On some notions of spectra for quaternionic operators and for <i>n</i>-tuples of operators

Fabrizio Colombo; Irene Sabadini

doi:10.1016/j.crma.2012.03.017

Functional Analysis

On some notions of spectra for quaternionic operators and for n-tuples of operators

Presented by: Jean-Pierre Kahane

Fabrizio Colombo ¹; Irene Sabadini ¹

¹ Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 399-402.

Abstract
Résumé

The S-spectrum has been introduced for the definition of the S-functional calculus that includes both the quaternionic functional calculus and a calculus for n-tuples of nonnecessarily commuting operators. The notion of right spectrum for right linear quaternionic operators has been widely used in the literature, especially in the context of quaternionic quantum mechanics. Moreover, several results in linear algebra, like the spectral theorem for quaternionic matrices, involve the right spectrum. In this Note we prove that the two notions of S-spectrum and of right spectrum coincide.

La notion de S-spectre a été introduite pour donner une définition de calcul formel utilisable pour des opérateurs linéaires quaternioniques et pour des n-uples dʼopérateurs non nécessairement commutatifs. La notion de spectre à droite pour des opérateurs linéaires a été largement utilisée dans la littérature, particulièrement dans le cadre de la mécanique quantique. Par la suite, on a établi que de nombreux résultats dʼalgèbre linéaire, comme le théorème spectral pour les matrices quaternioniques, sont reliés au spectre à droite. Dans cette Note on démontre que les deux notions de S-spectre et de spectre à droite coïncident.

Received: 2012-01-03
Accepted: 2012-03-22
Published online: 2012-04-01

DOI: 10.1016/j.crma.2012.03.017

Author's affiliations:

Fabrizio Colombo ¹; Irene Sabadini ¹

¹ Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

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     title = {On some notions of spectra for quaternionic operators and for \protect\emph{n}-tuples of operators},
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Fabrizio Colombo; Irene Sabadini. On some notions of spectra for quaternionic operators and for n-tuples of operators. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 399-402. doi : 10.1016/j.crma.2012.03.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.03.017/

References
Cited by

[1] S. Adler Quaternionic Quantum Field Theory, Oxford University Press, 1995

[2] F. Colombo; I. Sabadini The $F$ -spectrum and the $SC$ -functional calculus, Proc. Roy. Soc. Edinburgh Sect. A, Volume 142 (2012), pp. 1-22

[3] F. Colombo; I. Sabadini; D.C. Struppa Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions, Progress in Mathematics, vol. 289, Birkhäuser/Springer Basel AG, Basel, 2011 (vi+221 pp)

[4] S. De Leo; G. Scolarici Right eigenvalue equation in quaternionic quantum mechanics, J. Phys. A, Volume 33 (2000), pp. 2971-2995

[5] D.R. Farenick; B.A.F. Pidkowich The spectral theorem in quaternions, Linear Algebra Appl., Volume 371 (2003), pp. 75-102

[6] F. Zhang Quaternions and matrices of quaternions, Linear Algebra Appl., Volume 251 (1997), pp. 21-57

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