Bounded Sequences, Orbits and Invariant Subspaces

Let {xn} be a sequence of complex numbers and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^nx_j = \sum\nolimits_{k=0}^{n} (-1)^k\break\left(\begin{array}{l}n\\ k\\\end{array} \right)x_{n-k+j}}$$\end{document} . In this paper, we will show that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ |x_n| = O(n^k)}$$\end{document} , as n → ∞ for some positive integer k, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n|\Delta^n x_j|^{\frac{1}{n}} \to 0}$$\end{document} as n→ ∞, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{k+1} x_j = 0}$$\end{document} . More importantly, applications to the orbits of operators and invariant subspace problem are also given; this helps to improve former results obtained by Gelfand–Hille, Mbekhta–Zemánek and others.