Analytic 𝑚-isometries without the wandering subspace property

Anand, Akash; Chavan, Sameer; Trivedi, Shailesh

doi:10.1090/proc/14894

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.

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