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Transactions of the American Mathematical Society

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Projection orthogonale sur le graphe d’une relation linéaire fermé

Author: Yahya Mezroui
Journal: Trans. Amer. Math. Soc. 352 (2000), 2789-2800
MSC (1991): Primary 47H06
Published electronically: December 15, 1999
MathSciNet review: 1638254
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Abstract: Let ${LR(H)}$ denote the set of all closed linear relations on a Hilbert space $H$ (which contains all closed linear operators on $H$). In this paper, for every $E \in {\mathcal LR(H)}$ we define and study two associated linear operators on $H$, $\cos (E)$ and $\sin (E)$, which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of $E$, a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the $\cos (E)$ and $\sin (E)$ operators to operator theory will be given in a forthcoming paper.

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Additional Information

Yahya Mezroui
Affiliation: Laboratoire J.A. Dieudonné, UMR #6621 du CNRS, Université de Nice - Sophia Antipolis, 06108 Nice, Cedex 2, France

Keywords: Orthogonal projection, linear relation, gap metric
Received by editor(s): February 20, 1998
Published electronically: December 15, 1999
Article copyright: © Copyright 2000 American Mathematical Society