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 $\mathcal{CR}$ be the set of all bounded linear operators between Hilbert spaces $\mathcal{H}, \mathcal{K}$ with closed range. This paper is devoted to the study of the topological properties of $\mathcal{CR}$ if certain natural metrics are considered on it. We also define an action of the group $\mathcal{G}_\mathcal{H}\times\mathcal{K}$ on $\mathcal{CR}$ 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 $\mathcal{CR}$.

Keywords: Closed range, partial isometry, semi-Fredholm operators, positive operators, orbits, Moore-Penrose inverse

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