doi:10.1090/proc/15554engOliva Maza, J.On lie group representations and operator rangesART-2021-124447. In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R be an operator range of a Hilbert space H. Given the set Λ of R-invariant operators, and given a Lie group representation ρ : G → GL(H), we discuss the induced semigroup homomorphism ρ : ρ−1(Λ) → B(R) for the operator range topology on R. In one direction, we work under the assumption ρ−1(Λ) = G, so ρ : G → B(R) is in fact a group representation. In this setting, we prove that ρ is continuous (and smooth) if and only if the tangent map dρ is R-invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H, the set ρ−1(Λ) is neither a group for a large set of nonclosed operator ranges R nor closed for all nonclosed operator ranges R. Both results are proved by means of explicit counterexamples.2021http://zaguan.unizar.es/record/14845610.1090/proc/15554http://zaguan.unizar.es/record/148456oai:zaguan.unizar.es:148456info:eu-repo/grantAgreement/ES/MINECO/BES-2017-081552info:eu-repo/grantAgreement/ES/MINECO/MTM2016-77710-PPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 149, 10 (2021), 4317–4329by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/closedAccess