On the permutative equivalence of squares of unconditional bases

Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces

“…Even without having any information on the concavity of the space X , Proposition 7.5 below provides an improvement of [1,Proposition 5.6]. In addition to that, it shows that imposing some nontrivial concavity to the lattice structure allows to weaken the assumption on the lower democracy function.…”

“…Thus, the following theorem summarizes what can be rightfully called the "Casazza-Kalton paradigm" to tackle the uniqueness of unconditional basis problem extended to quasi-Banach lattices. To be able to prove it in this optimal form (even for locally convex spaces) has required the very recent solution in the positive of the "canceling squares" problem (see [1]). Theorem 3.12 (see [1,Theorem 3.9]) Let X be a quasi-Banach space with a normalized unconditional basis X .…”

“…In our notation, the Lorentz space d p, p (w) coincides with the Lorentz space denoted d(w, p) by Altshuler in [13] (see also [1,9,15]). We emphasize that Altshuler only considered non-increasing weights w ∈ c 0 \ 1 , but this restriction is unnecessary here.…”

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

“…Roughly speaking this result works when there is a "gap" between the lattice concavity of X and the lattice convexity of Y, i.e., in the case when there exist 1 ≤ q < r ≤ ∞ such that the lattice structure induced by their (unique) unconditional bases on X and Y satisfies a lower q-estimate and an upper r-estimate, respectively. This yields, for instance, that if X is either 1 or T , Y is either 2 or T (2) , and U is either c 0 or T * , then the spaces X ⊕ Y, X ⊕ U, Y ⊕ U, and X ⊕ Y ⊕ U all have a UTAP unconditional basis (see [1,Corollary 6.2] and [3,Theorem 4.7]).…”

“…And so are the spaces 1 ⊕ T and c 0 ⊕ T * . However, since 1 , c 0 , T , and T * are lattice anti-Euclidean spaces, it can be shown that any direct sum built with some of these spaces has a UTAP unconditional basis (see [1,Corollary 6.2]). Again, as 2 and T (2) are "highly" Euclidean, they are orthogonal to the methods from [1].…”