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Abstract

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.