  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.
