## Banach spaces widely complemented in each other

### Volume 133 / 2013

#### Abstract

Suppose that $X$ and $Y$ are Banach spaces that embed complementably into each other. Are $X$ and $Y$ necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if $X$ contains a complemented copy of its square $X^2$, then $X$ is isomorphic to $Y$ whenever there exists $p \in \mathbb N$ such that $X^p$ can be decomposed into a direct sum of $X^{p-1}$ and $Y$. Motivated by this fact, we introduce the concept of $(p, q, r)$
*widely complemented* subspaces in Banach spaces, where $p, q$ and $r \in \mathbb N$. Then, we completely determine when $X$ is isomorphic to $Y$ whenever $X$ is $(p, q, r)$ widely complemented in $Y$ and $Y$ is $(t, u, v)$ widely complemented in $X$. This new notion of complementability leads naturally to an extension of the Square-cube Problem for Banach spaces, the $p$-$q$-$r$ Problem.