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Structure of Cesàro function spaces. (English) Zbl 1200.46027

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 \(1<p\leq\infty\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
46B42 Banach lattices
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