Suppose that the Banach lattice \(E\) has order continuous norm. Each positive weak Dunford-Pettis operator \(E\rightarrow F\) is b-weakly compact if and only if one of \(E\) or \(F\) is a KB-space.
Theorem. Let \(E,F\) be Banach lattices such that \(F\) is Dedekind \(\sigma\)-complete. If each weak Dunford-Pettis operator \(T:E\rightarrow F\) is b-weakly compact, then \(E\) is a KB-space or \(F\) has order continuous norm.

MSC:

46B42 Banach lattices
47B07 Linear operators defined by compactness properties
47B60 Linear operators on ordered spaces
PDFBibTeX XMLCite
Full Text: DOI Link