On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsRn:X→X such thatRnRm=Rmin(n,m) ifn≠m and limn→∞Rnx=x for allx∈X. We prove that, ifRn−Rn−1 factors uniformly through somelp and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatLΛ=closed span\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ {z^k :k \in \Lambda } \right\} \subset L_1 \left( \mathbb{T} \right)$$ \end{document}, where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{T} = \left\{ {z \in \mathbb{C}:\left| z \right| = 1} \right\}$$ \end{document}, has an unconditional basis. Examples include the Hardy space\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$H_1 = L_{\mathbb{Z}_ + } $$ \end{document}.