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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations -
A. Melas, V. Nestoridis (2001)
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(i) The sequence ( T n ) n is called hypercyclic or universal if there exists a vector x 0 ∈ X such that the orbit { T n x 0 : n ∈ N } is dense in Y
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
- ISSN
- 1617-9447
- eISSN
- 2195-3724
- DOI
- 10.1007/s40315-022-00466-0
- Publisher site
- See Article on Publisher Site
Abstract
For a simply connected domain G⊂C\{0}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G\subset {\mathbb {C}}\smallsetminus \{0\}$$\end{document} and for a complex number α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}, with |α|≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\alpha |\ge 1$$\end{document}, we consider the sequence of operators Tα,n(f)(z)=∑j=0nf(j)(z)j!(αz)j,f∈H(G).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} T_{\alpha ,n}(f)(z)=\sum _{j=0}^{n}\frac{f^{(j)}(z)}{j!}(\alpha z)^j, \quad f\in H(G). \end{aligned}$$\end{document}We prove that this sequence of operators is hypercyclic. This problem was first investigated by Bernal-González et al. (J. Math. Anal. Appl. 474:480–491, 2019) and it is related to a question of V. Nestoridis.
Journal
Computational Methods and Function Theory – Springer Journals
Published: Sep 1, 2023
Keywords: Universality; Hypercyclicity; Universal Taylor series; 30K05; 47A16