# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Schroeder–Bernstein Quintuples for Banach Spaces

### Tom 54 / 2006

Bulletin Polish Acad. Sci. Math. 54 (2006), 113-124 MSC: 46B03, 46B20. DOI: 10.4064/ba54-2-3

#### Streszczenie

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 ${\mathbb N}$ for $X$ to be isomorphic to $Y$ whenever $$\cases{ X \sim X^p \oplus Y^q, \cr Y^t \sim X^r \oplus Y^{s}. }$$ 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 Pe/lczyński'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.

#### Autorzy

• Elói Medina GalegoDepartment of Mathematics – IME
University of São Paulo
São Paulo 05315-970, Brazil
e-mail

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