Math 642-600 Current Assignment — Spring 2021

Assignment 3 - Due Monday, 4/25/2021.

• Read section 7.2.1 and 7.2.3.
• Do the following problems.
  1. Let $L$ be a bounded linear operator on $\mathcal H$; that is, $L \in {\mathcal B}(\mathcal H)$. Show that $\rho(L)$ contains the exterior of the disk $\{\lambda \in {\mathbb C} \colon |\lambda| \le ||L||\}$. (Consequently, $\sigma(L)$ is contained in the disk.)

  2. Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find the norm and adjoint $L^\ast$. (You may use problem 1 above.)

  3. Let $L=L^\ast$ and suppose that for every $f\in D_L$ we have that $0\le \langle Lf,f\rangle \le 1$. Show that $\sigma(L)\subseteq [0,1]$.

  4. Let $Lu=-u''$, $D_L=\{u,u''\in L^2[0,1]\}: u(0)=u(1)=0\}$.
     1. You are given that the spectrum of $L$ is $\sigma(L)=\{n^2\pi^2\}_{n=1}^\infty$. Show that, for $\lambda \not\in \sigma(L)\cup \mathbb R_{\le 0}$, the Green's function for $L-\lambda I$ is given by \[ G(x,y;\lambda)=-\frac{1}{\sqrt{\lambda}\sin(\sqrt{\lambda})} \begin{cases} \sin(\sqrt{\lambda}(1-y))\sin(\sqrt{\lambda} x) & 0 \le x \le y \le 1\\ \sin(\sqrt{\lambda}(1-x))\sin(\sqrt{\lambda} y) & 0\le y \le x \le 1\end{cases}\]
     2. Let $\sigma_n=\{n^2\pi^2\}$. Show that $P_{\sigma_n}f= b_n\sin(n\pi x)$, where $b_n = 2\int_0^1 \sin(\pi y) f(y)dy$. (Hint: Use the residue theorem to find $\int_C G(x,y\lambda)d\lambda$, where $C$ is a simple closed curve with the point $\sigma_n$ inside.)

     3. Show that in $L^2[0,1]$ we have $f(x)=\lim_{N\to \infty}\sum_{n=1}^N b_n \sin(n\pi x)$.

  5. Find the Green's function for the operator $Lu = -u''$, with $D_L=\{u,u''\in L^2[0,\infty): u'(0)=0\}$. Use it and Stone's formula for the spectral family of $L$ to derive the Fourier cosine transform pair: \[ F(\xi)= \frac{2}{\pi}\int_0^\infty f(x)\cos(\xi x)dx \ \text{and}\ f(x)= \int_{0}^\infty F(\xi)\cos(\xi x)d\xi \]