Reducing subspaces for rank-one perturbations of normal operaators

dc.contributor.authorGallardo Gutiérrez, Eva A.
dc.contributor.authorGonzález Doña, Javier
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. At 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 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.
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.facultyInstituto de Ciencias Matemáticas (ICMAT)
dc.description.sponsorshipMinisterio de Ciencia e Innovación (MICINN)
dc.description.sponsorshipCentro de Excelencia Severo Ochoa
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dc.journal.titleProceedings of the Royal Society of Edinburgh: Section A Mathematics
dc.publisherCambridge University Press
dc.relation.projectIDSEV-2015-0554; 20205CEX001
dc.relation.projectIDPRE 2018-083669
dc.rightsAtribución-NoComercial-SinDerivadas 3.0 España
dc.rights.accessRightsopen access
dc.subject.keywordReducing subspaces
dc.subject.keywordRank-one perturbation of diagonal operators
dc.subject.keywordRank-one of normal operators
dc.subject.ucmAnálisis matemático
dc.subject.unesco1202 Análisis y Análisis Funcional
dc.titleReducing subspaces for rank-one perturbations of normal operaators
dc.typejournal article
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