
Journal of Convex Analysis 16 (2009), No. 2, 409421 Copyright Heldermann Verlag 2009 Maximal Monotone Operators with a Unique Extension to the Bidual Maicon Marques Alves Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460320, Brazil maicon@impa.br Benar Fux Svaiter Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460320, Brazil benar@impa.br [Abstractpdf] \newcommand{\tos}{\rightrightarrows} We study a sufficient condition under which a maximal monotone operator $T\colon X\tos X^*$ admits a unique maximal monotone extension to the bidual $\widetilde T\colon X^{**}\tos X^*$. We will prove that for nonlinear operators this condition is equivalent to uniqueness of the extension. The central tool in our approach is the $\mathcal{S}$function defined and studied previously by R. S. Burachik and B. F. Svaiter ["Maximal monotone operators, convex functions and a special family of enlargements", SetValued Analysis 10 (2002) 297316]. 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.\par 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. Keywords: Maximal monotone operators, extension, bidual, Banach spaces, BroendstedRockafellar property, Sfunction. MSC: 47H05, 49J52, 47N10 [ Fulltextpdf (137 KB)] for subscribers only. 