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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2009 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.