14 1. GENERALIZED ARGUMENT PRINCIPLE AND

ROUCH´

E’S THEOREM

Corollary 1.13. If the operator-valued functions A(z) and B(z) are normal

with respect to ∂V , then C(z) := A(z)B(z) is also normal with respect to ∂V , and

M(C(z); ∂V ) = M(A(z); ∂V ) + M(B(z); ∂V ).

The following general form of the argument principle will be useful. It can be

proven by the same argument as the one in Theorem 1.12.

Theorem 1.14. Suppose that A(z) is an operator-valued function which is nor-

mal with respect to ∂V . Let f(z) be a scalar function which is analytic in V and

continuous in V . Then

1

2

√

−1π

tr

∂V

f(z)A−1(z)

d

dz

A(z)dz =

σ

j=1

M(A(zj ))f(zj ),

where zj , j = 1, . . . , σ, are all the points in V which are either poles or characteristic

values of A(z).

1.3.2. Generalization of Rouch´ e’s Theorem. A generalization of Rouch´e’s

theorem to operator-valued functions is stated below.

Theorem 1.15 (Generalized Rouch´ e’s theorem). Let A(z) be an operator-

valued function which is normal with respect to ∂V . If an operator-valued func-

tion S(z) which is finitely meromorphic in V and continuous on ∂V satisfies the

condition

A−1(z)S(z)

L(B,B)

1, z ∈ ∂V,

then A(z) + S(z) is also normal with respect to ∂V and

M(A(z); ∂V ) = M(A(z) + S(z); ∂V ).

Proof. Let C(z) :=

A−1(z)S(z).

By Corollary 1.10, C(z) is finitely meromor-

phic in V . Suppose that z1, z2, . . . , zn, are all of the poles of C(z) in V and that

C(z) has the following Laurent series expansion in some neighborhood of each zj :

C(z) =

+∞

k=−νj

(z − zj

)kCkj).(

Let N be the intersection of the kernels Ker

Ckj) (

for j = 1, . . . , n and k =

1, . . . , νj . Then, dim B/N +∞ and the restriction C(z)|N of C(z) to N is holo-

morphic in V .

Let q := maxz∈∂V C(z) , which by assumption is less than 1. Since

∆z C(z)|N

2

= 4

∂

∂z

C(z)|N

2,

then C(z)|N is subharmonic in V , and hence we have from the maximum principle

max

z∈V

C(z)|N ≤ q.

It then follows that

(I + C(z))x ≥ (1 − q) x , x ∈ N, z ∈ V.