Maximal monotone operators with a unique extension to the bidual

Abstract

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 $\tilde{T}:X^{\ast \ast }\rightrightarrows X^{\ast }$. 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 Brondsted-Rockafellar condition. The central tool in our approach is the $\mathcal{S}$-function defined and studied by Burachik and Svaiter in 2000 \cite{BuSvSet02}(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.

Submitted for publication in JCA in March, 2 2008

References in corpus (3)

• Maximal monotonicity, conjugation and the duality product
• Monotone Linear Relations: Maximality and Fitzpatrick Functions
• Bronsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces

Cited by in corpus (7)

• Monotone Linear Relations: Maximality and Fitzpatrick Functions
• On Gossez type (D) maximal monotone operators
• New results on q-positivity
• Some results on the convexity of the closure of the domain of a maximally monotone operator
• Remarks on $p$-monotone operators
• Geometric properties of maximal monotone operators and convex functions which may represent them
• Construction of pathological maximally monotone operators on non-reflexive Banach spaces