Abstract
It is well known that a non-constant complex-valued function f defined on the open unit disk \({\mathbb{D}}\) of the complex plane is an analytic self-mapping of \({\mathbb{D}}\) if and only if Pick matrices \({\left[ (1 - f(z_i)\overline{f(z_j)})/\right.\left.(1 - z_i\overline{z}_j)\right]_{i,j=1}^n}\) are positive semidefinite for all choices of finitely many points \({z_{i} \in \mathbb{D}}\). A stronger version of the “if” part was established by Hindmarsh (Pac J Math 27:527–531, 1968): if all 3 × 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.
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The research was partly supported by the Summer Research Award from the College of William and Mary.
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Bolotnikov, V. Pick Matricies and Quaternionic Power Series. Integr. Equ. Oper. Theory 80, 293–302 (2014). https://doi.org/10.1007/s00020-014-2173-6
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DOI: https://doi.org/10.1007/s00020-014-2173-6