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\begin{document}
\thispagestyle{empty}

%%%%%%%   TITLE   %%%%%%
\begin{center}{\bf PM822 \qquad Assignment 2 
\qquad Due Tuesday, February 5}\end{center}  
\vskip 1em
 
\begin{Ex}
\item  Let $T\in \B(\H)$ with $\|T\|\le1$, and let $V \in\B(\K)$ be its minimal isometric coextension
and let $U \in \B(\L)$ be its minimal unitary dilation.
\begin{parts}
\item Show that $V$ is a pure isometry (i.e.\ no unitary summand) if and only if $T^{*n}$
converges to $0$ in the strong operator topology\ (i.e. $\lim_{n\to\infty} T^{*n}x = 0 $ for all $x \in \H$).

\item Define $M_+ = (\H \vee U\H) \ominus \H$ and $M_- = (\H \vee U^*\H) \ominus \H$.
Show that these subspaces are wandering for $U$, meaning that $U^nM$ are pairwise orthogonal for
all $n \in \bZ$. 
Hence deduce that the restrictions of $U$ to the reducing subspaces 
$\L_{\pm} = \bigvee_{n\in\bZ} U^n M_{\pm}$ are unitarily equivalent to the 
direct sum of copies of the bilateral shift.

\item Show that $N := (\L_+ \vee \L_-)^\perp$ is a subspace of $\H$ such that $T|_N$ is unitary.

\end{parts}


\item Let $A_1=A_2=[0]$ acting on $\H = \bC e_0$. Let $\K$ be a Hilbert space containing $\H$.
Let $V_1$ be any pure isometry such that $V_1^* e_0 =0$. 
Prove that there is an isometry $V_2$ commuting with $V_1$ so that $(V_1,V_2)$ is a minimal
isometric coextension of $(A_1,A_2)$.


\item 
\begin{parts}
\item If $T\in\B(\H)$ with $\spr(T) < 1$, show that $A = \sum_{n\ge0} T^{*n} T^n$ converges to a positive invertible operator with $A^{-1} \le I$.
\item (Rota) Show that $\|A^{1/2} T A^{-1/2} \| \le 1$. (Thus $T$ is similar to a contraction.)
\item Show that if $K$ is compact and $\spr(K) \le 1$, then $K$ is similar to $K_0 \oplus T$ where
$\spr(K_0) < 1$ and $T$ acts on a finite dimensional space. 
\item (Sz.Nagy) Show that if $K$ is compact, then $K$ is similar to a contraction if and only if it is power bounded.
\hint to deal with $T$ of part (c), use Jordan form.
\end{parts}


\item 
\begin{parts}
\item (Douglas) If $\left\| \begin{sbmatrix} A \\ C \end{sbmatrix} \right\| \le 1$, show that there is a 
contraction $L$ so that $C = L D_A$.\\
What is the analogue for $\| \big[ A \ B \big] \| \le 1$?

\item (Parrott) Suppose that $\left\| \begin{sbmatrix} A \\ C \end{sbmatrix} \right\| \le 1$ and
$\| \big[ A \ B \big] \| \le 1$. Prove that there is a matrix $X$ so that
$\left\| \begin{bmatrix} A & B \\ C & X \end{bmatrix} \right\| \le 1$.
\hint use $X = -LA^*K$ for certain $K,L$.
\end{parts}


\item 
\begin{parts}
\item Prove that $n \times n$ matrix $A_n$ below has norm
\[
 \strut\hspace{12mm}
 \left\| \begin{bmatrix} 
 a_0 & 0 & 0 & \dots & 0\\
 a_1 & a_0 & 0 & \ddots & 0\\
 a_2 & a_1 & a_0 &\ddots & 0\\
 \ddots & \ddots & \ddots & \ddots & 0\\
 a_{n-1} & a_{n-2} & \dots & a_1 & a_0
 \end{bmatrix} \right\| \le 
 \inf_{q \in \bC[z]}  \big\| a_0 + a_1 z + \dots + a_{n-1}z^{n-1} + z^n q(z) \big\|_\infty .
\]

\item Suppose that $\|A_n\| = 1$. Use 4(b) to find a constant $a_n$ so that the $(n+1) \times (n+1)$ matrix
$A_{n+1}$ defined with $a_0,\dots,a_n$ satisfies $\|A_{n+1}\|=1$.

\item (Carath\'eodory) Hence given $a_0,\dots,a_{n-1}$ with $\|A_n\|=1$, find a power series 
\[ h(z) = \sum_{n\ge0} a_n z^n = a_0 + a_1 z + \dots + a_{n-1}z^{n-1} + \text{higher order terms} \]
so that $\sup_{|z| < 1} |h(z)| = 1$.
\hint express $h$ as a limit of polynomials which converges uniformly on each disk $\ol{\bD}_r = \{z : |z| \le r\}$ for all $r<1$.
\end{parts}

\end{Ex}

\end{document}