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\begin{document}

\mytitle[Multiplication operator on the space of square-summable sequences]%
{Multiplication operator\\on the space of square-summable sequences}

\medskip

\begin{objectives}
Study some basic properties of the multiplication operator on the space $\ell^2$.
\end{objectives}

\begin{requirements}
Matrices diagonales,
producto de matrices diagonales,
the space $\ell^2$ of the square-summable sequences of complex numbers,
the space $\ell^\infty$ of the bounded sequences of complex numbers,
the supremum and the infimum of a set of real numbers,
the supremum and the infimum of a sequence of real numbers,
the closure of a set in a metric space,
bounded linear operators,
the norm of a bounded linear operator,
the adjoint (the Hermitian conjugate) of a bounded linear operator,
the invertibility of a bounded linear operator,
the spectrum of a bounded linear operator.
\end{requirements}

\noindent
Denote by $\NaturalNumbers$ the set of the natural numbers:
$\NaturalNumbers\eqdef\{0,1,2,\ldots\}$.
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 $\NaturalNumbers\to\ComplexNumbers$.

\begin{defn}[the space of the square-summable sequences of complex numbers]
\[
\ell^2\eqdef\ell^2(\NaturalNumbers)
\eqdef\biggl\{x\in\ComplexNumbers^\NaturalNumbers\colon\ 
\underbrace{\sum_{j\quad}\quad x_j\quad\hsp{3em}}_?\,\biggr\}.
\]
The space $\ell^2$ is a Hilbert space with respect to the inner product
\[
\langle x,y\rangle \eqdef \sum_{j\in\NaturalNumbers}\overline{x_j}\,y_j.
\]
This inner product is linear with respect to the second argument.
Many authors define the inner product to be linear with respect to the first argument.
\end{defn}

\begin{defn}[the space of the bounded sequences of complex numbers]
\[
\ell^\infty\eqdef\ell^\infty(\NaturalNumbers)
\eqdef\biggl\{x\in\ComplexNumbers^\NaturalNumbers\colon\ \bigubrace{9em}\,\biggr\}.
\]
\end{defn}


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

\medskip

\begin{defn}[canonical base of $\ell^2$]
For every $n\in\NaturalNumbers$, denote by $e_n$ the sequence
\[
e_n\eqdef\bigl(\delta_{n,j}\bigr)_{j\in\NaturalNumbers}.
\]
In the following exercises we shall see
that $(e_n)_{n\in\NaturalNumbers}$ is an orthonormal base of $\ell^2$.
\end{defn}

\bigskip

\begin{exercise}
Write the sequence $e_2$:\quad
$\displaystyle e_2=\bigl(\,\ubrace{2em}\,,\,\ubrace{2em}\,,\,\ubrace{2em}\,,\ubrace{2em}\,,\,\ubrace{2em}\,,\ \ldots\ \bigr)$.
\end{exercise}

\bigskip

\begin{exercise}
Write the following sequences:
\par\smallskip
\begin{align*}
-3e_0&=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr),
\\[3ex]
7e_1&=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr),
\\[3ex]
4e_0-5e_3&=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr).
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Write the following sequences in terms of the basic elements $e_n$:
\par\medskip
\[
\bigl(0,0,0,5,0,\ldots\bigr)=\ubrace{6em},
\qquad
\bigl(2,0,-4,0,0,\ldots\bigr)=\ubrace{6em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[orthonormality of the canonical base]
Let $m,n\in\NaturalNumbers$. Calculate the product:
\par\medskip
\[
\langle e_m,e_n\rangle = \hsp{20em}
\]
\end{exercise}

\clearpage

\begin{exercise}[norm of an element of the canonical base]
Let $n\in\NaturalNumbers$. Calculate the norm of $e_n$:
\par\medskip
\[
\|e_n\|_2=\ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}[norm of a multiple of an element of the canonical base]
Let $n\in\NaturalNumbers$ and $\lambda\in\ComplexNumbers$.
Calculate the norm of $\lambda e_n$:
\par\medskip
\[
\|\lambda e_n\|_2=\ubrace{4em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$ and $n\in\NaturalNumbers$. Calculate:
\par\medskip
\[
\langle e_n,x\rangle = \hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Suppose that $x\in\ell^2$ and $\langle e_n,x\rangle=0$ for all $n\in\NaturalNumbers$.
Then
\par\medskip
\[
\ubrace{6em}
\]
According to general criteria of orthonormal basis in a Hilbert space,
it implies that the sequence $(e_n)_{n\in\NaturalNumbers}$ is an orthonormal base of $\ell^2$.
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$.
Write the decomposition of $x$ in the base $(e_n)_{n\in\NaturalNumbers}$:
\par\medskip
\[
x=\sum_{n\in\NaturalNumbers}\ubrace{4em} e_n.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x\in\ell^2$ and $m\in\NaturalNumbers$ such that
\[
\forall n\in\NaturalNumbers\setminus\{m\}\qquad \langle e_n,x\rangle = 0.
\]
What can you say about $x$?
Write $x$ in terms of some basic elements $e_n$ and some inner products.
\end{exercise}

\vspace{6em}

\clearpage
\section*{Multiplication operator: definition and examples}

\medskip
\begin{defn}[component-wise product of sequences]
Let $x,y\in\ComplexNumbers^\NaturalNumbers$.
Denote by $xy$ the \emph{component-wise product} of the sequences $x$ and $y$:
\[
\forall n\in\NaturalNumbers\qquad (xy)_n\eqdef x_n y_n.
\]
In other words,
\[
xy\eqdef\bigl(x_n y_n\bigr)_{n\in\NaturalNumbers}.
\]
\end{defn}

\bigskip

\begin{defn}[multiplication operator]
Let $a\in\ell^\infty$.
Define $M_a\colon\ell^2\to\ell^2$ by
\[
M_a x \eqdef a x,
\]
that is,
\[
\forall x\in\ell^2\qquad\forall j\in\NaturalNumbers\qquad
(M_a x)_j\eqdef a_j x_j.
\]
\end{defn}

\bigskip

\begin{exercise}
Let $a\colon\NaturalNumbers\to\ComplexNumbers$ be defined by
\[
a\eqdef\bigl(1+(-1)^n\bigr)_{n\in\NaturalNumbers}
=\bigl(\,\ubrace{2em}\,,\,\ubrace{2em}\,,\,\ubrace{2em}\,,\ubrace{2em}\,,\,\ubrace{2em}\,,\ \ldots\ \bigr).
\]
Consider
\[
x\eqdef\left(\,\frac{1}{n+1}\,\right)_{n\in\NaturalNumbers}
=\biggl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \biggr).
\]
Calculate $M_a x$:
\[
M_a x = \biggl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \biggr).
\]
\end{exercise}

\bigskip

\clearpage
\section*{Multiplication operator\\
acting on the canonical base of $\boldsymbol{\ell^2}$}

\medskip

\begin{exercise}
Let $a\in\ell^\infty$ be a general bounded sequence:
\[
a\eqdef\bigl(a_0,a_1,a_2,a_3,a_4,\ldots\bigr).
\]
Consider $x=e_3$:
\[
e_3\eqdef\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr).
\]
Calculate $M_a e_3$:
\[
M_a e_3=\bigl(\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\hsp{2em},\ \ldots\ \bigr).
\]
\par\bigskip\noindent
Write the answer in terms of some basic elements:
\quad $\displaystyle M_a e_3=\ubrace{4em}$.
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$ and $n\in\NaturalNumbers$.
Write a general formula for $M_a e_n$ and prove it.
\end{exercise}

\vspace{25ex}

\begin{exercise}
Let $a\in\ell^\infty$ and $n\in\NaturalNumbers$.
Calcule the $\ell^2$-norm of the sequence $M_a e_n$:
\par\medskip
\[
\|M_a e_n\|_2 =\ubrace{5em}.
\]
\end{exercise}



\clearpage
\section*{Supremum and infimum (review)}

\medskip

\begin{defn}
Denote by $\ExtendedRealNumbers$ the \emph{extended real line}
$\RealNumbers\cup\{-\infty,+\infty\}$ with the canonical order.
The additional elements $-\infty$ and $+\infty$ satisfy
\[
\forall\alpha\in\RealNumbers\qquad-\infty<a,\qquad\qquad\qquad
\forall\alpha\in\RealNumbers\qquad\alpha<+\infty,\qquad\qquad
-\infty<+\infty.
\]
\end{defn}

\medskip

\begin{exercise}[upper bound of a set of real numbers]
Let $A\subset\ExtendedRealNumbers$ and $\beta\in\ExtendedRealNumbers$.
Recall the definition:
\par\medskip
\[
\beta\quad\text{is an upper bound of}\quad A
\qquad\Longleftrightarrow\qquad
\ubrace{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $A\subset\ExtendedRealNumbers$ and $\gamma\in\ExtendedRealNumbers$. Then,
\par\medskip
\[
\gamma\quad\text{is not an upper bound of}\quad A
\qquad\Longleftrightarrow\qquad
\ubrace{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $A\subset\ExtendedRealNumbers$ and $\beta\in\ExtendedRealNumbers$.
What does mean the phrase ``$\beta$ is the supremum of $A$''?
Write the definition using the concept of the upper bounds.
(It is known that for every $A\subset\ExtendedRealNumbers$
there exists a unique supremum in $\ExtendedRealNumbers$.
We shall not prove this fact here.)
\par\medskip
\[
\beta=\sup(A)
\qquad\Longleftrightarrow\qquad
\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $A\subset\ExtendedRealNumbers$ and $\beta\in\ExtendedRealNumbers$.
What does mean the phrase ``$\beta$ is the supremum of $A$''?
Write the answer in terms of the quantifications $\forall$ and $\exists$
and some of the inequalities $<,>,\le,\ge$,
without mentioning explicitly the concept of upper bounds.
\par\medskip
\[
\beta=\sup(A)
\qquad\Longleftrightarrow\qquad
\begin{cases}
1)\quad\forall \alpha\in A\quad\ldots\hsp{10em}\vertstrut
\\[1ex]
2)\quad\forall \gamma<\beta\quad\ldots\hsp{5em}\vertstrut
\end{cases}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x=(x_n)_{n\in\NaturalNumbers}$ be a sequence in $\RealNumbers$ and $\beta\in\ExtendedRealNumbers$.
What does mean the phrase ``$\beta$ is the supremum of the sequence $x$''?
Write the answer in terms of the quantifications $\forall$ and $\exists$
and some of the inequalities $<,>,\le,\ge$.
\par\medskip
\[
\beta=\sup_{n\in\NaturalNumbers}x_n
\qquad\Longleftrightarrow\qquad
\begin{cases}
1)\quad\forall n\in\NaturalNumbers\quad\ldots\hsp{5em}\vertstrut
\\[1ex]
2)\ \hsp{15em}\vertstrut
\end{cases}
\]
\end{exercise}

\clearpage

\section*{Multiplication operator: boundness and norm}

\medskip

\begin{exercise}[definition of the supremum norm, review]
Let $a\in\ell^\infty$. Put
\[
\nu\eqdef\|a\|_\infty=\sup_{n\in\NaturalNumbers}|a_n|.
\]
Write the definition of $\nu$ using the quantifications $\forall$ and $\exists$
and some of the inequalities $<,>,\le,\ge$,
without mentioning explicitly the concept of upper bounds.
\end{exercise}

\vspace{14ex}

\begin{exercise}[boundness of the multiplication operator]
Let $a\in\ell^\infty$ and $x\in\ell^2$. Prove that $\|M_a x\|_2\le\|a\|_\infty\|x\|_2$.
\par\medskip
\[
\|M_a x\|_2^2 = \|\,\ubrace{3em}\,\|_2^2
=\sum_{j\in\NaturalNumbers}\ubrace{4em}\hsp{15em}
\]
\end{exercise}

\vspace{8ex}

\begin{exercise}
Let $a\in\ell^\infty$. Put $\nu\eqdef\|a\|_\infty$.
Given a number $\gamma<\nu$,
construct a sequence $x\in\ell^2$ such that
\[
\|x\|_2=1\qquad\text{and}\qquad \|M_a x\|_2^2 \ge \gamma.
\]
\end{exercise}

\vspace{10ex}

\begin{exercise}[norm of a bounder linear operator, review]
Let $T\colon\ell^2\to\ell^2$ be a bounded linear operator.
Recall the definition of the norm of $T$:
\par\medskip
\[
\|T\|\eqdef\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}[norm of the multiplication operator]
Let $a\in\ell^\infty$. Calculate $\|M_a\|$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\clearpage
\noindent
In the previous exercises we computed the norm of $\|M_a\|$
using the definition of supremum.
There another way to prove one part of this result.

\medskip

\begin{exercise}[monotonicity of the image of a subset under a function]
Let $f\colon X\to Y$ be an arbitrary function and $A\subset B\subset X$.
Compare the images of $A$ and $B$ under the function $f$
(choose $\subset$ or $\supset$):
\par\medskip
\[
f(A)\ \ubrace{2em}\ f(B).
\]
\end{exercise}

\medskip

\begin{exercise}
Let $a\in\ell^\infty$.
Compare the following sets (put $\subset$ or $\supset$):
\par\medskip
\begin{align*}
\bigl\{e_n\colon\quad n\in\NaturalNumbers\bigr\}
&\ \ubrace{3ex}\ 
\bigl\{x\in\ell^2\colon\quad\|x\|_2=1\bigr\}
\\[2ex]
\bigl\{M_a e_n\colon\quad n\in\NaturalNumbers\bigr\}
&\ \ubrace{3ex}\
\bigl\{M_a x\colon\quad x\in\ell^2,\quad\|x\|_2=1\bigr\},
\\[2ex]
\bigl\{\|M_a e_n\|_2\colon\quad n\in\NaturalNumbers\bigr\}
&\ \ubrace{3ex}\
\bigl\{\|M_a x\|_2\colon\quad x\in\ell^2,\quad\|x\|_2=1\bigr\}.
\end{align*}
\end{exercise}

\medskip

\begin{exercise}[monotonicity of the supremum of a set]
Let $A,B\in\ExtendedRealNumbers$ such that $A\subset B$.
Compare the supremums of $A$ and $B$:
\par\medskip
\[
\sup(A)\ \ubrace{2em}\ \sup(B).
\]
\end{exercise}

\medskip

\begin{exercise}
Applying the results of two previous exercises
deduce an inequality between $\|M_a\|$ and $\|a\|_\infty$.
\end{exercise}

\bigskip\bigskip\bigskip


\clearpage
\section*{Linear operations with multiplication operators}

\medskip

\begin{exercise}[linear operations with bounded linear operators, review]
Recall the definition of the algebraic operations with bounded linear operators.
If $T\colon\ell^2\to\ell^2$ and $S\colon\ell^2\to\ell^2$
are bounded linear operators
and $\lambda\in\ComplexNumbers$, then for all $x\in\ell^2$
\par\medskip
\[
(T+S)(x)\eqdef \ubrace{6em},\qquad\qquad
(\lambda T)(x)\eqdef\ubrace{6em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a,b\in\ell^\infty$. Calculate $M_a+M_b$.
\end{exercise}

\begin{solution}
Let $x$ be an arbitrary element of $\ell^2$.
Then
\par\medskip
\[
(M_a+M_b)(x)=\hsp{15em}.
\]
\par\bigskip\noindent
On the other hand,
\par\medskip\noindent
\[
M_{a+b}(x)=\hsp{15em}
\]
\par\bigskip\noindent
Since $x$ is arbitrary,
\par\medskip
\[
M_a+M_b=\ubrace{4em}.
\qedhere
\]
\end{solution}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$ and $\lambda\in\ComplexNumbers$.
Calculate $\lambda M_a$.
\end{exercise}

\vspace{15ex}

\clearpage
\section*{Product of multiplication operators}

\medskip

\begin{exercise}[product of bounded linear operators, review]
Recall the definition of the product of bounded linear operators.
If $T\colon\ell^2\to\ell^2$ and $S\colon\ell^2\to\ell^2$
are bounded linear operators,
then for all $x\in\ell^2$
\par\medskip
\[
(TS)(x)\eqdef \ubrace{6em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $a,b\in\ell^\infty$. Calculate $M_a M_b$.
\end{exercise}

\vspace{14ex}


\section*{Adjoint operator to the multiplication operator}

\medskip

\begin{exercise}[adjoint to a bounded linear operator, review]
Recall the definition of the adjoint to a bounded linear operator.
Let $T\colon\ell^2\to\ell^2$ is a bounded linear operator.
A bounded linear operator $S\colon\ell^2\to\ell^2$ is called
\emph{adjoint} to $T$ if for all $x,y\in\ell^2$
\par\medskip
\[
\langle x,Ty\rangle = \ubrace{6em}.
\]
It is known that there exist a unique operator $S$ satisfying this condition.
This operator is denoted by $T^\ast$.
\end{exercise}

\bigskip

\begin{exercise}
Let $a\in\ell^\infty$.
Calculate $M_a^\ast$, that is, the adjoint to the operator $M_a$.
\end{exercise}

\begin{solution}
For all $x,y\in\ell^2$,
\par\medskip
\[
\langle x,M_a y\rangle = \hsp{25em}
\]
\par\vspace{8ex}\noindent
Conclusion:\quad $M_a^\ast = \ubrace{4em}$.
\end{solution}



\label{endlabel}
\end{document}
