1.1. DEFINITIONS AND PRELIMINARIES 9 Next, we define the index of A ∈ Fred(B, B ) to be ind A = dim Ker A − codim Im A. In finite dimensions, the index depends only on the spaces and not on the operator. The following proposition shows that the index is stable under compact per- turbations [164]. Proposition 1.4. If A : B → B is Fredholm and K : B → B is compact, then their sum A + K is Fredholm, and ind (A + K) = ind A. Proposition 1.4 is a consequence of the following fundamental result about the index of Fredholm operators. Proposition 1.5. The mapping A → ind A is continuous in Fred(B, B ) i.e., ind is constant on each connected component of Fred(B, B ). 1.1.3. Characteristic Value and Multiplicity. We now introduce the no- tions of characteristic values and root functions of analytic operator-valued func- tions, with which the readers might not be familiar. We refer, for instance, to the book by Markus [175] for the details. Let U(z0) be the set of all operator-valued functions with values in L(B,B ) which are holomorphic in some neighborhood of z0, except possibly at z0. The point z0 is called a characteristic value of A(z) ∈ U(z0) if there exists a vector-valued function φ(z) with values in B such that (i) φ(z) is holomorphic at z0 and φ(z0) = 0, (ii) A(z)φ(z) is holomorphic at z0 and vanishes at this point. Here, φ(z) is called a root function of A(z) associated with the characteristic value z0. The vector φ0 = φ(z0) is called an eigenvector. The closure of the linear set of eigenvectors corresponding to z0 is denoted by KerA(z0). Suppose that z0 is a characteristic value of the function A(z) and φ(z) is an associated root function. Then there exists a number m(φ) ≥ 1 and a vector-valued function ψ(z) with values in B , holomorphic at z0, such that A(z)φ(z) = (z − z0)m(φ)ψ(z), ψ(z0) = 0. The number m(φ) is called the multiplicity of the root function φ(z). For φ0 ∈ KerA(z0), we define the rank of φ0, denoted by rank(φ0), to be the maximum of the multiplicities of all root functions φ(z) with φ(z0) = φ0. Suppose that n = dim KerA(z0) +∞ and that the ranks of all vectors in KerA(z0) are finite. A system of eigenvectors φj 0 , j = 1, . . . , n, is called a canonical system of eigenvectors of A(z) associated to z0 if their ranks possess the following property: for j = 1, . . . , n, rank(φj) 0 is the maximum of the ranks of all eigenvectors in the direct complement in KerA(z0) of the linear span of the vectors φ1, 0 . . . , φj−1. 0 We call N(A(z0)) := n j=1 rank(φj) 0 the null multiplicity of the characteristic value z0 of A(z). If z0 is not a characteristic value of A(z), we put N(A(z0)) = 0.

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