Structure of Cesàro function spaces
type: Item
instance of: scholarly article
title: Structure of Cesàro function spaces
zbMATH Open document ID: 1200.46027
DOI: 10.1016/S0019-3577(10)00002-9
author: Lech Maligranda
author: Sergey Astashkin
published in: Indagationes Mathematicae. New Series
publication date: 2010-04-29T00:00:00Z
review text: Let \(1\leq p \leq \infty\). The Cesàro sequence space \(\text{ces}_p\) is the set of real sequences \(x = \{x_k\}\) such that \[ \|x\|_{c(p)} = \left[\sum_{n=1}^\infty\left(\frac{1}{n} \sum_{k=1}^n|x_k|\right)^p\right]^{1/p} < \infty,\quad 1\leq p < \infty, \] and \[ \|x\|_{c(\infty)} = \sup_{n\in\mathbb{N}}\frac{1}{n} \sum_{n=1}^n |x_k| < \infty, \quad p=\infty. \] The Cesàro function spaces Ces\(_p = \text{Ces}_p(I)\) are classes of Lebesgue measurable real functions \(f\) on \(I =[0,1]\) or \(I=[0,\infty)\) such that \[ \|f\|_{c(p)} = \left[\int_I\left(\frac{1}{x} \int_{0}^x|f(t)|\,dt\right)^p \,dx\right]^{1/p} < \infty,\quad 1\leq p < \infty, \] and \[ \|f\|_{c(\infty)} = \sup_{x\in I, x>0}\frac{1}{x} \int_{0}^x |f(t)|\,dt < \infty, \quad p=\infty. \] The authors investigate several isomorphic structure properties of Cesàro sequence and function spaces. They find their dual spaces up to equivalence of norm, with different description for \([0,1]\) and \([0,\infty)\). They prove that Ces\(_p\), \(1<p<\infty\), is strictly convex but not reflexive. It is also shown that these spaces are not isomorphic to any space \(L^q\) with \(1\leq q\leq \infty\), and do not enjoy the Dunford-Pettis property while they have the weak Banach-Saks property. It is characterized when the Cesàro spaces contain isomorphic and complemented copies of \(\ell^p\), and a description when Ces\(_p[0,1]\) contains an isomorphic subspace of \(\ell^q\) is given. Finally, the authors prove that Ces\(_p[0,1]\) and Ces\(_p[0,\infty)\) are isomorphic for \(1<p\leq\infty\).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 5702858
zbMATH Keywords: Cesàro sequence spaces
zbMATH Keywords: Cesàro function spaces, Köthe dual
zbMATH Keywords: associated space
zbMATH Keywords: dual space
zbMATH Keywords: \(L^p\) spaces
zbMATH Keywords: copies of \(\ell^p\)
zbMATH Keywords: weak Banach-Saks property
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: Rademacher functions
zbMATH Keywords: type
zbMATH Keywords: cotype
zbMATH Keywords: isomorphism
zbMATH Keywords: subspaces
zbMATH Keywords: complemented subspaces
reviewed by: Anna Kamińska
MaRDI profile type: Publication
cites work: Q5317928
cites work: Q3748810
cites work: A Fixed Point Free Nonexpansive Map
cites work: Cesàro function spaces fail the fixed point property
cites work: Rademacher functions in Cesàro type spaces
cites work: Banach-Saks property in Marcinkiewicz spaces
cites work: Q4720203
cites work: Factorizing the classical inequalities
cites work: Q4039801
cites work: Geometric properties related to fixed point theory in some Banach function lattices
cites work: Some geometric properties related to fixed point theory in Cesàro spaces
cites work: Q4937960
cites work: Packing constant for Cesaro sequence spaces.
cites work: Q4508963
cites work: Q4386398
cites work: Banach-Saks property and property \((\beta)\) in Cesàro sequence spaces
cites work: Q4850055
cites work: Q4954113
cites work: The Banach–Saks property in rearrangement invariant spaces
cites work: Every nonreflexive subspace of $L_1[0,1]$ fails the fixed point property
cites work: Renormings of \(\ell^1\) and \(c_0\) and fixed point properties
cites work: Q5573683
cites work: Functional analysis and infinite-dimensional geometry
cites work: Q4771727
cites work: Q4774562
cites work: Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$
cites work: Q3218622
cites work: Q3769379
cites work: On the Brudnyĭ-Krugljak duality theory of spaces formed by the \(\mathcal K\)-method of interpolation
cites work: Q5787654
cites work: Q5439977
cites work: Q3684631
cites work: Q5645909
cites work: Q4136031
cites work: Q4188284
cites work: On some Banach lattices. IV
cites work: Some examples of normed Köthe spaces
cites work: Q4348185
cites work: Q5692555
cites work: On the James constant and \(B\)-convexity of Cesàro and Cesàro--Orlicz sequence spaces
cites work: Q3876469
cites work: On the structure of subspaces of the spaces Λ_{𝑝}(𝜇)
cites work: Banach-Saks exponent of certain Banach spaces of sequences
cites work: Boundedness of classical operators on classical Lorentz spaces
cites work: Q5617991
cites work: Q5617992
cites work: The level function in rearrangement invariant spaces
cites work: Transferring monotonicity in weighted norm inequalities
cites work: Q3791686
cites work: Sur les suites faiblement convergentes dans l'espace L

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cites work: Q5830862

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cites work: Q3327152

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cites work: Q3985412

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cites work: Q4259426