Homework 2

Updated until Sep 19; due Sep 26 in class.

tags: 224a 2020

1. Prove that an operator
   $T:X\to Y$ between two Banach spaces is continuous with respect to their norms if and only if it is bounded.
2. Reed and Simon VII.9a:Prove that if
   $A=A^{\ast }\in \mathcal{L}(H)$ then
   
   $$\Vert A\Vert =\underset{\Vert x\Vert =1}{sup}(x,Ax).$$
   Also do 9b: give an example showing that this is not true in general for non self-adjoint operators.
3. Prove the polarization identity (Reed and Simon Problem II.4a). Use this to show that
   $A\ge 0$ implies that
   $A=A^{\ast }$.
4. Reed and Simon VI.7.
5. Show that for any subspace
   $M\subset H$,
   $(M^{\perp }{)}^{\perp }=\bar{M}$.
6. Read the proof of the BLT theorem in Reed and Simon, Theorem I.7.
7. Reed and Simon VI.14a.
8. Reed and Simon VI.25. Assume that the underlying measure is Lebesgue measure on
   $(0,1{)}^2$.
9. Reed and Simon VI.41.
10. Prove that if
    $T_n$ is a Cauchy sequence of compact operators, then
    $T_n\to T$ for some compact
    $T$. You may use the fact that every compact operator is the norm limit of finite rank operators. hint: use a diagonalization (in the sense of logic, not linear algebra) argument.
11. Prove that if
    $T\ge 0\in K(H)$ with eigenvalues
    ${\lambda }_1\ge {\lambda }_2\ge \dots ,$ then
    
    $${\lambda }_k=\underset{V:dim(V)=k}{max}\underset{x\in V\setminus \{0\}}{min}(x,Tx)/\Vert x{\Vert }^2.$$
    This is half of the Courant-Fisher theorem.
12. Prove that if
    $T$ is trace class then there are Hilbert-Schmidt operators
    $T_1,T_2$ such that
    $T=T_1{T}_2$ (hint: use the SVD). Mimicking the proof in class for positive operators, use this to show that for such
    $T$, the trace
    $${\mathrm{tr}}_{\varphi }(T)=\sum _n({\varphi }_n,T{\varphi }_n)$$
    is invariant of the choice of ONB
    $\varphi$. Conclude that
    $\mathrm{tr}(T)=\sum _n{\lambda }_n$ for all self-adjoint trace class
    $T$.
13. Give an example of a compact operator which is not trace class.