Linear mappings on ordered vector spaces

Abstract

Before we can start to analyze the generalized Riccati operators derived in the previous chapter, we have to deal with generalized Lyapunov operators of the type occuring e.g. in Theorem 1.5.3. It is crucial to observe that these Lyapunov operators possess certain positivity properties, which will be discussed in the following. The present chapter is split in two parts. In the first part, we introduce the general notions of ordered Banach spaces and resolvent positive operators, which is the set-up to be used in Chapter 4. In the second part, we focus on the ordered vector space $\mathcal{H}^n$ of Hermitian matrices, which is relevant for our further investigations in Chapter 5.