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\begin{document}
\begin{center}
{\bf Homework 2}\\
(due Wednesday, February 1)
\end{center}
\vspace*{.3in}

\noindent{\bf Book Problems:}
\begin{description}
 \item[Section 6.2, pages 163-4] 2.3, 2.6 (Note that you get the second eigenvector for almost free), 2.10 
\end{description}

\begin{prob}
Let $V$ be the (complex) vector space of continuous functions $f:\R\to\C$ such that $f(x+1) =f(x)$ for every $x\in\R$.  Fix a number $a\in [0,1)$ and let $T:V\to V$ be given by $T(f) =g$ where $g(x) =f(x+a)$.
Let us give $V$ the inner product
$$
\pair{f}{g} = \int_0^1 f(x)\overline{g(x)}\,dx.
$$
\begin{enumerate}
 \item Show that $T$ has an adjoint by finding one.
\item Decide whether $T$ is normal, unitary, and/or self-adjoint.
\item Show for each $n\in\Z$ that $e^{2\pi ina}$ is an eigenvalue of T, and find an eigenvector for this
eigenvalue.
\item (Extra credit) As in the finite dimensional case, eigenvectors corresponding to different eigenvalues are orthogonal.  Use this and the previous item to show that $f$ has no other eigenvectors for $T$.
\end{enumerate}
\end{prob}

\begin{prob}
Let $V$ be a finite dimensional inner product space.  Prove or disprove each of the following.
\begin{enumerate}
 \item If $T:V\to V$ is an invertible operator, then $(T^{-1})^* = (T^*)^{-1}$.
\item If $S,T:V\to V$ are self-adjoint operators, then so is $S\circ T$.
\item If $S,T:V\to V$ are orthogonal operators, then $S\circ T = T\circ S$.
\item The set of orthogonal operators $S:V\to V$ is a vector space.
\end{enumerate}
\end{prob}

\begin{prob} (Axler, page 150 \#24) Show that a sum of two positive operators $S,T:V\to V$ is positive.
\end{prob}


\begin{prob} (Axler, page 148 \#5)
Let $T:V\to V$ be a linear operator (not necessarily normal or anything else special) on a finite dimensional inner product space.  Show that if $\lambda$ is an eigenvalue of $T$, then $\bar\lambda$ is an eigenvalue of $T^*$.
\end{prob}







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