Maximal monotonicity criteria for the composition and the sum under weak interiority conditions

The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality (X × X*, X* × X).