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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