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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Projection orthogonale sur le graphe d’une relation linéaire fermé
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by Yahya Mezroui PDF
Trans. Amer. Math. Soc. 352 (2000), 2789-2800 Request permission

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
  • Email: mezroui@math.unice.fr
  • Received by editor(s): February 20, 1998
  • Published electronically: December 15, 1999
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 2789-2800
  • MSC (1991): Primary 47H06
  • DOI: https://doi.org/10.1090/S0002-9947-99-02410-1
  • MathSciNet review: 1638254