Abstract
In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operatorsU i inH = ℝn 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]).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N.N. Vakhania and N.P. Kandelaki, On orthogonal random vectors in Hilbert space. (Russian)Dokl. Akad. Nauk SSSR 294 (1987), No. 3, 528–531;English transl. in Soviet Math. Dokl. 35 (1987), No. 3.
—, A generalization of the Hurwitz-Radon-Eckmann theorem and orthogonal random vectors. (Russian)Dokl. Akad. Nauk SSSR 296 (1987), No. 2, 265–266;English transl. in Soviet Math. Dokl. 36 (1988), No. 2.
—, Orthogonal random vectors in Hilbert space. (Russian)Trudy Inst. Vychisl. Mat. Akad. Nauk Gruz. SSR 28:1 (1988), 11–37.
J.T. Schwartz, Differential Geometry and Topology. Gordon and Breach, New York, 1968.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1. Academic Press, New York/London, 1972.
N.N. Vakhania, V.I. Tarieladze, and S.A. Chobanyan, Probability distributions on Banach spaces. D. Reidel, Dordrecht/Boston/Lancaster/Tokyo, 1987, (transl. from Russian, Nauka, Moscow, 1985).
N.P. Kandelaki, I.N. Kartsivadze, and T.L. Chantladze, On orthogonal multiplication in Hilbert space. (Russian)Trudy Tbilis. Univ. Mat. Mech. Astron. 179 (1976), 43–57.