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 (X, Y) formed by a
quasi-Banach space X and a Banach space Y which guarantee that every unconditional basis of their direct sum X ⊕ Y
splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces
Hp(Td) ⊕ T (2) and Hp(Td) ⊕ 2, for p ∈ (0, 1) and d ∈ N,
have a unique unconditional basis (up to equivalence and permutation).F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximación. P. Wojtaszczyk was supported by National Science Centre, Poland grant UMO-2016/21/B/ST1/00241
