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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