Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type $$\{\begin{array}{l}{\rho }_1\ddot{\phi }+a{A}^{\frac{1}{2}}(A^{\frac{1}{2}}\phi +\psi )=0\\ {\rho }_2\ddot{\psi }+bA\psi +a(A^{\frac{1}{2}}\phi +\psi )-\delta A^{\gamma }\theta =0\\ {\rho }_3\dot{\theta }+cA\theta +\delta A^{\gamma }\dot{\psi }=0\end{array}$$ where the operator $A$ is strictly positive selfadjoint. For any fixed $\gamma \in \mathbb{R}$, the stability properties of the related solution semigroup $S(t)$ are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of $S(t)$ when the spectrum of the leading operator $A$ is not made by eigenvalues only.