Contractive and optimal sets in modular spaces

Abstract

We study contractive, existence and optimal sets in modular spaces. We define the analogous classes of sets with respect to a modular and generalize a number of results due to Beauzamy and Maurey and Brack to modular spaces. We then apply them to show that in certain Köthe sequence spaces all of these notions are equivalent, extending a result of Davis and Enflo for lp spaces, 1 < p < ∞. We also show that in class of Orlicz sequence spaces the theorem that any optimal set in lp spaces, 1 < p < ∞, is an intersection of half-spaces determined by one-complemented hyperplanes in lp, is no longer true. We then find a class of optimal sets, called strongly p-optimal, such that this characterization holds true hi Musielak-Orlicz sequence spaces. We provide examples which show that all Families of sets discussed in the paper like optimal, existence or contractive with respect to a norm or a modular or their strong counterparts, are not equivalent in general. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim.

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Mathematische Nachrichten

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