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We present a new sufficient condition under which a maximal monotone operator$T:X\tos X^*$ admits a unique maximal monotone extension to the bidual$\widetilde T:X^{**} \rightrightarrows X^*$. For non-linear operators thiscondition is equivalent to uniqueness of the extension.  The class of maximal monotone operators which satisfy this new conditionincludes class of Gossez type D maximal monotone operators, previously definedand studied by J.-P. Gossez, and all maximal monotone operators of this newclass satisfies a restricted version of Brondsted-Rockafellar condition.  The central tool in our approach is the $\mathcal{S}$-function defined andstudied by Burachik and Svaiter in 2000 \cite{BuSvSet02}(submission date, July2000). For a generic operator, this function is the supremum of all convexlower semicontinuous functions which are majorized by the duality product inthe graph of the operator.  We also prove in this work that if the graph of a maximal monotone operatoris convex, then this graph is an affine linear subspace.

Maximal monotone operators with a unique extension to the bidual