16 1. GENERALIZED ARGUMENT PRINCIPLE AND ROUCH´ THEOREM 1.4. Concluding Remarks In this chapter, we have reviewed the main results in the theory of Gohberg and Sigal on meromorphic operator-valued functions. These results concern the generalization of the argument principle and the Rouch´ e theorem to meromorphic operator-valued functions. Some of these results have been extended to very gen- eral operator-valued functions in [46, 170] and with other types of spectrum than isolated eigenvalues in [174]. Throughout this book, the theory of Gohberg and Sigal will be applied to perturbation theory of eigenvalues. Other interesting applications include the in- vestigation of scattering resonances and scattering poles [118, 57] and the study of the regularity of the solutions of elliptic boundary value problems near conical points [154].
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