Lieb-Thirring estimates for non self-adjoint Schr\'odinger operators

For general non-symmetric operators $A$, we prove that the moment of order $\gamma \ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $\gamma$ of negative eigenvalues of its symmetric part $H = {1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schr\"odinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schr\"odinger self-adjoint operators.