Estimate for the spectrum of an operator bundle and its application to stability problems

Simple estimates are obtained for the spectrum of the operator bundle $R(\lambda)=\sum_{i=0}^nA_{n-i}\lambda^i$ in terms of estimates of the maximum and minimum eigenvalues of the operators $\frac12(A_{n-i}+A_{n-i}^*)$ $(i=0,1,2,\dots,n)$ and the norms of the operators $\frac12(A_{n-i}-A_{n-i}^*)$ $(i=0,1,2,\dots,n)$. We formulate a criterion of the asymptotic stability of the differential equations
$$ \sum_{i=0}^nA_{n-i}\frac{d^{(i)}x}{dt^i}=0 $$
We present examples of the stability conditions for equations with $n=2$ and $n=3$.