Regular Maximal Monotone Operators

Abstract

The purpose of this paper is to introduce a class of maximal monotone operators on Banach spaces that contains all maximal monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all maximal monotone operators that verify the simplest possible sum theorem. Dually strongly maximal monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then $$\overline {{\text{dom(}}T{\text{)}}} $$ (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of $$\overline {{\text{dom(}}T{\text{)}}} $$ at which T is locally bounded, and T is maximal monotone locally, as well as other results.