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Journal of Convex Analysis 08 (2001), No. 1, 223--240
Copyright Heldermann Verlag 2001

Self Dual Operators on Convex Functionals; Geometric Mean and Square Root of Convex Functionals

Marc Atteia
IREM, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse, France

Mustapha Raïssouli
Dép. de Mathématiques et Informatique, Faculté des Sciences, Université Moulay Ismail, 4010 Zitoune Meknès, Marocco

Let Conv(X) be the set of the convex functionals defined on a linear space X, with values in the union of R and the point of positive infinity. We give an extension of the notion of duality for (convex) functionals to mappings which operate from Conv(X) × Conv(X) into Conv(X). Afterwards, we present an algorithm which associates, under convenient assumptions, a self-dual operator to a given operator and its dual. Finally, we give some examples which prove the generality and interest of our approach.

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