On differential and difference Fredholm operators

The results concerning Fredholm abstract parabolic and difference operators defined by a family of evolution operators and some examples of operators and equations to which these results can be applied are presented. It was assumed the family of evolution operator admits an exponential dichotomy at particular points with numbers defining dichotomy at these points and the corresponding pairs of projection-valued functions. A Fredholm operator representing a linear operator with a finite-dimensional kernel with a closed range of values of finite codimension was also defined. It was proved that a linear operator is Fredholm if and only if the node operator is Fredholm. It was also proved that if a difference operator is Fredholm, it is invertible.