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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Perturbation of normal quaternionic operators


Authors: Paula Cerejeiras, Fabrizio Colombo, Uwe Kähler and Irene Sabadini
Journal: Trans. Amer. Math. Soc. 372 (2019), 3257-3281
MSC (2010): Primary 47A10, 47A60
DOI: https://doi.org/10.1090/tran/7749
Published electronically: January 4, 2019
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Abstract: 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 $ S$-resolvent operators and the $ S$-spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the $ S$-spectrum of a quaternionic linear operator $ T$ 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 $ S$-spectrum and of slice hyperholomorphicity of the $ S$-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.


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Additional Information

Paula Cerejeiras
Affiliation: CIDMA, Departamento de Matemática, Universidade de Aveiro, Campus de Santiago, P-3810-193 Aveiro, Portugal
Email: pceres@ua.pt

Fabrizio Colombo
Affiliation: Politecnico di Milano, Dipartimento di Matematica, Via Edoardo Bonardi, 9, 20133 Milano, Italy
Email: fabrizio.colombo@polimi.it

Uwe Kähler
Affiliation: CIDMA, Departamento de Matemática, Universidade de Aveiro, Campus de Santiago, P-3810-193 Aveiro, Portugal
Email: ukaehler@ua.pt

Irene Sabadini
Affiliation: Politecnico di Milano, Dipartimento di Matematica, Via Edoardo Bonardi, 9, 20133 Milano, Italy
Email: irene.sabadini@polimi.it

DOI: https://doi.org/10.1090/tran/7749
Keywords: Perturbation of quaternionic operators, normal operators, $S$-spectrum, $S$-resolvent operators, slice hyperholomorphic functions
Received by editor(s): October 29, 2017
Received by editor(s) in revised form: July 14, 2018
Published electronically: January 4, 2019
Additional Notes: The work of the first and third authors was partially supported by Portuguese funds through CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2019.
Article copyright: © Copyright 2018 American Mathematical Society