On the existence of subspace-hypercyclic operators and a new criteria for subspace-hypercyclicity

A bounded linear operator T on a Banach space X is called subspace-hypercyclic if there is a subspace M⊊X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \subsetneq X$$\end{document} and a vector x∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in X$$\end{document} such that orb(x,T)∩M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{orb}\,}}{(x,T)} \cap M$$\end{document} is dense in M. We show that every Banach space supports subspace-hypercyclic operators and, if the space is separable, the operator obtained is also weakly mixing. Additionally, we provide a new criteria for subspace-hypercyclicity, generalizing a previous result from Le.