Metric and homogeneous structure of closed range operators

Abstract

Let CR be the set of all bounded linear operators between Hilbert spaces H, K with closed range. This paper is devoted to the study of the topological properties of CR if certain natural metrics are considered on it. We also define an action of the group GH × GK on CR and determine the orbits of this action. These orbits, which are strongly related to the connected components for the topology defined by the metrics mentioned above, determine a stratification of the set of Fredholm and semi-Fredholm operators. Finally, we calculate the distance, with respect to some of the metrics mentioned above, between different orbits of CR.