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. We show that the norm of the corresponding resolvent operator, $\mathrm{\parallel }(G-\lambda {)}^{-1}\mathrm{\parallel }$, is approximately constant as $\mathrm{\mid }\lambda \mathrm{\mid }\to +\mathrm{\infty }$ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, ${\bar{\mathbb{C}}}_{-}:=\{\lambda \in \mathbb{C}:\mathrm{Re}\lambda \le 0\}$. Our proof rests on a precise asymptotic analysis of the norm of the inverse of $T(\lambda )$, the quadratic operator associated with $G$.