Spectral Theory for Operator Matrices Related to Models in Mechanics

We derive various properties of the operator matrix
$$\mathscr A=\begin{vmatrix} 0&I \\ -A_0&-D \end{vmatrix},$$
where $A_0$ is a uniformly positive operator and $A_0^{-1/2}DA_0^{-1/2}$ is a bounded nonnegative operator in a Hilbert space $H$. Such operator matrices are associated with second-order problems of the form $\ddot z(t)+A_0z(t)+D\dot z(t)=0$, which are used as models for transverse motions of thin beams in the presence of damping.