A spectral countability condition for almost automorphy of solutions of differential equations

We consider the almost automorphy of bounded mild solutions to equations of the form

$(*)\quad\qquad\qquad\qquad\qquad\qquad dx/dt = A(t)x + f(t) \quad\qquad\qquad\qquad\qquad\qquad\qquad$

with (generally unbounded) $\tau$-periodic $A(\cdot )$ and almost automorphic $f(\cdot )$ in a Banach space $\mathbb{X}$. Under the assumption that $\mathbb{X}$ does not contain $c_0$, the part of the spectrum of the monodromy operator associated with the evolutionary process generated by $A(\cdot )$ on the unit circle is countable. We prove that every bounded mild solution of $(*)$ on the real line is almost automorphic.