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

)kCkj)|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 +

C0j)) (

= ind(I +

+∞

k=0

(z − zj

)kCkj)) (

= ind(I + C(z)) = 0.

Thus, (I +

C0j)) (

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.