Some remarks on spreading models and mixed Tsirelson spaces

Manoussakis, A.

doi:10.1090/S0002-9939-02-06832-6

Some remarks on spreading models and mixed Tsirelson spaces
HTML articles powered by AMS MathViewer

by A. Manoussakis PDF

Proc. Amer. Math. Soc. 131 (2003), 2515-2525 Request permission

Abstract:

We prove that if a Banach space with a bimonotone shrinking basis does not contain $\ell _{1}^{\omega }$ spreading models but every block sequence of the basis contains a further block sequence which is a $c-\ell _{1}^{n}$ spreading model for every $n\in \mathbb {N}$, then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space $T[(\mathcal {S}_{n},\theta _{n})_{n}]$, such that $\theta _{n}\searrow 0$, does not contain $\ell _{1}^{\omega 2}$ spreading models.

References

Dale E. Alspach and Spiros Argyros, Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.) 321 (1992), 44. MR 1191024
G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125–149. MR 1679593, DOI 10.1007/BF02775031
S. A. Argyros and I. Deliyanni, Examples of asymptotic $l_1$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 973–995. MR 1390965, DOI 10.1090/S0002-9947-97-01774-1
S. A. Argyros, I. Deliyanni, D. N. Kutzarova, and A. Manoussakis, Modified mixed Tsirelson spaces, J. Funct. Anal. 159 (1998), no. 1, 43–109. MR 1654174, DOI 10.1006/jfan.1998.3310
S.A. Argyros, I. Deliyanni, A. Manoussakis, Distortion and Spreading models in Modified mixed Tsirelson spaces, preprint.
S. A. Argyros and I. Gasparis, Unconditional structures of weakly null sequences, Trans. Amer. Math. Soc. 353 (2001), no. 5, 2019–2058. MR 1813606, DOI 10.1090/S0002-9947-01-02711-8
S. A. Argyros, S. Mercourakis, and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157–193. MR 1658551, DOI 10.1007/BF02764008
S.A. Argyros, A. Tolias, Methods in the Theory of Hereditarily Indecomposable Banach Spaces, preprint.
I. Gasparis, A Continuum of totally incomparable Hereditarily Indecomposable Banach spaces, to appear in Studia Math.
I. Gasparis, Strictly Singular non-compact operators on Hereditarily Indecomposable Banach spaces, to appear in Proc. Amer. Math. Soc.
Denka Kutzarova and Pei-Kee Lin, Remarks about Schlumprecht space, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2059–2068. MR 1654081, DOI 10.1090/S0002-9939-99-05248-X
A. Manoussakis, On the structure of a certain class of mixed Tsirelson spaces, Positivity 5 (2001), no. 3, 193–238. MR 1836747, DOI 10.1023/A:1011456204116
E. Odell and Th. Schlumprecht, On the richness of the set of $p$’s in Krivine’s theorem, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 177–198. MR 1353459
E. Odell and Th. Schlumprecht, A problem on spreading models, J. Funct. Anal. 153 (1998), no. 2, 249–261. MR 1614578, DOI 10.1006/jfan.1997.3191
E.Odell, N. Tomczak-Jaegermann, R.Wagner, Proximity to $\ell _{1}$ and distortion in asymptotic $\ell _{1}$ spaces, Journal of Funct. Analysis 150 (1997), 101-145.
Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. MR 1177333, DOI 10.1007/BF02782845
B.Tsirelson, Not every Banach space contains $\ell _{p}$ or $c_0$, Funct. Anal. Appl. 8 (1974), 138-141.