1.3. MAIN RESULTS OF THE GOHBERG AND SIGAL THEORY 15 This implies that (I + C(z))|N has a closed range and Ker(I + C(z))|N = 0. Therefore, I + C(z) has a closed range and a kernel of finite dimension for z V \ {z1, . . . , zn}. By a slight extension of Proposition 1.5 [241], I(z) defined by I(z) = dimKer(I + C(z)) codim Im(I + C(z)) is continuous for z V \ {z1, . . . , zn}. Thus, ind(I + C(z)) = 0 for z V \ {z1, . . . , zn}. Moreover, since the Laurent series expansion of (I + C(z))|N in a neighborhood of zj is given by (1.12) (I + C(z))|N = I|N + +∞ k=0 (z zj)kC(j)| k N , it follows that (I+C0j))|N ( has a closed range and a trivial kernel. Using Propositions 1.4 and 1.5, we have ind(I + C(j)) 0 = ind(I + +∞ k=0 (z zj)kC(j)) k = ind(I + C(z)) = 0. Thus, (I + C(j)) 0 is Fredholm. By Lemma 1.11, we deduce that I + C(z) is normal with respect to ∂V . Now we claim that M(I + C(z) ∂V ) = 0. To see this, we note that I + tC(z) is normal with respect to ∂V for 0 t 1. Let f(t) := M(I + tC(z) ∂V ). Then f(t) attains integers as its values. On the other hand, since (1.13) f(t) = 1 2 −1π tr ∂V t(I + tC(z))−1 d dz C(z) dz and (I + tC(z))−1 is continuous in [0, 1] in operator norm uniformly in z ∂V , f(t) is continuous in [0, 1]. Thus, f(1) = f(0) = 0. Finally, with the help of Corollary 1.13, we can conclude that the theorem holds. 1.3.3. Generalization of Steinberg’s theorem. Steinberg’s theorem as- serts that if K(z) is a compact operator on a Banach space, which is analytic in z for z in a region V in the complex plane, then I + K(z) is meromorphic in V . See [238]. A generalization of this theorem to finitely meromorphic operators was first given by Gohberg and Sigal in [114]. The following important result holds. Theorem 1.16 (Generalized Steinberg’s theorem). Suppose that A(z) is an operator-valued function which is finitely meromorphic and of Fredholm type in the domain V . If the operator A(z) is invertible at one point of V , then A(z) has a bounded inverse for all z V , except possibly for certain isolated points.
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