Abstract. We consider local symmetric semigroups of Hilbert space operators. For an open semigroup ${\eufm S}$ in some topological group and a dense subsemigroup ${\eufm S}'$ of ${\eufm S}$, these are semigroups of unbounded selfadjoint operators $(H(t))_{t \in{\eufm S}'}$ that admit local continuous extensions to open subsets of ${\eufm S}$. We study the possibility to continuously extend $H(\cdot )$ to a semigroup of selfadjoint operators defined for all $t \in{\eufm S}$ in several settings. Integral representation formulae for the extended semigroups $(H(t))_{t \in{\eufm S}}$ by means of real characters of ${\eufm S}$ are established. Our proofs rely on graph limits of selfadjoint operators, commutativity of unbounded operators and semigroup techniques, among others.

AMS Subject Classification (1991): 47D03, 47B15, 47B25

Keyword(s): local semigroups of operators, integral representation, selfadjoint operators

Received May 14, 2010, and in revised form December 17, 2010. (Registered under 35/2010.)