Positive and Generalized Positive Real Lemma for Slice Hyperholomorphic Functions

  • Daniel Alpay
  • Fabrizio Colombo
  • Izchak Lewkowicz
  • Irene SabadiniEmail author


In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball.


Positive real functions Kalman–Yakubovich–Popov lemma Slice hyperholomorphic functions Unit ball Half space Negative squares 

Mathematics Subject Classification

Primary 30C40 Secondary 30G35 93B15 



It is a pleasure to thank Prof. Rovnyak for comments on Theorem 5.2.


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Copyright information

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Authors and Affiliations

  1. 1.Faculty of Mathematics, Physics, and Computation, Schmid College of Science and TechnologyChapman UniversityOrangeUSA
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  3. 3.Department of Electrical EngineeringBen-Gurion University of the NegevBeer-ShevaIsrael

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