Notes on Lieb-Thirring type inequality for a complex perturbation of fractional Schr\"odinger operator

For $s\text{textgreater}0$, let $H\mathrm{\_}0=(-\mathrm{\Delta }{)}^s$ be the fractional Laplacian. In this paper, we obtainLieb-Thirring type inequalities for the fractional Schr"odinger operator defined as $H=H\mathrm{\_}0+V$,where $V \in L^p($\mathbb{R}$^d), p\ge 1, d\ge 1,$ is a complex-valued potential.Our methods are based on results of articles by Borichev-Golinskii-Kupin \cite{BoGoKu} and Hansmann \cite{Ha1}.