The reconstruction formula for Banach frames and duality

TL;DR

This paper establishes conditions under which Banach frames allow for unconditional reconstruction formulas, and explores their implications for atomic decompositions and classical Banach space properties.

Contribution

It proves that Banach frames for subspaces of $L_p$ or $L_{p,q}$ always satisfy an unconditional reconstruction formula, leading to new James-type results.

Findings

Banach frames for certain spaces satisfy unconditional reconstruction formulas

Unconditional atomic decompositions are shrinking iff they lack an $ ext{ell}_1$ copy

Unconditional Schauder frames are boundedly complete iff they lack a $c_0$ copy

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, $c_0$).