[Paper Review] Maximal monotone operators with a unique extension to the bidual

M. Marques Alves, B. F. Svaiter|ArXiv.org|May 29, 2008

Optimization and Variational Analysis28 references17 citations

TL;DR

This paper introduces a new sufficient condition for a maximal monotone operator on a Banach space to have a unique maximal monotone extension to the bidual space, using the S-function as a central tool. The condition generalizes Gossez's type D condition and implies a restricted Brønsted-Rockafellar property, with the key result that such operators must have affine linear graphs if their S-function is finite-valued.

ABSTRACT

We present a new sufficient condition under which a maximal monotone operator $T:X os X^*$ admits a unique maximal monotone extension to the bidual $\widetilde T:X^{**} ightrightarrows X^*$. 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 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.

Motivation & Objective

• To identify a new sufficient condition under which a maximal monotone operator on a Banach space admits a unique maximal monotone extension to the bidual space.
• To generalize Gossez's type D condition, which ensures uniqueness of extension, by introducing a broader class of operators.
• To establish a connection between the finiteness of the S-function and the structure of the graph of the extension, particularly its convexity and linearity.
• To prove that operators satisfying the new condition satisfy a restricted version of the Brønsted-Rockafellar property.
• To show that if the graph of a maximal monotone operator is convex, then it must be an affine linear subspace.

Proposed method

• The S-function, defined as the supremum of all convex lower semicontinuous functions majorized by the duality product on the operator's graph, is used as the central analytical tool.
• The conjugate of the S-function, denoted $(\mathcal{S}_T)^*$, is analyzed to characterize the domain where the function is finite, linking it to the structure of the bidual extension.
• The proof uses the duality between the operator and its inverse via the map $\Lambda$, which preserves the duality product and allows transfer of properties to the dual space.
• A key step involves showing that if $(\mathcal{S}_T)^*$ is finite at a point, then a segment of points in the dual space lies in the domain of the extended operator's inverse, implying monotonicity and convexity of the extended graph.
• The authors use the fact that a maximal monotone and convex set in a product space must be affine linear, leading to a contradiction if the original operator is not affine.
• The proof of uniqueness relies on showing that the extended operator $\widetilde{T}$ is the only maximal monotone extension by verifying that its inverse satisfies the required inequalities and is maximal.

Experimental results

Research questions

• RQ1 Under what conditions does a maximal monotone operator on a Banach space admit a unique maximal monotone extension to the bidual space?
• RQ2 How does the finiteness of the S-function relate to the structure of the graph of the extended operator?
• RQ3 Can the new condition for uniqueness of extension be shown to imply a restricted version of the Brønsted-Rockafellar property?
• RQ4 Is there a structural characterization of the graph of the extension when the S-function is finite-valued?
• RQ5 What is the relationship between the S-function and the duality product in the context of bidual extensions?

Key findings

• The paper establishes a new sufficient condition for unique maximal monotone extension to the bidual, which generalizes Gossez's type D condition.
• All maximal monotone operators satisfying the new condition satisfy a restricted version of the Brønsted-Rockafellar property.
• If the S-function is finite-valued at a point in $X^* \times X^{**}$, then a segment of points in the dual space lies in the domain of the extended operator's inverse.
• The graph of the maximal monotone extension $\widetilde{T}$ is shown to be convex and maximal monotone, hence affine linear, under the new condition.
• The paper proves that if the graph of a maximal monotone operator is convex, then it must be an affine linear subspace, which is a structural result in its own right.
• The unique extension $\widetilde{T}$ is characterized via the conjugate of the S-function, with $\mathcal{S}_T^*(x^*,x^{**}) \geq \langle x^*, x^{**} \rangle$ for all $(x^*,x^{**}) \in X^* \times X^{**}$.