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.
