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Reducing subspaces for rank-one perturbations of normal operators

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Análisis Aplicadoes
dc.contributor.authorGallardo Gutiérrez, Eva A.
dc.contributor.authorGonzález Doña, Francisco Javier
dc.contributor.funderMinisterio de Ciencia e Innovación (España)es
dc.contributor.funderConsejo Superior de Investigaciones Científicas (España)es
dc.contributor.funderAgencia Estatal de Investigación (España)es
dc.date.accessioned2024-07-24T10:54:57Z
dc.date.available2024-07-24T10:54:57Z
dc.date.issued2023-08-01
dc.description.abstractWe study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator T=D+u¿v (uniquely determined by such expression) is contained in a line, since in such a case T has a reducing subspace if and only if T is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators T=D+u¿v with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of T is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces M of T such that the restriction T¿M is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.es
dc.description.sponsorshipBoth authors are partially supported by Plan Nacional I+D grant no. PID2019105979GB-I00, Spain, the Spanish Ministry of Science and Innovation, through the `Severo Ochoa Programme for Centres of Excellence in R&D' (SEV-20150554) and from the Spanish National Research Council, through the `Ayuda extraordinaria a Centros de Excelencia Severo Ochoa' (20205CEX001). The second author also acknowledges support of the Grant SEV-2015-0554-18-3 funded by: MCIN/AEI/10.13039/501100011033.en
dc.format.extent33
dc.identifier.bibliographicCitationGallardo-Gutiérrez, E. A., & González-Doña, F. J. (2023). Reducing subspaces for rank-one perturbations of normal operators. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 153(4), 1391–1423. doi:10.1017/prm.2022.51en
dc.identifier.doihttps://doi.org/10.1017/prm.2022.51
dc.identifier.issn0308-2105
dc.identifier.issn1473-7124 (eISSN)
dc.identifier.publicationfirstpage1391
dc.identifier.publicationissue4
dc.identifier.publicationlastpage1423
dc.identifier.publicationtitleProceedings of the Royal Society of Edinburgh. Section A: Mathematicsen
dc.identifier.publicationvolume153
dc.identifier.urihttps://hdl.handle.net/10016/44213
dc.identifier.uxxiAR/0000033345
dc.relation.projectIDGobierno de España. PID2019-105979GB-I00es
dc.relation.projectIDConsejo Superior de Investigaciones Científicas. 20205CEX001es
dc.relation.projectIDGobierno de España. SEV-2015-0554-18-3es
dc.rights© The Author(s), 2022en
dc.rightsPublished by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.en
dc.rights.accessRightsopen accessen
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en
dc.subject.ecienciaMatemáticas
dc.subject.otherReducing subspacesen
dc.subject.otherRank-one perturbation of diagonal operatorsen
dc.subject.otherRank-one perturbation of normal operatorsen
dc.titleReducing subspaces for rank-one perturbations of normal operatorses
dc.typeresearch articleen
dc.type.hasVersionVoRen
dspace.entity.typePublicationen
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