The Properties of b-AM-compact Operators on Banach Lattice

Several characterizations of b-AM-compact operators are considered in this paper, and we show that: 1) If F is an infinite-dimensional Banach lattice, then E is a KB-space if and only if every AM-compact operator from E into F is b-AM-compact. 2) The Banach lattice E is a discrete KB-space if and only if every continuous operator from E into Banach lattice F is b-AM-compact. 3) If the topological dual E' is discrete, then every b-weakly compact operator from Banach E into Banach space X is b-AM-compact. Moreover, following properties about the problems of domination in the class of positive b-AM-compact operators are established: 1) If E and F are two Banach lattices, then for all operators S, T: E → F such that 0 \leqslant \rmS \leqslant \rmT and T is b-AM-compact, the operator S is b-AM-compact if and only if the norm of F is order continuous or E' is discrete. 2) If S, T are two operators from E into F with 0 \leqslant \rmS \leqslant \rmT , if T is b-AM-compact, then S^2 is likewise b-AM-compact.

Then we give some necessary conditions and some sufficient conditions on Banach lattices E and F for the duality properties for b-AM-compact operators: (i)If T: E → F is a regular b-AM-compact operator, and the norm on E' is order continuous, then T\rm':\rmF\rm' \to E is also b-AM-compact operator (ii)If for every positive operator T: E→F with T\rm':\rmF\rm' \to E\rm' b-AM-compact, the operator T is b-AM-compact operator, then either E' is discrete or F has order continuous norm. Last, we give several equivalent conditions characterizing the case when K_\rmb - \rmAM^\rmr\left( \rmE,\rmF \right) is Dedekind σ-complete.