The Kato decomposition of quasi-Fredholm relations

Quasi-Fredholm relations of degree d is an element of N in Hilbert spaces are defined in terms of conditions on their ranges and kernels. They are completely characterized in terms of an algebraic decomposition with a quasi-Fredholm relation of degree 0 and a nilpotent operator of degree d. The adjoint of a quasi-Fredholm relation of degree d is an element of N is shown to be quasi-Fredholm relation of degree d is an element of N. The class of quasi-Fredholm relations contains the semi-Fredholm relations. Earlier results for quasi-Fredholm operators and semi-Fredholm operators are included.