# Homework 3, Math 224a Fall 2020
### Updated until Oct 8; due Oct 15.
###### tags: `224a 2020`

1. Show that if $T\in L(H)$ is finite rank, then
$$ \dim\ker(I-T)=\dim\ker(I-T^*)<\infty.$$
1. Show that a general second order ODE
$$ a_2(x)u''(x)+a_1(x)u'(x)+a_0(x)u(x)=f(x)$$
with $a_2(x)\neq 0, u\in C^2[a,b]$
has the same solution space as one of the form
$$ -(p(x)u'(x))' + q(x)u(x) = F(x)$$
This is called *Sturm-Liouville Form*. How does this transformation affect the boundary conditions?
7. Read Section 2 of http://lie.math.okstate.edu/~binegar/4233/4233-l17.pdf, which gives an example of how Sturm-Liouville problems arise naturally from physical PDE via separation of variables. Note that these are exampels of *singular* SL problems, which refers to any problem not satisfying all of the hypotheses: (1) $[a,b]$ is a finite interval (2) $p(x)$ is continuous and bounded on $(a,b)$ (3) $p(x)\neq 0$ on $[a,b]$. Similar conclusions can be reached for such problems, but it is much more involved and delicate.
1. Consider the left shift operator $L(x_1,x_2,\ldots) = L(x_2,x_3,\ldots)$ on $\ell^2(\mathbb{N})$. Compute the continuous spectrum of $L$ and $R=L^*$.
2. Reed and Simon VI.10
3. Reed and Simon VI.11
4. Reed and Simon VII.3
5. Reed and Simon VII.8
8. Reed and Simon VII.14
10. Reed and Simon VII.22b
11. Reed and Simon VII.24
12. Consider the operator $(A\psi)_n = \psi_{n-1}+\psi_{n+1}$ on $\ell^2(\mathbb{Z})$. We showed via Fourier series that $A$ is unitarily equivalent to $M_{2\cos(x)}$ on $L^2([0,2\pi),dx/2\pi)$ and has spectrum $[-2,2]$.
Let $(Bf)(x)=xf(x)$ on $L^2([-2,2],dx)$.
(a) Calculate the point and continuous spectrum of $A$ and $B$.
(b) Show that $\|f(A)\|=\|f(B)\|$ for every $f\in C([-2,2])$.
(c ) Show that $A$ is unitarily equivalent to $B\oplus B$, i.e., there is a decomposition $L^2([-2,2],dx)=H_1\oplus H_2$ and an isometry $U$ such that $UAU^{-1}=B\oplus B$ on $H_1\oplus H_2$. Find the isometry.