Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces

zbMath1509.47024arXiv1603.07836MaRDI QIDQ6271855

Masatoshi Enomoto, Yasuo Watatani

Publication date: 25 March 2016

Abstract: 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.

Mathematics Subject Classification ID

Invariant subspaces of linear operators (47A15) Representations of quivers and partially ordered sets (16G20) Structure theory of linear operators (47A65) Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.) (46C07)

This page was built for publication: Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces