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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 themoment of order $\gamma$ of negative eigenvalues of its symmetric part $H ={1/2} [A + A^*].$ As an application, we obtain Lieb-Thirring estimates for nonself-adjoint Schr\"odinger operators. In particular, we recover recent resultsby Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment ofresonances of Schr\"odinger self-adjoint operators.

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