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Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

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Abstract

We study several classes of indecomposable representations of quivers on infinite-dimensional Hilbert spaces and their relation. Many examples are constructed using strongly irreducible operators. Some problems in operator theory are rephrased in terms of representations of quivers. We shall show two kinds of constructions of quite non-trivial indecomposable Hilbert representations (H, f) of the Kronecker quiver such that \({End(H,f) = \mathbb{C} I}\) which is called transitive. One is a perturbation of a weighted shift operator by a rank-one operator. The other one is a modification of an unbounded operator used by Harrison,Radjavi and Rosenthal to provide a transitive lattice.

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Authors and Affiliations

  1. Institute of Education and Research, Koshien University, Takarazuka, Hyogo, 665-0006, Japan

    Masatoshi Enomoto

  2. Department of Mathematical Sciences, Kyushu University, Motooka, Fukuoka, 819-0395, Japan

    Yasuo Watatani

Authors
  1. Masatoshi Enomoto
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  2. Yasuo Watatani
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Correspondence to Yasuo Watatani.

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

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Enomoto, M., Watatani, Y. Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces. Integr. Equ. Oper. Theory 83, 563–587 (2015). https://doi.org/10.1007/s00020-015-2228-3

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  • DOI: https://doi.org/10.1007/s00020-015-2228-3

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