
Journal of Convex Analysis 13 (2006), No. 3, 603622 Copyright Heldermann Verlag 2006 On NonEnlargeable and Fully Enlargeable Monotone Operators Regina Sandra Burachik regi@cos.ufrj.br Alfredo Noel Iusem Inst. Matématica Pura e Aplicada, Estrada Dońa Castorina 110, Rio de Janeiro, CEP 22460320, Brazil iusp@impa.br [Abstractpdf] \def\ve{\varepsilon} We consider a family of enlargements of maximal monotone operators in a reflexive Banach space. Each enlargement, depending on a parameter $\ve\ge 0$, is a continuous pointtoset mapping $E(\ve,x)$ whose graph contains the graph of the given operator $T$. The enlargements are also continuous in $\ve$, and they coincide with $T$ for $\ve=0$. The family contains both a maximal and a minimal enlargement, denoted as $T^e$ and $T^{se}$ respectively. We address the following questions: \newline a) which are the operators which are not enlarged by $T^e$, i.e., such that $T(\cdot)=T^e(\ve,\cdot)$ for some $\ve>0$? \newline b) same as (a) but for $T^{se}$ instead of $T^e$. \newline c) Which operators are fully enlargeable by $T^e$, in the sense that for all $x$ and all $\ve>0$ there exists $\delta>0$ such that all points whose distance to $T(x)$ is less than $\delta$ belong to $T^e(\ve,x)$? \newline We prove that the operators not enlarged by $T^e$ are precisely the pointtopoint affine operators with skew symmetric linear part; those not enlarged by $T^{se}$ are the pointtopoint and affine operators, and the operators fully enlarged by $T^e$ are those operators $T$ whose Fitzpatrick function is continuous in its second argument at pairs belonging to the graph of $T$. Keywords: Maximal monotone operators, enlargements. MSC: 46N10, 47H05 [ Fulltextpdf (558 KB)] for subscribers only. 