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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10479-3

Published electronically:
August 6, 2010

MathSciNet review:
2729088

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Abstract: Suppose that is a Banach space, is a regular method of summability and is a bounded sequence in indexed by the dyadic tree . We prove that there exists a subtree such that: either (a) for any chain of the sequence is summable with respect to or (b) for any chain of the sequence is not summable with respect to . Moreover, in case (a) we prove the existence of a subtree such that if is any chain of , then all the subsequences of are summable to the same limit. In case (b), provided that is the Cesàro method of summability and that for any chain of the sequence is weakly null, we prove the existence of a subtree such that for any chain of any spreading model for the sequence has a basis equivalent to the usual -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:
https://doi.org/10.1090/S0002-9939-2010-10479-3

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.