Strongly damped pencils of operators and solvability of the corresponding operator-differential equations

An investigation is made of the operator pencil $L(\lambda)=A+\lambda B+\lambda^2C$ under the assumption that the selfadjoint operators $A$, $B$, and $C$ satisfy the strong damping condition $(Bx,x)^2>4(Ax,x)(Cx,x)$. Such operator pencils have been studied thoroughly in the literature under the condition that their spectral zones are separated. The present article is a study of the spectral properties of the linear factors into which the pencil splits when the spectral zones adjoin. The results carry over to the case of pencils of unbounded operators and are used to prove the existence and uniqueness of solutions of equations of the form $Fu''+iGu'+Hu=0$ or $-Fu''+Gu'+Hu=0$ on the semi-axis $(0,\infty)$, where $H\gg0$ and $F\geqslant0$ are selfadjoint operators whose domains satisfy the inclusion $D(F)\supseteq D(H)$, and $G$ is a symmetric operator such that $D(G)\supseteq D(H)$, and $(Gy,y)\ne0$ for $y\in\operatorname{Ker}F\cap D(H^{1/2})$, $y\ne0$.
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