Analytic vectors and integrability of Lie algebra representations
Let G be a real finite-dimensional Lie group with Lie algebra \({\mathfrak G}\) and let (T,D) be a representation of \({\mathfrak G}\) in a Banach space F. This means, in part, that T(\(\xi)\) is a linear operator in F for every \(\xi\in {\mathfrak G}\) and D is a dense subset of F which is contained in the domain of T(\(\xi)\)T(\(\eta)\) for every \(\xi\),\(\eta\in {\mathfrak G}\). The representation (T,D) is said to be integrable if there exists a strongly continuous local representation R of G in F such that \(D\subset \{f\in F:\) the map \(x\to R(x)f\) is C \(1\}\) and for every \(f\in D\), \(dR(e;\xi)f=T(\xi)f.\) The question entertained by the author is: What conditions are sufficient for (T,D) to be integrable? Theorem. Suppose T(\(\xi)\)D\(\subset D\) for each \(\xi\in {\mathfrak G}\) and there exists a set S generating \({\mathfrak G}\) such that (i) for each \(\alpha\in S\), T(\(\alpha)\) is closable and T(\(\alpha)\) is the generator of a one-parameter group of operators exp(tT(\(\alpha)\)); (ii) given \(\alpha\in S\), each f in D is an analytic vector for T(\(\alpha)\). Then the representation (T,d) is integrable.
strongly continuous local representation
scientific article; zbMATH DE number 4047981
scientific article