Isomorphic structure of Ces\`aro and Tandori spaces

TL;DR

This paper explores the isomorphic relationships between Cesàro and Tandori spaces, revealing surprising equivalences and distinctions, and examines their properties such as Schur and Dunford-Pettis, with implications for related function and sequence spaces.

Contribution

It proves that Cesàro function and sequence spaces are isomorphic, answering an open question, and analyzes their structural properties and relations to classical spaces.

Findings

Cesàro and Tandori spaces are not isomorphic to certain classical spaces.

Cesàro and Tandori spaces share isomorphic structure despite lacking natural lattice preduals.

Many Cesàro-Marcinkiewicz and Cesàro-Lorentz spaces possess the Dunford-Pettis property.

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.