This is an announcement for the paper "The reconstruction formula for Banach frames and duality" by Daniel Carando, Silvia Lassalle, and Pablo Schmidberg.
Abstract: We study conditions on a Banach frame that ensures the validity of a reconstruction formula. In particular, we show that any Banach frames for (a subspace of) $L_p$ or $L_{p,q}$ ($1\le p < \infty$) with respect to a solid sequence space always satisfies an unconditional reconstruction formula. The existence of reconstruction formulae allows us to prove some James-type results for atomic decompositions: an unconditional atomic decomposition (or unconditional Schauder frame) for $X$ is shrinking (respectively, boundedly complete) if and only if $X$ does not contain an isomorphic copy of $\ell_1$ (respectively, $c_0$).
Archive classification: math.FA math.CA
Mathematics Subject Classification: 41A65, 42C15, 46B10, 46B15
Remarks: 16 pages
Submitted from: slassall@dm.uba.ar
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1101.2430
or