\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*{objectives}{Objectives}
\newtheorem*{requirements}{Requirements}

\newcommand{\eqdef}{\coloneqq}
\newcommand{\NaturalNumbers}{{\mathbb{N}}}
\newcommand{\RealNumbers}{{\mathbb{R}}}
\newcommand{\PositiveRealNumbers}{{\mathbb{R}_+}}
\newcommand{\ComplexNumbers}{{\mathbb{C}}}
\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{\bigubrace}[1]{\underbrace{\mbox{}\vphantom{\sum_0^0}\hspace{#1}\mbox{}}_?}
\newcommand{\vertstrut}{{\displaystyle\vphantom{\int}}}
\newcommand{\bigvertstrut}{{\displaystyle\vphantom{\int_0^1}}}
\newcommand{\Range}[1]{{\mathcal{R}(#1)}}
\newcommand{\ClosedRange}[1]{{\mathcal{CR}(#1)}}
\DeclareMathOperator{\spectrum}{Sp}
\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}

\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

\begin{document}

\mytitle[Criteria of multiplication operator on $\ell^2$]%
{Criteria of multiplication operator\\on the space of square-summable sequences}

\medskip

\begin{objectives}
Establish some criteria of the multiplication operator on the space $\ell^2$.
\end{objectives}

\begin{requirements}
The space $\ell^2$ of the square-summable sequences of complex numbers,
multiplication operator by a bounded sequence on the space $\ell^2$,
basic properties of the multiplication operator.
\end{requirements}

\noindent
Denote by $\NaturalNumbers$ the set of the natural numbers.
The notation $\ComplexNumbers^\NaturalNumbers$ will be used
for the set of all the sequences of complex numbers, that is,
for the set of all the functions $\NaturalNumbers\to\ComplexNumbers$.

\section*{The space of the square-summable sequences (review)}

\begin{exercise}{}
Recall the definition of $\ell^2(\NaturalNumbers)$
and the definition of the inner product in $\ell^2(\NaturalNumbers)$.
We suppose that the inner product is linear with respect to the second argument.
\par\smallskip
\[
\ell^2\eqdef\ell^2(\NaturalNumbers)
\eqdef\biggl\{x\in\ComplexNumbers^\NaturalNumbers\colon\ 
\sum_{\phantom{j}}\quad \hsp{5em}\,\biggr\},
\qquad\qquad
\langle x,y\rangle \eqdef \sum_{\phantom{j\in\NaturalNumbers}}\hsp{3em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[the space of the bounded sequences of complex numbers]
\par\smallskip
\[
\ell^\infty\eqdef\ell^\infty(\NaturalNumbers)
\eqdef\biggl\{x\in\ComplexNumbers^\NaturalNumbers\colon\ \hsp{7em}\,\biggr\}.
\]
\end{exercise}

\bigskip

\clearpage
\section*{Canonical base of $\boldsymbol{\ell^2}$ (review)}

\medskip

\begin{exercise}[canonical base of $\ell^2$]
For every $n\in\NaturalNumbers$, the sequence $e_n$ is defined by:
\par\smallskip
\[
e_n\eqdef\bigl(\,\ubrace{3em}\,\bigr)_{j\in\NaturalNumbers}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$ and $n\in\NaturalNumbers$. Calculate:
\quad
$\langle e_n,x\rangle = \hsp{4em}$.
\end{exercise}

\bigskip

\begin{exercise}
It is known that $(e_n)_{n\in\NaturalNumbers}$
is an orthonormal base of $\ell^2$.
Therefore every $x\in\ell^2$ can be decomposed in the following manner:
\par\medskip
\[
x
=\sum_{n\in\NaturalNumbers}\langle\quad,\quad\rangle e_n
=\sum_{n\in\NaturalNumbers}\ubrace{2em}e_n.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $b\in\ell^2$ and $k\in\NaturalNumbers$ such that
\[
\forall j\in\NaturalNumbers\qquad \langle e_j,b\rangle = 0.
\]
Find a formula for $b$.
\end{exercise}

\vspace{20em}


\clearpage
\section*{Definition and basic properties\\of the multiplication operator (review)}

\medskip
\begin{exercise}[point-wise product of sequences]
Let $x,y\in\ComplexNumbers^\NaturalNumbers$.
The \emph{point-wise product} or \emph{component-wise product}
of the sequences $x$ and $y$ is defined by:
\par\smallskip
\[
\forall n\in\NaturalNumbers\qquad (xy)_n\eqdef \ubrace{3em},
\qquad\text{or}\qquad
xy\eqdef\bigl(\,\ubrace{3em}\,\bigr)_{n\in\NaturalNumbers}.
\]
\end{exercise}

\bigskip

\begin{defn}[definition of the multiplication operator]
Let $a\in\ell^\infty$.
The \emph{multiplication operator} by the sequence $a$
is an operator $\ell^2\to\ell^2$ defined by:
\par\smallskip
\[
\forall x\in\ell^2\qquad
M_a x \eqdef \ubrace{4em}
\]
that is,
\[
\forall x\in\ell^2\qquad\forall j\in\NaturalNumbers\qquad
(M_a x)(j)\eqdef \ubrace{4em}.
\]
\end{defn}

\bigskip

\begin{exercise}[norm of the multiplication operator]
Let $a\in\ell^\infty$. Then $M_a$ is bounded and
\par\smallskip
\[
\|M_a\|=\ubrace{4em}
\]
\end{exercise}

\bigskip

\begin{exercise}[algebraic operations with multiplication operators, review]
Let $a,b\in\ell^\infty$ and $\lambda\in\ComplexNumbers$.
Recall the formulas:\par
\begin{align*}
M_a + M_b &= \hsp{4em}
&
\lambda M_a &= \hsp{4em}
\\[3ex]
\bigl(M_a\bigr)^\ast &= \hsp{4em}
&
M_a M_b &= \hsp{4em}
\end{align*}
\end{exercise}

\bigskip

\clearpage
\section*{Invertibility and spectrum\\of the multiplication operator (review)}

\medskip

\begin{defn}
Let $a\in\ell^\infty$.
Denote by $\ClosedRange{a}$ the closure of the range of $a$.
\end{defn}

\bigskip

\begin{exercise}[criterion of invertibility of the multiplication operator]
Let $a\in\ell^\infty$.
Then the following three conditions are equivalent:
\par\medskip
\[
M_a\ \text{is invertible}
\qquad\Longleftrightarrow\qquad
\inf_{n\in\NaturalNumbers}|a_n|\,\ubrace{3em}
\qquad\Longleftrightarrow\qquad
0\notin\,\ubrace{4em}.
\]
\par\medskip\noindent
If these conditions are fulfilled, then
\par\medskip
\[
M_a^{-1}=\ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[spectrum of the multiplication operator]
Let $a\in\ell^\infty$. Then
\par\medskip
\[
\spectrum(M_a)=\ubrace{4em}.
\]
\end{exercise}

\bigskip

\clearpage
\section*{Multiplication operator and canonical base}

\medskip

\begin{exercise}
Let $a\in\ell^\infty$, $x\in\ell^2$ and $n\in\NaturalNumbers$.
Calculate:
\par\medskip
\[
\langle e_n,M_a x\rangle = \hsp{25em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$ and $x\in\ell^2$.
Prove that
\[
M_a x = \sum_{j\in\NaturalNumbers} a_j\langle e_j,x\rangle\,e_j.
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ and $j\in\NaturalNumbers$.
Calculate:
\par\medskip
\[
\langle e_j,M_a e_j\rangle=\hsp{20em}
\]
\end{exercise}

\vspace{6ex}

\begin{exercise}
Let $a\in\ell^\infty$ and the sequence $(\lambda_j)_{j\in\NaturalNumbers}$ is defined by:
\[
\forall j\in\NaturalNumbers\qquad\lambda_j\eqdef\langle e_j,M_a e_j\rangle.
\]
Express $a$ through $\lambda_j$.
\end{exercise}

\vspace{10ex}

\begin{exercise}
Let $a\in\ell^\infty$ and $j,k\in\NaturalNumbers$ such that $j\ne k$.
Calculate:
\par\medskip
\[
\langle e_j,M_a e_k\rangle=\hsp{20em}
\]
\end{exercise}

\vspace{10ex}



\clearpage
\section*{Criteria of multiplication operator on $\boldsymbol{\ell^2}$}

\medskip

\begin{exercise}
Let $k\in\NaturalNumbers$ and $A\colon\ell^2\to\ell^2$ be a bounded linear operator
such that
\[
\forall j\in\NaturalNumbers\setminus\{k\}\qquad \langle e_j,Ae_k\rangle\,=\,0.
\]
Find a formula for $Ae_k$.
\end{exercise}

\vspace{10em}

\begin{exercise}
Let $A\colon\ell^2\to\ell^2$ be a bounded linear operator.
Prove that the following conditions are equivalent:
\begin{itemize}
\item[(a)] There exists $a\in\ell^\infty$ such that $A=M_a$.
\item[(b)] There exists $a\in\ell^\infty$ such that for all $x\in\ell^2$,
\[
A x = \sum_{j\in\NaturalNumbers} a_j\langle e_j,x\rangle\,e_j.
\]
\item[(c)] For all $j,k\in\NaturalNumbers$ with $j\ne k$,
\[
\langle e_j,Ae_k\rangle\,=\,0.
\]
\end{itemize}
\end{exercise}



\label{endlabel}
\end{document}
