On Two Properties of Enlargements of Maximal Monotone Operators

We give an answer to an open problem regarding the full enlargeability of a maximal monotone operator $S:X\rightrightarrows X^*$ by $S^{se}$, the smallest enlargement belonging to a certain class of enlargements associated to $S$. Moreover, we prove the weak$^*$ closedness of the set $S_{h_S}(\varepsilon_1,x)+T_{h_T} (\varepsilon_2,x)$ under a weak generalized interior regularity condition.