Journal of Convex Analysis 21 (2014), No. 2, 495--505
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 22460-320, Brazil
obueno@impa.br
Juan Enrique Martínez-Legaz
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 22460-320, Brazil
benar@impa.br
Fitzpatrick proved that maximal monotone operators in topological vector spaces are representable by lower semi-continuous convex functions. A monotone operator is representable if it can be represented by a lower-semicontinuous 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
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