# Journal of Operator Theory

Volume 61, Issue 1, Winter 2009 pp. 171-190.

Metric and homogeneous structure of closed range operators**Authors**: Gustavo Corach (1), Alexandra Maestripieri (2), and Mostafa Mbekhta (3)

**Author institution:**(1) Departamento de Matematica, Facultad de Ingenieria, UBA, Buenos Aires, 1063 , Argentina

(2) Departamento de Matematica, Facultad de Ingenieria, UBA, Buenos Aires, 1063 , Argentina

(3) Departement de Mathematiques, Universite Lille 1, F-59655, France

**Summary:**Let $\CR$ be the set of all bounded linear operators between Hilbert spaces $\cH, \cK$ 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 $\G_\cH\times\G_\cK$ 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 $\CR$.

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