Hypercyclicity of translation operators in a reproducing kernel Hilbert space of entire functions induced by an analytic Hilbert-space-valued kernel
In this article, the hypercyclicity of non-trivial translation operators in some general reproducing kernel Hilbert spaces of entire functions is studied. These spaces are obtained by duality in a complex separable Hilbert space \(\mathcal{H}\) by means of an analytic \(\mathcal{H}\)-valued kernel, namely, \(\mathcal{H}_K\). This space inherits the Hilbertian structure from \(\mathcal{H}\). The result is partially based on a result by \textit{K. C. Chan} and \textit{J. H. Shapiro} [Indiana Univ. Math. J. 40, No. 4, 1421--1449 (1991; Zbl 0771.47015)] that states the hypercyclicity from the density of the set of polynomials on \(\mathbb{C}\). In some cases, the space \(\mathcal{H}_K\) can be isometrically isomorphic to a de~Branges space where the set of polynomials is dense. Then, the hypercyclicity of the non-trivial translation operators can be stated. The results are illustrated with an example taken from Hamburger moment problem theory.
scientific article; zbMATH DE number 6502586
scientific article