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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Banach spaces with a unique nontrivial decomposition


Authors: Spiros A. Argyros and Theocharis Raikoftsalis
Journal: Proc. Amer. Math. Soc. 136 (2008), 3611-3620
MSC (2000): Primary 46B20, 46B26
Published electronically: June 2, 2008
MathSciNet review: 2415045
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Abstract: Motivated by a problem of P. Koszmider we introduce the class of quasi-prime Banach spaces. This class lies between the classes of prime and primary Banach spaces. It is shown that for every $ 1<p<\infty$ there exists a strictly quasi-prime separable reflexive Banach space $ \mathfrak{X}_p$ such that $ \ell_p$ is a complemented subspace of $ \mathfrak{X}_p$. A similar result also holds for the case of $ \ell_1$ and $ c_0$. More generally, for every separable decomposable prime Banach space $ Y$ not containing $ \ell_1$ there exists a strictly quasi-prime $ \mathfrak{X}_Y$ containing $ Y$ as a complemented subspace. We also investigate the operators acting on these spaces as well as the complemented subspaces of their finite powers.


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Additional Information

Spiros A. Argyros
Affiliation: Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, National Technical University of Athens, 157 80, Athens, Greece
Email: sargyros@math.ntua.gr

Theocharis Raikoftsalis
Affiliation: Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, National Technical University of Athens, 157 80, Athens, Greece
Email: th-raik@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09368-4
PII: S 0002-9939(08)09368-4
Keywords: Prime Banach space, primary Banach space, Schauder decomposition, hereditarily indecomposable Banach space, strictly singular operators, complemented subspaces
Received by editor(s): July 12, 2007
Received by editor(s) in revised form: September 11, 2007
Published electronically: June 2, 2008
Additional Notes: This work was partially supported by the Leukippos NTUA Research programme.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society