8 1. GENERALIZED ARGUMENT PRINCIPLE AND
ROUCH´
E’S THEOREM
matrix-valued function A(z) is holomorphic in a neighborhood of V and is invertible
in V except possibly at a point z0 ∈ V , then by Gaussian eliminations we can write
(1.2) A(z) = E(z)D(z)F (z) in V,
where E(z), F (z) are holomorphic and invertible in V and D(z) is given by
D(z) =
⎛
⎜
⎝
(z −
z0)k1
0
...
0 (z − z0)kn
⎞
⎟
⎠
.
Moreover, the powers k1, k2, . . . , kn are uniquely determined up to a permutation.
Let tr denote the trace. By virtue of the factorization (1.2), it is easy to produce
the following identity:
1
2π
√
−1
tr
∂V
A(z)−1
d
dz
A(z) dz
=
1
2π
√
−1
tr
∂V
E(z)−1
d
dz
E(z) +
D(z)−1
d
dz
D(z) + F
(z)−1
d
dz
F (z) dz
=
1
2π
√
−1
tr
∂V
D(z)−1
d
dz
D(z) dz
=
n
j=1
kj ,
which generalizes (1.1).
In the next sections, we will extend the above identity as well as the factoriza-
tion (1.2) to infinite-dimensional spaces under some natural conditions.
1.1. Definitions and Preliminaries
In this section we introduce the notation we will use in the text, gather a few
definitions, and present some basic results, which are useful for the statement of
the generalized Rouch´ e theorem.
1.1.1. Compact Operators. If B and B are two Banach spaces, we denote
by L(B,B ) the space of bounded linear operators from B into B . An operator
K ∈ L(B,B ) is said to be compact provided K takes any bounded subset of B to
a relatively compact subset of B , that is, a set with compact closure.
The operator K is said to be of finite rank if Im(K), the range of K, is finite-
dimensional. Clearly every operator of finite rank is compact.
The next result is called the Fredholm alternative. See, for example, [164].
Proposition 1.3 (Fredholm alternative). Let K be a compact operator on the
Banach space B. For λ ∈ C, λ = 0, (λI − K) is surjective if and only if it is
injective.
1.1.2. Fredholm Operators. An operator A ∈ L(B,B ) is said to be Fred-
holm provided the subspace Ker A is finite-dimensional and the subspace Im A is
closed in B and of finite codimension. Let Fred(B, B ) denote the collection of
all Fredholm operators from B into B . We can show that Fred(B, B ) is open in
L(B,B ).
