\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage}

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\interior{\operatorname{int}}
\def\dist{\operatorname{dist}}
\def\Ker{\operatorname{Ker}}
\def\Range{\operatorname{Range}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\T{{\mathbb T}}
\def\D{{\mathcal D}}

\begin{document}

\title{Functional Analysis}
\author{Homework 6}
\date{due April 1, 2009}
\maketitle

\begin{enumerate}

\item Let $E, F$ be Banach spaces and $A \colon \D(A) \subset E \to F$
  an unbounded operator.  Show that the following are equivalent.

  \begin{enumerate}
  \item $A$ is closeable.  (I.e., $A$ has a closed extension.)
  \item $\overline{\Gamma(A)}$ is the graph of a closed operator
  $\overline A$, the smallest closed extension of $A$.
  \item $(0,v) \in \overline{\Gamma(A)}$ implies that $v=0$.
  \end{enumerate}


\item Let $E$ be a Banach space and $A, B$ operators from $E$ into
  itself with domains $\D(A)$ and $\D(B)$.
  \begin{enumerate}
  \item What is $\D(AB)$?
  \item Show that $A$ closed and $B$ bounded implies $AB$ closed.
  \item Show that $A$ bounded and $B$ closed does not imply that $AB$
  is closed.
  \end{enumerate}


\item Consider $\partial_r$ as an unbounded operator on
  $L^2((0,\infty),r^2 \, \d r)$ with domain
\[
  \D(\partial_r) = \{ u \in L^2((0,\infty),r^2 \, \d r) \colon
  u \in AC((0,\infty)), \partial_r u \in
  L^2((0,\infty),r^2 \, \d r) \} \,.
\]
\begin{enumerate}
\item Show that $r^{1/2} \, u(r) \to 0$ as $r \to \infty$ for every $u
  \in \D(\partial_r)$.
\item Compute the adjoint of $\partial_r$ and its domain.
\end{enumerate}

\item Consider $\partial_x$ acting on $L^2(\T)$ with
\[
  \D(\partial_x) = \{u \in L^2(\T) \colon u \in AC(\T), \partial_x u
  \in L^2(\T) \} \,.
\]
  \begin{enumerate}
  \item Show that $\partial_x^*=-\partial_x$ on $\D(\partial_x)$ and
  that $\D(\partial_x^*) \supset \D(\partial_x)$.
\item Find $\Range(\partial_x)$ and $\Ker(\partial_x)$, then conclude
  that $\D(\partial_x^*) = \D(\partial_x)$.
  \end{enumerate}
\end{enumerate}
\end{document}
