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 $$ \sum_{\lambda \in \sigma_d (B)} \dist(\lambda, \overline{\num}(A))^p \leq \| B-A \|_{\mathcal{S}_p}^p$$ and apply it to recover and improve some Lieb-Thirring type inequalities for non-selfadjoint Jacobi and Schr\"odinger operators.