Isomorphic structure of Cesàro and Tandori spaces

Abstract

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

References in corpus (1)

• On the dual of Cesàro function space

Cited by in corpus (2)

• Quotients, ${\ell }_{\infty }$ and abstract Cesàro spaces
• Banach spaces of sequences arising from infinite matrices