Pick matricies and quaternionic power series

Abstract

It is well known that a non-constant complex-valued function $f$ defined on the open unit disk $\mathbb D$ is an analytic self-mapping of $\D$ if and only if Pick matrices $\left[ (1-f(z_i)\overline{f(z_j)})/(1-z_i\overline{z}_j)\right]_{i,j=1}^n$ are positive semidefinite for all choices of finitely many points $z_i\in\D$. A stronger version of the "if" part was established by Alan Hindmarsh: if all $3\times 3$ Pick matrices are positive semidefinite, then $f$ is an analytic self-mapping of $\mathbb D$. In this paper, we extend this result to the non-commutative setting of power series over quaternions.