# 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.

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

- Let K be a compact, self-adjoint operator and let M be the span
of the set of eigenvectors {φ
_{j}} corresponding to all
eigenvalues λ_{j} ≠ 0. (Note: both M and
M^{⊥} may be infinite dimensional.)
- Show that M and M
^{⊥} are both invariant under K.
- Show that K restricted to M
^{⊥} is compact.
- Show that either M
^{⊥} = {0} or that it is the
eigenspace for λ = 0.
- 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.