Regular methods of summability on treesequences in Banach spaces
Author:
Costas Poulios
Journal:
Proc. Amer. Math. Soc. 139 (2011), 259271
MSC (2010):
Primary 40C05, 46B99; Secondary 05D10, 05C55
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 treesequences. This also allows us to improve the result of case (b).
 1.
Bernard
Beauzamy, BanachSaks properties and spreading models, Math.
Scand. 44 (1979), no. 2, 357–384. MR 555227
(81a:46018)
 2.
Antoine
Brunel and Louis
Sucheston, On 𝐵convex Banach spaces, Math. Systems
Theory 7 (1974), no. 4, 294–299. MR 0438085
(55 #11004)
 3.
Nelson
Dunford and Jacob
T. Schwartz, Linear Operators. I. General Theory, With the
assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics,
Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers,
Ltd., London, 1958. MR 0117523
(22 #8302)
 4.
P.
Erdős and M.
Magidor, A note on regular methods of
summability and the BanachSaks property, Proc.
Amer. Math. Soc. 59 (1976), no. 2, 232–234. MR 0430596
(55 #3601), 10.1090/S00029939197604305961
 5.
Fred
Galvin and Karel
Prikry, Borel sets and Ramsey’s theorem, J. Symbolic
Logic 38 (1973), 193–198. MR 0337630
(49 #2399)
 6.
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, SpringerVerlag, New York, 1995. MR 1321597
(96e:03057)
 7.
H. P. Rosenthal, Weakly independent sequences and the BanchSaks property, in Proceedings of the Durham Symposium on the relations between infinite and finite dimensional convexity. Durham, July 1975, pp. 2224.
 8.
Jacques
Stern, A Ramsey theorem for trees, with an application to Banach
spaces, Israel J. Math. 29 (1978), no. 23,
179–188. MR 0476554
(57 #16114)
 9.
A.
Tsarpalias, A note on the Ramsey
property, Proc. Amer. Math. Soc.
127 (1999), no. 2,
583–587. MR 1458267
(99c:04005), 10.1090/S0002993999045189
 1.
 B. Beauzamy, BanachSaks properties and spreading models, Math. Scand. 44 (1979), 357384. MR 555227 (81a:46018)
 2.
 A. Brunel and L. Sucheston, On Bconvex Banach spaces, Math. System Theory 7 (1974), 294299. MR 0438085 (55:11004)
 3.
 N. Dunford and J.T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 0117523 (22:8302)
 4.
 P. Erdös and M. Magidor, A note on regular methods of summability and the BanachSaks property, Proc. Amer. Math. Soc. 59 (1976), 232234. MR 0430596 (55:3601)
 5.
 F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, J. Symbolic Logic 38 (1973), 193198. MR 0337630 (49:2399)
 6.
 A. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995. MR 1321597 (96e:03057)
 7.
 H. P. Rosenthal, Weakly independent sequences and the BanchSaks property, in Proceedings of the Durham Symposium on the relations between infinite and finite dimensional convexity. Durham, July 1975, pp. 2224.
 8.
 J. Stern, A Ramsey theorem for trees with an application to Banach spaces, Israel J. Math. 29 (1978), 179188. MR 0476554 (57:16114)
 9.
 A. Tsarpalias, A note on the Ramsey property, Proc. Amer. Math. Soc. 127 (1999), 583587. MR 1458267 (99c:04005)
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Additional Information
Costas Poulios
Affiliation:
Department of Mathematics, University of Athens, 15784, Athens, Greece
Email:
kpoulios@math.uoa.gr
DOI:
http://dx.doi.org/10.1090/S000299392010104793
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.
