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Register a new userA new old class of maximal monotone operators
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In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatricks family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brondsted-Rockafellar property. In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons. We will also prove that an auxiliary condition used by the authors to prove the restricted Brondsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.
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We present a new sufficient condition under which a maximal monotone operator $T:Xtos X^*$ admits a unique maximal monotone extension to the bidual $widetilde T:X^{**} rightrightarrows X^*$. 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.
We are concerned with surjectivity of perturbations of maximal monotone operators in non-reflexive Banach spaces. While in a reflexive setting, a classical surjectivity result due to Rockafellar gives a necessary and sufficient condition to maximal monotonicity, in a non-reflexive space we characterize maximality using a ``enlarged version of the duality mapping, introduced previously by Gossez.
We study the relations between some geometric properties of maximal monotone operators and generic geometric and analytical properties of the functions on the associate Fitzpatrick family of convex representations. We also investigate under which conditions a convex function represents a maximal monotone operator with bounded range and provide an example of a non type (D) operator on this class.
We observe that if f is a continuous function on an interval I and x_0 in I, then f is operator monotone if and only if the function (f(x) - f(x_0)/(x - x_0) is strongly operator convex. Then starting with an operator monotone function f_0, we construct a strongly operator convex function f_1, an (ordinary) operator convex function f_2, and then a new operator monotone function f_3. The process can be continued to obtain an infinite sequence which cycles between the three classes of functions. We also describe two other constructions, similar in spirit. We prove two lemmas which enable a treatment of those aspects of strong operator convexity needed for this paper which is more elementary than previous treatments. And we discuss the functions phi such that the composite phi circ f is operator convex or strongly operator convex whenever f is strongly operator convex.
Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation.
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