On the Spectrum of Orthomorphisms and Barbashin Operators

Abstract

The paper is concerned with the spectrum of an operator $A=C+K$, where $C$ is an orthomorphism and $K$ is a compact operator. The proofs are in a certain sense constructive. The results are applied to Barbashin equations $\frac{dx}{dt}=Ax$, where $A=C+K$ with a multiplication operator $C$ and an integral operator $K$. In some particular cases even necessary and sufficient conditions for stability are given.