On the largest analytic set for cyclic operators
The author gives a complete discription of the largest analytic set for cyclic Hilbert-space operators and explains more about bounded point evaluations from the point of view of local spectral theory. Also, the author proves for every cyclic Banach-space operator \(T\in B(X)\) possessing Bishop's property \((\beta)\), that the set \(\sigma(T)\setminus \sigma_{ap}(T)\) is the open set of complex numbers \(\lambda\in\mathbb{C}\) for which there exists a nonzero analytic function \(\varphi:V\to X^*\) on some open disc \(V\) centered at \(\lambda\) satisfying \((T^*- \alpha)\phi(\alpha)= 0\) for all \(\alpha\) in \(V\). By this result, it is shown that densely similar cyclic Banach-space operators possessing Bishop's property \((\beta)\) have the same approximate point spectra, which generalizes Theorem 4 of the article by \textit{M. Raphael} [Indiana Univ. Math. J. 31, 243--246 (1982; Zbl 0455.47020)].
scientific article; zbMATH DE number 2012526