Analytic $m$-isometries without the wandering subspace property
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- by Akash Anand, Sameer Chavan and Shailesh Trivedi PDF
- Proc. Amer. Math. Soc. 148 (2020), 2129-2142 Request permission
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.References
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Additional Information
- Akash Anand
- Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
- MR Author ID: 888355
- Email: akasha@iitk.ac.in
- Sameer Chavan
- Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
- MR Author ID: 784696
- Email: chavan@iitk.ac.in
- Shailesh Trivedi
- Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
- MR Author ID: 1064875
- Email: shailtr@iitk.ac.in
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 2129-2142
- MSC (2010): Primary 47B37; Secondary 47A15, 05C20
- DOI: https://doi.org/10.1090/proc/14894
- MathSciNet review: 4078097