Stability and ergodicity of dominated semigroups. II: The strong case (Q1317996)

Let \({\mathfrak S}=(S_ \lambda)_{\lambda\in\Lambda}\) and \({\mathfrak T}=(T_ \lambda)_{\lambda\in\Lambda}\) be nets of positive operators on a Banach lattice \(E\) such that \(0\leq S_ \lambda\leq T_ \lambda\) for all \(\lambda\in\Lambda\). We are looking for conditions on \(\mathfrak S\), \(\mathfrak T\) or \(E\) which ensure that strong stability of \(\mathfrak T\), i.e., \(\lim_ \lambda T_ \lambda x\) exists for all \(x\in E\), is inherited by the dominated family \(\mathfrak S\). If \(\mathfrak S\) and \(\mathfrak T\) are so-called \(M\)- nets this turns out to be true. As a consequence strong stability is preserved under domination for various operator families, including Cesaro and Abel means of one-parameter semigroups, ergodic nets for arbitrary semigroups of operators, and Abel means of pseudo-resolvents. If the families \(\mathfrak S\) and \(\mathfrak T\) are the powers \((S^ n)_{n\in\mathbb{N}}\) and \((T^ n)_{n\in\mathbb{N}}\) of operators \(S\) and \(T\), respectively, then the inheritance of strong stability is much harder to achieve. It is shown that on a Banach lattice \(E\) with order continuous norm strong convergence of \((T^ n)_{n\in\mathbb{N}}\) towards an operator of rank \(r<\infty\) implies strong convergence of \((S^ n)_{n\in\mathbb{N}}\) to an operator of rank \(q\leq r\). This can be applied to obtain inheritance of strong stability for representations of certain Abelian semigroups by positive operators on a Banach lattice. [Part I is to appear in Math. Z.].