Isomorphic Structure of Cesàro and Tandori Spaces

Author:

Astashkin Sergey V.,Lesnik Karol,Maligranda Lech

Abstract

AbstractWe investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $\text{Ces}_{\infty }$ and its sequence counterpart $\text{ces}_{\infty }$ are isomorphic. This is rather surprising since $\text{Ces}_{\infty }$ (like Talagrand’s example) has no natural lattice predual. We prove that $\text{ces}_{\infty }$ is not isomorphic to $\ell _{\infty }$ nor is $\text{Ces}_{\infty }$ isomorphic to the Tandori space $\widetilde{L_{1}}$ with the norm $\Vert f\Vert _{\widetilde{L_{1}}}=\Vert \widetilde{f}\Vert _{L_{1}}$ , where $\widetilde{f}(t):=\text{ess}\,\sup _{s\geqslant t}|f(s)|$ . Our investigation also involves 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.

Publisher

Canadian Mathematical Society

Subject

General Mathematics

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Banach spaces of sequences arising from infinite matrices;Annals of Functional Analysis;2024-05-08

2. On weakly almost square Banach spaces;Proceedings of the Edinburgh Mathematical Society;2023-10-05

3. Quotients, $$\ell _\infty $$ and abstract Cesàro spaces;Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas;2022-06-04

4. Isomorphic and isometric structure of the optimal domains for Hardy-type operators;Studia Mathematica;2021

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