Fredholm operators on Hilbert modules

Let A be a unital not necessarily commutative C*-algebra. In this manuscript, we define that a bounded adjointable operator between two Hilbert A-modules is a Fredholm operator if it has finitely generated projective kernel and cokernel as a straight forward generalization of the classical definition. We generalize the classical Atkinson theorem and a version of Atiyah–Janich theorem to show that the index holds the homotopy invariance property for this class of operators.