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 ${\mathbb{X}}_1\oplus \cdots \oplus {\mathbb{X}}_n$ as direct sums of unconditional bases of its 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.