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We give a generalization of the Beurling-Lax theorem both in the complex andquaternionic settings. We consider in the first case functions meromorphic inthe right complex half-plane, and functions slice hypermeromorphic in the rightquaternionic half-space in the second case. In both settings we also discuss aunified framework, which includes both the disk and the half-plane for thecomplex case and the open unit ball and the half-space in the quaternionicsetting.

Beurling-Lax type theorems in the complex and quaternionic setting: the half-space case