Projection orthogonale sur le graphe d'une relation linéaire fermé

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.