Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces

In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of $L_p$ for $1<p<\mathrm{\infty }$, $p\mathrm{\ne }2$. We extend their construction to rearrangement-invariant spaces and Hardy spaces. For any separable rearrangement-invariant space $X$ with non-trivial Boyd indices, we show that these subspaces are complemented, previously obtained by Ghawadrah via a different argument, and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to $X$. To this end, we introduce an ordinal index that detects whether $X$ embeds complementably into an arbitrary separable Banach space. On the other hand, we give sufficient conditions which imply that in certain spaces close to $L_1$ and $H^1$, these subspaces fail to be complemented.