Rosenthal's space revisited
Volume 262 / 2022
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 $0 \lt \alpha \le 1$.
