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 notions of monotonicity have been introduced: Gossez's maximal monotonicity of dense type, Fitzpatrick and Phelps's local maximal monotonicity, and Simons's monotonicity of type (NI). While these monotonicities are automatic for maximal monotone operators in reflexive Banach spaces and for subdifferentials 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 operators, all these notions coincide and are equivalent to the monotonicity of the conjugate operator. This condition is further analyzed and illustrated by examples.