Stability and almost periodicity of asymptotically dominated semigroups of positive operators (Q2716106)

Let \(E\) be an ordered Banach space with a (closed, generating, normal) positive cone \(E_+\), such that the norm is monotone on \(E_+\). For a fixed subset \(D\subset E_+\), every homomorphism \({\mathcal S}= (S_t)_{t\in J}\) from \(J\) into \(D^D\) is said to be a representation of \(J\) in \(D^D\), where \(J\) is either (the additive semigroup) \(\mathbb{N}\) or \(\mathbb{R}_+\). A representation \({\mathcal S}= (S_t)_{t\in J}\) in \(D^D\) is asymptotically dominated by the representation \({\mathcal T}= (T_t)_{t\in J}\) in \(D^D\) if \(\lim_{t\to\infty} d(T_t x- S_tx, E_+)= 0\) for all \(x\in D\).NEWLINENEWLINENEWLINEThe aim of the paper is to study asymptotic properties of asymptotically dominated representations. The authors' techniques allow them to show that on a Banach lattice with order continuous norm strong stability and almost periodicity of a (discrete or strongly continuous) semigroup of positive operators is preserved under asymptotic domination.