2023-12-02T15:38:09Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/719552023-08-04T04:32:51Zcom_20.500.14352_14col_20.500.14352_15
Reducing subspaces for rank-one perturbations of normal operaators
Gallardo Gutiérrez, Eva A.
González Doña, Javier
517
Reducing subspaces
Rank-one perturbation of diagonal operators
Rank-one of normal operators
Análisis matemático
1202 Análisis y Análisis Funcional
We 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.
Ministerio de Ciencia e Innovación (MICINN)
Centro de Excelencia Severo Ochoa
FPI
Depto. de Análisis Matemático y Matemática Aplicada
Fac. de Ciencias Matemáticas
Instituto de Ciencias Matemáticas (ICMAT)
TRUE
pub
2023-06-22T10:58:14Z
2023-06-22T10:58:14Z
2022-08-09
journal article
https://hdl.handle.net/20.500.14352/71955
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0308-2105
10.1017/prm.2022.51
https://doi.org/10.1017/prm.2022.51
eng
PID2019-105979GB-I00
SEV-2015-0554; 20205CEX001
PRE 2018-083669
Atribución-NoComercial-SinDerivadas 3.0 España
https://creativecommons.org/licenses/by-nc-nd/3.0/es/
open access
application/pdf
Cambridge University Press