Exponential dichotomies and Fredholm operator for parabolic equations

It proved true for infinite-dimensional systems that the operator F ≔ ddt − A(t) is Fredholm when A(t), roughly speaking, admits exponential dichotomies on both R+ and R−. It is also interesting to study its converse problem. Although in [1,2] results were given to the question when the Fredholm alternative for bounded solutions of parabolic equations implies exponential dichotomies, but neither of them gave a direct answer to the operator F. In this paper, we try to obtain a direct answer for parabolic equations defined on a Banach space and to give an approach in the space of functions of compact support.