Livshits, LeoMacDonald, GordonMarcoux, Laurent W.Radjavi, Heydar2018-07-112018-07-112018-08-15https://dx.doi.org/10.1016/j.jfa.2018.04.002http://hdl.handle.net/10012/13469The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.jfa.2018.04.002 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/Given a complex, separable Hilbert space H, we characterize those operators for which ‖PT(I−P)‖=‖(I−P)TP‖ for all orthogonal projections P on H. When H is finite-dimensional, we also obtain a complete characterization of those operators for which rank(I−P)TP=rankPT(I−P) for all orthogonal projections P. When H is infinite-dimensional, we show that any operator with the latter property is normal, and its spectrum is contained in either a line or a circle in the complex plane.enAttribution-NonCommercial-NoDerivatives 4.0 InternationalCornersNormal operatorsReductive operatorsUnitary operatorsArticle