Bounded point evaluation for operators with the wandering subspace property

TL;DR

This paper generalizes the concept of bounded point evaluations from cyclic operators to those with the wandering subspace property, providing characterizations and examples for such operators, especially left-invertible ones.

Contribution

It extends the characterization of bounded point evaluations to a broader class of operators with the wandering subspace property, including explicit descriptions for left-invertible cases.

Findings

Characterization of bpe(T) via invertibility of projections

Determination of bpe(T) and abpe(T) for left-invertible operators

Examples illustrating the relationship between abpe(T) and bpe(T)

Abstract

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 $bpe(T)$ of all bounded point evaluations for an operator $T$ with the wandering subspace property in terms of the invertibility of certain projections. This result generalizes the earlier established characterization of $bpe(T)$ for a finitely cyclic operator $T$. Further, if $T$ is a left-invertible operator with the wandering subspace property, then we determine the $bpe(T)$ and the set $abpe(T)$ of all analytic bounded point evaluations for $T$. We also give examples of left-invertible operator $T$ with the wandering subspace property for which $\mathbb D\big(0, r(T')^{-1}\big) \subsetneqq abpe(T) \subseteq bpe(T)$, where $r(T')$ is the spectral radius of the Cauchy dual $T'$ of $T$.