Diagonalization of Operator functions by algebraic methods

Abstract

We give conditions for local diagonalization of an analytic operator family $L\left(\epsilon \right)$ according to $L\left(\epsilon \right)=\psi \left(\epsilon \right)\cdot \Delta \left(\epsilon \right)\cdot {\varphi }^{-1}\left(\epsilon \right)$ with diagonal operator polynomial $\Delta \left(\epsilon \right)$ and analytic near identity bijections $\psi \left(\epsilon \right)$ and $\varphi \left(\epsilon \right)$. The family $L\left(\epsilon \right)$ is acting between real or complex Banach spaces B and $\overline{B}$.

The basic assumption is given by stabilization of the Jordan chains at length k in the sense that no root elements with finite rank above k are allowed to exist. Jordan chains with infinite rank may appear. Decompositions of the linear spaces B and $\overline{B}$ are constructed with corresponding subspaces assumed to be closed. These assumptions ensure finite pole order equal to k of the generalized inverse $L^{-1}\left(\epsilon \right)$ at $\epsilon =0$. The Smith form and smooth continuation of kernels and ranges of $L\left(\epsilon \right)$ to appropriate limit spaces at $\epsilon =0$ arise immediately.

An algebraically oriented and self-contained approach is used, based on a recursion that allows for construction of power series solutions of $L\left(\epsilon \right)\cdot b=0$. The power series solutions are convergent, as soon as analyticity of $L\left(\epsilon \right)$ and continuity of related projections are assumed.