Perturbation of normal quaternionic operators

• Published in: Transactions of the American Mathematical Society 2019-09, Vol.372 (5), p.3257-3281
• Main Authors: Cerejeiras, Paula, Colombo, Fabrizio, Kähler, Uwe, Sabadini, Irene
• Format: Article
• Language: English
• Subjects: Research article
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7749

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two SS-resolvent operators and the SS-spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the SS-spectrum of a quaternionic linear operator TT is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of SS-spectrum and of slice hyperholomorphicity of the SS-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.