CHAPTER 1 Generalized Argument Principle and Rouch´ e’s Theorem In this chapter we review the results of Gohberg and Sigal in [114] concerning the generalization to operator-valued functions of two classical results in complex analysis, the argument principle and Rouch´ e’s theorem. To state the argument principle, we first observe that if f is holomorphic and has a zero of order n at z0, we can write f(z) = (z−z0)ng(z), where g is holomorphic and nowhere vanishing in a neighborhood of z0, and therefore f (z) f(z) = n z z0 + g (z) g(z) . Then the function f /f has a simple pole with residue n at z0. A similar fact also holds if f has a pole of order n at z0, that is, if f(z) = (z z0)−nh(z), where h is holomorphic and nowhere vanishing in a neighborhood of z0. Then f (z) f(z) = n z z0 + h (z) h(z) . Therefore, if f is holomorphic, the function f /f will have simple poles at the zeros and poles of f, and the residue is simply the order of the zero of f or the negative of the order of the pole of f. The argument principle results from an application of the residue formula. It asserts the following. Theorem 1.1 (Argument principle). Let V C be a bounded domain with smooth boundary ∂V positively oriented and let f(z) be a meromorphic function in a neighborhood of V . Let P and N be the number of poles and zeros of f in V , counted with their multiplicities. If f has no poles and never vanishes on ∂V , then (1.1) 1 −1 ∂V f (z) f(z) dz = N P. Rouch´ e’s theorem is a consequence of the argument principle [237]. It is in some sense a continuity statement. It says that a holomorphic function can be perturbed slightly without changing the number of its zeros. It reads as follows. Theorem 1.2 (Rouch´ e’s theorem). With V as above, suppose that f(z) and g(z) are holomorphic in a neighborhood of V . If |f(z)| |g(z)| for all z ∂V , then f and f + g have the same number of zeros in V . In order to explain the main results of Gohberg and Sigal in [114], we begin with the finite-dimensional case which was first considered by Keldy˘ s in [152] see also [183]. We proceed to generalize formula (1.1) in this case as follows. If a 7
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