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Cantor–Bernstein Sextuples for Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Elói M. Galego
Elói M. Galego*
Affiliation:
Department of Mathematics - IME, University of São Paulo, São Paulo 05508-090, Brazil e-mail: [email protected]
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Abstract

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Let $X$ and $Y$ be Banach spaces isomorphic to complemented subspaces of each other with supplements $A$ and $B$. In 1996, W. T. Gowers solved the Schroeder–Bernstein (or Cantor–Bernstein) problem for Banach spaces by showing that $X$ is not necessarily isomorphic to $Y$. In this paper, we obtain a necessary and sufficient condition on the sextuples $\left( p,\,q,\,r,\,s,\,u,\,v \right)$ in $\mathbb{N}$ with $p\,+\,q\,\ge \,1$, $r+s\ge 1$ and $u,\,v\,\in \,{{\mathbb{N}}^{*}}$, to provide that $X$ is isomorphic to $Y$, whenever these spaces satisfy the following decomposition scheme

$${{A}^{u}}\,\sim \,{{X}^{p}}\,\oplus \,{{Y}^{q}},\,{{B}^{v}}\,\sim \,{{X}^{r}}\,\oplus \,{{Y}^{s}}.$$

Namely, $\Phi \,=\,\left( p\,-\,u \right)\left( s\,-\,v \right)-\left( q\,+\,u \right)\left( r\,+\,v \right)$ is different from zero and $\Phi $ divides $p\,+\,q$ and $r\,+\,s$. These sextuples are called Cantor–Bernstein sextuples for Banach spaces. The simplest case $\left( 1,0,0,1,1,1 \right)$ indicates the well-known Pełczyński's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder– Bernstein problem become evident.

Keywords

46B0346B20Pełczyński's decomposition methodSchroeder-Bernstein problem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 2 , 01 June 2010 , pp. 278 - 285
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Casazza, P. G., The Schroeder-Bernstein property for Banach spaces. In: Banach Space Theory. Contemp. Math. 85. American Mathematical Society, Providence, RI, 1989, pp. 61–77.Google Scholar
[2] Ferenczi, V. and Galego, E. M., Some results on the Schroeder-Bernstein property for separable Banach spaces. Canad. J. Math. 59(2007), no. 11 63–84.Google Scholar
[3] Galego, E. M., On solutions to the Schroeder-Bernstein problem for Banach spaces. Arch. Math. (Basel) 79(2002), no. 4, 299–307.Google Scholar
[4] Galego, E. M., Banach spaces complemented in each other without isomorphic finite sums. Bull. Pol. Acad. Sci. Math. 50(2002), no. 1, 1–9.Google Scholar
[5] Galego, E. M., The Schroeder-Bernstein index for Banach spaces. Studia Math. 164(2004), no. 1, 29–38. doi:10.4064/sm164-1-2Google Scholar
[6] Galego, E. M., On pairs of Banach spaces which are isomorphic to complemented subspaces of each other. Colloq. Math. 101(2004), no. 2, 279–287. doi:10.4064/cm101-2-10Google Scholar
[7] Galego, E. M., A remark on non-separable solutions to the Schroeder-Bernstein problem for Banach spaces. Results Math. 47(2005), no. 1–2, 55–60.Google Scholar
[8] Galego, E. M., Schroeder-Bernstein quintuples for Banach spaces. Bull. Pol. Acad. Sci. Math. 54(2006), no. 2, 113–124. doi:10.4064/ba54-2-3Google Scholar
[9] Galego, E. M., A note on extensions of Pełczyński's decomposition method in Banach spaces. Studia Math. 180(2007), no. 1, 27–40. doi:10.4064/sm180-1-3Google Scholar
[10] Galego, E. M., Some Schroeder-Bernstein type theorems for Banach spaces. J. Math. Anal. Appl. 338(2008), no. 1, 653–661. doi:10.1016/j.jmaa.2007.04.078Google Scholar
[11] Gowers, W. T., The unconditional basic sequence problem. J. Amer. Math. Soc. 6(1993), 851–874. doi:10.2307/2152743Google Scholar
[12] Gowers, W. T., A solution to the Schroeder-Bernstein problem for Banach spaces. Bull. London Math. Soc. 28(1996), no. 3, 297–304. doi:10.1112/blms/28.3.297Google Scholar
[13] Gowers, W. T., and Maurey, B., Banach spaces with small spaces of operators. Math. Ann. 307(1997), no. 4, 543–568. doi:10.1007/s002080050050Google Scholar