Abstract

We study a sufficient condition under which a maximal monotone operator T ⁣:X⇉X∗T\colon X\tos X^* admits a unique maximal monotone extension to the bidual T~ ⁣:X∗∗⇉X∗\widetilde T\colon X^{**}\tos X^*. We will prove that for non-linear operators this condition is equivalent to uniqueness of the extension. The central tool in our approach is the S\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", Set-Valued Analysis 10 (2002) 297–316]. 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Maicon Marques Alves

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

maicon@impa.br

Benar Fux Svaiter

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

benar@impa.br

Keywords

  • Maximal monotone operators
  • extension
  • bidual
  • Banach spaces
  • Brøndsted-Rockafellar property
  • S-function

Mathematics Subject Classification

47H05, 49J52, 47N10

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Suggested citation

M. Marques Alves, B. F. Svaiter. “Maximal Monotone Operators with a Unique Extension to the Bidual.” Journal of Convex Analysis 16 (2009), No. 2, 409–421. https://doi.org/10.68381/jca16022

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.