Self-Mappings of the Quaternionic Unit Ball: Multiplier Properties, the Schwarz-Pick Inequality, and the Nevanlinna-Pick Interpolation Problem

摘要

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball B into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space H-2 (B). In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case. As an application, we establish the Schwarz-Pick inequality for slice regular self-mappings of B.