000148456 001__ 148456 000148456 005__ 20250117162507.0 000148456 0247_ $$2doi$$a10.1090/proc/15554 000148456 0248_ $$2sideral$$a124447 000148456 037__ $$aART-2021-124447 000148456 041__ $$aeng 000148456 100__ $$0(orcid)0000-0001-8546-5883$$aOliva Maza, J.$$uUniversidad de Zaragoza 000148456 245__ $$aOn lie group representations and operator ranges 000148456 260__ $$c2021 000148456 5203_ $$a. 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. 000148456 536__ $$9info:eu-repo/grantAgreement/ES/MINECO/BES-2017-081552$$9info:eu-repo/grantAgreement/ES/MINECO/MTM2016-77710-P 000148456 540__ $$9info:eu-repo/semantics/closedAccess$$aby-nc-nd$$uhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/ 000148456 590__ $$a0.971$$b2021 000148456 591__ $$aMATHEMATICS$$b166 / 333 = 0.498$$c2021$$dQ2$$eT2 000148456 591__ $$aMATHEMATICS, APPLIED$$b207 / 267 = 0.775$$c2021$$dQ4$$eT3 000148456 592__ $$a0.891$$b2021 000148456 593__ $$aMathematics (miscellaneous)$$c2021$$dQ1 000148456 593__ $$aApplied Mathematics$$c2021$$dQ1 000148456 594__ $$a1.7$$b2021 000148456 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/acceptedVersion 000148456 7102_ $$12006$$2015$$aUniversidad de Zaragoza$$bDpto. Matemáticas$$cÁrea Análisis Matemático 000148456 773__ $$g149, 10 (2021), 4317–4329$$pProc. Am. Math. Soc.$$tPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY$$x0002-9939 000148456 8564_ $$s249378$$uhttps://zaguan.unizar.es/record/148456/files/texto_completo.pdf$$yPostprint 000148456 8564_ $$s1731176$$uhttps://zaguan.unizar.es/record/148456/files/texto_completo.jpg?subformat=icon$$xicon$$yPostprint 000148456 909CO $$ooai:zaguan.unizar.es:148456$$particulos$$pdriver 000148456 951__ $$a2025-01-17-14:36:09 000148456 980__ $$aARTICLE