Rosenthal's space revisited

TL;DR

This paper explores the properties of generalized Rosenthal's spaces linked to rearrangement invariant spaces, examining their connection to independent random variables and the conditions under which these spaces are isomorphic or not.

Contribution

It introduces a new class of rearrangement invariant spaces close to L^, which are not isomorphic to spaces on (0,), and analyzes their properties using disjoint sequences.

Findings

Identifies a wide class of r.i. spaces close to L^ that are not isomorphic to spaces on (0,)

Establishes connections between Rosenthal's spaces and independent symmetrically distributed variables

Shows Lorentz spaces b1=1 are among the spaces with these properties

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's space, corresponding to the space $Z_E$, and the behaviour of independent symmetrically distributed random variables in $E$. The results obtained are applied to consider the problem of the 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 $0<\alpha\le 1$.