Clausuras de operadores monótonos en espacios vectoriales topológicos

Abstract

Fitzpatrick en [1] demostró que todo operador monótono maximal es un operador re- presentable en espacios vectoriales topológicos. Por lema de Zorn, todo operador monótono tiene extensión monótona maximal; por lo tanto, todo operador monótono posee extensión representable y la menor de todas las extensiones representables es llamada la clausura representable. Por otro lado, la clausura polar monótona de un operador monótono, que la podemos ver como la intersección de todas las extensiones monótonas maximales, tiene la propiedad de ser representable y además contiene a la clausura representable. El objetivo de la tesis es saber cuándo estas dos clausuras son iguales en espacios vectoriales topológicos. Además, demostrar que estas dos clausuras son iguales, sin ninguna hipótesis, en espacios de dimensión finita.

A monotone operator is representable if it can be represented by a lower semi-continuous convex function. It is difficult by definition to know if a monotone operator is representable. Fitzpatrick in [1] proved that every maximal monotone operator is representable in topological vector spaces. By Zorn’s lemma, every monotone operator has a maximal monotone extension. So that, every monotone operator has a representable extension and the smallest representable extension is called representable closure. Then a way to know if a monotone operator is representable is studying its representable closure but its geometry is difficult to know. On the other hand, the intersection of all maximal monotone extensions is called the monotone polar closure that is representable, this definition has a geometric interpretation. Martínez-Legaz and Svaiter in [2] prove that these two clousures are equals in finite dimension spaces. A natural question is to know if these closures are equals in infinite dimension spaces. Simons presents an example in [3] where these two closures are not equals. The goal of the thesis is to study the properties of these two closures and to know when they are equals in topological vector spaces. Furthermore, to give another proof of the equality of these closures in finite dimension spaces.