Reducing subspaces for rank-one perturbations of normal operaators



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Gallardo Gutiérrez, Eva A. and González Doña, Javier (2022) Reducing subspaces for rank-one perturbations of normal operaators. Proceedings of the Royal Society of Edinburgh: Section A Mathematics . pp. 1-33. ISSN 0308-2105

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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.

Item Type:Article
Uncontrolled Keywords:Reducing subspaces; Rank-one perturbation of diagonal operators; Rank-one of normal operators
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:74481
Deposited On:09 Sep 2022 07:36
Last Modified:13 Sep 2022 10:11

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