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arXiv · 1805.00691

Duality problem for disjointly homogeneous rearrangement invariant spaces

Sergey V. Astashkin
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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 $c_0$.

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Sergey V. Astashkin. 2018-05-02. Duality problem for disjointly homogeneous rearrangement invariant spaces. https://doi.org/10.1016/j.jfa.2018.08.020

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