Orthogonal Random Vectors and the Hurwitz-radon-eckmann Theorem

In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operators Ui in H = Rn with the property UiUj = −UjUi (i 6= j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operators Ui must commute with a given arbitrary self-adjoint operator and H can be infinite-dimensional. Our second main result deals with the conditions for almost sure orthogonality of two random vectors taking values in a finite or infinite-dimensional Hilbert space H. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors in H with the same covariance operator and each pair having a linear support in H ⊕H. The paper is based on the results obtained jointly with N.P.Kandelaki (see [1,2,3]).