Uniqueness of unconditional basis of ℓ₂⊕ T⁽²⁾

2020/12/12 by Albiac, Fernando, Ansorena, Jose L.
#46A16 #46A35 #46A40 #46A45 (Secondary) #46B15 (Primary) 46B20 #46B42 #46B45 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2012.06783

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 \mathbbX₁⊕…⊕\mathbbX_n as direct sums of unconditional bases of its summands. The general splitting principle we obtain yields, in particular, that if each \mathbbX_i has a unique unconditional basis (up to equivalence and permutation), then \mathbbX₁⊕ ⋯⊕\mathbbX_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 ℓ₂⊕ T⁽²⁾ has a unique unconditional basis.

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