Uniqueness of unconditional basis of $H_p(\mathbb{T})\oplus {\ell }_2$ and $H_p(\mathbb{T})\oplus {\mathcal{T}}^{(2)}$ for $0<p<1$

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).