Gallardo Gutiérrez, Eva A.González Doña, Francisco Javier2024-07-242024-07-242023-08-01Gallardo-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.2022.510308-21051473-7124 (eISSN)https://hdl.handle.net/10016/44213We 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.33© The Author(s), 2022Published 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.Reducing subspacesRank-one perturbation of diagonal operatorsRank-one perturbation of normal operatorsresearch articleMatemáticashttps://doi.org/10.1017/prm.2022.51open access139141423Proceedings of the Royal Society of Edinburgh. Section A: Mathematics153AR/0000033345