),(()( rk krnkr nARvvvFAR ????? where the),..., 2,1 )((rV???? are given by Eq. (3.2). Denoting byρ(A) the spectral radius of A, for every λ,μ such that ρ(A)<min(| λ|,|μ|), the Hilbert identity holds true (see [1]):)()()()()(ARARARAR ????????? Therefore for every λ?Σ A, we have )( )( 2ARd AdR ????? and in general )A,...; 3,2,1k( ),(!)1( )( 1??????????ARkd ARd kkk k so, for every λ0∈ D,Rλ(A) can be expanded in the Taylor series ??????? 0 0 1) )(()1()( 0k kkkARAR???? which is absolutely and uniformly convergent in D. Defining )(: ),..., (: 0011?? rrvvvv ????),,..., (: 1,, ???rnknkvvFF? where the v?(λ) are defined by Eq. (3.2) , we can prove the following theorem. Theorem 3.2. The Taylor expansion (3.7) of the resolvent Rλ(A)in a neighborhood of any regular point λ0 can be written in the form
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