Generating Fréchet-Montel spaces that are not Schwartz by closed linear operators (Q1058104)

Given a densely defined linear operator \(T: X\supseteq D(T)\to X\) on a Banach space X with non-empty resolvent set, a natural object of study is the Fréchet space \(D_{\infty}:=\cap_{n\in {\mathbb{N}}}D(T^ n)\) generated by T. We construct operators T such that \(D_{\infty}(T)\) is Montel but not Schwartz. This especially means that T is not allowed to have a (power) compact resolent. On the other hand it follows from a previous result of the author [Spectral properties of operators generating (FM)-spaces. Preprint, Kiel (1984)] that T must have a Riesz resolvent whenever \(D_{\infty}(T)\) is a Montel space.