Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y . Let (p, q, r, s) be a quadruple in with p + q  ≥  2 and r + s ≥  2. Suppose that for every pair of Banach spaces X and Y isomorphic to complemented subspaces of each other and satisfying the following Decomposition Scheme we conclude that X^m is isomorphic to Yⁿ for some . In this paper, we show that the discriminant of this quadruple is different from zero. This result completes the characterization of quadruples in which are nearly Schroeder-Bernstein Quadruples for Banach spaces. Received: 10 September 2005