On the approximate point spectrum of the Bergman space Cesàro operator
\textit{A. G. Siskakis} [J. Lond. Math. Soc., II. Ser. 36, 153-164 (1987; Zbl 0634.47038)] obtained the spectra and norm bounds for the operators \(C|_{H^p}\) on the Hardy space \(H^p\). The authors (1) identify the spectrum of the Cesàro operator \(\sigma ( C|_{L^p_a})\) on the Bergman spaces \(L^p_a(D)\) for \(p>1\); NEWLINENEWLINENEWLINE(2) prove that for \(p>2\) the approximate point spectrum of \(C|_{L^p_a}\) is the boundary of the spectrum; i.e., \(\sigma_{ap}(C|_{L^p_a}) = \partial B(\frac p4, \frac p4)\); NEWLINENEWLINENEWLINE(3) for \(p \geq 4\), give a growth condition on the resolvent and obtain as a consequence that \(C|_{L^p_a}\) has Bishop's property \((\beta)\). The basic tool used to investigate the Cesàro operator is the semigroup analysis of Siskakis.
scientific article; zbMATH DE number 1652732