 

Masatoshi Enomoto and
Yasuo Watatani
Unbounded strongly irreducible operators and transitive representations of quivers
on infinitedimensional Hilbert spaces
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print


Published: 
October 8, 2019. 
Keywords: 
unbounded strongly irreducible operators, transitive operators,
quiver, indecomposable representation, Hilbert space. 
Subject: 
Primary 47A65, Secondary 46C07, 47A15, 16G20. 


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 infinitedimensional Hilbert spaces.
We regard the theory of Hilbert representations of quivers as a generalization of the theory of unbounded
operators. A nonzero 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 Γ be a quiver whose underlying undirected graph is an extended
Dynkin diagram. Then there exists an infinitedimensional transitive Hilbert representation of Γ
if and only if Γ is not an oriented cyclic quiver. 

Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers 23654053 and 25287019.


Author information
Masatoshi Enomoto:
Institute of Education and Research
Koshien University
Takarazuka, Hyogo 6650006, Japan
enomotoma@hotmail.co.jp
Yasuo Watatani:
Department of Mathematical Sciences
Kyushu University
Motooka, Fukuoka, 8190395, Japan
watatani@math.kyushuu.ac.jp

