A localization criterion for the eigenvalues of a spectrally unstable operator

A study was conducted to investigate an operator with non-self-adjoint property and that a nontrivial information about its spectrum was obtained in such an unstable situation. It was demonstrated that operators of these form arose in studying regularized determinants for elliptic operators on manifolds with boundary and in linearizing continuity equations for systems with nonequilibrium thermodynamics. Another result for a second-order operator was obtained under the assumption that the eigenvalues of the matrix A varied along fixed rays. It was found that it was comparatively simple to study the case of a piecewise constant matrix A when the eigenvalues of the matrix A had nonconstant arguments.