Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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'Història Econòmica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
and: MOVE (Markets, Organizations and Votes in Economics)

Benar F. Svaiter

Instituto de Matématica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil

benar@impa.br

Keywords

  • Monotone operator
  • representable operator
  • monotone polar
  • closure
  • topological vector space

Mathematics Subject Classification

46A99, 47H05, 47N10

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Suggested citation

O. Bueno, J. E. Martínez-Legaz, B. F. Svaiter. “On the Monotone Polar and Representable Closures of Monotone Operators.” Journal of Convex Analysis 21 (2014), No. 2, 495–505. https://doi.org/10.68381/jca21026

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.