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%version 0.3,\ \   13.02.2018 taken from 4
%version 1.0,\ \   20.02.2018 now I will send it to KA
%version 1.1,\ \   24.02.2018 Mnogo ochepyatok ot Ivana Frolova
%version 1.2,\ \   07.03.2018 Kolya Konovalov nashel oshibku. 
%version 2.0,\ \   15.05.2018 Vasya Krylov i Lesha Gorinov nashli ser'eznuyu oshibku (takzhe i v lekciyakh)



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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{5}{Hodge theory 5: Fredholm operators}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\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{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
$C^{-1}|v| < |E(v)| < C |v|$ 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 closure of the image of an open ball
cannot contain a neighbourhood of 0.
\ez

\exercise[!]\label{_compa_never_surj_Exercise_}
Prove that a compact operator of Hilbert spaces is never 
surjective.
\ez

\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



\exercise
Let $F:\; H_1\arrow H_2$ be a continuous operator
on Hilbert spaces, and $F^*:\; H_2\arrow H_1$ 
an operator which satisfies $(x, F(h)) = (F^*(x), h)$
for any $x\in H_2, h\in H_1$.
\enum
\ite[!] Prove that such an operator always exists.
\ite Prove that it is uniquely defined.
\ite Prove that $F^{**}=F$.
\ite[*] Prove that $F^*$ is compact if and only if $F$ is
compact.
\ee
\ez

\definition
In this case $F^*$ is called {\bf adjoint} to $F$.
\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 $A(V_1)^\bot\cap A(V_2)^\bot=\ker A^*$.
Prove that $(A^*)^{-1}(V_1) = A(V_2)^\bot$.
\ite Suppose that $A$ is bijective. Prove that $A^*$
is injective, and $(A^*)^{-1}$ applied to the decomposition
$V=V_1 \oplus V_2$ gives a decomposition $W=A(V_1)^\bot\oplus A(V_2)^\bot$.
\ite[!] Suppose that $A$ is bijective. Prove that
$A(V_1)^\bot+ A(V_2)^\bot= W$ 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{Fredholm operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
A continuous map $F:\; H_1\arrow H_2$
of Hilbert spaces is called {\bf Fredholm}
if its kernel is finite-dimensional, image
closed, and cokernel finite-dimensional.
\ed

\exercise
Let  $F:\; H_1\arrow H_2$ be a Fredholm operator.
Prove that $F$ induces a homeomorphism
from $H_1/\ker F$ to $\im F$.
\ez

\definition
An operator on Hilbert spaces
{\bf has finite rank} if its image
is finite-dimensional.
\ed

\exercise �\label{_Fredho_invertible_Zadacha_}
Let $F:\; H_1\arrow H_2$ be a Fredholm operator.
\enum
\ite Prove that there exists a Fredholm operator
$F_1:\; H_2\arrow H_1$ such that  $\Id_{H_1}- FF_1$ 
has finite rank.
\ite Prove that in this case
$\Id_{H_2}- F_1F$ has finite rank.
\ee
\ez


%\exercise
%Let $F:\; H_1\arrow H_2$,  $G:\; H_2\arrow H_3$
%be continuous maps.
%\enum
%\ite Prove that a composition of Fredholm maps is
%Fredholm.
%\ite[!] Suppose that $FG$ and $GF$ is Fredholm. Prove that
%$F$ and $G$ are Fredholm.
%\ite Suppose that $FG$ is Fredholm.  Prove that
%$F$ and $G$ are Fredholm or find a counterexample.
%\ee
%\ez

\exercise
Let $K:\; H \arrow H$ be a compact operator. 
\enum 
\ite Prove that
the image of $\Id_H + K$ is closed.
\ite[!] Prove that $\Id_H + K$ is Fredholm.
\ee
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Calkin algebra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $R$ be a $k$-algebra. {\bf Two-sided ideal}
in $R$ is a subspace $I\subset R$ such that
$IR=I$ and $RI=I$. 
\ed

\exercise
Let $A$ be the algebra of continuous operators
$\Hom(H,H)$ from a Hilbert space to itself. Prove that
the space $K$ of compact operators is a two-sided ideal in
$A$.
\ez

\definition 
The quotient algebra $\Hom(H,H)/K$ is called {\bf the
  Calkin algebra} of $H$.
\ed

\exercise[**]
Prove that the Calkin algebra admits a continuous
homeomorphism to a closed subalgebra of 
$\Hom(H,H)$, or find a counterexample.
\ez

\exercise[**]
Prove that the Calkin algebra does not contain non-trivial
closed two-sided ideals.
\ez

\exercise
Let $A$ be an invertible operator on a Hilbert space $H$.
Prove that $A + B$ is invertible for any $B \in \End(H)$
such that $\|B\| < \|A^{-1}\|^{-1}$.
\ez

\exercise[!]\label{_Fredholm_open_Exercise_}
Let $A$ be a Fredholm operator on a Hilbert space $H$,
and $A_0:\; H/\ker A \arrow \im A$ the corresponding
invertible operator. Prove that $A + B$ is Fredholm for any $B \in \End(H)$
such that $\|B\| < \|A_0^{-1}\|^{-1}$.
\ez

\exercise\label{_add_f_rank_Fredholm_Exercise_}
Let $A$ be a Fredholm operator on a Hilbert space $A$,
and $R$ a finite rank operator. Prove that $A+R$ is
Fredholm.
\ez

\exercise[!]
Let $A$ be a Fredholm operator on a Hilbert space $H$,
and $K$ a compact operator. Prove that $A+K$ is
Fredholm.
\ez

\hint
Use Exercises \ref{_Fredholm_open_Exercise_}
and \ref {_add_f_rank_Fredholm_Exercise_}.
\eh

\exercise[*]
Let $A$ be a Fredholm operator on a Hilbert space $A$.
Prove that there exists a compact operator $K$
such that $A+K$ is invertible, or find a counterexample.
\ez

\exercise[!]
Let $A$ be an operator on a Hilbert space $H$.
Prove that $A$ is Fredholm if and only if
its class in Calkin algebra is invertible in Calkin algebra.
\ez

\exercise[**]
Prove that the Calkin algebra $C$ is equipped with
a norm, that is, a function $\nu:\; C \arrow \R^{\geq 0}$
such that $\nu(c) > 0 $ for all non-zero $c$,
$\nu(x+y) \leq \nu(x) + \nu(y)$ and $\nu(xy) \leq
\nu(x)\nu(y)$.
\ez


\end{document}
