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4 Bulletin of the Polish Academy of Sciences. Mathematics

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Schroeder-Bernstein quintuples for Banach spaces

Galego E. M.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

Vol. 54, no 2

113-124

Let X and Y be two Banach spaces, each 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. In this paper, we obtain necessary and sufficient conditions on the quintuples (p, q, r, s, t) in N for X to be isomorphic to Y whenever [...]. Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition methods in Banach spaces involving finite sums of X and Y, similar to Pelczynski's decomposition method. Inspired by this result, we also introduce the notion of Schroeder-Bernstein sextuples for Banach spaces and pose a conjecture which would complete their characterization.

An arithmetic characterization of decomposition methods in Banach spaces similar to Pełczyńsk's decomposition method

Galego E. M.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 3

273-282

Inspired by Pełczyński's decomposition method in Banach spaces, we introduce the notion of Schroeder-Bernstein quadruples for Banach spaces. Then we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

Banach spaces complemented in each other without isomorphic finite sums

Galego E. M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 1

1-9

We show that the first solution of W.T. Gowers to the Schroeder-Bernstein problem for Banach spaces (unpublished) also provides the first example of two Banach spaces Z and W such that each of them is isomorphic to a complemented subspace of the other, but [Z sup m] is not isomorphic to [W sup n] for every m, n [belongs to] N*.

How to generate new Banach spaces non-isomorphic to their cartesian square

Galego E.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 1

22-25

Let alpha be a nondenumerable regular ordinal, fi any ordinal, X a Banach space. We introduce a closed subspace X[...] of the X[s] defined in [7] and of the X[alfa] defined in [6] to get a way to generate new Banach spaces non-isomorphic to their cartesian squares. In particular we obtain new Banach spaces X and Y non-isomorphic to their cartesian squares but having the property that X [...] Y is isomorphic to its cartesian square, see [5], and we give infinite solutions to Schroeder-Bernstein problem, see [4].