Cyclic properties of Volterra operator.
An operator \(T\) on a Hilbert space \(H\) is called supercyclic if there exists a vector \(x \in H\) such that the projective orbit \(\{ {\lambda} T^{n}x: {\lambda} \in \mathbb C\), \(n \in \mathbb N \}\) is dense in \(H\). An operator \(T\) is called hypercyclic if there exists a vector whose orbit under \(T\) is dense. An operator \(T\) is called cyclic if the linear span of some orbit is dense. In this paper, the authors deal with a question on supercyclicity of the Volterra operator on \(L^{2}[0,1]\) in \textit{N. H. Salas} [Stud. Math. 135, 55--74 (1999; Zbl 0940.47005)]. The paper consists of two main sections. In section \(2\), they introduce the elementary facts on hypercyclicity and cyclicity of the Volterra operator. In section \(3\), they prove that Volterra and identity plus Volterra operators are not supercyclic on \(L^{2}[0,1]\).
scientific article; zbMATH DE number 2101104