% LaTeX file -- PM 810, Winter 2019, Assignment 1

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 810 \hfill Assignment 1 \hfill
Due Thursday January 31.}} 
\vskip 1em 

\begin{Ex}

\item Let $\Omega$ be an open subset of $\bC$. Show that a function $f:\Omega \to X$ 
into a Banach space $X$ is weakly analytic if and only if it is strongly analytic.\\
\hint (i) If $\ol{b_r(z_0)} \subset \Omega$, show that $\big\{\frac1h(f(z_0+h)-f(z_0)): |h|\le r\big\}$ is bounded.\\
\strut\qquad(ii) Show that $f$ is continuous. \\
\strut\qquad(iii) Define $x_n = \frac1{2\pi} \int_0^{2\pi} f(z_0+re^{it})(re^{it})^{-n}\,dt$. Show that $f(z_0+z) = \sum_{n\ge0} x_n z^n$.

\item Let $a$ be an element of a unital Banach algebra $\fA$.
Suppose that $f(z)$ is analytic in a neighbourhood of
$\sigma(a)$ and that $g(w)$ is analytic in a neighbourhood of
$f(\sigma(a))$. Show that $(g\circ f)(a) = g(f(a))$.


\item Define $e^a = \sum_{n\ge0} \frac1{n!} a^n$.
\begin{parts} 
\item If $ab=ba$, show that $e^{a+b} = e^a e^b$.
\item Let $a \in \fA^{-1}$ such that $0$ is in the unbounded component of the resolvent.
(This includes special case $\|1-a\|<1$.) 
Prove that there is an element $b \in \fA$ such that $a=e^b$.
\end{parts}

\item \textit{Kakutani's shift.} Let $\{e_n : n \ge 0\}$ be an orthonormal basis
for a Hilbert space $\H$.\\ Let $a_n = \gcd(n,2^n)^{-1}$ for $n \ge 1$.  
Define $A \in \B(\H)$ by $Ae_n = a_{n+1} e_{n+1}$ for $n \ge 0$.
\begin{parts}
\item Compute $\spr(A)$. \hint Compute $\|A^{2^n}\|$.
\item Show $A$ is unitarily equivalent to $\lambda A$ for $|\lambda|=1$.
What does this say about $\sigma(A)$?
\item Define $A_k\in\B(\H)$ by $A_ke_n = a_{n,k} e_{n+1}$, 
where $a_{n,k}=a_{n+1}$ if $a_{n+1} \ge 2^{-k}$ and $a_{n,k}=0$
otherwise.  Find $\sigma(A_k)$.  Show that $A = \lim\limits_{k\to\infty} A_k$.\vspace{-.4ex}
\item Show that $\sigma(A)$ is the disk of radius $\spr(A)$.
\hint if $(\lambda -A)^{-1}e_0 = \sum c_n e_n$, solve for $c_n$ in terms of $\lambda$ and
$b_n =  \| A^n e_0\|$. Use info about $b_n$ from 3(a).
\item Conclude that the spectrum and spectral radius are not continuous functions.
\end{parts}

\item {\it Upper semicontinuity of the spectrum.}
Let $\fA$ be a unital Banach algebra; $a \in \fA$.
\begin{parts}
\item Given $\ep>0$, find $\delta>0$ so that $\|a-b\|<\delta$ implies
that $\sigma(b)$ is contained in $ \sigma(a)_\ep := \{z : \dist(z,\sigma(a)) < \ep \}$.
\hint Enclose $\sigma(a)$ inside a curve $\C$ contained in $\sigma(a)_\ep$.
Then $\|R(a,z)\|$ is bounded on $\C$.
\item Suppose  $\sigma(a) = \sigma_1 \cup \sigma_2$, where $\sigma_i$ are disjoint 
non-empty clopen subsets of $\sigma(a)$.   Enclose $\sigma_1$ inside curves $\C_i$
so that $\ol{\Omega_i} = \C_i \cup \{z : \ind_{\C_i}(z) \ne 0\}$ are disjoint (how?).
Find $\delta>0$ so that $\|a-b\|<\delta$ implies that 
$\sigma(b) \cap \Omega_i$ is not empty.\\
\hint Use part (a) to get $\delta$. Let $b_t = (1-t)a + tb$ for $0 \le t \le 1$. 
Consider the Riesz projection $e_t$ for the operator $b_t$ and
its spectrum in $\Omega_1$.  This is a continuous path.
\end{parts}

\item \begin{parts}
\item Let $\fA_0^{-1}$ denote the set of all invertible elements of a
Banach algebra $\fA$ which may be written as a finite product of
exponentials $e^{b_i}$. Show that this is a normal subgroup of the
group $\fA^{-1}$ of invertible elements of $\fA$. 

\item Show that $\fA_0^{-1}$ consists of all invertible elements which
can be connected to the identity by a continuous path of invertible
elements.  
Hence conclude that it is an open subgroup (and thus also closed).
\hint problem 3(b).

\item Let $\fA = \CT$.  Describe $\CT_0^{-1}$ and $\CT^{-1}/\CT_0^{-1}$.
\end{parts}

\end{Ex}
\end{document}
 

 