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Let $ H_0=-\dd+V_0 $ be a multidimensional Schr\"odinger ope\-rator with areal-valued potential and infinite band spectrum, and $H=H_0+V$ be itsnon-selfadjoint perturbation defined with the help of Kato approach. We proveLieb--Thirring type inequalities for the discrete spectrum of $H$ in the casewhen $V_0\in L^\infty(\br^d)$ and $V\in L^p(\br^d)$, $p>\max(d/2, 1)$.

On non-selfadjoint perturbations of infinite band Schr\"odinger operators and Kato method