Block-diagonalisation of matrices and operators (Q1030752)

Let \(\left\{ A(\rho )\right\} _{\rho \in \mathbb{C}}\) be a family of complex \(m\times m\) matrices. Suppose \(A(\rho )\) has a full asymptotic expansion near zero (\(\rho \rightarrow 0\)): \[ A(\rho )\sim A_{0}+\rho A_{1}+\rho ^{2}A_{2}+\cdots. \] Further, suppose that the spectrum of the matrix \(A_{0}\) is decomposed into groups of eigenvalues as follows \[ \text{spec}A_{0}=\bigcup_{i\in \mathcal{I}}\mathcal{G}_{i},\quad\text{conv}\mathcal{G}_{i}\cap \text{conv}\mathcal{G}_{j}=\emptyset ,\quad i\neq j \] where \(\mathcal{G}_{i}\)s are groups of eigenvalues. The author presents a constructive scheme to decompose the above family \(\{A(\rho )\} _{\rho \in \mathbb{C}}\) asymptotically into blocks corresponding to the groups \(\mathcal{G}_{i}\)s of the limit matrix \(A_{0}=A(0)\). He also generalizes this constructive scheme to families of matrices depending upon additional parameters and operators on Banach and Hilbert spaces.