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.
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