On Frame Properties of Iterates of a Multiplication Operator

Dynamical sampling that is a relatively new research topic in applied harmonic analysis has attracted considerable attention in recent years. One of the central problems in dynamical sampling is investigation of frame properties for families of elements obtained by iterates of operators. Note that investigation of basicity properties of iterates of operators is problematic even in the case of well known “standard” operators. In this note we consider iterates of the multiplication operator Tφx(t)=φ(t)x(t),x∈L2(a,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\varphi x(t)=\varphi (t)x(t), x \in L_2(a,b)$$\end{document} and show that it cannot form a frame for the space L2(a,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2(a,b)$$\end{document}.