On Non-Enlargeable and Fully Enlargeable Monotone Operators

\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 point-to-set 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 point-to-point affine operators with skew symmetric linear part; those not enlarged by $T^{se}$ are the point-to-point 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 $T$.