Theory of Orthogonally Additive Operators

Abstract

An orthogonally additive operator is a map that satisfies the additivity property under the disjointness condition. This thesis focuses on the theory of orthogonally additive operators. The concept of fragments plays a significant role in constructing the theory of orthogonally additive operators, and it is also studied in this thesis. The first chapter is dedicated to the study of fragments. The concept is explored in the context of vector lattices and lattice-normed vector spaces. The conclusions derived from Sections 2.2 and 2.3 of Chapter 2 can be found in [1]. The second chapter introduces the classes of orthogonally additive operators defined on vector lattices and a novel class of vector lattice known as C-complete. This chapter also addresses the extension problems associated with orthogonally additive maps. Various examples and conclusions are provided to support the findings. In the last chapter, orthogonally additive operators are examined in the context of lattice normed spaces.