Convergence of arithmetic means of operators in Fréchet spaces
A well-known result of \textit{M. Lin} [Proc. Am. Math. Soc. 46, 217--225 (1974; Zbl 0291.47006)] states that, if \(T\) is a bounded operator on a Banach space \(X\) which satisfies the assumption \(\lim_{n\to+\infty}\frac{||T^{n}||}{n}=0\), then its arithmetic means \[ T_{[n]}=\frac{1}{n}\sum_{j=1}^{n}T^{j} \] converge uniformly to a bounded projection \(P\) on \(X\) (\(T\) is then called uniformly mean ergodic) if and only if \(I-T\) has closed range, in which case \(X\) can be decomposed as the direct sum \(X=\ker(I-T)\oplus (I-T)X\). The purpose of the paper under review is to study in a systematic way how far this result may be extended to Fréchet spaces, given that it has been observed in [\textit{A. Albanese, J. Bonet} and \textit{W. Ricker}, Ann. Acad. Sci. Fenn., Math. 34, No. 2, 401--436 (2009; Zbl 1194.47012)] that it fails on the space \(s\) of rapidly decreasing sequences. It is shown that on every Montel Köthe space not isomorphic to a countable product of copies of the scalar field, there exists a continuous power-bounded linear operator \(T\) which is uniformly mean ergodic but such that \(I-T\) does not have closed range. On the other hand, it is shown that Lin's result can be extended to a special class of Fréchet spaces, called prequojections. The arithmetic means of the Cesàro operator on some classical sequence spaces are then studied, and finally Browder's equality, which states that for a power-bounded operator \(T\) on a reflexive Banach space \(X\), \[ (I-T)X=\left\{x\in X : \sup_{n\geq 1}\left\|\sum_{j=1}^{n}T^{j}x\right\|<+\infty\right\} \] is investigated in the Fréchet space setting.
scientific article; zbMATH DE number 6143657