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# Isomorphic structure of Cesàro and Tandori spaces

• Sergey V. Astashkin,
Department of Mathematics and Mechanics, Samara State University, Acad. Pavlova 1, 443011 Samara, Russia
• Karol Lesnik,
Institute of Mathematics of Electric Faculty, Poznań University of Technology, ul. Piotrowo 3a, 60-965 Poznań, Poland
• Lech Maligranda,
Department of Engineering Sciences and Mathematics, LuleåUniversity of Technology, SE-971 87 Luleå, Sweden
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## 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 $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers the question posted previously. This is rather surprising since $Ces_{\infty}$ (like the known Talagrand's example) has no natural lattice predual. We prove that $ces_{\infty}$ is not isomorphic to ${\ell}_{\infty}$ nor is $Ces_{\infty}$ isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \operatorname{esssup}_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro-Marcinkiewicz and Cesàro-Lorentz spaces have the latter property.
 Keywords: Cesàro and Tandori sequence spaces, Cesàro and Tandori function spaces, Cesàro operator, Banach ideal space, symmetric space, Schur property, Dunford-Pettis property, isomorphism
 MSC Classifications: 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B20 - Geometry and structure of normed linear spaces 46B42 - Banach lattices [See also 46A40, 46B40] 46B45 - Banach sequence spaces [See also 46A45]

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