Closedness under addition for families of quasimonotone operators

In the last two decades, several properties of operators that are weaker than monotonicity have received attention by researchers from many areas including mathematical economics, with the goal to develop new tools applicable in convex analysis and related topics. This paper puts in perspective notions that are extensions of monotoniticity but not beyond quasimonotonicity like pseudomonotonicity, semistrict quasimonotonicity, strict quasimonotonicity and proper quasimonotonicity, and discusses systematically when the sum of two operators satisfying one of those properties, inherits the same property. The case of properly quasimonotone operators deserves a special attention since this notion, being stronger than quasimonotonicity, suffices to obtain many results, including the solvability of variational inequality problems. Several examples showing the optimality in some sense of our results, are presented.