1.2. FACTORIZATION OF OPERATORS 11

1.2. Factorization of Operators

We say that A(z) ∈ U(z0) admits a factorization at z0 if A(z) can be written

as

(1.5) A(z) = E(z)D(z)F (z),

where E(z), F (z) are regular at z0 and

(1.6) D(z) = P0 +

n

j=1

(z −

z0)kj

Pj .

Here, Pj ’s are mutually disjoint projections, P1, . . . , Pn are one-dimensional opera-

tors, and I −

n

j=0

Pj is a finite-dimensional operator.

Theorem 1.8. A(z) ∈ U(z0) admits a factorization at z0 if and only if A(z) is

finitely meromorphic and of Fredholm type of index zero at z0.

Proof. Suppose that A(z) is finitely meromorphic and of Fredholm type of

index zero at z0. We shall construct E, F, and D such that (1.5) holds. Write the

Laurent series expansion of A(z) as follows:

A(z) =

+∞

j=−ν

(z −

z0)j

Aj

in some neighborhood U of z0. Since indA0 = 0, then by the Fredholm alternative

B0 := A0 + K0 is invertible for some finite-dimensional operator K0. Consequently,

B(z) := K0 +

+∞

j=0

(z −

z0)j

Aj

is invertible in some neighborhood U1 of z0 and

(1.7) A(z) = C(z) + B(z) = B(z)[I +

B−1(z)C(z)],

where

C(z) =

−1

j=−ν

(z −

z0)j

Aj − K0.

Since K(z) :=

B−1(z)C(z)

is finitely meromorphic, we can write K(z) in the

form

K(z) =

ν

j=1

(z −

z0)−j

Kj + T1(z),

where Kj , j = 1, . . . , ν, are finite-dimensional and T1 is holomorphic.

Since the operators Aj and Kj are finite-dimensional, there exists a subspace

N of B of finite codimension such that

⎧

⎪N

⎪

⎨

⎪

⎪

⎩N

⊂ Ker Aj , j = −ν, . . . , −1,

N ⊂ Ker Kj , j = 0, . . . , ν,

Im Kj = {0}, j = 1, . . . , ν.