This is an announcement for the paper "A Bohl--Bohr--Kadets type theorem characterizing Banach spaces not containing $c_0$" by Balint Farkas.

Abstract: We prove that a separable Banach space $E$ does not contain a copy of the space $\co$ of null-sequences if and only if for every doubly power-bounded operator $T$ on $E$ and for every vector $x\in E$ the relative compactness of the sets ${T^{n+m}x-T^nx: n\in \NN}$ (for some/all $m\in\NN$, $m\geq 1$) and ${T^nx:n\in \NN}$ are equivalent. With the help of the Jacobs--de Leeuw--Glicksberg decomposition of strongly compact semigroups the case of (not necessarily invertible) power-bounded operators is also handled.

Archive classification: math.FA

Mathematics Subject Classification: 47A99, 46B04, 43A60

Submitted from: farkasb@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1301.6250

or

http://arXiv.org/abs/1301.6250