Finite Blaschke products over quaternions: unitary realizations and zero structure

Abstract

We consider power series over the skew field $\mathbb{H}$ of real quaternions which are analogous to finite Blaschke products in the classical complex setting. Several intrinsic characteriztions of such series are given in terms of their coefficients as well as in terms of their left and right values. We also discuss the zero structure of finite Blaschke products including left/right zeros and their various multiplicities. We show how to construct a finite Blaschke product with prescribed zero structure. In particular, given a quaternion polynomial $p$ with all zeros less then one in modulus, we explicitly construct a power series $R$ with quaternion coefficients with no zeros such that $pR$ is a finite Blaschke product.