\documentclass[12pt]{article} \usepackage{url} %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} \newcommand{\firstdate}{14.02.2018} %\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{4}{Hodge theory 4: Compact 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{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}