  We introduce unbounded strongly irreducible operators and transitive
operators. These operators are related to a certain class of indecomposable
Hilbert representations of quivers on infinite-dimensional Hilbert spaces. We
regard the theory of Hilbert representations of quivers is a generalization of
the theory of unbounded operators. A non-zero Hilbert representation of a
quiver is said to be transitive if the endomorphism algebra is trivial. If a
Hilbert representation of a quiver is transitive, then it is indecomposable.
But the converse is not true. Let $\Gamma$ be a quiver whose underlying
undirected graph is an extended Dynkin diagram. Then there exists an
infinite-dimensional transitive Hilbert representation of $\Gamma$ if and only
if $\Gamma$ is not an oriented cyclic quiver.
