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

 

Regular methods of summability on tree-sequences in Banach spaces


Author: Costas Poulios
Journal: Proc. Amer. Math. Soc. 139 (2011), 259-271
MSC (2010): Primary 40C05, 46B99; Secondary 05D10, 05C55
Published electronically: August 6, 2010
MathSciNet review: 2729088
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Abstract: Suppose that $ X$ is a Banach space, $ \langle a_{ij}\rangle$ is a regular method of summability and $ (x_{s})_{s\in S}$ is a bounded sequence in $ X$ indexed by the dyadic tree $ S$. We prove that there exists a subtree $ S'\subseteq S$ such that: either (a) for any chain $ \beta$ of $ S'$ the sequence $ (x_{s})_{s\in\beta}$ is summable with respect to $ \langle a_{ij}\rangle$ or (b) for any chain $ \beta$ of $ S'$ the sequence $ (x_{s})_{s\in\beta}$ is not summable with respect to $ \langle a_{ij}\rangle$. Moreover, in case (a) we prove the existence of a subtree $ T\subseteq S'$ such that if $ \beta$ is any chain of $ T$, then all the subsequences of $ (x_{s})_{s\in\beta}$ are summable to the same limit. In case (b), provided that $ \langle a_{ij}\rangle$ is the Cesàro method of summability and that for any chain $ \beta$ of $ S'$ the sequence $ (x_{s})_{s\in\beta}$ is weakly null, we prove the existence of a subtree $ T\subseteq S'$ such that for any chain $ \beta$ of $ T$ any spreading model for the sequence $ (x_{s})_{s\in\beta}$ has a basis equivalent to the usual $ l_{1}$-basis. Finally, we generalize the theory of spreading models to tree-sequences. This also allows us to improve the result of case (b).


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Additional Information

Costas Poulios
Affiliation: Department of Mathematics, University of Athens, 15784, Athens, Greece
Email: k-poulios@math.uoa.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10479-3
PII: S 0002-9939(2010)10479-3
Received by editor(s): December 11, 2009
Received by editor(s) in revised form: February 16, 2010, and March 1, 2010
Published electronically: August 6, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.