On the Regularity of Maximal Monotone Operators and Related Results

TL;DR

This paper investigates the conditions under which the domain and range closures of maximal monotone operators are convex, characterizes regularity of such operators, and improves existing sum theorem results.

Contribution

It establishes the necessity of Simons' convexity condition for bounded maximal monotone operators and provides new characterizations of regularity, including for operators of type (FPV).

Findings

Simons' condition is necessary and sufficient for convexity of domain/range closures in bounded maximal monotone operators.

Provides several new characterizations for the regularity of maximal monotone operators.

Shows that maximal monotone operators of type (FPV) are regular and improves sum theorem results.

Abstract

In the first part of the note we prove that a sufficient condition (due to Simons) for the convexity of the closure of the domain/range of a monotone operator is also necessary when the operator has bounded domain and is maximal. Simons' condition is closely related to the notion of regular maximal monotone operator. In the second part of the note we give several characterizations for the regularity of a maximal monotone operator, show that a maximal monotone operator of type (FPV) is regular and improve a previous sum theorem type result.