An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schr\"odinger operators

For bounded linear operators $A,B$ on a Hilbert space $\mathcal{H}$ we show the validity of the estimate $$\sum _{\lambda \in {\sigma }_d(B)}\text{dist}(\lambda ,\overline{\text{num}}(A){)}^p\le \mathrm{\parallel }B-A{\mathrm{\parallel }}_{{\mathcal{S}}_p}^p$$ and apply it to recover and improve some Lieb-Thirring type inequalities for non-selfadjoint Jacobi and Schr"odinger operators.