1.3. MAIN RESULTS OF THE GOHBERG AND SIGAL THEORY 13 Proof. To prove that A is normal with respect to ∂V , it suffices to prove that A(z) is invertible except at a finite number of points in V . To this end choose a connected open set U with U V so that A(z) is invertible in V \ U. Then, for each ξ U, there exists a neighborhood of ξ in which the factorization (1.5) holds. In Uξ, the kernel of A(z) has a constant dimension except at ξ. Since U is compact, we can find a finite covering of U, i.e., U 1 ··· k , for some points ξ1, . . . , ξk U. Therefore, dim Ker A(z) is constant in V \{ξ1, . . . , ξk}, and so A(z) is invertible in V \ {ξ1, . . . , ξk}. Now, if A(z) is normal with respect to the contour ∂V and zi, i = 1, . . . , σ, are all its characteristic values and poles lying in V , we put M(A(z) ∂V ) = σ i=1 M(A(zi)). (1.9) The full multiplicity M(A(z) ∂V ) of A(z) in V is the number of characteristic values of A(z) in V , counted with their multiplicities, minus the number of poles of A(z) in V , counted with their multiplicities. Theorem 1.12 (Generalized argument principle). Suppose that the operator- valued function A(z) is normal with respect to ∂V . Then we have M(A(z) ∂V ) = 1 2 −1π tr ∂V A−1(z) d dz A(z)dz. (1.10) Proof. Let zj, j = 1, . . . , σ, denote all the characteristic values and all the poles of A lying in V . The key of the proof lies in using the factorization (1.5) in each of the neighborhoods of the points zj. We have 1 2 −1π tr ∂V A−1(z) d dz A(z)dz = σ j=1 1 2 −1π tr ∂Vj A−1(z) d dz A(z)dz, (1.11) where, for each j, Vj is a neighborhood of zj. Moreover, in each Vj, the following factorization of A holds: A(z) = E(j)(z)D(j)(z)F (j) (z), D(j)(z) = P (j) 0 + nj i=1 (z zj)kij P (j) i . As for the matrix-valued case at the beginning of this chapter, it is readily verified that 1 2 −1π tr ∂Vj A−1(z) d dz A(z)dz = 1 2 −1π tr ∂Vj (D(j)(z))−1 d dz D(j)(z)dz = nj i=1 kij = M(A(zj)). Now, (1.10) follows by using (1.11). The following is an immediate consequence of Lemma 1.11, identity (1.10), and (1.4).
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