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 operatorsU _i inH = ℝ ⁿ with the propertyU _i U _j = −U _j U _i (i≠j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operatorsU _i must commute with a given arbitrary self-adjoint operator andH 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 spaceH. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors inH with the same covariance operator and each pair having a linear support inH⊕H. The paper is based on the results obtained jointly with N.P. Kandelaki (see 1,2,3]).