On Lie group representations and operator ranges.

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. [ABSTRACT FROM AUTHOR]