This paper is devoted to study the \(A_ X\)-operator associated to a Killing vector field (resp. infinitesimal analytic transformation) on a complete but non-compact Riemannian (resp. Kaehler) manifold. Recall that in the compact case Kostant (resp. Lichnerowicz) proved that this operator belongs to the holonomy algebra at any point. We first consider simply-connected irreducible and non-compact manifolds where such a result does not hold and we give examples of all these cases, which essentially correspond to hyperkaehler structures. The second part of this paper is dedicated to give sufficient conditions in order for \(A_ X\) lying in the holonomy algebra, in real and complex cases, on complete but non-compact Riemannian manifolds. These conditions refer to vector fields whose norms are either pointwise or globally bounded.