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Let Z 0 be a bounded operator in a Banach space X with purely essential spectrum and K a nuclear operator in X. Using methods of complex analysis we study the discrete spectrum of Z 0 + K and derive a Lieb-Thirring type inequality. We obtain estimates for the number of eigenvalues in certain regions of the complex plane and an estimate for the asymptotics of the eigenvalues approaching to the essential spectrum of Z 0 .Mathematics Subject Classifictaion (2010): 47A75, 47A10, 47A55, 47B10.

Abstract. Let Z 0 be a bounded operator in a Banach space X with purely essential spectrum and K a nuclear operator in X . Using methods of complex analysis we study the set of accumulation points of the discrete spectrum of the operator Z := Z 0 + K . We formulate conditions for Z to exclude certain points or subsetes of the essential spectrum of Z to be accumulation points of the discrete spectrum. These results are applied to the operator of multiplication perturbed by integral operators with continuous kernel and to the discrete Laplacian perturbed by nuclear Jacobi operators.Mathematics subject classification (2010): 26D15, 26A51, 32F99, 41A17.

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