On the permutative equivalence of squares of unconditional bases

On the number of non permutatively equivalent sequences in a Banach space

This paper contains results concerning the Borel reduction of the relation $E_0$ of eventual agreement between sequences of 0's and 1's, to the relation of permutative equivalence between basic sequences in a Banach space. For more clarity in this abstract, we state these results in terms of classification by real numbers. If $R$ is some (analytic) equivalence relation on a Polish space $X$, it is said that $R$ is classifiable (by real numbers) if there exists a Borel map $g$ from $X$ into the real line such that $x R x'$ if and only if $g(x)=g(x')$. If $R$ is not classifiable, there must be $…

The complexity of classifying separable Banach spaces up to isomorphism

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomo…

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

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 $\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$ 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 st…

Uniqueness of unconditional basis of $H_p(\mathbb{T})\oplus\ell_{2}$ and $H_p(\mathbb{T})\oplus\mathcal{T}^{(2)}$ for $0<p<1$

Our goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair $(\mathbb{X}, \mathbb{Y})$ formed by a quasi-Banach space $\mathbb{X}$ and a Banach space $\mathbb{Y}$ which guarantee that every unconditional basis of their direct sum $\mathbb{X}\oplus\mathbb{Y}$ splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces $H_p(\mathbb{T}^d) \oplus\mathcal{T}^{(2)}$ and $H_p(\mathbb{T}^d)\oplus\ell_2$, for $p\in(0,1)$ and $d\in\mathbb{N}$,…

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

This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits us to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from [F. Albiac and C. Leránoz, Uniqueness of unconditional bases in nonlocally convex $\ell_1$-products, J. Math. Anal. Appl. 374 (2011), no. 2, 394--401] we show that if $X$ is quasi-Banach space with a str…

0<p≤1时向量值$H_p$空间之间的同构与无条件结构的唯一性

Isomorphisms between vector-valued $H_p$-spaces for $0<p\le 1$ and uniqueness of unconditional structure

本文通过识别Banach值Hardy空间并发展新格估计技术，证明$0<p<1$时$H_p(\mathbb{D},\ell_2)$具有唯一原子格结构。