Metric and homogeneous structure of closed range operators

TL;DR

This paper investigates the topological and metric properties of the set of bounded linear operators with closed range between Hilbert spaces, including orbit structures under group actions and stratification of Fredholm operators.

Contribution

It introduces natural metrics on the space of closed range operators, analyzes the group action and orbit structure, and computes distances between orbits, enriching the understanding of operator topology.

Findings

Determined the orbits of the group action on closed range operators.

Established a stratification of Fredholm and semi-Fredholm operators.

Calculated distances between different orbits under various metrics.

Abstract

Let $\CR$ be the set of all bounded linear operators between Hilbert spaces $\cH, \cK$. 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 $\G_\cH\times\G_\cK$ on $\CR$ and determine the orbits of this action. These orbits 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 $\CR$.