14 1. GENERALIZED ARGUMENT PRINCIPLE AND ROUCH´ THEOREM Corollary 1.13. If the operator-valued functions A(z) and B(z) are normal with respect to ∂V , then C(z) := A(z)B(z) is also normal with respect to ∂V , and M(C(z) ∂V ) = M(A(z) ∂V ) + M(B(z) ∂V ). The following general form of the argument principle will be useful. It can be proven by the same argument as the one in Theorem 1.12. Theorem 1.14. Suppose that A(z) is an operator-valued function which is nor- mal with respect to ∂V . Let f(z) be a scalar function which is analytic in V and continuous in V . Then 1 2 −1π tr ∂V f(z)A−1(z) d dz A(z)dz = σ j=1 M(A(zj))f(zj), where zj, j = 1, . . . , σ, are all the points in V which are either poles or characteristic values of A(z). 1.3.2. Generalization of Rouch´ e’s Theorem. A generalization of Rouch´e’s theorem to operator-valued functions is stated below. Theorem 1.15 (Generalized Rouch´ e’s theorem). Let A(z) be an operator- valued function which is normal with respect to ∂V . If an operator-valued func- tion S(z) which is finitely meromorphic in V and continuous on ∂V satisfies the condition A−1(z)S(z) L(B,B) 1, z ∂V, then A(z) + S(z) is also normal with respect to ∂V and M(A(z) ∂V ) = M(A(z) + S(z) ∂V ). Proof. Let C(z) := A−1(z)S(z). By Corollary 1.10, C(z) is finitely meromor- phic in V . Suppose that z1, z2, . . . , zn, are all of the poles of C(z) in V and that C(z) has the following Laurent series expansion in some neighborhood of each zj: C(z) = +∞ k=−νj (z zj)kC(j). k Let N be the intersection of the kernels Ker C(j) k for j = 1, . . . , n and k = 1, . . . , νj. Then, dim B/N +∞ and the restriction C(z)|N of C(z) to N is holo- morphic in V . Let q := maxz∈∂V C(z) , which by assumption is less than 1. Since ∆z C(z)|N 2 = 4 ∂z C(z)|N 2 , then C(z)|N is subharmonic in V , and hence we have from the maximum principle max z∈V C(z)|N≤ q. It then follows that (I + C(z))x≥ (1 q) x , x N, z V.
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