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Isomorphic Structure of Cesàro and Tandori Spaces

Part of: Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 January 2019

Sergey V. Astashkin ,
Karol Lesnik  and
Lech Maligranda
Sergey V. Astashkin
Affiliation:
Department of Mathematics and Mechanics, Samara State University, Acad. Pavlova 1, 443011 Samara, Russia Samara State Aerospace University (SSAU), Moskovskoye shosse 34, 443086, Samara, Russia Email: astash56@mail.ru
Karol Lesnik
Affiliation:
Institute of Mathematics, Poznań University of Technology, ul. Piotrowo 3a, 60-965 Poznań, Poland Email: klesnik@vp.pl
Lech Maligranda
Affiliation:
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden Email: lech.maligranda@ltu.se
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Abstract

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We 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.

Keywords

Cesàro and Tandori sequence spacesCesàro and Tandori function spacesCesàro operatorBanach ideal spacesymmetric spaceSchur propertyDunford–Pettis propertyisomorphism

MSC classification

Primary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 46B20: Geometry and structure of normed linear spaces 46B42: Banach lattices 46B45: Banach sequence spaces
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 3 , June 2019 , pp. 501 - 532
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Author S. V. A. was partially supported by the Ministry of Education and Science of the Russian Federation (project 1.470.2016/1.4) and author K. L. was partially supported by the grant 04/43/DSPB/0086 from the Polish Ministry of Science and Higher Education.

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