This chapter considers \( A \in \mathcal{M}_N \left( \mathbb{C} \right) \) and uses the Dunford integral formula to construct the spectral projections of ℂN onto N[(AλI)ν(λ)] for every λσ(A). It also shows that, for each λσ(A), the algebraic ascent ν(λ) equals the order of λ as a pole of the associated resolvent operator
$$ \begin{array}{*{20}c} {\mathcal{R}\left( {z;A} \right): = \left( {zI - A} \right)^{ - 1} ,} & {z \in \rho \left( A \right): = \mathbb{C}\backslash \sigma \left( A \right)} \\ \end{array} . $$
Precisely, this chapter is structured as follows. Section 3.1 gives a universal estimate for the norm of the inverse of a matrix in terms of its determinant and its norm. From this estimate it will become apparent that the eigenvalues of A are poles of the resolvent operator (3.1). The necessary analysis to show this feature will be carried out in Section 3.3. Section 3.2 gives a result on Laurent series valid for vector-valued holomorphic functions. Finally, Section 3.4 constructs the spectral projections associated with the direct sum decomposition (1.6), whose validity was established by the Jordan Theorem 1.2.1.


Laurent Series Spectral Projection Invertible Matrix Resolvent Operator Removable Singularity 
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© Birkhäuser Verlag AG 2007

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