% LaTeX file -- H^p Spaces, Fall 2017, Assignment 3

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\begin{document}
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\vspace*{1cm}
%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 950 $H^p$ Spaces \hfill Assignment 3 \hfill Due Thursday November 23.}} 

\vskip1em 

\begin{Ex}


\item
\begin{parts}
\item Let $V$ be an isometry on a Hilbert space $\H$. Let %\\[1ex]\strut\qquad
$ \W = \ran(V)^\perp \qand  \K = \bigcap_{k\ge1} \ran(V^k) .$ %\\[1ex]
Show that $\W$ is a wandering space (i.e. $\ip{V^k x,V^ly} = 0$ for all $x,y\in\W$ and $k>l\ge0$), 
and that $\K^\perp = \sum_{k\ge0}^\oplus V^k\W$.

\item Show that $\K$ is a reducing subspace for $V$,  $V|_\K$ is unitary and $V|_{\K^\perp}$ is unitarily equivalent to a direct sum of $\Dim\W$ copies of the unilateral shift.

\item Let $0 \ne x\in\K^\perp$, and let $\M$ be the cyclic invariant subspace for $V$ determined by $x$. Prove that $V|_\M$ is unitarily equivalent to the unilateral shift. 
\hint show that the wandering space for $V|_\M$ is 1-dimensional and there is no unitary part.
\end{parts}



\item
\begin{parts}
\item A weak-$*$ linear functional on $H^\infty$ is given by $\Phi(h) = \int_\bT hf\,dm$ for some $f\in L^1$.
Find $f_1, f_2$ in $L^2$ so that $\Phi(h) = \ip{M_h f_1,f_2}$.
Hence for any $\ep>0$, find $g_1,g_2$ in $H^2$ so that $\Psi(h) = \ip{T_h g_1,g_2}$ satisfies $\|\Phi-\Psi\| < \ep$.

\item Find sequences $G_i= (g_i^{(1)}, g_i^{(2)}, \dots)$ in $H^2$ so that \\[1ex]\strut\qquad
$ \sum_{k\ge1} \|g_i^{(k)}\|_2^2 < \infty \FOR i=1,2 \qand \Phi(h) = \sum_{k\ge1} \ip{M_h g_1^{(k)}, g_2^{(k)}} .$ 

\item Think of $G_i$ as vectors in the direct sum of countably many copies of $H^2$. 
Let $\M$ be the cyclic invariant subspace for shift generated by $G_1$.
Use 1(c) to find $g_1,g_2$ in $H^2$ so that $\Phi(h) = \ip{T_h g_1,g_2}$.

\item Show that the weak-$*$ closed ideals of $H^\infty$ are of the form $\omega H^\infty$ for $\omega$ inner.\\
\hint if $\I$ is a weak-$*$ closed ideal, its closure in $H^2$ is an invariant subspace, say $\omega H^2$.  If $I\ne \omega H^\infty$, separate $I$ from $\omega$ with a weak-$*$ continuous functional.
\end{parts}


\item \begin{parts}
\item Suppose $A$ is a function algebra on $X$, $f \in A$, and $E$
is a (clopen) subset of $X$ so that $\re f \ge 1$ on $E$ and $\re f \le -1$ 
on $X \setminus E$.  Prove that $\upchi_E$ belongs to $A$.
\hint \vspace{.3ex} use Runge's Theorem to find polynomials $p_n$ 
that converge uniformly on 
$(\|f\|+1)\ol{\bD} \cap  \big\{ |\re z| \ge 1/2 \big\}$ to a characteristic function.

\item Show that $H^\infty + \ol{H^\infty }$ is not dense in $L^\infty$.
\hint First show this is equivalent to $\re H^\infty$ being dense in $L^\infty_\bR$.
Let $E$ be a measureable subset of $\bT$ with $0 < |E| < 2\pi$,
and set $f = 2\upchi_E - 1$.  Suppose that there is an  $h \in H^\infty$ such that
$\| f - \re h\| < 1/2$.  Apply part (a) with $X = \M(L^\infty)$. 
\end{parts}


\item Let $A$ be a closed subalgebra of $L^\infty$ containing $H^\infty$.
\begin{parts}
\item Show that the restriction map from $\M(A)$ to $\M(H^\infty)$ is injective.
Thus $\M(A)$ can be considered as a subset of $\M(H^\infty)$.

\item Show that if $H^\infty \subsetneq A \subset L^\infty$, then $A \supset H^\infty + C$.
\hint follow the strategy of Wermer's theorem, either evaluation at $0$ is in $\M(A)$ or it isn't.

\item If $g$ is invertible in $A$, show that there is an invertible $h$ in $H^\infty$ 
and an invertible unimodular function $u \in A$ so that $g=uh$.  Hence show that
$A$ is generated by $H^\infty$ and $\U = \{ u \in A^{-1} : |u|=1 \ae \text{ on }\bT\}$.

\item Show that if $A$ is generated as a closed algebra 
by $H^\infty$ and a \textit{group} $\U$ of unimodular functions, 
then $\M(A) = \{ \phi \in \M(H^\infty) : |\phi(u)|=1 \FORAL u \in \U \}$.

\item What is $\M(H^\infty + C)$?
\end{parts}

%\item Use the maximal function to give a different proof of (part of) Fatou's theorem:
%if $f \in L^1$ and $u = P * f$ is its harmonic extension, show that 
%$\ntlim_{z\to e^{it}} u(z) = f(t) \ae$ as follows.
%
%\begin{parts}
%\item WLOG $f \in L^1_\bR$.  Show that 
%$ \Omega(t) = \ntlimsup_{z\to e^{it}} u(z) -  \ntliminf_{z\to e^{it}} u(z)$
%is bounded by $C\, M\!f(t)$,
%where $C$ is a constant only depending on $\alpha$.
%
%\item Hence show that $|\{t : \Omega(t) > \ep \}| < C/\ep \|f\|_1$.  Approximate $f$ by
%trig polynomials to improve this estimate.
%
%\end{parts}
%
%\item
%\begin{parts}
%\item Show that a conformal map of $\bD$ onto $\bH = \{z : \im z > 0 \}$
%induces an isometric map of $H^\infty$ onto $H^\infty(\bH)$; and it takes
%interpolating sequences to interpolating sequences.
%
%\item Consider the points $z_n = i + 10 n$ in $\bH$ for $n \in \bN$.  
%If $a=(a_n) \in \ell^\infty$ with $\|a\|_\infty \le 1$, define
%$f_1(z) = \sum_{n\ge1} \dfrac{-4a_n}{(z-\bar z_n)^2}$.
%Prove that $f_1 \in H^\infty(\bH)$ with $\|f_1\|_\infty \le 10$ and $|f_1(z_n) - a_n| \le 1/6$ for $n \ge1$.
%
%\item Repeat this procedure with  $a^{(1)} = \big(a_n - f_1(z_n)\big)$ 
%to get a function $f_2$ with better estimates, etc.
%Hence construct an infinite series which converges to an interpolating function for $a$.
%Thus showing that $\{z_n : n \ge1\}$ is an interpolating sequence.
%\end{parts}

\end{Ex}
\end{document}
