Journal of Operator Theory

Volume 90, Issue 1, Summer 2023 pp. 191-208.

Bounded point evaluation for operators with the wandering subspace property

Authors: Shailesh Trivedi
Author institution: Department of Mathematics, Birla Institute of Technology and Science, Pilani-333031, India

Summary: We extend and study the notion of bounded point evaluation introduced by Williams for a cyclic operator to the class of operators with the wandering subspace property. We characterize the set $\mathrm{bpe}(T)$ of all bounded point evaluations for an operator $T$ with the wandering subspace property in terms of the invertibility of certain projections. Further, we determine $\mathrm{bpe}(T)$ and $\mathrm{abpe}(T)$ for a left-invertible operator $T$ with the wandering subspace property. We also give examples of left-invertible operators $T$ with the wandering subspace property for which $\mathbb D (0, r(T')^{-1} ) \subsetneqq \mathrm{abpe}(T) \subseteq \mathrm{bpe}(T)$, where $r(T')$ is the spectral radius of the Cauchy dual $T'$ of $T$.

DOI: http://dx.doi.org/10.7900/jot.2021nov05.2354
Keywords: Bounded point evaluation, wandering subspace property, weighted\break shift, directed graph.

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