The Baillon-Haddad Theorem Revisited

J.-B. Baillon and G. Haddad ["Quelque propriétés des opérateurs angle-bornés et n-cycliquement monotones", Israel J. Math. 26 (1977) 137–150] proved that if the gradient of a convex and continuously differentiable function is nonexpansive, then it is actually firmly nonexpansive. This result, which has become known as the Baillon-Haddad theorem, has found many applications in optimization and numerical functional analysis. In this note, we propose short alternative proofs of this result and strengthen its conclusion.