We introduce the concept of ρ-semimonotone point-to-set operators in Hilbert spaces. This notion is symmetrical with respect to the graph of T, as is the case for monotonicity, but not for other related notions, like e.g. hypomonotonicity, of which our new class is a relaxation. We give a necessary condition for ρ-semimonotonicity of T in terms of Lispchitz continuity of [T + ρ-11]-1 and a sufficient condition related to expansivity of T. We also establish surjectivity results for maximal ρ-semimonotone operators.
hypomonotonicity; surjectivity; prox-regularity; semimonotonicity
