Cantor–Bernstein Sextuples for Banach Spaces

Abstract

AbstractLet X and Y be Banach spaces isomorphic to complemented subspaces of each other with supplements A and B. In 1996, W. T. Gowers solved the Schroeder–Bernstein (or Cantor–Bernstein) problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain a necessary and sufficient condition on the sextuples (p, q, r, s, u, v) in ℕ with p + q ≥ 1, r + s ≥ 1 and u, v ∈ ℕ*, to provide that X is isomorphic to Y, whenever these spaces satisfy the following decomposition schemeNamely, Φ = (p–u)(s–v)–(q + u)(r + v) is different from zero and Φ divides p + q and r + s. These sextuples are called Cantor–Bernstein sextuples for Banach spaces. The simplest case (1, 0, 0, 1, 1, 1) indicates the well-known Pełczyński's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder– Bernstein problem become evident.