Geodesic
Estimate for the spectrum of an operator bundle and its application to stability problems
Matematičeskie zametki, Tome 19 (1976) no. 4, pp. 611-614 
V. I. Frolov. Estimate for the spectrum of an operator bundle and its application to stability problems. Matematičeskie zametki, Tome 19 (1976) no. 4, pp. 611-614. http://geodesic.mathdoc.fr/item/MZM_1976_19_4_a14/
@article{MZM_1976_19_4_a14,
     author = {V. I. Frolov},
     title = {Estimate for the spectrum of an operator bundle and its application to stability problems},
     journal = {Matemati\v{c}eskie zametki},
     pages = {611--614},
     year = {1976},
     volume = {19},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_19_4_a14/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $n=3$.