On transformations of elliptic spaces

In this work, we deal with a Hermitian space V over an involutory division ring D with valuation. We will require further a generalized Cauchy-Schwartz inequality relative to the given valuation. A parallel between linear transformations of V and infinite matrices over D is given. This allows us to give a criterion on the entries of the matrices of the group C of unitary transformations module infinitesimals. Relative to a given orthogonal basis, the space V is such that for each unitary transformation phi of C, the matrix of phi has entries from a fixed additive subgroup of D. Also, a matrix representation of the group C over the residue division ring is given. (C) 1999 Elsevier Science Inc. All rights reserved.