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A hyponormal weighted shift on a directed tree whose square has trivial domain


Authors: Zenon Jan Jabłoński, Il Bong Jung and Jan Stochel
Journal: Proc. Amer. Math. Soc. 142 (2014), 3109-3116
MSC (2010): Primary 47B37, 47B20; Secondary 47A05
Published electronically: May 21, 2014
MathSciNet review: 3223367
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Abstract: It is proved that, up to isomorphism, there are only two directed trees that admit a hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable (i.e., countably infinite) directed trees, one with root, the other without, whose every vertex has an enumerable set of successors. An example of a nonzero hyponormal composition operator in an $ L^2$-space whose square has trivial domain is established.


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Additional Information

Zenon Jan Jabłoński
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland
Email: Zenon.Jablonski@im.uj.edu.pl

Il Bong Jung
Affiliation: Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
Email: ibjung@knu.ac.kr

Jan Stochel
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland
Email: Jan.Stochel@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2014-12112-5
Keywords: Directed tree, weighted shift on a directed tree, hyponormal operator, trivial domain of square
Received by editor(s): May 6, 2011
Received by editor(s) in revised form: September 18, 2012
Published electronically: May 21, 2014
Additional Notes: The research of the first and third authors was supported by the MNiSzW (Ministry of Science and Higher Education) grant No. NN201 546438 (2010-2013)
The second author was supported by the WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (grant No. R32-2009-000-20021-0)
Communicated by: Marius Junge
Article copyright: © Copyright 2014 American Mathematical Society