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\newtheorem*{objectives}{Objectives}

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\begin{document}

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

\bigskip\noindent
This text is a rough draft.

\begin{objectives}
Establish criteria of ``vertical'' operators
(in other words, operators that are invariant under horizontal shifts)
on the Bergman space on the upper half-plane.
\end{objectives}

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

\bigskip

\subsection*{Bergman kernel on the upper half-plane (short review)}

\medskip

\begin{fact}[evaluation functional is bounded]
Given a point $w\in\Pi$, denote by $\eval_w$ the evaluation functional at $w$
on the Bergman space $\BergmanSpace{\Pi}$:
\par\smallskip
\[
\eval_w(f)\eqdef f(w).
\]
It is known that this functional is bounded.
\end{fact}

\bigskip

\begin{defn}[Bergman kernel on the upper half-plane]
Given a point $w\in\Pi$,
denote by $K_{\Pi,w}$ the function corresponding
to the evaluation functional at the point $w$
(this function is called sometimes the \emph{Bergman kernel}
of the space $\BergmanSpace{\Pi}$):
\[
\forall f\in\BergmanSpace{\Pi}\qquad \eval_w(f)=\langle K_{\Pi,w},f\rangle.
\]
\end{defn}

\bigskip

\begin{fact}[explicit formula for the Bergman kernel on the upper half-plane]
Let $w\in\Pi$. It is known that
\[
K_{\Pi,w}(\xi)=-\frac{1}{\pi(\overline{w}-\xi)^2}.
\]
\end{fact}

\bigskip

\begin{exercise}[norm of the Bergman kernel on the upper half-plane]
Let $w\in\Pi$.
Calculate the norm $\|K_{\Pi,w}\|_2$ of the function $K_{\Pi,w}$.
\end{exercise}

\bigskip

\clearpage
\section*{Berezin transform of bounded linear operators\\(short review)}

\medskip

\begin{defn}[Berezin transform of a bounded linear operator]
Let $S\colon\BergmanSpace{\Pi}\to\BergmanSpace{\Pi}$ be a bounded linear operator.
The \emph{Berezin transform} of $S$ is defined by
\[
\BerezinTransform{S}(w)\eqdef\frac{\langle SK_{\Pi,w},K_{\Pi,w}\rangle}{\langle K_{\Pi,w},K_{\Pi,w}\rangle}.
\]
\end{defn}

\bigskip

\begin{exercise}
Prove that the Berezin transform of bounded linear operators is a linear map:
\par\smallskip
\begin{align*}
\BerezinTransform{S_1+S_2}(w)&=\hsp{25em}
\\[3ex]
\BerezinTransform{\lambda S}(w)&=\hsp{25em}
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Let $S\colon\BergmanSpace{\Pi}\to\BergmanSpace{\Pi}$ be a bounded linear operator
and $S^\ast$ be its adjoint operator.
Prove that $\BerezinTransform{S^\ast}=\overline{\BerezinTransform{S}}$.
\par\smallskip
\[
\BerezinTransform{S^\ast}(w)=\hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Find a proof of the fact that the Berezin transform
of bounded linear operators is an injective map:
if $\BerezinTransform{S}=0$, then $S=0$.
\end{exercise}

\bigskip

\bigskip

\clearpage
\subsection*{Multiplication operator on the real line}

\medskip

\begin{defn} \label{defn:multiplication_operator_real_line}
Let $g\in L^\infty(\RealNumbers)$.
Define $M_g\in\BoundedLinearOperators(L^2(\RealNumbers))$ by
\[
(M_g f)(x)\eqdef g(x)f(x).
\]
\end{defn}

\bigskip

\begin{exercise}[product of the multiplication operators]
Let $g_1,g_2\in L^\infty(\RealNumbers)$.
Calculate the product of the operators $M_{g_1}$ and $M_{g_2}$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}[multiplication operators commute]
Let $g_1,g_2\in L^\infty(\RealNumbers)$.
Prove that
\[
M_{g_1}M_{g_2}=M_{g_2}M_{g_1}.
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{defn}
Given a number $h\in\RealNumbers$
denote by $\Theta_h$ the following function $\RealNumbers\to\ComplexNumbers$:
\[
\Theta_h(x)\eqdef\econstant^{\imagunit h x}.
\]
\end{defn}

\bigskip

\begin{fact}[criterion of multiplication operator on the real line]
Let $S\in\BoundedLinearOperators(L^2(\RealNumbers))$.
Then the following conditions are equivalent:
\begin{itemize}
\item[(a)] $S$ is invariant under multiplication by $\Theta_h$ for all $h\in\RealNumbers$:
\[
\forall h\in\RealNumbers\qquad S M_{\Theta_h}=M_{\Theta_h} S.
\]
\item[(b)] $S$ is the multplication operator by a bounded function:
\[
\exists m\in L^\infty(\RealNumbers)\qquad S=M_m.
\]
\end{itemize}
\end{fact}

\bigskip

\begin{exercise}
Prove that (b) implies (a).
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Find in the literature a proof that (a) implies (b).
Write here an exact reference to the proof.
Study the proof.
\end{exercise}

\bigskip\bigskip

\clearpage
\subsection*{Multiplication operator on the positive half-line}

\medskip

\begin{defn} \label{defn:multiplication_operator_positive_halfline}
Let $\sigma\in L^\infty(\PositiveRealNumbers)$.
Define $M_\sigma\in\BoundedLinearOperators(L^2(\PositiveRealNumbers))$ by
\[
(M_\sigma f)(x)\eqdef\sigma(x)f(x).
\]
\end{defn}

\bigskip

\begin{exercise}[product of the multiplication operators]
Let $\sigma_1,\sigma_2\in L^\infty(\PositiveRealNumbers)$.
Calculate the product of the operators $M_{\sigma_1}$ and $M_{\sigma_2}$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{defn}
Given a number $h\in\RealNumbers$
denote by $\Theta_h^+$ the restriction of $\Theta_h$ onto the positive half-line:
\[
\Theta_h^+(x)\eqdef\econstant^{\imagunit h x}\qquad(x\in\PositiveRealNumbers).
\]
\end{defn}

\bigskip

\begin{exercise}[criterion of multiplication operator on the positive half-line]
\label{exer:criterium_of_multiplication_operator}
Let $S\in\BoundedLinearOperators(L^2(\PositiveRealNumbers))$.
Prove that the following conditions are equivalent:
\begin{itemize}
\item[(a)] $S$ is invariant under multiplication by $\Theta_h^+$ for all $h\in\RealNumbers$:
\[
\forall h\in\RealNumbers\qquad
S M_{\Theta_h^+} = M_{\Theta_h^+} S.
\]
\item[(b)] $S$ is the multiplication operator by a bounded function:
\[
\exists\sigma\in L^\infty(\PositiveRealNumbers)\qquad S=M_\sigma.
\]
\end{itemize}
\end{exercise}

\bigskip

\clearpage
\subsection*{Vertical operators on the positive half-plane}

\medskip

\begin{defn}[horizontal shift operator]
Let $h\in\RealNumbers$.
Define $H_h\colon A^2(\Pi)\to A^2(\Pi)$ by
\[
(H_h f)(w)\eqdef f(w-h).
\]
\end{defn}

\bigskip

\begin{exercise} \label{exer:kernel_shifted}
Let $h\in\RealNumbers$ and $w\in\Pi$.
Calculate $H_h K_{\Pi,w}$.
\[
(H_h K_{\Pi,w})(z)=\hsp{20em}
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\bigskip\noindent
The following fact is taken from papers of N.~Vasilevski.

\begin{fact}
Define $R\colon\BergmanSpace{\Pi}\to L^2(\PositiveRealNumbers)$ by
\[
R\colon\BergmanSpace{\Pi}\to L^2(\PositiveRealNumbers),\qquad
(R\phi)(x)=\sqrt{x}\frac{1}{\sqrt{\pi}}\int_\Pi \phi(w)\econstant^{-\imagunit\overline{w}x}\,d\mu(w),
\]
This operator $R$ is an isometrical isomorphism,
and its inverse $R^\ast\colon L^2(\PositiveRealNumbers)\to\BergmanSpace{\Pi}$ is given by:
\[
(R^\ast f)(z)=\frac{1}{\sqrt{\pi}}\int_{\PositiveRealNumbers}\sqrt{\xi}f(\xi)\econstant^{\imagunit z\xi}\,d\xi.
\]
\end{fact}

\bigskip

\begin{exercise} \label{exer:Rast_Theta}
Let $h\in\RealNumbers$ and $f\in L^2(\PositiveRealNumbers)$.
Prove that
\begin{equation} \label{eq:Rast_Theta}
R^\ast(\Theta_h^+ f)=H_h R^\ast(f).
\end{equation}
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\clearpage
\begin{exercise} \label{exer:vertical_operator_criterium}
Let $S\in\BoundedLinearOperators(\BergmanSpace{\Pi})$.
Prove that the following conditions are equivalent:
\begin{itemize}
\item[(a)] $S$ is invariant under horizontal shifts:
\[
\forall h\in\RealNumbers\qquad S H_h = H_h S.
\]
\item[(b)] $RSR^\ast$ is invariant under multiplication by $\Theta_h^+$ for all $h\in\RealNumbers$:
\[
\forall h\in\RealNumbers\qquad
RSR^\ast M_{\Theta_h^+}=M_{\Theta_h^+} RSR^\ast.
\]
\item[(c)] There exists a function $\sigma\in L^\infty(\PositiveRealNumbers)$ such that
\[
S=R^\ast M_\sigma R.
\]
\item[(d)] The Berezin transform of $S$ is a vertical function:
\[
\forall z\in\Pi\qquad \BerezinTransform{S}(z)=\BerezinTransform{S}(\Im(z)).
\]
\end{itemize}
\end{exercise}




\label{endlabel}
\end{document}
