A Remark on non-separable solutions to the Schroeder-Bernstein Problem for Banach spaces

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We show that the geometric structure of Banach spaces which are solutions to the Schroeder-Bernstein Problem is very complex. More precisely, we prove that there exists a non-separable solution E to this problem such that (a) E is isomorphic to each one of its finite codimensional subspaces. (b) E has no complemented Hereditarily Indecomposable subspace. (c) E has no complemented subspace isomorphic to its square. (d) E has no non-trivial divisor.