On the permutative equivalence of squares of unconditional bases

TL;DR

This paper proves that if the squares of two unconditional bases are permutation equivalent, then the bases themselves are permutatively equivalent, resolving a long-standing open question and impacting the study of unconditional bases in quasi-Banach spaces.

Contribution

It establishes that permutation equivalence of squares of unconditional bases implies the bases are permutatively equivalent, providing a new approach to basis uniqueness in quasi-Banach spaces.

Findings

Proves bases are permutatively equivalent if their squares are permutation equivalent.

Provides examples illustrating the theoretical framework.

Shows the result applies to finite sums of quasi-Banach spaces with unique bases.

Abstract

We prove that if the squares of two unconditional bases are equivalent up to a permutation, then the bases themselves are permutatively equivalent. This settles a twenty year-old question raised by Casazza and Kalton in [Uniqueness of unconditional bases in Banach spaces, Israel J. Math. 103 (1998), 141--175]. Solving this problem provides a new paradigm to study the uniqueness of unconditional basis in the general framework of quasi-Banach spaces. Multiple examples are given to illustrate how to put in practice this theoretical scheme. Among the main applications of this principle we obtain the uniqueness of unconditional basis up to permutation of finite sums of quasi-Banach spaces with this property.