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\begin{document}
\title{Assignment 7: Compact operators}

\date{Assigned Wed 05/16. Due Wed 05/23.}
\maketitle

\begin{questions}
%\item Suppose $T \in B(X, X)$ is a compact operator. Show that there exist $T_n \in B(X,X)$ such that $\im(T_n)$ is finite dimensional, and $(T_n) \to T$ in $B(X, X)$.

\item An operator $T \in B(X, X)$ is called \textit{Hilbert-Schmidt} if there exists an orthonormal complete system $\{e_n\}$ of $X$, such that the series $\sum \norm{T e_n}^2$ converges.
\begin{parts}
\item Show that Hilbert-Schmidt operators are compact. \hintb{Let $T_N(x) = \sum_1^N \ip{x}{e_n} T(e_n)$. Show that $T_N$ is compact, and $\norm{T - T_N}_{B(X, X)}$ converges to $0$.}
\item Find an example of an operator which is compact but not Hilbert-Schmidt.
\end{parts}

\item Let $\lap\inv:\Lpspace{2}([-\pi, \pi]) \to \Lpspace{2}([-\pi, \pi])$ be defined by $\lap\inv f = -\sum_{n\neq0} \frac{\hat f(n)}{n^2} e_n$. \noteb{If you choose to assume your functions are periodic on $[0,1]$ (instead of $[-\pi, \pi]$), and define $e_n(x) = e^{2\pi i nx}$ (instead of $e_n(x) = e^{i n x}$), then you should define $\lap\inv f = \frac{-1}{4\pi^2} \sum_{n\neq0} (\cdots)$ instead.}
\begin{parts}
\item If $f \in C^2$, and $\int f = 0$, then show that $(\lap\inv) \lap f = f$. What happens if $\int f \neq 0$? \noteb{By $C^2$ here we mean functions which are periodic and have continuous second derivative. For simplicity, we've assume that $f$ is a function of one variable. In this case, the Laplacian $\lap f$ is just the second derivative $f''$.}
\item If $f \in H^s$ with $s > \frac{1}{2}$ with $\int f = 0$, then show that $\lap\inv f \in C^2$, thus $\lap (\lap\inv f)$ is well defined. Further show $\lap( \lap\inv f) = f$.
\item Show that $\lap\inv: \Lpspace{2} \to \Lpspace{2}$ is compact and Hermitian.
\item Compute $\sigma(\lap\inv)$, and show that $\sigma(\lap\inv)$ consists of only eigenvalues. What are the eigenfunctions of $\lap\inv$?
\item If you replace $\lap$ with some elliptic differential operator $L$ (see homework 5 question 1), show that you can define $L\inv$ in a similar manner to make $L\inv:\Lpspace{2} \to \Lpspace{2}$ compact (but not necessarily hermitian), such that $L\inv$ is the inverse of $L$ on some nice subset of $\Lpspace{2}$. Compute $\sigma(L\inv)$.
\end{parts}



\item
\begin{parts}
\item Let $1 \leq p < q \leq \infty$. Show that the inclusion map $i:\lpspace{p} \to \lpspace{q}$ is not compact. \noteb{Homework 2 4(d) shows that $\lpspace{p} \subset \lpspace{q}$, and that the inclusion is continuous.}
\item Let $1 \leq p < q \leq \infty$. Show that the inclusion map $i:\Lpspace{q}([0,1]) \to \Lpspace{p}([0,1])$ is not compact. \noteb{See homework 2 4(b) for the definition of this map. It also shows up in a few subsequent homeworks. One possible solution would be to let $\chi_n(x) = 1$ if the $n\Th$ digit in the binary expansion of $x$ is $1$, and let $\chi_n(x) = -1$ otherwise. Then $\Lpnorm{\chi_n}{p} = \Lpnorm{\chi_n}{q} = 1$ for all $n$, but $\{\chi_n\}$ is discrete in $\Lpspace{p}$. The $\chi_n$'s are called the Radamacher functions, and like the trigonometric basis $e_n$, the Radamacher functions form a orthogonal complete set in $\Lpspace{2}$. Simpler solutions to this problem exist.}
\end{parts}

\item Let $A \in B(X, X)$. We define $e^A = \sum_0^\infty \frac{A^n}{n!}$.
\begin{parts}
\item Show that the series $\sum \frac{A^n}{n!}$ converges in $B(X,X)$.
\item If $A, B \in B(X, X)$ are such that $A B = B A$, show that $e^A e^B = e^{A+B}$.
\item We say $U \in B(X, X)$ is unitary if $U^* = U\inv$. If $A$ is Hermitian, show that $e^{iA}$ is unitary.
\end{parts}
\item Let $K:[0,1]\times[0,1] \to \C$ be continuous and periodic, and define $T:\Lpspace{p} \to \Lpspace{p}$ by $Tf(x) = \int K(x,y) f(y) dy$ (see homework 6, 2(a)).
\begin{parts}
\item When $p = 2$, show that $T$ is Hilbert-Schmidt (hence compact). \hintb{Let $e_n$ be the Fourier basis of $\Lpspace{2}$. Show that $\sum \Lpnorm{T e_n}{2}^2 = \int \abs{K(x,y)}^2 \,dx\,dy$.}
\uplevel{\noindent $T$ is still compact (though not Hilbert-Schmidt) when $p \neq 2$. We prove that here.}
\item Let $(K_n)$ be a sequence of continuous functions such that $(K_n) \to K$ uniformly (on $[0,1]^2$). Let $T_n\in( \Lpspace{p}, \Lpspace{p} )$ be defined by $T_n f (x) = \int K_n(x,y) f(y) \, dy$. Show that $(T_n) \to T$ in $B(\Lpspace{p}, \Lpspace{p})$.
\item Suppose $K_N \in \Span\{e_n \st n \in \Z^2, \abs{n} \leq N \}$ (recall $e_n = e^{2 \pi i \ip{n}{x}}$). Show that the image of $T_N$ is finite dimensional. \noteb{Here $T_N$ is defined as in the previous subpart.}
\item Show that $T:\Lpspace{p} \to \Lpspace{p}$ is compact.
\end{parts}

\item If $A, B \in B(X, X)$, show that $\sigma(AB) - \{0\} = \sigma(BA) - \{0\} $. \hintb{Show that $I - AB$ is invertible if and only if $I - BA$ is invertible. Use some clever geometric series trick to do this.}

\end{questions}
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