We prove that a family of $q$-periodic continuous matrix valued function $\{A(t)\}_{t\in \mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $\int_0^t e^{i\mu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $\mu$ and the solution of the Cauchy operator Problem \begin{gather*} \dot{Y}(t)=-Y(t)A(t)+ e^{i \mu t}(I-P) ,\quad t\geq s \\ Y(s)=0, \end{gather*} has a limit in $\mathcal{L}(\mathbb{C}^n)$ as $s$ tends to $-\infty$ which is bounded uniformly with respect to the parameter $\mu$. Here, $\{ U(t,s): t, s\in\mathbb{R}\}$ is the evolution family generated by $\{A(t)\}_{t\in \mathbb{R}}$, $\mu$ is a real number and $q$ is a fixed positive number.