Monotone Operators without Enlargements

TL;DR

This paper explores conditions under which monotone operators are non-enlargeable in nonreflexive Banach spaces, providing new characterizations and proofs related to maximal monotonicity and operator sums.

Contribution

It offers a new characterization of non-enlargeable operators in nonreflexive Banach spaces with a closedness assumption and provides novel proofs for the maximality of sums of certain monotone operators.

Findings

Characterization of non-enlargeable operators in nonreflexive Banach spaces.

New proof of maximality of the sum of two maximally monotone linear relations.

New proof of maximality of the sum involving a linear relation and a normal cone operator.

Abstract

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone.