Semi-Fredholm theory on Hilbert $C^{*}$ -modules

Abstract

We establish the semi-Fredholm theory on Hilbert C∗-modules as a continuation of Fredholm theory on Hilbert C∗-modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert C∗-module, and we prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open, and many other results that generalize their classical counterparts.