Stably Dominated and Stably Regular Operators

This chapter is mainly devoted to studying those properties of Banach lattices and operators on them whose description requires the vector lattice structure as well as the pure Banach space structure. In particular, we study the properties of the operators acting between vector lattices and remaining to be order bounded under multiplication by arbitrary Banach endomorphisms in these lattices. In spite of being natural, such classes of operators have attracted very little attention of mathematicians. At present, the theory of p-absolutely summing operators plays a key role in solving the arising problems; in Sections 3.1–3.3 and 3.5, we briefly expose those results of the theory which are most important for us. The reader interested in a more complete presentation of the theory will refer to the monographs [42, 32, 45] and the articles [20, 22, 39].