12 1. GENERALIZED ARGUMENT PRINCIPLE AND ROUCH´ THEOREM Let C be a direct finite-dimensional complement of N in B and let P be the pro- jection onto C satisfying P (I − P ) = 0. Set P0 := I − P . We have I + K(z) = I + P K(z)P + P0K(z)P = I + P K(z)P + P0T1(z)P, and therefore, (1.8) I + K(z) = (I + P K(z)P )(I + P0T1(z)P ). Since P (I + K(z))P can be viewed as an operator from C into itself and C is finite-dimensional, it follows from Gaussian elimination that P (I + K(z))P = E1(z)D1(z)F1(z), where D1(z) is diagonal and E1(z) and F1(z) are holomorphic and invertible. In view of (1.8), this implies that A(z) = B(z)(P0 + P (I + K(z))P )(I + P0T1(z)P ) = B(z)(P0 + E1(z)D1(z)F1(z))(I + P0T1(z)P ) = B(z)(P0 + E1(z))(P0 + D1(z))(P0 + F1(z))(I + P0T1(z)P ). Here I + P0T1(z)P is holomorphic and invertible with inverse I − P0T1(z)P . Thus, taking E(z) := B(z)(P0 + E1(z)), F (z) := (P0 + F1(z))(I + P0T1(z)P ) yields the desired factorization for A since E(z) and F (z), given by the above formulas, are holomorphic and invertible at z0. The converse result, that A(z) = E(z)D(z)F (z) with E(z), F (z) regular at z0 and D(z) satisfying (1.6) is finitely meromorphic and of Fredholm type of index zero at z0, is easy. Corollary 1.9. A(z) is normal at z0 if and only if A(z) admits a factorization such that I = n j=0 Pj in (1.6). Moreover, we have M(A(z0)) = k1 + ··· + kn for k1, ··· , kn, given by (1.6). Corollary 1.10. Every normal point of A(z) is a normal point of A−1(z). 1.3. Main Results of the Gohberg and Sigal Theory We now tackle our main goal of this chapter, which is to generalize the argument principle and Rouch´ e’s theorem to operator-valued functions. 1.3.1. Argument Principle. Let V be a simply connected bounded domain with rectifiable boundary ∂V . An operator-valued function A(z) which is finitely meromorphic and of Fredholm type in V and continuous on ∂V is called normal with respect to ∂V if the operator A(z) is invertible in V , except for a finite number of points of V which are normal points of A(z). Lemma 1.11. An operator-valued function A(z) is normal with respect to ∂V if it is finitely meromorphic and of Fredholm type in V , continuous on ∂V , and invertible for all z ∈ ∂V .

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