UNIQUENESS OF UNCONDITIONAL BASIS OF l2 ⊕ T(2)

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 X-1 circle plus center dot center dot center dot circle plus X-n as direct sums of unconditional bases of their summands. The general splitting principle we obtain yields, in particular, that if each X-i has a unique unconditional basis ( up to equivalence and permutation), then X-1 circle plus center dot center dot center dot circle plus 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 l(2) circle plus T-(2) has a unique unconditional basis.