Uniqueness of unconditional basis of $H_p(\mathbb{T})\oplus\ell_{2}$ and $H_p(\mathbb{T})\oplus\mathcal{T}^{(2)}$ for $0<p<1$

TL;DR

This paper establishes conditions under which the direct sum of certain quasi-Banach and Banach spaces has a unique unconditional basis, and applies these results to specific function and sequence spaces for 0<p<1.

Contribution

It provides new general criteria ensuring the uniqueness of unconditional bases in direct sums of quasi-Banach and Banach spaces, with applications to specific spaces like $H_p( ext{T}^d)$ and $ ext{T}^{(2)}$.

Findings

Spaces $H_p( ext{T}^d) igoplus ext{T}^{(2)}$ and $H_p( ext{T}^d) igoplus ext{l}_2$ have unique unconditional bases.

General conditions are established for the splitting of unconditional bases in direct sums of quasi-Banach and Banach spaces.

The results apply to spaces with $0<p<1$, extending the understanding of unconditional basis uniqueness in quasi-Banach settings.

Abstract

Our goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair $(\mathbb{X}, \mathbb{Y})$ formed by a quasi-Banach space $\mathbb{X}$ and a Banach space $\mathbb{Y}$ which guarantee that every unconditional basis of their direct sum $\mathbb{X}\oplus\mathbb{Y}$ splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces $H_p(\mathbb{T}^d) \oplus\mathcal{T}^{(2)}$ and $H_p(\mathbb{T}^d)\oplus\ell_2$, for $p\in(0,1)$ and $d\in\mathbb{N}$, have a unique unconditional basis (up to equivalence and permutation).