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)jAj 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)jAj 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)jAj − 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)−jKj + 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 ⎨ ⎪ ⎩ ⊂ Ker Aj, j = −ν, . . . , −1, N ⊂ Ker Kj, j = 0, . . . , ν, N Im Kj = {0}, j = 1, . . . , ν.

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