Some remarks about disjointly homogeneous symmetric spaces

TL;DR

This paper investigates the properties of symmetric spaces on [0,1], specifically focusing on $p$-disjointly homogeneous spaces, and constructs examples showing distinctions between restricted and full $p$-disjoint homogeneity.

Contribution

It constructs, for each $1 \,\leq p<\infty$, a restricted $p$-disjointly homogeneous space that is not $p$-disjointly homogeneous, answering a recent open question.

Findings

Constructed examples for each $p$ showing the difference between restricted and full $p$-disjoint homogeneity.

Proved that $p$-disjoint homogeneity is preserved under Banach isomorphisms.

Established that the property of $p$-disjoint homogeneity is invariant under Banach isomorphisms.

Abstract

Let $1\le p<\infty$. A symmetric space $X$ on $[0,1]$ is said to be $p$-disjointly homogeneous (resp. restricted $p$-disjointly homogeneous) if every sequence of normalized pairwise disjoint functions from $X$ (resp. characteristic functions) contains a subsequence equivalent in $X$ to the unit vector basis of $l_p$. Answering a question posed recently, we construct, for each $1\le p<\infty$, a restricted $p$-disjointly homogeneous symmetric space, which is not $p$-disjointly homogeneous. Moreover, we prove that the property of $p$-disjoint homogeneity is preserved under Banach isomorphisms.