8 1. GENERALIZED ARGUMENT PRINCIPLE AND ROUCH´ THEOREM matrix-valued function A(z) is holomorphic in a neighborhood of V and is invertible in V except possibly at a point z0 V , then by Gaussian eliminations we can write (1.2) A(z) = E(z)D(z)F (z) in V, where E(z), F (z) are holomorphic and invertible in V and D(z) is given by D(z) = (z z0)k1 0 ... 0 (z z0)kn . Moreover, the powers k1, k2, . . . , kn are uniquely determined up to a permutation. Let tr denote the trace. By virtue of the factorization (1.2), it is easy to produce the following identity: 1 −1 tr ∂V A(z)−1 d dz A(z) dz = 1 −1 tr ∂V E(z)−1 d dz E(z) + D(z)−1 d dz D(z) + F (z)−1 d dz F (z) dz = 1 −1 tr ∂V D(z)−1 d dz D(z) dz = n j=1 kj, which generalizes (1.1). In the next sections, we will extend the above identity as well as the factoriza- tion (1.2) to infinite-dimensional spaces under some natural conditions. 1.1. Definitions and Preliminaries In this section we introduce the notation we will use in the text, gather a few definitions, and present some basic results, which are useful for the statement of the generalized Rouch´ e theorem. 1.1.1. Compact Operators. If B and B are two Banach spaces, we denote by L(B,B ) the space of bounded linear operators from B into B . An operator K L(B,B ) is said to be compact provided K takes any bounded subset of B to a relatively compact subset of B , that is, a set with compact closure. The operator K is said to be of finite rank if Im(K), the range of K, is finite- dimensional. Clearly every operator of finite rank is compact. The next result is called the Fredholm alternative. See, for example, [164]. Proposition 1.3 (Fredholm alternative). Let K be a compact operator on the Banach space B. For λ C, λ = 0, (λI K) is surjective if and only if it is injective. 1.1.2. Fredholm Operators. An operator A L(B,B ) is said to be Fred- holm provided the subspace Ker A is finite-dimensional and the subspace Im A is closed in B and of finite codimension. Let Fred(B, B ) denote the collection of all Fredholm operators from B into B . We can show that Fred(B, B ) is open in L(B,B ).
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