\documentclass[12pt]{article} \usepackage{url} %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) \newcommand{\version}{version 2.0,\ \ 15.05.2018} \newcommand{\firstdate}{21.02.2018} %\addtolength{\topmargin}{-12mm} %\addtolength{\textheight}{25mm} %\addtolength{\oddsidemargin}{-10mm} %\addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Hogde theory, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 Á\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}