Faber-hypercyclic operators
Let \(X\) be a complex infinite-dimensional separable Banach space and \(T\) be a bounded linear operator on \(X\). Let \(\Omega\) be a bounded domain of the complex plane whose boundary is a closed Jordan curve and \((F_n^{\Omega})_{n\geq 0}\) be the sequence of Faber polynomials of \(\Omega\). Then \(T\) is said to be \(\Omega\)-hypercyclic if there exists a vector \(x\) of \(X\) such that \(\{F_n^{\Omega}(T)x: n\geq 0\}\) is dense in \(X\). If \(\Omega\) is the unit disk \(\mathbb D\), then \(F_n^{\mathbb D}\) is just the basic Taylor polynomials associated to \(\mathbb D\), that is, \(F_n^{\mathbb D}(z)=z^n\). Thus \(\mathbb D\)-hypercyclicity is exactly the same notion as hypercyclicity, which has been studied extensively in recent years. The aim of this paper is to extend the spectral theory of \(\mathbb D\)-hypercyclicity to \(\Omega\)-hypercyclicity. Several remarkable results are established: 1. Under the additional assumption that the boundary of \(\Omega\) is a rectifiable Jordan curve, the authors prove that if \(T\) is \(\Omega\)-hypercyclic, then every connected component of the spectrum of \(T\) meets the boundary of \(\Omega\), which is the analogue of Kitai's necessary condition [\textit{C. Kitai}, Ph.D. Thesis, University of Toronto (1982)]. 2. A form of the Godefroy-Shapiro Criterion [\textit{G. Godefroy} and \textit{J. H. Shapiro}, J. Funct. Anal. 98, No.~2, 229--269 (1991; Zbl 0732.47016)] is derived in this new setting under the only assumption of the rectifiability of the boundary of the domain. 3. The influence of eigenvectors associated to eigenvalues belonging to the boundary of \(\Omega\) on the \(\Omega\)-hypercyclicity of the operator is studied and a ``unimodular counterpart'' of the Godefroy-Shapiro Criterion [\textit{F. Bayart} and \textit{S. Grivaux}, J. Funct. Anal. 226, No.~2, 281--300 (2005; Zbl 1089.47008)] is obtained in this setting for domains with \(C^{1+\alpha}\) boundary. 4. It is proved that, under some reasonable assumptions on the \(\partial\Omega\)-eigenvectors of \(T\) and on the smoothness of \(\partial\Omega\), \(T\) is frequently \(\Omega\)-hypercyclic. This is the exact parallel of a statement of [\textit{F. Bayart} and \textit{S. Grivaux}, Proc. Lond. Math. Soc. (3) 94, No.~1, 181--210 (2007; Zbl 1115.47006)], but its proof does not rely on ergodic theory. An interesting feature in this study is the influence of the geometry of the domain \(\Omega\) and of the smoothness of its boundary on \(\Omega\)-hypercyclicity.
scientific article; zbMATH DE number 5320452
scientific article