The Mathieu–Hill operator equation with dissipation and estimates of its instability index

The behavior as $t\to\infty$ of solutions of the equation
$$ \partial^2_tu+\gamma\partial_tu+Au+f(t)u=0 $$
in a Hilbert space is studied, where $A=A^*$ is a positive definite operator with compact inverse and the operator $f$ is periodic in $t$. The notion of instability index is introduced for this equation; we prove that the instability index is finite under natural assumptions ($f$ must be dominated by $A$). Asymptotic estimates of the instability index are obtained as $\gamma\to0$, and an example is constructed showing that they cannot be improved. Furthermore, we study the qualitative characteristics of the spectrum of the monodromy operator and the existence of the Floquet representation for this problem.