Expressing infinite matrices over rings as products of involutions

Abstract

Let K be an arbitrary field and R be an arbitrary associative ring with identity 1. Słowik in [12] proved that each matrix of ±UT(∞,K) (the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or −1) can be expressed as a product of at most five involutions. In this article, we extend the investigate to an arbitrary associative ring R with identity 1. Our conclusion is that every element of ±UT(∞,R) can be expressed as a product of at most four involutions. We also prove that for the complex field every element of ΩT(∞,C) (the group of upper triangular infinite matrices whose entries lying on the main diagonal satisfy aa‾=1) can be expressed as a product of at most three coninvolutions (matrices satisfying AA‾=I).