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) L"p or L"p","q ([email protected]?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 X is shrinking (respectively, boundedly complete) if and only if X does not contain an isomorphic copy of @?"1 (respectively, c"0).

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