The Existence Of Strongly Non-Regular Operators Between Banach Lattices

In this paper, the orthomorphisms on the Archimedean Riesz space R " with the usual coordinatewise ordering are characterized. Also, the direct sum decomposition of an order bounded operator with respect to the orthomorphisms is obtained. In addition, a counterexample is constructed to show that the characterization and the related results do not hold for the non-Archimedean Riesz space R# with the lexicographical ordering.Then the order bound norm imposed on the order bounded operators between two Banach lattices is fully studied. The results include the relationships between the order bound norm and the other two types of norms of a regular operator; respectively ,and a condition under which the space of order bounded operators is a Dedekind complete Banach lattice.The main purpose of the thesis is to study the existence of strongly non-regular operators between classical Banach lattices. Alternatively, we investigate the relationships between the space of bounded operators and its regular operator subspace with respect to the operator norm topology, thereby answering partially the question of how big the regular operator subspace is and discussing the existence of strongly non-regular operators between some classical Banach lattices.