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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 950 \hfill Assignment 3 \hfill
Due Thursday November 27}} 
\vskip 2em 

\begin{Ex}

\item Suppose that $\fA$ is a unital subalgebra of $\M_n$ such that $\Lat\fA=\{0,\bC^n\}$ is trivial. 
(Such an algebra is called \textit{transitive}.)  Show that $\fA=\M_n$ as follows:
\begin{parts}
\item Show that if $T$ commutes with $\fA$, then $T$ is scalar.
\hint consider $\ker(T-\lambda I)$.
\item Show that if $x$ and $y$ are linearly independent
vectors, then there is an $A\in\fA$ such that $Ax=0$ and $Ay\ne0$.  \\
\hint otherwise $T(Ax)=Ay$ defines a well defined operator in $\fA'$.
\item Construct all of the matrix units of $\M_n$ in $\fA$.   
\end{parts}

\item Show that every unital subalgebra of $\M_2$ is similar to one of five possible algebras, 
and determine which are reflexive.

\item Let $P$ be the $n\times n$ projection matrix with $p_{ij}=1/n$ for all $1\le i,j \le n$.
Compute $\dist(P,\T_n)$, and find an explicit upper triangular matrix $T$ so that $\|P-T\| = \dist(P,\T_n)$.

\item 
\begin{parts}
\item  Let $\N$ be a nest of order type $\omega = \bN \cup \{\infty\}$. Given $K\in\K(\H)$,
find $T\in \T(\N) \cap \K$ such that $\|K-T\| = \dist(K,\T(\N))$.\\
\hint find $N_0 < \H$ in $\N$ so that  $\| P_{N_0}^\perp K \| \le \frac12\dist(K,\T(\N))$, and use
this to reduce the problem to a finite nest.

\item Let $\N$ be the Volterra nest on $L^2(0,1)$, and let $C= 1 1^*$. Compute $\dist(C,\T(\N))$
and show that there is no closest compact operator.\\
\hint suppose that $T\in\T(\N)$ satisfies $\|C-T\| = \dist(C,\T(\N))$. Let $x = \sqrt2 \upchi_{(0,1/2)}$. 
Show that $P_{N_t}^\perp T P_{N_t}^\perp x = \frac12 P_{N_t}^\perp  x$ for all $t\in[0,1/2)$.

\item Show that one closest element of $\T(\N)$ to $C$ is $V+D$ where $V$ is the Volterra operator
$Vf(t) = \int_t^1 f(x)\,dx$ and $D = M_h$ where $h(x) = \min\{x,1-x\}$.
\hint $C = V+V^*$. Chop the nest into $2^n$ equal pieces. Use Q3 to find a block upper triangular operator which achieves the distance to the block upper triangulars. Take a limit.
\end{parts}


\item  Let $\M = \{M_n\}_{n\ge1} \cup\{\H\}$ and $\N = \{N_n\}_{n\ge1} \cup \{\H\}$ be two
order $\omega$ nests such that $\Dim M_n  = \Dim N_n  = n$ for $n\ge1$ and
$\lim_{n\to\infty} \|P_{M_n} - P_{N_n}\| = 0$.

\begin{parts}
\item Show explicitly how to construct a sequence of unitaries $U_k$ such that $U_k-I$ are finite rank so that $\lim_{k\to\infty} \sup_{n\ge1} \| U_kP_{M_n} - P_{N_n}U_k \| = 0.$
\item Hence show that there is an invertible operator
$S$ with $S-I$ compact such that $SM_n = N_n$ for all $n\ge1$.
\item Also show that there is a unitary operator $U$ such that
$UM_n = N_n$ for all $n\ge1$. 
\item \textbf{Bonus} Are there examples where you cannot take $U-I$ in the compacts?
%\item Finally, explain why there need not be a unitary such that
%$U-I$ is compact with this property (see Example 12.19).
\end{parts}

\item  Let $\N$ be the Volterra nest.  Let $\D(\N) = \T(\N) \cap \T(\N)^*$.
\begin{parts} 
\item Show that there are operators $A$ and $B$ in $\T(\N)$ such that $(AB-BA)^2=I$.  \\
\hint find $A$ and $B$ in $\T(\N\oplus\N)$ first, where $\N\oplus\N = \{ N_t \oplus N_t : 0 \le t \le 1\}$.
\item Show that there is no ideal $\I$ of $\T(\N)$ such that $\T(\N) = \D(\N)+\I$.
\end{parts}


\end{Ex}
\end{document}
 

 