Abstract: We consider the evolution equation $\ddot{u}(t)+Au(t)+2aB\dot{u}(t)=0$, where $B$ is comparable to $A^{\alpha}$ for some $1/2\le\alpha\le1$. This is a model for an elastic system with structural damping, and it is known that the system operator associated with this model is the infinitesimal generator of an analytic semigroup on the natural energy space. However, except for the case of so-called Kelvin-Voigt damping ($\alpha=1$), this operator is neither sectorial nor associated with a coercive sesquilinear form. We show that these properties can be obtained via a different inner product on the energy space.