Uniqueness of unconditional basis of [formula omitted] and [formula omitted] 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 (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)⊕ℓ2, for p∈(0,1) and d∈N, have a unique unconditional basis (up to equivalence and permutation).