Generalized Argument Principle
and Rouch´ e’s Theorem
In this chapter we review the results of Gohberg and Sigal in  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) =
where g is holomorphic
and nowhere vanishing in a neighborhood of z0, and therefore
z − z0
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 −
where h is
holomorphic and nowhere vanishing in a neighborhood of z0. Then
z − z0
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
dz = N − P.
Rouch´ e’s theorem is a consequence of the argument principle . 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 , we begin
with the finite-dimensional case which was first considered by Keldy˘ s in ;
see also . We proceed to generalize formula (1.1) in this case as follows. If a