16 1. GENERALIZED ARGUMENT PRINCIPLE AND
ROUCH´
E’S 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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