In this paper, we define trace-like operators on a subspace of the space of derivations of the free Lie algebra generated by the first homology group H of a surface Σ. This definition depends on the choice of a Lagrangian of H, and we call these operators the Lagrangian traces. We suppose that Σ is the boundary of a handlebody with first homology group , and we show that, in degree greater than 2, the Lagrangian traces corresponding to the Lagrangian vanish on the image by the Johnson homomorphisms of the elements of the Johnson filtration that extend to the handlebody.