On the Construction of Invariant Measure over the Orthogonal Group on the Hilbert Space by the Method of Cayley Transformation

Abstract

The purpose of this paper is to construct some invariant measures over the infinite dimensional rotation group, analogously to the Haar measure in the finite dimensional case. In this direction there are some results making use of Haar measure on compact groups or Gaussian measure on Hilbert spaces. See, [5], [6], [9] and [10]. But it seems to me that the treatment of the whole group is complicated and difficult. On the other hand, the Cayley transformation in the finite dimensional Euclid space gives a correspondence between the special orthogonal group and the set of skew-symmetri c operators, and still may be useful for the infinite dimensional case. Thus, we restrict our consideration to a subgroup which is included in the domain of Cayley transformation. Then the problem is transformed as follows. To the rotationally invariant measure on this subgroup what measure corresponds on a suitable class of infinite dimensional skew-symmetric operators'? In order to solve it we first consider the Cayley image of Haar measure in the ^-dimensional case and second construct a finitely additive measure as the limit of n—>oo. Lastly we discuss the countably additive extension of so obtained measure. I like to express my thanks to Prof. H. Yoshizawa for the constant encouragement. Also I thank deeply to Prof. Y. Yamasaki and Prof. T. Hirai for their useful suggestions. § 1. Some properties of Cayley transformation in the finite dimensional case.