Preregular Operators On Banach Lattices

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 it is known that 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.In this thesis we devote to investigate preregularity of operators and some related properties. The thesis mainly consists of three parts.In the first part we will first present some characterizations of preregularity of operators, then we will deduce some sufficient conditions for preregular operators being regular and order bounded, and finally we try to establish some relations among the regular operators, order bounded operators and preregular operators.In the second part we consider the topological and order properties of spaces of preregular operators. One of main results in this part is that under the so-called preregular norm the space of preregular operators is a Banach space. Also we will deduce some conditions for the equivalent of operator norm, regular norm and preregular norm respectively. Some results involved the Riesz decomposition properties of preregular operator spaces are also included.In the third part we mainly point out that compact operators may not be preregular. Some positive results for all compact operators being preregular are included here, especially we construct two concrete counterexamples of compact operators which are not preregular. These examples also show that M-and L-weakly compact may not be preregular.