Normed linear relations: domain decomposability, adjoint subspaces, and selections

Abstract

This paper is a study of normed linear relations with emphasis on results and properties that do not necessarily hold, or have no counterparts, for arbitrary normed convex processes. Sufficient conditions are given under which the norm of a linear relation M is equal to the norm of some operator part (i.e., a single-valued linear selection) of M. Generalizations of the closed-graph theorem and the open-mapping principle for linear relations are established in terms of linear selections. The norm of the adjoint subspace of a closed linear relation M is shown to be equal to the norm of M without any restrictions on the domain of M. The rich structure of normed linear relations is uncovered via four fundamental concepts and tools: algebraic, topological, and proximinal operator parts, and the adjoint subspace of a linear relation. Notions of algebraic, topological, and orthogonal domain decomposability of linear relations in Banach spaces are studied, and the relationships between these notions of domain decomposability and the corresponding notions of operator parts are determined.