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Invertible weighted shift operators which are $ m$-isometries


Authors: Muneo Chō, Schôichi Ôta and Kôtarô Tanahashi
Journal: Proc. Amer. Math. Soc. 141 (2013), 4241-4247
MSC (2010): Primary 47B37
DOI: https://doi.org/10.1090/S0002-9939-2013-11701-6
Published electronically: August 5, 2013
MathSciNet review: 3105867
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Abstract: For a bounded linear operator $ T$ on a complex Hilbert space $ {\mathcal H}$, let $ \Delta _{T,m} = \sum _{k=0}^m (-1)^k \ \begin {pmatrix}m \\ k \end{pmatrix} \, T^{*m-k}T^{m-k} \ \ $$ \mbox {for} \ m \in {\mathbb{N}}$. $ T$ is called an $ m$-isometry if $ \Delta _{T,m}=0$. In this paper, for every even number $ m$, we give an example of invertible $ (m+1)$-isometry which is not an $ m$-isometry. Next we show that if $ T$ is an $ m$-isometry, then the operator $ \Delta _{T, m-1}$ is not invertible.


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Additional Information

Muneo Chō
Affiliation: Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan
Email: chiyom01@kanagawa-u.ac.jp

Schôichi Ôta
Affiliation: Department of Content and Creative Design, Kyushu University, Fukuoka 815-8540, Japan
Email: ota@design.kyushu-u.ac.jp

Kôtarô Tanahashi
Affiliation: Department of Mathematics, Tohoku Pharmaceutical University, Sendai 981-8558, Japan
Email: tanahasi@tohoku-pharm.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2013-11701-6
Keywords: Hilbert space, $m$-isometry, bilateral weighted shift
Received by editor(s): October 2, 2011
Received by editor(s) in revised form: January 31, 2012
Published electronically: August 5, 2013
Additional Notes: The first author’s research was partially supported by Grant-in-Aid for Scientific Research, No. 20540192
The second author’s research was partially supported by Grant-in-Aid for Scientific Research, No. 20540178
The third author’s research was partially supported by Grant-in-Aid for Scientific Research, No. 20540184
Communicated by: Richard Rochberg
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.