Borwein, “Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators

The concept of a monotone operator — which covers both linear positive semi-definite operators and subdifferentials of convex functions — is fundamental in various branches of mathematics. Over the last few decades, several stronger no-tions of monotonicity have been introduced: Gossez’s maxi-mal monotonicity of dense type, Fitzpatrick and Phelps’s lo-cal maximal monotonicity, and Simons’s monotonicity of type (NI). While these monotonicities are automatic for maximal monotone operators in reflexive Banach spaces and for sub-differentials of convex functions, their precise relationship is largely unknown. Here, it is shown — within the beautiful framework of Convex Analysis — that for continuous linear monotone op-erators, all these notions coincide and are equivalent to the monotonicity of the conjugate operator. This condition is fur