Regular methods of summability on tree-sequences in Banach spaces
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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).References
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Additional Information
- Costas Poulios
- Affiliation: Department of Mathematics, University of Athens, 15784, Athens, Greece
- Email: k-poulios@math.uoa.gr
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 259-271
- MSC (2010): Primary 40C05, 46B99; Secondary 05D10, 05C55
- DOI: https://doi.org/10.1090/S0002-9939-2010-10479-3
- MathSciNet review: 2729088