Evaluation points for cyclic operators with Bishop's property (\(\beta\))
Let \(T\) a bounded operator on a Hilbert space \(\mathcal H\). Let \(B(T)\) denote the set of bounded point evaluations, of \(T\) and \(B_a(T)\) the set of analytic point evaluations, of \(T\). \textit{L. R. Williams} [Dyn. Syst. Appl. 3, 103-112 (1994; Zbl 0805.47019)] showed that for any cyclic operator, \(B_a(T) \subset B(T) \setminus \sigma_{ap}(T)\), where \(\sigma_{ap}(T)\) denotes the approximate spectrum. The authors of the present work prove the following precise result: If \(T\) is cyclic and satisfies the Bishop property \(\beta\), then \(B_a(T)=B(T)\setminus \sigma_{ap}(T)\). As a corollary, they also obtain a local spectral version of their theorem. The proof, which is quite elegant, is carried out by relating the analytic point evaluations to a certain generalized resolvent set. Operators that satisfy the Bishop property (\(\beta\)) are, for instance, subnormal operators and decomposable operators. A particular case of the main result for the operator of multiplication by \(z\) in the Hardy space was proved by \textit{T. T. Trent} [Pac. J. Math. 80, 279-292 (1979; Zbl 0405.46043)]. The authors also give sufficient conditions on \(B_a(T)\) under which the spectrum of \(T\) and the local spectrum of \(T\) at any nonzero \(y\in \mathcal H\) coincide. Other interesting results in the paper under review are certain descriptions of the spectral pictures of quasi-similar cyclic operators. For instance, if \(T\) and \(S\) are quasi-similar cyclic operators satisfying Bishop's property (\(\beta\)), then \(T\) and \(S\) have the same approximate spectrum and the same essential spectrum.
scientific article; zbMATH DE number 2057775
scientific article