Symmetric closed operators commuting with a unitary type I representation of finite multiplicity are self-adjoint (Q1203578)

In the first part of this paper we prove the following: Theorem. Let \(\tau\) be a type I unitary representation of a group \(G\) whose direct integral decomposition has (a.e.) finite multiplicity. Let \((T, {\mathcal D}_T)\) be a densely defined symmetric operator such that \(\tau (x) {\mathcal D}_T= {\mathcal D}_T\) and \(\tau (x) Tf= T\tau (x)f\) for all \(x\in G\) and \(f\in {\mathcal D}_T\). Then \(T\) is essentially selfadjoint. In the second part of the paper we analyze commutative algebras of unbounded invariant operators such as occur in the papers [\textit{E. P. van den Ban}, Ark. Mat. 25, 175-187 (1987; Zbl 0645.43009)]\ and [\textit{L. Corwin} and \textit{F. P. Greenleaf}, Commun. Pure Appl. Math. 45, No. 6, 681-748 (1992; Zbl 0812.43004)].