×

zbMATH — the first resource for mathematics

On the geometric properties of Cesàro spaces. (English. Russian original) Zbl 1253.46022
Sb. Math. 203, No. 4, 514-533 (2012); translation from Mat. Sb. 203, No. 4, 61-80 (2012).
By definition, the Cesàro space \(\mathrm{Ces}_p[0,1]\) consists of all measurable functions \(f\) on \([0,1]\) with \[ \|f\|_{C_p}= \left[\int_0^1\left(\frac{1}{t}\int_0^1|f(s)|ds\right)^pdt\right]^{1/p}<\infty, \quad 1\leq p<\infty. \] S. V. Astashkin and L. Maligranda [Indag. Math., New Ser. 20, No. 3, 329–379 (2009; Zbl 1200.46027)] give a description of the set of all \(q\) for which isomorphic copies of \(\ell^q\) are contained in \(\mathrm{Ces}_p[0,1]\). The author considers complemented copies of \(\ell^q\). He shows that \(\mathrm{Ces}_p[0,1]\) contains a complemented copy of \(\ell^q\) if and only if either \(q=1\) or \(q=p\). As a corollary, he obtains that \(\mathrm{Ces}_p[0,1]\), \(p>1\), contains no complemented copy of \(L_q[0,1]\), \(q>1\).

MSC:
46B25 Classical Banach spaces in the general theory
46B42 Banach lattices
PDF BibTeX XML Cite
Full Text: DOI