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 ).