Schatten class and Berezin transform of quaternionic linear operators Fabrizio Colombo Jonathan Gantner Politecnico di Milano Politecnico di Milano Dipartimento di Matematica Dipartimento di Matematica Via E. Bonardi, 9 Via E. Bonardi, 9 20133 Milano, Italy 20133 Milano, Italy
[email protected] [email protected] Tim Janssens University of Antwerp Department of Mathematics Middelheimlaan, 1 2020 Antwerp, Belgium
[email protected] Abstract In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum for quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S-functional calculus. AMS Classification: 47A10, 47A60, 30G35. Key words: Schatten class of operators, Spectral theorem and the S-spectrum, Berezin transform of quater- nionic operators, Weighted Bergman spaces of slice hyperholomorphic functions. arXiv:1511.07308v1 [math.FA] 23 Nov 2015 1 Introduction The study of quaternionic linear operators was stimulated by the celebrated paper of G. Birkhoff and J. von Neumann [13] on the logic of quantum mechanics, where they proved that there are essentially two possible ways to formulate quantum mechanics: the well known one using complex numbers and also one using quaternions.