On lie group representations and operator ranges

Oliva Maza, J. (Universidad de Zaragoza)

Resumen: . 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.

Idioma: Inglés

DOI: 10.1090/proc/15554

Año: 2021

Publicado en: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 149, 10 (2021), 4317–4329
ISSN: 0002-9939

Factor impacto JCR: 0.971 (2021)

Categ. JCR: MATHEMATICS rank: 166 / 333 = 0.498 (2021) - Q2 - T2
Categ. JCR: MATHEMATICS, APPLIED rank: 207 / 267 = 0.775 (2021) - Q4 - T3

Factor impacto CITESCORE: 1.7 - Mathematics (Q3)

Factor impacto SCIMAGO: 0.891 - Mathematics (miscellaneous) (Q1) - Applied Mathematics (Q1)

Financiación: info:eu-repo/grantAgreement/ES/MINECO/BES-2017-081552
Financiación: info:eu-repo/grantAgreement/ES/MINECO/MTM2016-77710-P

Tipo y forma: Article (PostPrint)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)