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## Colloquium Mathematicum

2013 | 133 | 2 | 283-291
Tytuł artykułu

### Banach spaces widely complemented in each other

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Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that $X^{p}$ can be decomposed into a direct sum of $X^{p-1}$ and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and r ∈ ℕ. Then, we completely determine when X is isomorphic to Y whenever X is (p,q,r) widely complemented in Y and Y is (t,u,v) widely complemented in X. This new notion of complementability leads naturally to an extension of the Square-cube Problem for Banach spaces, the p-q-r Problem.
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Tom
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283-291
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wydano
2013
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autor
• Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
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