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

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\i{{\mathrm i}}
\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\Z{{\mathbb Z}}
\def\L{{\mathcal L}}
\def\K{{\mathcal K}}
\def\D{{\mathcal D}}

\begin{document}

\title{Functional Analysis}
\author{Homework 8}
\date{due May 4, 2009}
\maketitle

\begin{enumerate}

\item Let $\Omega \subset \R^n$ be open and $K \colon L^2(\Omega) \to
L^2(\Omega)$ a Hilbert--Schmidt operator, i.e.\
\[
  Kf(x) = \int_\Omega k(x,y) \, f(y) \, \d y 
\]
with $k \in L^2(\Omega \times \Omega)$.  Recall that
\[
  \lVert K \rVert_{\text{HS}}^2 = \sum_{j \in \N} \lVert K e_j
  \rVert^2 
\]
for any orthonormal basis $\{e_j\}$ of $L^2(\Omega)$. 

$A \colon L^2(\Omega) \to L^2(\Omega)$ is of \emph{trace class},
if
\[
  \sum_{j \in \N} \lvert \langle A e_j,e_j \rangle \rvert < \infty
  \,.
\]
\begin{enumerate}
\item 
Show that the composition of two Hilbert--Schmidt operators is of
trace class.

\item Assume, in addition, that $k(x,y)=\overline{k(y,x)}$.  Show that
$K$ is self-adjoint and its eigenvalues $\lambda_j$ satisfy
\[
  \sum_{j \in \N} \lvert \lambda_j \rvert^2 < \infty \,.
\]

\end{enumerate}


\item Let
\[
  V^* \supset H^* \equiv H \supset V
\]
be a Gelfand-triple of Hilbert spaces, with $V$ dense in $H$ and the
injection $i \colon V \to H$ continuous.  Let $a \colon V \times V \to
\R$ be a continuous, coercive bilinear form, define $A \colon V
\to V^*$ by $\langle A u, v \rangle_{V^*,V} = a(u,v)$ for all $u,v \in
V$, and set
\[
  \D(A) = \{ u \in V \colon Au \in H \} \,.
\]
\begin{enumerate}
\item Show that $A \colon \D(A) \to H$ is closed.
\item Show that $\lVert Au \rVert_H$ defines a norm on $\D(A)$ which
is equivalent to the graph norm
\[
  \Bigl( \lVert Au \rVert_H^2 + \lVert u \rVert_H^2 \Bigr)^{\tfrac12}
  \,.
\]
\item Show that $\D(A)$ is a Hilbert space with any of the norms from
(ii) and that $A \colon \D(A) \to H$ is an isomorphism.
\end{enumerate}

\item Recall that $L^2(\T)$ can be endowed with the norm
\[
  \lVert u \rVert^2 = \sum_{k \in \Z} \lvert u_k \rvert^2 \,,
\]
where $u_k$ are the Fourier coefficients of $u$, i.e.,
\[
  u(x) = \sum_{k\in \Z} u_k \, \e^{\i k x} \,.
\]
Let 
\[
  H^1(\T) = \{ u \in L^2(\T) \colon u' \in L^2(\T) \} \,,
\]
endowed with norm
\[
  \lVert u \rVert_1^2 = \sum_{k \in \Z} (1+k^2) \, \lvert u_k \rvert^2 \,.
\]
Show that the embedding $L^2 \supset H^1$ is compact.
\end{enumerate}
\end{document}
