% LaTeX file -- H^p Spaces, Fall 2017, Assignment 2

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 950 $H^p$ Spaces \hfill Assignment 2 \hfill Due Thursday October 26.}} 

\vskip3em 

\begin{Ex}


\item Let $Z(B)$ denote the zeros of a Blaschke product $B$ including
multiplicity, and let $S_i$ be singular inner functions corresponding to
singular measures $\mu_i \in M(\bT)_+$.  Show that $B_1S_1$ divides
$B_2S_2$ if and only if $Z(B_1) \subseteq Z(B_2)$ and $\mu_1 \le \mu_2$.

\item 
\begin{parts}
\item Let $\omega(z) = \exp\big( \frac{z+1}{z-1} \big)$, a singular inner function.  Describe the
sets $\{z : |\omega(z)| = r \}$.  
Hence show that for $a \in \bD$, the zeros of 
$\omega_a(z) = \dfrac{\omega(z)-a}{1-\bar{a}\omega(z)}$ approach 1 tangentially.

\item Let $T$ be the operator on $\K=H^2 \ominus \omega H^2$ given by
$Tf = P_\K zf$ for $f \in \K$.  Find all invariant subspaces of $T$.

\item \textbf{Bonus.} Is $\omega_a$ a Blaschke product for
all $a \in\bD\setminus\{0\}$?  I suspect this is true.
\end{parts}

\item \begin{parts}
\item If $f(z)$ and $1/f(z)$ are both in $H^1(\bD)$, prove that $f$ is outer.

\item If $f\in H^1(\bD)$ and $\re f(z) > 0$ on $\bD$, prove that $f$ is outer.
\hint $f\!+\!\ep$ is outer.  Split $\int \log|f+\ep| \,dt$ over $E=\{t : |f(e^{it})| \ge 1/2 \}$ and $\bT\setminus E$. 

\item If $\omega(z)$ is a non-constant inner function, 
for which $a \in \bC$ is $\omega(z) - a$ outer?
\end{parts}

\item If $f,g \in H^1$, $f$ is outer and $h=g/f \in L^1$, show that $h \in H^1$.
\hint reduce to the case $g$ outer and use the integral formula.

\item Suppose that $f(z) = \sum_{n\ge0} a_n z^n$ belongs to $H^1(\bD)$.
Prove that $\sum_{n=1}^\infty \frac1n |a_n| \le \pi \|f\|_1$.
\hint Factor $f$ into two $H^2$ functions, replace their Taylor coefficients
by their absolute values, so that the product $F$ has positive coefficients
and $\|f\|_1=\|F\|_1$.  Compute $\int_{-\pi}^{\pi} t \im F(re^{it}) \,dt$.

\item Let $\tau(z) = \lambda \dfrac{z-a}{1-\bar a z}$ for $a \in \bD$ and $|\lambda|=1$. 
\begin{parts}
\item Show that if $B$ is a Blaschke product, so is $B \circ \tau$;
and if $S$ is a singular inner function, so is $S \circ \tau$.
\item Hence deduce that if $F$ is an outer function in $H^p$, then $F(\tau(z))$ is outer.
\hint By Assignment 1, 5(a), if $f \in H^p$, so is $f(\tau(z))$.
\item Show that $\alpha_\tau(f) = f \circ \tau$ defines a continuous automorphism of $\AD$.
\item Let $\alpha$ be an automorphism of $\AD$.  \\
(i) Show that $\ran \alpha(f) = \ran f$ for $f\in \AD$.\\
(ii) Show that $\tau = \alpha(z)$ must be a conformal map of $\bD$ onto itself.\\
(iii) Show that $\alpha = \alpha_\tau$.
\end{parts}


\end{Ex}
\end{document}

