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%version 1.0,\ \   12.02.2018 
%version 1.1,\ \   14.02.2018 melkie ispravleniya ot K. A. 
%version 1.2,\ \   17.02.2018 Hilbert cube, errors in 
%                  4.6, 4.7 found by Ivan Frolov.
%version 1.3,\ \   07.03.2018  more errors from Kolya Konovalov


\newcommand{\version}{version 1.3,\ \   07.03.2018}
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\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{4}{Hodge theory 4: Compact operators}

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{\scriptsize
  {\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra stuff to work.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Compact operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
A set is called {\bf precompact} if its closure is compact.
A subset $B$ of a topological vector space is called
{\bf bounded} if for any neighbourhood $U$ of 0,
there exists a constant $\lambda >0$ such that $\lambda U$
contains $B$. An operator on topological vector spaces
is called {\bf compact} if the image of any bounded set
is precompact.
\ed

\exercise
Prove that an open set in a Hilbert space is never precompact.
\ez

\exercise[*]  (Riesz theorem)
Prove that an open set in a normed infinite-dimensional
vector space is never precompact.
\ez

\exercise[**]
Construct an infinite-dimensional locally convex topological
vector space $H$ such that any bounded subset of $H$
is precompact.
\ez

\definition
Let $H$ be a Hilbert space, and $e_1, ..., e_n, ...$ an 
orthonormal basis. Then any point in $H$ can be expressed
as $\sum \alpha_i e_i$ with $\alpha_i \in \R$ and
$\sum |\alpha_i|^2 < \infty$. Let $\{x_i\}$ be a sequence
of positive numbers with $\sum x_i^2 <0$
{\bf The Hilbert cube} is the set of all 
vectors $\sum \alpha_i e_i\in H$ satisfying
$|\alpha_i| \leq x_i$.
\ed

\exercise Prove that the Hilbert
cube is compact.
\ez

\exercise
Let $K:\; H\arrow H_1$ be a operator
on Hilbert spaces. 
\enum
\ite Suppose that $K$ is compact.
Prove that for any $\epsilon >0$ there exists
a subspace $W \subset H$, closed and of finite
codimension, such that the operator norm of 
the restriction $K\restrict W$ satisfies
$\left\|K\restrict W\right\| < \epsilon.$
\ite[*] Prove the converse: any operator with this property is compact.
\ee
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Banach inverse map theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\exercise
%Let $E;\; V_1\arrow V_2$ be a bijective linear map
%of normed spaces. Prove that $E$ is a homeomphism
%if and only if there exist a constant $C>0$ such that
%$C^{-1}|v| < |å(v)| < C |v|$ for any $v\in V_1$.
%\ez
%
%
%\exercise[!]
%Let $E= E_1+K$ be a bijective linear map
%on Hilbert spaces, with $E_1$ a homeomorphism
%and $K$ compact. Prove that $E$ is a homeomorphism.
%\ez
%
%\hint Use the previous exercise.
%\eh
%
%\exercise
%Let $K:\; H_1 \arrow H_2$ be a compact operator
%with zero kernel; consider its action on the projective spaces
%$K_{\Bbb P}:\; {\Bbb P}H_1 \arrow {\Bbb P}H_2$.
%Prove that the image of $K_{\Bbb P}$ is compact.
%\ez
%
%\exercise\label{_compa_never_surj_Exercise_}
%Prove that a compact operator of Hilbert spaces is never 
%surjective.
%\ez
%
%\hint Use the previous exercise.
%\eh
%
%\exercise
%Let $H$ be a Hilbert space, and $H^*$ the space of continuous
%linear functionals on $H$. Consider the map $\tau:\; H\arrow H^*$
%defined by the scalar product $g$, $x \arrow g(x, \cdot)$.
%Prove that $\tau$ is an isomorphism.
%\ez
%
%
%\definition
%Let $A:\; H \arrow H_1$ be a continuous operator on Hilbert spaces.
%The {\bf adjoint operator} is the corresponding map on dual
%spaces $A^*:\; H_1^* \arrow H^*$. Identifying spaces and their
%duals, we consider $A^*$ as a map from $H_1$ to $H$.
%\ed
%
%\exercise
%Let $A:\; H \arrow H_1$ be a continuous operator on Hilbert spaces.
%Prove that $\ker A^*$ is equal to $(\im A)^\bot$.
%\ez
%
%\exercise\label{_direct_su_ima_Exercise_}
%Let $A:\; V \arrow W$ be a continuous operator on Hilbert spaces,
%and $V= V_1 \oplus V_2$ be an orthogonal decomposition.
%Denote by $A_1$ the map which is equal to $A$ on $V_1$
%and 0 on $V_2$ and $A_2$ the map which is equal to $A$ on $V_2$
%and 0 on $V_1$.
%\enum
%\ite Prove that $\ker A_i^* = A(V_i)^\bot$.
%\ite Prove that $A(V_1)^\bot\cap A(V_2)^\bot=\ker A^*$.
%\ite Suppose that $A$ is bijective. Prove that $A^*$
%is injective. Prove that $A^*(\ker A_1^*+\ker A_2^*)=V$.
%\ite[!] Suppose that $A$ is bijective. Prove that
%$A(V_1)^\bot\cup A(V_2)^\bot= W$ and 
%$A(V_1)^\bot\cap A(V_2)^\bot=0$.
%Prove that the spaces $A(V_i)$ are closed in $W$.
%\ee
%\ez
%
%\exercise[!]\label{_non_con_Exercise_}
%Let $A:\; V \arrow W$ be a continuous bijective
%operator on Hilbert spaces. Prove that either $A^{-1}$
%is continuous or there exists a subspace
%$V_1\subset V$ such that $A\restrict{V_1}$ is compact.
%\ez
%
%\exercise[!]
%Let $A$ be a continuous bijective
%operator on Hilbert spaces.
%Prove that $A^{-1}$ is continuous.
%\ez
%
%\hint Use
%Exercises \ref{_compa_never_surj_Exercise_},
%\ref{_direct_su_ima_Exercise_} and 
%\ref{_non_con_Exercise_}. \eh
%
%
%\exercise[**]
%Let $A:\; V_1 \arrow V_2$ be a bijective,
%continuous map of locally convex topological vector spaces.
%Prove that $A^*$ is injective. Prove that $A^*$ is surjective
%or find a counterexample.
%\ez
%
%\exercise[**]
%Let $A:\; V_1 \arrow V_2$ be a bijective,
%continuous map of Banach spaces.
%Prove that $A^{-1}$ is continuous.
%\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weak topology}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $x_i \in H$ be a sequence of points in a Hilbert space $H$.
We say that $x_i$ {\bf  weakly converges} to $x\in H$
if for any $z\in H^*$ one has $\lim_i \langle x_i, z\rangle = 
\langle x, z\rangle$.
{\bf Weak topology} on $H$ is topology with subbase
of open sets given by $\lambda^{-1}(]a, b[)$,
where $\lambda$ is any continuous functional.
\ed

\exercise
Prove that an open ball in a Hilbert space 
is not open in the weak topology.
\ez

\exercise
Let $y(i)= \alpha_j(i) e_j$ be a sequence of points in 
a unit ball in the Hilbert space with orthonormal basis $e_i$.
Prove that $y(i)$ converges to $y=\sum \alpha_i y_i$ in weak topology
if and only if $\lim_i  \alpha_j(i) =\alpha_j$ for all $j$.
\ez

\exercise
Prove that the closure of the 
unit ball in a Hilbert space is compact in weak topology.
\ez

\exercise[!]
Let $A:\; H \arrow H$ be a continuous operator on
a Hilbert space. Prove that $A$ is compact if and only
if it maps weakly converging sequences to converging
sequences, and their weak limits to limits.
\ez

\exercise
Let $K:\; H \arrow H$ be a compact operator on
a Hilbert space, and $B$ a closure of the unit ball.
Prove that $K(B)$ is compact.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Von Neumann spectral theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf Spectrum} $\Spec(A)$ of a continuous operator $A:\; H \arrow H$
on a Banach space is the set of all $\lambda\in \C$ such that
$A-\lambda\Id$ is not invertible.
\ed

\exercise[!]
Let $A$ be a  continuous operator $A:\; H \arrow H$
on a Banach space.
Prove that its spectrum is closed in $\C$.
\ez

\exercise[*]
Prove that the spectrum of any operator is non-empty.
\ez

\exercise
Find an operator on a Hilbert space 
with spectrum a unit circle in $\C$.
\ez

\exercise[**]
Let $\epsilon$ be a positive number,
and $K$ a compact operator on a Hilbert space.
Prove that the set
$\{ \lambda\in \Spec K\ \ | \ \ |\lambda| > \epsilon\}$
is finite.
\ez

\definition
Let $K:\; H \arrow H$ be an  operator
on a Banach space, and 
$H_{\lambda}:=\bigcup_n \ker(K-\lambda \Id_H)^n$.
The space $H_{\lambda}$ is called
{\bf the root space} of $H$.
\ed

\exercise
Let $K:\; H \arrow H$ be a compact  operator
on Hilbert spaces, and $H_\lambda$ its root spaces. Prove that
\enum
\ite for any $\lambda\neq 0$, the space
$H_{\lambda}$ is finite-dimensional.
\ite[*] Prove that $\Spec(K)$ is countable.
\ite[**]
Prove that $H$ is the closure of
$\bigoplus_{\lambda\in \Spec(K)} H_{\lambda}$, or find a counterexample.
\ee
\ez

\exercise[**]
Construct an injective compact operator
$K$ with \\ $\Spec(K)=\{0\}$, or find a counterexample.
\ez

\definition
An operator $A:\; H \arrow H$ on a Hilbert space is called
{\bf self-adjoint} if $A^*=A$.
\ed

\exercise[!]
Let $A:\; H \arrow H$ be a compact self-adjoint operator.
Prove that there exists $x\in H$ such that
$\frac{|A(x)|}{|x|}= \sup_h \frac{|A(h)|}{|h|}$.
\ez

\hint Use the weak compactness of the closed ball.
\eh

\exercise
Let $A:\; H \arrow H$ be a self-adjoint operator, and 
$z$ a unit vector such that 
$\frac{|A(z)|}{|z|}= \sup_h \frac{|A(h)|}{|h|}$.
\enum
\ite Prove that $\|A^2\|=\|A\|^2$, and 
$\frac{|A^2(z)|}{|z|}= \sup_h \frac{|A^2(h)|}{|h|}$.
\ite[!] Prove that $z$ is an eigenvector for $A^2$.
\ee
\ez

\hint
Prove that $g(A^2(z), z)=|z||A^2(z)| \cos \phi$,
where  $\phi$ is an angle between
$z$ and $A^2(z)$.
\eh


\exercise[!]
Let $A:\; H \arrow H$ be a compact self-adjoint operator,
and $z$ a unit vector such that 
$\frac{|A(z)|}{|z|}= \sup_h \frac{|A(h)|}{|h|}$.
Prove that $z^\bot$ is $A^2$-invariant, and use
this to show that $A^2$ is diagonalizable.
\ez

\hint Use the previous exercise.
\eh

\exercise[!]
Let $A:\; H \arrow H$ be a compact self-adjoint operator
on a Hilbert space.
Prove that $A$ is diagonalizable 
in an orthonormal basis.
\ez

\hint Use the previous exercise.
\eh


\end{document}
