Maximal monotone operators with a unique extension to the bidual

We present a new sufficient condition under which a maximal monotone operator T : X ⇉ X∗ admits a unique maximal monotone extension to the bidual T̃ : X∗∗ ⇉ X∗. For non-linear operators this condition is equivalent to uniqueness of the extension. The class of maximal monotone operators which satisfy this new condition includes class of Gossez type D maximal monotone operators, previously defined and studied by J.-P. Gossez, and all maximal monotone operators of this new class satisfies a restricted version of Brøndsted-Rockafellar condition. The central tool in our approach is the S-function defined and studied by Burachik and Svaiter in 2000 [4](submission date, July 2000). For a generic operator, this function is the supremum of all convex lower semicontinuous functions which are majorized by the duality product in the graph of the operator. We also prove in this work that if the graph of a maximal monotone operator is convex, then this graph is an affine linear subspace. 2000 Mathematics Subject Classification: 47H05, 49J52, 47N10.