Preregularity of Compact Operators

The regular operators on Banach lattices play very important and interesting role in the literature of Banach lattice and operator theory.There are number of results concerning the regularity of operators,but there is no exact way in the literature to conclude the regularity of continuous linear operators.It is natural and interesting to consider a weaker property,so-called Preregularity of operators on Banach lattices.We first show the existence of non-preregular compact operators on Banach lattices.Then,we present the counterexamples respectively for discrete and continuous domains and range spaces.These examples also show that weakly compact may not be preregular.