Selecting basic sequences in $\varphi $-stable Banach spaces

Tadeusz Figiel¹ ; Ryszard Frankiewicz¹ ; Ryszard A. Komorowski² ; Czesław Ryll-Nardzewski²

¹Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
²Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Studia Mathematica, Tome 159 (2003) no. 3, pp. 499-515

Résumé

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).

Détail

DOI : 10.4064/sm159-3-10

Keywords: paper make concept varphi stability banach spaces where varphi function banach space function varphi satisfy natural conditions saturated subspaces varphi stable lemma corollary varphi stable banach space easily construct basic sequences which have property varphi defined terms varphi theorem leads appropriate functions varphi results existence unconditional basic sequences special properties proofs known results particular get proof gowers dichotomy theorem which produces best unconditionality constant complex