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On the convexity of the quaternionic essential numerical range

Published online by Cambridge University Press:  15 May 2024

LuÍs Carvalho
Affiliation:
ISCTE - Lisbon University Institute, Lisbon, Portugal Center for Mathematical Analysis, Geometry, and Dynamical Systems Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
Cristina Diogo*
Affiliation:
ISCTE - Lisbon University Institute, Lisbon, Portugal Center for Mathematical Analysis, Geometry, and Dynamical Systems Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
Sérgio Mendes
Affiliation:
ISCTE - Lisbon University Institute, Lisbon, Portugal Centro de Matemática e Aplicações, Universidade da Beira Interior, Covilhã, Portugal
Helena Soares
Affiliation:
ISCTE - Lisbon University Institute, Lisbon, Portugal Centro de Investigação em Matemática e Aplicações, Universidade de Évora, Évora, Portugal
*
Corresponding author: Cristina Diogo, email: cristina.diogo@iscte-iul.pt

Abstract

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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Footnotes

The first and second authors were partially supported by FCT through CAMGSD, projects UIDB/04459/2020 and UIDP/04459/2020. The third author was partially supported by FCT through CMA-UBI, project number UIDB/00212/2020. Lastly, the fourth author was partially supported by FCT through CIMA, project number UIDB/04674/2020.

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