In this paper we make use of a new concept of φ-stability for Banach spaces, where φ is a function. If a Banach space X and the function φ satisfy some natural conditions, then X is saturated with subspaces that are φ-stable (cf. Lemma 2.1 and Corollary 7.8). In a φ-stable Banach space one can easily construct basic sequences which have a property P(φ) defined in terms of φ (cf. Theorem 4.5). This leads us, for appropriate functions φ, to new results on the existence of unconditional basic sequences with some special properties as well as new proofs of some known results. In particular, we get a new proof of the Gowers dichotomy theorem which produces the best unconditionality constant (also in the complex case).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-10,
author = {Tadeusz Figiel and Ryszard Frankiewicz and Ryszard A. Komorowski and Czes\l aw Ryll-Nardzewski},
title = {Selecting basic sequences in $\phi$-stable Banach spaces},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {499-515},
zbl = {1065.46012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-10}
}
Tadeusz Figiel; Ryszard Frankiewicz; Ryszard A. Komorowski; Czesław Ryll-Nardzewski. Selecting basic sequences in φ-stable Banach spaces. Studia Mathematica, Tome 157 (2003) pp. 499-515. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-10/