Uniqueness of unconditional basis of $\ell _{2}\oplus \mathcal T^{(2)}$

Abstract

We provide a new extension of Pitt’s theorem for compact operators between quasi-Banach lattices which permits to describe unconditional bases of finite direct sums of Banach spaces $\mathbb {X}_{1}\oplus \dots \oplus \mathbb {X}_{n}$ as direct sums of unconditional bases of their summands. The general splitting principle we obtain yields, in particular, that if each $\mathbb {X}_{i}$ has a unique unconditional basis (up to equivalence and permutation), then $\mathbb {X}_{1}\oplus \cdots \oplus \mathbb {X}_{n}$ has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $\ell _2\oplus \mathcal {T}^{(2)}$ has a unique unconditional basis.