ExamplesOfBoundedAndUnboundedOperators 2005-05-21 13:10:00 2007-09-15 04:53:12 Example operator norm % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\F}{\mathbbmss{F}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} The aim of this page is to list examples of \PMlinkname{bounded}{BoundedOperator} and unbounded linear operators. \subsubsection*{Bounded} \begin{itemize} \item Identity operator, Zero operator \item Shift operators on $\ell^p$ \item A linear operator is continuous if and only if it is bounded (see \PMlinkname{this page}{ContinuousLinearMapping}). \item Any isometry is bounded. \item A multiplication operator $h(t) \mapsto f(t) h(t)$, where $f(t)$ is continuous and $h\in L^p[0,1]$. \item An integral operator $h(t) \mapsto \int_0^1 K(t,s) h(s)\,ds$, where $\int_0^1\int_0^1 \abs{K(s,t)}^2\,ds\,dt < \infty$ and $h\in L^2[0,1]$. In fact this is a Hilbert-Schmidt operator. \item The Volterra operator $h(t) \mapsto \int_0^t h(s)\,ds$, where $h\in L^p[0,1]$. \end{itemize} \subsubsection*{Unbounded} \begin{itemize} \item The derivative is an unbounded operator on the vector space of smooth functions equipped with the $\operatorname{sup}$-norm. \end{itemize}