Abstract view
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, 60965 Poznań, Poland
Lech Maligranda,
Department of Engineering Sciences and Mathematics, LuleåUniversity of Technology, SE971 87 Luleå, Sweden
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 DunfordPettis properties of Cesàro and Tandori
spaces.
In particular, using results of Bourgain we show that a wide
class of CesàroMarcinkiewicz and
CesàroLorentz 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, DunfordPettis property, isomorphism
Cesàro and Tandori sequence spaces, Cesàro and Tandori function spaces, Cesàro operator, Banach ideal space, symmetric space, Schur property, DunfordPettis property, isomorphism

MSC Classifications: 
46E30, 46B20, 46B42, 46B45 show english descriptions
Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Geometry and structure of normed linear spaces Banach lattices [See also 46A40, 46B40] Banach sequence spaces [See also 46A45]
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]
