Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems

We prove that a family of $q$-periodic continuous matrix valued function ${A(t)}_{tin mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $int_0^t e^{imu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $mu$ and the solution of the Cauchy operator Problem $$displaylines{ dot{Y}(t)=-Y(t)A(t)+ e^{i mu t}(I-P) ,quad tgeq s cr Y(s)=0, }$$ 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, ${ U(t,s): t, sinmathbb{R}}$ is the evolution family generated by ${A(t)}_{tin mathbb{R}}$, $mu$ is a real number and q is a fixed positive number.