10 1. GENERALIZED ARGUMENT PRINCIPLE AND

ROUCH´

E’S THEOREM

Suppose that

A−1(z)

exists and is holomorphic in some neighborhood of z0,

except possibly at z0. Then the number

M(A(z0)) = N(A(z0)) −

N(A−1(z0))

is called the multiplicity of z0. If z0 is a characteristic value and not a pole of A(z),

then M(A(z0)) = N(A(z0)) while M(A(z0)) = −N(A−1(z0)) if z0 is a pole and not

a characteristic value of A(z).

1.1.4. Normal Points. Suppose that z0 is a pole of the operator-valued func-

tion A(z) and the Laurent series expansion of A(z) at z0 is given by

A(z) =

j≥−s

(z −

z0)j

Aj . (1.3)

If in (1.3) the operators A−j , j = 1, . . . , s, have finite-dimensional ranges, then A(z)

is called finitely meromorphic at z0.

The operator-valued function A(z) is said to be of Fredholm type (of index zero)

at the point z0 if the operator A0 in (1.3) is Fredholm (of index zero).

If A(z) is holomorphic and invertible at z0, then z0 is called a regular point of

A(z). The point z0 is called a normal point of A(z) if A(z) is finitely meromorphic,

of Fredholm type at z0, and regular in a neighborhood of z0 except at z0 itself.

1.1.5. Trace. Let A be a finite-dimensional operator acting from B into itself.

There exists a finite-dimensional invariant subspace C of A such that A annihilates

some direct complement of C in B. We define the trace of A to be that of A|C ,

which is given in the usual way. It is desirable to recall some results about the

trace operator.

Proposition 1.6. The following results hold:

(i) tr A is independent of the choice of C, so that it is well-defined.

(ii) tr is linear.

(iii) If B is a finite-dimensional operator from B to itself, then

tr AB = tr BA.

(iv) If M is a finite-dimensional operator from B × B to itself, given by

M =

A B

C D

,

then tr M = tr A + tr D.

Recall that if an operator-valued function C(z) is finitely meromorphic in the

neighborhood V of z0, which contains no poles of C(z) except possibly z0, then

∂V

C(z) dz is a finite-dimensional operator. The following identity will also be

used frequently.

Proposition 1.7. Let A(z) and B(z) be two operator-valued functions which

are finitely meromorphic in the neighborhood V of z0, which contains no poles of

A(z) and B(z) other than z0. Then we have

(1.4) tr

∂V

A(z)B(z) dz = tr

∂V

B(z)A(z) dz.