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Rosenthal's space revisited

Volume 262 / 2022

Sergey V. Astashkin, Guillermo P. Curbera Studia Mathematica 262 (2022), 197-224 MSC: Primary 46E30, 46B09; Secondary 46B15. DOI: 10.4064/sm201011-4-1 Published online: 19 September 2021

Abstract

Let $E$ be a rearrangement invariant (r.i.) function space on $[0,1]$, and let $Z_E$ consist of all measurable functions $f$ on $(0,\infty )$ such that $f^*\chi _{[0,1]}\in E$ and $f^*\chi _{[1,\infty )}\in L^2$. We reveal close connections between properties of the generalized Rosenthal space, corresponding to the space $Z_E$, and the behaviour of independent symmetrically distributed random variables in $E$. The results obtained are applied to the problem of existence of isomorphisms between r.i. spaces on $[0,1]$ and $(0,\infty )$. Exploiting particular properties of disjoint sequences, we identify a rather wide new class of r.i. spaces on $[0,1]$, “close” to $L^\infty $, which fail to be isomorphic to r.i. spaces on $(0,\infty )$. In particular, this property is shared by the Lorentz spaces $\Lambda _2(\log ^{-\alpha }(e/u))$ with $0 \lt \alpha \le 1$.

Authors

  • Sergey V. AstashkinDepartment of Mathematics
    Samara National Research University
    Moskovskoye shosse 34
    443086, Samara, Russia
    e-mail
  • Guillermo P. CurberaFacultad de Matemáticas
    & Instituto de Matemáticas (IMUS)
    Universidad de Sevilla
    Calle Tarfia s/n
    41012 Sevilla, Spain
    e-mail

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