\(C_{0}\)-semigroups and mean ergodic operators in a class of Fréchet spaces
The authors prove that there is an important class of Fréchet spaces, namely the quojections, in which important results concerning \({\mathcal C}_0\)-semigroups on Banach spaces continue to hold. They prove that if \(X\) is a quojection Fréchet space and \((T(t))_{t\geq 0}\) is an exponentially equicontinuous, uniformly continuous semigroup on \(X\), then the infinitesimal generator \(A\) of \((T(t))_{t\geq 0}\) is everywhere defined, i.e., \(D(A)=X\) and hence, \(A\in {\mathcal L}(X)\). They also prove that, if \(X\) is a quojection GDP-Fréchet space and \((T(t))_{t\geq 0}\) is an exponentially equicontinuous \({\mathcal C}_0\)-semigroup on \(X\), then \((T(t))_{t\geq 0}\) is uniformly continuous and its infinitesimal generator belongs to \({\mathcal L}(X)\) (a GDP-space is a Grothendieck locally convex Hausdorff space with the Dunford-Pettis property). A result concerning any mean ergodic operator satisfying \({\frac{1}{n}}T^n\rightarrow 0\) in \({\mathcal L}_b(X)\) and defined in a GDP-space is presented. One proves that there exists a projection \(P\in {\mathcal L}(X)\) such that the sequence of squares \((T_{[n]})^2\rightarrow P\) in \({\mathcal L}_b(X)\) as \(n\rightarrow \infty\) (here, \(T_{[n]}:={\frac{1}{n}}\sum_{m=1}^nT^m, \; n\in {\mathbb N}\)). If, in addition, \(X\) is a quojection Fréchet space, then \(\lim_{n\rightarrow \infty}T_{[n]}= P\) in \({\mathcal L}_b(X)\), i.e., \(T\) is uniformly mean ergodic. For GDP-Banach spaces, one recovers a classical result of H.\,P.\thinspace Lotz.
scientific article; zbMATH DE number 5673263
scientific article