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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 E 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.

Projection orthogonale sur le graphe d’une relation linéaire fermé