Letters in Mathematical Physics

, Volume 98, Issue 1, pp 79–95 | Cite as

An Eigenvalue Estimate and Its Application to Non-Selfadjoint Jacobi and Schrödinger Operators



For bounded linear operators A, B on a Hilbert space \({\mathcal{H}}\) we show that \({ \sum_{\lambda}{\rm dist}(\lambda, {\rm Num}(A))^p}\) is bounded from above by the Schatten-p-norm of BA. Here, the sum is taken over all discrete eigenvalues of B and Num(A) denotes the numerical range of A. We apply this estimate to recover and improve some Lieb–Thirring type inequalities for non-selfadjoint Jacobi and Schrödinger operators.

Mathematics Subject Classification (2000)

47A75 47A12 47B10 35J10 47B36 


Eigenvalue estimates numerical range Lieb–Thirring inequalities Schrödinger operators complex-valued potentials Jacobi operators 


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© Springer 2011

Authors and Affiliations

  1. 1.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany

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