Maximally monotone operators with ranges whose closures are not convex and an answer to a recent question by Stephen Simons

Abstract

In his recent Proceedings of the AMS paper, “Gossez’s skew linear map and its pathological maximally monotone multifunctions”, Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is not convex whenever the scalar is between 0 0 and 4 4 . The problem of the convexity of that range when the scalar is equal to 4 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4 4 . We derive this result from an abstract framework that allows us to also obtain a corresponding result for the Fitzpatrick-Phelps integral operator.