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Absolutely summing convolution operators in L p ( G )
Author(s) -
Okada S.,
Ricker W. J.,
RodríguezPiazza L.
Publication year - 2011
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdq042
Subject(s) - mathematics , convolution (computer science) , combinatorics , pure mathematics , computer science , artificial neural network , machine learning
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 .