# Math 641-600 — Fall 2015

## Current Assignment

Assignment 10 - Due Monday, November 30.

• Read sections 3.5, 3.6, and 4.1.
• Do the following problems.

1. Section 3.4: 2(c), 6
2. Section 3.5: 1(b), 2(a)
3. (This is a variant of problem 3.4.3 in Keener.) Consider the operator $Ku(x) = \int_{-1}^1 (1-|x-y|)u(y)dy$ and the eigenvalue problem $\lambda u = Ku$.
1. Show that $K$ is a self-adjoint, Hilbert-Schmidt operator.
2. Let $f\in C[-1,1]$. If $v= Kf$, show that $-v"=2f$, $v(1)+v(-1)=0$, and $v'(1)+v'(-1)$.
3. Use the previous part to convert the eigenvalue problem $\lambda u = Ku$ into this eigenvalue problem: \left\{ \begin{align} u"+&\frac{2}{\lambda} u =0,\\ u(1)+&u(-1) =0 \\ u'(1)+ &u'(-1)=0. \end{align} \right.
4. Solve the eigenvalue above to get the eigenvalues and eigenvectors of $\lambda u = Ku$. Explain why the eigenvectors form a complete set for $L^2[-1,1]$.

4. Let K be a compact, self-adjoint operator and let M be the span of the set of eigenvectors {φj} corresponding to all eigenvalues λj ≠ 0. (Note: both M and M may be infinite dimensional.)
1. Show that M and M are both invariant under K.
2. Show that K restricted to M is compact.
3. Show that either M = {0} or that it is the eigenspace for λ = 0.
4. Show that one may choose a complete, orthonormal set from among the eigenvectors of K. (Use Proposition 2.4 in Spectral Theory for Compact Operators.)

Updated 11/20/2015.