Three-space problems and bounded approximation properties

Wolfgang Lusky¹

¹ Institute for Mathematics University of Paderborn Warburger Str. 100 D-33098 Paderborn, Germany

Studia Mathematica, Tome 159 (2003) no. 3, pp. 417-434

Résumé

Let $ \{ R_n \}_{n=1}^{ \infty} $ be a commuting approximating sequence of the Banach space $X$ leaving the closed subspace $A \subset X$ invariant. Then we prove three-space results of the following kind: If the operators $R_n$ induce basis projections on $X/A$, and $X$ or $A$ is an ${\cal L}_p$-space, then both $X $ and $A $ have bases. We apply these results to show that the spaces $C_{ {\mit\Lambda}} = \overline{ \hbox{span}} \{ z^k : k \in {\mit\Lambda} \} \subset C( \mathbb T)$ and $L_{ {\mit\Lambda}} = \overline{ \hbox{span}} \{ z^k : k \in {\mit\Lambda} \} \subset L_1( \mathbb T)$ have bases whenever $ {\mit\Lambda} \subset \mathbb Z$ and $ \mathbb Z \setminus {\mit\Lambda}$ is a Sidon set.

Détail

DOI : 10.4064/sm159-3-6

Keywords: infty commuting approximating sequence banach space leaving closed subspace subset invariant prove three space results following kind operators induce basis projections cal p space have bases apply these results spaces mit lambda overline hbox span mit lambda subset mathbb mit lambda overline hbox span mit lambda subset mathbb have bases whenever mit lambda subset mathbb mathbb setminus mit lambda sidon set