## Cantor–Schroeder–Bernstein quadruples for Banach spaces

### Volume 111 / 2008

#### Abstract

Two Banach spaces $X$ and $Y$ are symmetrically complemented in each other if there exists a supplement of $Y$ in $X$ which is isomorphic to some supplement of $X$ in $Y$. In 1996, W. T. Gowers solved the Schroeder–Bernstein (or Cantor–Bernstein) Problem for Banach spaces by constructing two non-isomorphic Banach spaces which are symmetrically complemented in each other. In this paper, we show how to modify such a symmetry in order to ensure that $X$ is isomorphic to $Y$. To do this, first we introduce the notion of Cantor–Schroeder–Bernstein Quadruples for Banach spaces. Then we characterize them by using some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997. This new insight into the geometry of Banach spaces complemented in each other leads naturally to the Strong Square-hyperplane Problem which is closely related to the Schroeder–Bernstein Problem.