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 \(R(\lambda ) = \sum\nolimits_{i = 0}^n {A_{n - i} \lambda ^i }\) in terms of estimates of the maximum and minimum eigenvalues of the operators \(\frac{1}{2}(A_{n - i} - A_{n - i}^* )(i = 0,1,2, \ldots n)\) and the norms of the operators \(\frac{1}{2}(A_{n - i} - A_{n - i}^* )(i = 0,1,2, \ldots n)\) We formulate a criterion of the asymptotic stability of the differential equations $$\sum\nolimits_{i = 1}^n {A_{n - i} } \frac{{d^{(i)} x}}{{dt^i }} = 0.$$ We present examples of the stability conditions for equations with n=2 and n=3.