Projections onto translation-invariant subspaces of 𝐻¹(𝑅)

Recently I. Klemes has characterized the complemented translation-invariant subspaces of H 1 ( T ) {H^1}(\mathbb {T}) . In this paper we investigate the case of H 1 ( R ) {H^1}(\mathbb {R}) . The main results are that the hull of a complemented translation-invariant subspace is ε \varepsilon -separated for some ε > 0 \varepsilon > 0 , and that an ε \varepsilon -separated subset of R + {\mathbb {R}^ + } which is in the ring generated by cosets of closed subgroups of R \mathbb {R} (intersected with R + {\mathbb {R}^ + } ) and lacunary sequences is the hull of a complemented ideal.