\documentclass[12pt]{article}
\usepackage[width=16cm,height=22cm]{geometry}
\usepackage[intlimits]{amsmath}
\usepackage{amsfonts,amssymb,amsthm,mathtools}
\usepackage{ifthen}

\theoremstyle{definition}
\newtheorem{exercise}{Exercise}
\newtheorem{defn}{Definition}
\newtheorem{note}{Note}
\newtheorem*{requirements}{Requirements}
\newtheorem*{objectives}{Objectives}

\newcommand{\eqdef}{\coloneqq}
\newcommand{\PositiveRealNumbers}{\mathbb{R}_+}
\newcommand{\RealNumbers}{\mathbb{R}}
\newcommand{\ComplexNumbers}{\mathbb{C}}
\newcommand{\UnitDisk}{\mathbb{D}}
\newcommand{\eps}{\varepsilon}
\newcommand{\hsp}[1]{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}
\newcommand{\ubrace}[1]{\underbrace{\mbox{}\vphantom{0_0^0}\hspace{#1}\mbox{}}_?}
\newcommand{\BerezinTransform}[1]{\mathcal{B}(#1)}
\newcommand{\BergmanSpace}[1]{\mathcal{A}^2(#1)}
\newcommand{\BoundedLinearOperators}[1]{\mathcal{L}(#1)}
\DeclareMathOperator{\imagunit}{i}
\DeclareMathOperator{\econstant}{e}

\makeatletter
\newcommand{\mytitle}[2][empty]{
\newcommand{\titlename}{#2}
\newcommand{\shorttitlename}{\ifthenelse{\equal{#1}{empty}}{#2}{#1}}
\renewcommand{\@oddfoot}{\hfill{}\shorttitlename{},\quad{}page~{\thepage}~of~\pageref{endlabel}\hfill}
\begin{center}\bfseries\LARGE\titlename\end{center}\par\medskip}
\makeatother

\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}

\begin{document}

\mytitle[Vertical Toeplitz operators]{Vertical Toeplitz operators\\on the Bergman space\\on the upper half-plane}

\bigskip\noindent
This text is just a stub.

\begin{objectives}
Denote by $\BergmanSpace{\Pi}$ the Bergman space
of the analytic square integrable functions
on the upper half-plane $\Pi$ of the complex plane.
Given a function $g\in L^\infty(\Pi)$ denote by $T_g$ the Toeplitz operator
with generating symbol $g$ acting in the space $\BergmanSpace{\Pi}$.

Prove that $T_g$ is invariant under horizontal shifts if and only if
$g$ depends only on the imaginary part of the argument.
\end{objectives}

\begin{requirements}
Bergman space of the analytic square integrable functions on the upper half-plane,
Berezin transform, shift operator.
\end{requirements}

\bigskip

\begin{exercise} \label{exer:Toeplitz_injective}
Find the following theorem for Toeplitz operators in $\BergmanSpace{\UnitDisk}$:
If $a\in L^\infty(\UnitDisk)$ and $T_a=0$, then $a=0$ almost everywhere.
This theorem can be found in the book of Vasilevski or in other papers.
\end{exercise}

\bigskip

\begin{exercise}
Let $\phi\colon\UnitDisk\to\Pi$ be a biholomorphism
and $E\subset\UnitDisk$ be a set of zero Lebesgue measure: $\mu(E)=0$.
Prove that $\mu(\phi(E))=0$.
\end{exercise}

\bigskip

\begin{exercise}
Try to translate the theorem cited in the Exercise \ref{exer:Toeplitz_injective}
to the case of Toeplitz operators in $\BergmanSpace{\Pi}$.
\end{exercise}

\bigskip

\begin{defn}[horizontal shift]
Let $h\in\RealNumbers$.
Define $H_h\in\BoundedLinearOperators{\BergmanSpace{\Pi}}$ by
\[
(H_h f)(w)\eqdef f(w-h).
\]
\end{defn}

\bigskip

\begin{exercise}
Let $h\in\RealNumbers$ and $a\in L^\infty(\Pi)$.
Express the product
\[
H_{-h} T_a H_h
\]
as $T_b$ for some $b\in L^\infty(\Pi)$.
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in L^\infty(\Pi)$ such that
\[
\forall h\in\RealNumbers\qquad T_a H_h=H_h T_a.
\]
Prove that $a(w+h)=a(w)$ for all $h\in\RealNumbers$ and for almost all $w\in\Pi$. \\
Prove that $a(w)=a(\imagunit \Im(w))$ for almost all $w\in\Pi$.
\end{exercise}


\label{endlabel}
\end{document}


