Pick matricies and quaternionic power series

TL;DR

This paper extends the classical Pick matrix characterization of analytic self-maps of the disk to the quaternionic setting, establishing positivity conditions for quaternionic power series.

Contribution

It generalizes the Pick matrix positivity criterion from complex functions to quaternionic power series, providing a non-commutative analogue.

Findings

Positive semidefiniteness of quaternionic Pick matrices characterizes quaternionic self-maps.

Extension of Hindmarsh's 3x3 Pick matrix result to quaternions.

Framework for analyzing quaternionic power series using Pick matrices.

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.