This is an announcement for the paper "Isomorphic structure of Ces`aro and Tandori spaces" by Sergey V. Astashkin, Karol Lesnik, and Lech Maligranda.

Abstract: We investigate the isomorphic structure of the Ces`aro spaces and their duals, the Tandori spaces. The main result states that the Ces`aro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers to the question posted in \cite{AM09}. This is rather surprising since $Ces_{\infty}$ has no natural lattice predual similarly as the known Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ is isomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $|f|_{\widetilde{L_1}} =|\widetilde{f}|_{L_1},$ where $\widetilde{f}(t): \esssup_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Ces`aro and Tandori spaces. In particular, using Bourgain's results we show that a wide class of Ces{`a}ro-Marcinkiewicz and Ces{`a}ro-Lorentz spaces have the latter property.

Archive classification: math.FA

Mathematics Subject Classification: 46E30, 46B20, 46B42

Submitted from: lech.maligranda@ltu.se

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

http://front.math.ucdavis.edu/1512.03336

or

http://arXiv.org/abs/1512.03336