Selecting basic sequences in $\varphi $-stable Banach spaces
In this paper we make use of a new concept of $\varphi $-stability for Banach spaces, where $\varphi $ is a function. If a Banach space $X$ and the function $\varphi $ satisfy some natural conditions, then $X$ is saturated with subspaces that are $\varphi $-stable (cf. Lemma 2.1 and Corollary 7.8). In a $\varphi $-stable Banach space one can easily construct basic sequences which have a property $P(\varphi )$ defined in terms of $\varphi $ (cf. Theorem 4.5).
This leads us, for appropriate functions $\varphi $, 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).