Absolutely summing convolution operators in L p ( G )

Let G be an infinite, compact, abelian group. We investigate the Banach space∑ 1 ( p )consisting of all absolutely summing convolution operators from L p ( G ) to itself. For 1⩽ p ⩽2 these spaces were identified by Beauzamy (and in C ( G ) by Lust‐Piquard). So, we concentrate on 2 < p ⩽ ∞. More tractable than∑ 1 ( p )is the subspace∑ 1 ( 1 , p )consisting of those operators from∑ 1 ( p )which have an absolutely summing L p ( G )‐valued extension to L 1 ( G ), the reason being the availability of a factorization theorem (via L 2 ( G )) for operators from∑ 1 ( 1 , p ). The Banach space S p consisting of all f ∈ L p ( G ) with an unconditionally convergent Fourier series, introduced by Bachelis, plays a crucial role. Whereas both∑ 1 ( 1 , p )and S p turn out to be reflexive Banach lattices (with an unconditional basis if G is metrizable), this is not so for∑ 1 ( p ). It is shown that Σ 1 1 , p ⫋ S p ⫋ L p . Moreover,∑ 1 ( 1 , p )has cotype 2 whereas S p has cotype p but fails to be r ‐concave for every 1 ⩽ r < p . We also establish that∑ 1 ( 1 , p ) ⫋ ∑ 1 ( p ) ⫋ L pand∑ 1 ( p ) ⊊ S p .