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

Let Conv(X) be the set of the convex functionals defined on a linear space X, with values in $\mathbb{R}\cup\{+\infty\}$. 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