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
Cited by
3 articles.
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1. On weakly almost square Banach spaces;Proceedings of the Edinburgh Mathematical Society;2023-10-05
2. Quotients, $$\ell _\infty $$ and abstract Cesàro spaces;Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas;2022-06-04
3. Isomorphic and isometric structure of the optimal domains for Hardy-type operators;Studia Mathematica;2021