Property \((\omega)\) and the single-valued extension property
Let \(X\) be a complex Banach space and \(T: X\to X\) be a continuous linear operator. Following \textit{V. Rakočević} [Mat. Vesn. 37, 423--426 (1985; Zbl 0596.47001)], the operator \(T\) is said to satisfy the property \((\omega)\) if \(\sigma_a(T)\setminus \sigma_{ea}(T) =\pi_{00} (T)\), where \[\sigma_a(T)=\{\lambda\in\mathbb{C}: T-\lambda I \textrm{ is not a bounded below operator}\},\] \[\sigma_{ae}=\{\lambda\in\mathbb{C}: T -\lambda I \text{ is not an upper semi-Weyl operator}\}\] and \[\pi_{00}(T) =\{\lambda\in\sigma(T): \lambda \text{ is an isolated point of } \sigma(T) \text{ and } 0<{\dim}\ker(T-\lambda I)<\infty\}.\] In the paper under review, the authors first introduce a new notion of spectrum inspired by the single-valued extension property. Using this new spectrum, they then give necessary and sufficient conditions for a continuous linear operator defined on a complex Banach space to satisfy property \((\omega)\). Moreover, the authors give examples which illustrate that there is no relationship between property \((\omega)\) for operators and hypercyclic (or supercyclic) property. They also present some results on the relationship between hypercyclic property (or supercyclic property) and property \((\omega)\).
scientific article