Duality problem for disjointly homogeneous rearrangement invariant spaces

TL;DR

This paper constructs new examples of reflexive p-disjointly homogeneous rearrangement invariant spaces, answering a question about their duals and introducing a Tsirelson type space with unique properties.

Contribution

It provides the first construction of reflexive p-disjointly homogeneous spaces with non-disjointly homogeneous duals and introduces a novel Tsirelson type space with specific structural features.

Findings

Constructed reflexive p-disjointly homogeneous spaces with non-disjointly homogeneous duals.

Developed new examples of disjointly homogeneous rearrangement invariant spaces using interpolation.

Presented a Tsirelson type space that contains no subspace isomorphic to l_p or c_0.

Abstract

Let $1\le p<\infty$. A Banach lattice $E$ is said to be disjointly homogeneous (resp. $p$-disjointly homogeneous) if two arbitrary normalized disjoint sequences from $E$ contain equivalent in $E$ subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in $E$ to the unit vector basis of $l_p$). Answering a question raised in 2014 by Flores, Hernandez, Spinu, Tradacete, and Troitsky, for each $1<p<\infty$, we construct a reflexive $p$-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to $l_p$, $1\le p<\infty$, or $c_0$.