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Analytic $ m$-isometries without the wandering subspace property


Authors: Akash Anand, Sameer Chavan and Shailesh Trivedi
Journal: Proc. Amer. Math. Soc. 148 (2020), 2129-2142
MSC (2010): Primary 47B37; Secondary 47A15, 05C20
DOI: https://doi.org/10.1090/proc/14894
Published electronically: February 4, 2020
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Abstract: The wandering subspace problem for an analytic norm-increasing $ m$-isometry $ T$ on a Hilbert space $ \mathcal {H}$ asks whether every $ T$-invariant subspace of $ \mathcal {H}$ can be generated by a wandering subspace. An affirmative solution to this problem for $ m=1$ is ascribed to Beurling-Lax-Halmos, while that for $ m=2$ is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic $ 3$-isometries which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one-dimensional space, their norms can be made arbitrarily close to $ 1$. We also show that if the wandering subspace property fails for an analytic norm-increasing $ m$-isometry, then it fails miserably in the sense that the smallest $ T$-invariant subspace generated by the wandering subspace is of infinite codimension.


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Additional Information

Akash Anand
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
Email: akasha@iitk.ac.in

Sameer Chavan
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
Email: chavan@iitk.ac.in

Shailesh Trivedi
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
Email: shailtr@iitk.ac.in

DOI: https://doi.org/10.1090/proc/14894
Keywords: Wandering subspace property, Wold-type decomposition, weighted shift, one-circuit directed graphs
Received by editor(s): July 31, 2019
Received by editor(s) in revised form: October 2, 2019
Published electronically: February 4, 2020
Additional Notes: The work of the third author was supported through the Inspire Faculty Fellowship DST/INSPIRE/04/2018/000338
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2020 American Mathematical Society