Math 6420 • PDE

Define
\[
  Ku(x)=\sum_{n=0}^{10}\left(\int_{-1}^{1}e^{-ny}u(y)\,dy\right)x^n,
  \qquad x\in[-1,1].
\]
\begin{enumerate}
\item[(a)] Find the adjoint \(K^*\) of the operator \(K\).

\item[(b)] Assuming \(\lambda\ne0\) is fixed and real, use the Fredholm Alternative to
characterize the set of functions \(g\) such that the problem \((K-\lambda I)u=g\) on \([-1,1]\)
has a solution.

\item[(c)] From the structure of the operator \(K\), what can be said about the set
\[
  \sigma_p=\{\lambda\in\mathbb{C}:(K-\lambda I)u=0\text{ has a solution }u\ne0\}?
\]
There is no need to actually calculate \(\sigma_p\).
\end{enumerate}