Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

We study the generator $G$ of the one-dimensional damped wave equation with unbounded damping at infinity. We show that the norm of the corresponding resolvent operator, $\Vert {(G-\lambda )}^{-1}\Vert$, is approximately constant as $\left|\lambda \right|\to +\infty$ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, ${\overline{ℂ}}_{-}≔\{\lambda \in ℂ:Re\lambda \le 0\}$. Our proof rests on a precise asymptotic analysis of the norm of the inverse of $T\left(\lambda \right)$, the quadratic operator associated with $G$.