On isomorphisms of $L^1$ spaces of analytic functions onto $l^1$

Miroslav Pavlovi{\cj}

Abstract: It is proved that an $L^1_{\f}$ space of analytic functions in the unit disc, with the weight $\f'(1-|z|)$, is isomorphic to the Lebesgue sequence space $l^1$ only if $\f$ is ``normal''. The converse is known from the papers of Shields and Williams [{\bf 13}] and Lindenstrauss and Pelczynski [{\bf 4}]. The key of our proof are three classical results: Paley's theorem on lacunary series, Pelczynski's theorem on complemented subspaces of $l^1$ and Lindenstrauss-Pelczynski's theorem on the equivalence of unconditional bases in $l^1$.

Classification (MSC2000): 46E15, 46B20

Full text of the article:

• Compressed PostScript file (48 kilobytes)
• PDF file (130 kilobytes)