Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We consider a complement of Gabriel’s theorem for these representations. Let $\mathrm{\Gamma }$ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams $\widetilde{A_n}(n\ge 0)$, $\widetilde{D_n}(n\ge 4)$, $\widetilde{E_6}$,$\widetilde{E_7}$ and $\widetilde{E_8}$, then there exists an indecomposable representation of $\mathrm{\Gamma }$ on separable infinite-dimensional Hilbert spaces.