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Journal of Operator Theory

Volume 41, Issue 1, Winter 1999  pp. 69-92.

Le complete des operateurs fermes a domaine dense pour la metrique du gap

Authors Yahya Mezroui
Author institution: Laboratoire J.A. Dieudonne, Universite de Nice - Sophia - Antipolis, UMR no. 6621 du CNRS, 06108 NICE Cedex 2, France

Summary:  The set ${\cal LR}(H)$ of closed linear relations on a separable Hilbert space $H$ (i.e. the set of all closed linear subspaces of $H \oplus H$ of infinite di\-mension and codimension) contains the set ${\cal G}(H)$ of the graphs of all closed densely defined linear operators on $H$. Equipped with the ``gap" metric $g$, ${\cal LR}(H)$ is a complete metric space. In this paper we establish a certain number of properties of ${\cal LR}(H)$ and we caracterize the closure of ${\cal G}(H)$ in ${\cal LR}(H)$, providing thus a completion of the set ${\cal C}(H)$ of all closed densely defined linear operators on $H$.

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