The reconstruction formula for Banach frames and duality

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) LpLp or Lp,qLp,q (1≤p<∞1≤p<∞) with respect to a solid sequence space always satisfies an unconditional reconstruction formula. The existence of reconstruction formulas allows us to prove some James-type results for atomic decompositions: an unconditional atomic decomposition (or unconditional Schauder frame) for XX is shrinking (respectively, boundedly complete) if and only if XX does not contain an isomorphic copy of ℓ1ℓ1 (respectively, c0c0).