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.)
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