
Journal of Convex Analysis 21 (2014), No. 2, 495505 Copyright Heldermann Verlag 2014 On the Monotone Polar and Representable Closures of Monotone Operators Orestes Bueno Instituto de Matématica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460320, Brazil obueno@impa.br Juan Enrique MartínezLegaz Dep. d'Economia i d'Histňria Econňmica, Universitat Autňnoma de Barcelona, 08193 Bellaterra, Spain Benar F. Svaiter Instituto de Matématica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460320, Brazil benar@impa.br Fitzpatrick proved that maximal monotone operators in topological vector spaces are representable by lower semicontinuous convex functions. A monotone operator is representable if it can be represented by a lowersemicontinuous convex function. The smallest representable extension of a monotone operator is its representable closure. The intersection of all maximal monotone extensions of a monotone operator, its monotone polar closure, is also representable. A natural question is whether these two closures coincide. In finite dimensional spaces they do coincide. The aim of this paper is to analyze such a question in the context of topological vector spaces. In particular, we prove in this context that if the convex hull of a monotone operator is not monotone, then the representable closure and the monotone polar closure of such operator do coincide. Keywords: Monotone operator, representable operator, monotone polar, closure, topological vector space. MSC: 46A99, 47H05, 47N10 [ Fulltextpdf (127 KB)] for subscribers only. 