Marcoux, L.W.Omladič, M.Popov, A.I.Radjavi, H.Yahaghi, B.2020-04-012020-04-012016https://doi.org/10.1007/s00233-015-9772-7http://hdl.handle.net/10012/15730This is a post-peer-review, pre-copyedit version of an article published in Semigroup Forum. The final authenticated version is available online at: https://doi.org/10.1007/s00233-015-9772-7Let 𝜑 be a linear functional of rank one acting on an irreducible semigroup S of operators on a finite- or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of 𝜑 cannot be a singleton. We start a study of possible finite ranges for such functionals. In particular, we prove that in certain cases, the existence of a single such functional 𝜑 with a two-element range yields valuable information on the structure of S.enirreducible operator semigroupsranges of vector statessemigroups of small rankcompact groups of unitary matricesselfadjoint semigroupsArticle