On the Maximal Extensions of Monotone Operators and Criteria for Maximality A. Eberhard * and R. Wenczel October 21, 2013 Abstract Within a nonzero, real Banach space we study the problem of characterising a maximal extension of a monotone operator in terms of minimality properties of representative functions that are bounded by the Penot and Fitzpatrick functions. We single out a property of this space of representative functions that enable a very compact treatment of maximality and pre-maximality issues. This Paper is Dedicated to the memory of Charles Pearce. Introduction Monotone operators have found wide application in areas of optimization, control theory and the theory of partial differential equations. Many applications require maximality of a sum of monotone operators. Thus early on in the development of this theory the issue of when a sum of monotone operators is maximal was asked and was eventually resolved for reflexive Banach spaces by Rockafellar [14]. In this paper we single out a fundamental property of the space of representative functions that enables such a theorem to be efficiently derived. This leads to a particularly compact way of discussing issues of maximality and pre-maximality, at least in reflexive spaces. Suppose X is a Banach space and X * its dual. We may view X× X * paired with X * × X using the coupling 〈(z,z * ), (x * ,x)〉 = 〈z,x * 〉 + 〈x, z * 〉 and the norm ‖(z,z * )‖ 2 = ‖z‖ 2 + ‖z * ‖ 2 . Denote by by PC (X × X * ) the set of all proper convex functions f : X × X * → R + := (−∞, +∞]. When we pair with the topologies s(X) × σ (X * ,X) (product of the strong and weak * ) with σ (X * ,X) × s(X) (product of the weak * and strong) the associated conjugation operation of a proper closed convex function f ∈ Γ s×w * (X × X * ) is denoted by f * ∈ Γ w * ×s (X * × X) and given by f * (z * ,z) := sup (x,x * ) {〈(x, x * ) , (z * ,z)〉− f (x, x * )} . (1) Definition 1 We call a proper convex function H T ∈ PC (X × X * ) a representative function of a monotone mapping T on X when H T (y,y * ) ≥〈y,y * 〉 for all (y,y * ) ∈ X × X * with H T (y,y * )= 〈y,y * 〉 if y * ∈ T (y). If T is not specified we say a proper convex function f is representative when f (y,y * ) ≥〈y,y * 〉 for all (y,y * ) ∈ X × X * and then say that it represents (the monotone set) M f := {(x, x * ) ∈ X × X * | f (x, x * )= 〈x, x * 〉} . Define the ‘transpose’ operator †:(x * ,x) ↔ (x, x * ), and c T (·, ·) := δ T (·, ·)+ 〈·, ·〉, where δ T denotes the indicator function of the graph of T . Fitzpatrick [9] showed that F T := c *† T and P T := F *† T induce functions that represent T , when T is maximal monotone. In [6] it is shown that P T is representative when T is just monotone, while F T may fail to be so when T is not maximal. * The author’s research was funded by ARC grant DP120100567. 2000 Mathematics Subject Classification: 47H05, 46N10, 47H04, 49J53. Keywords and Phrases: sum theorems, maximal extensions, monotone operators, representative functions. 1