Geodesic
On the Monotone Polar and Representable Closures of Monotone Operators
Journal of convex analysis, Tome 21 (2014) no. 2, pp. 495-505

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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.
Classification : 46A99, 47H05, 47N10
Mots-clés : Monotone operator, representable operator, monotone polar, closure, topological vector space 
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     title = {On the {Monotone} {Polar} and {Representable} {Closures} of {Monotone} {Operators}},
     journal = {Journal of convex analysis},
     pages = {495--505},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2014},
     url = {http://geodesic.mathdoc.fr/item/JCA_2014_21_2_JCA_2014_21_2_a9/}
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O. Bueno; J. E. Martínez-Legaz; B. F. Svaiter. On the Monotone Polar and Representable Closures of Monotone Operators. Journal of convex analysis, Tome 21 (2014) no. 2, pp. 495-505. http://geodesic.mathdoc.fr/item/JCA_2014_21_2_JCA_2014_21_2_a9/