Existence of Hypercyclic Algebras in Fréchet Algebras

Abstract

We prove that every separable infinite-dimensional Fréchet algebra X admits a continuous linear operator T supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} of X can be prescribed as an invariant hypercyclic algebra. When X admits a continuous norm, the operator can additionally be chosen in the form T=I+K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=I+K$$\end{document}, where K is nuclear, so that A=spanOrb(a,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}= {{\,\textrm{span}\,}}{{\,\textrm{Orb}\,}}(a,T)$$\end{document} for any prescribed a∈A\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in \mathcal {A}\backslash \{0\}$$\end{document}.