CHAPTER 1

Generalized Argument Principle

and Rouch´ e’s Theorem

In this chapter we review the results of Gohberg and Sigal in [114] concerning

the generalization to operator-valued functions of two classical results in complex

analysis, the argument principle and Rouch´ e’s theorem.

To state the argument principle, we first observe that if f is holomorphic and

has a zero of order n at z0, we can write f(z) =

(z−z0)ng(z),

where g is holomorphic

and nowhere vanishing in a neighborhood of z0, and therefore

f (z)

f(z)

=

n

z − z0

+

g (z)

g(z)

.

Then the function f /f has a simple pole with residue n at z0. A similar fact also

holds if f has a pole of order n at z0, that is, if f(z) = (z −

z0)−nh(z),

where h is

holomorphic and nowhere vanishing in a neighborhood of z0. Then

f (z)

f(z)

= −

n

z − z0

+

h (z)

h(z)

.

Therefore, if f is holomorphic, the function f /f will have simple poles at the zeros

and poles of f, and the residue is simply the order of the zero of f or the negative

of the order of the pole of f.

The argument principle results from an application of the residue formula. It

asserts the following.

Theorem 1.1 (Argument principle). Let V ⊂ C be a bounded domain with

smooth boundary ∂V positively oriented and let f(z) be a meromorphic function in

a neighborhood of V . Let P and N be the number of poles and zeros of f in V ,

counted with their multiplicities. If f has no poles and never vanishes on ∂V , then

(1.1)

1

2π

√

−1

∂V

f (z)

f(z)

dz = N − P.

Rouch´ e’s theorem is a consequence of the argument principle [237]. It is in

some sense a continuity statement. It says that a holomorphic function can be

perturbed slightly without changing the number of its zeros. It reads as follows.

Theorem 1.2 (Rouch´ e’s theorem). With V as above, suppose that f(z) and

g(z) are holomorphic in a neighborhood of V . If |f(z)| |g(z)| for all z ∈ ∂V , then

f and f + g have the same number of zeros in V .

In order to explain the main results of Gohberg and Sigal in [114], we begin

with the finite-dimensional case which was first considered by Keldy˘ s in [152];

see also [183]. We proceed to generalize formula (1.1) in this case as follows. If a

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http://dx.doi.org/10.1090/surv/153/02