Math 6420 • PDE

Define the operator \(K\) on \(L^2[0,1]\) by
\[
  Ku(x)=\int_0^1 k(x,y)u(y)\,dy,
\]
for \(x\in[0,1]\), where \(k(x,y)\) is a real-valued, bounded, continuous function, with
\(k(x,y)>0\).
\begin{enumerate}
\item[(a)] Find the \(L^2\) adjoint \(K^*\) for \(K\), and find a condition on \(k\) which guarantees
that \(K\) is self-adjoint.

\item[(b)] Describe the properties of the set
\[
  \{\lambda\in\mathbb{C}:(K-\lambda I)u=0\text{ for some }u\ne0\}
\]
(cardinality, bounds, etc..).

\item[(c)] Consider the problem \(Ku=f\). Let \(X\) be the set of all \(f\in L^2[0,1]\) such that a
unique solution \(u=K^{-1}f\) exists. Can we be assured that there is a constant \(C\) such that
\(\lVert K^{-1}f\rVert\le C\lVert f\rVert\), for all \(f\in X\)? Why or why not?
\end{enumerate}