Math 6420 • PDE

Let \(K:H\to H\) be a compact linear operator on the infinite-dimensional Hilbert
space \(H\).
\begin{enumerate}
\item[(a)] Prove that \(K^{-1}\), if it exists, is not bounded.
\item[(b)] Given \(T>0\), let \(H=L^2(0,T)\). For \(u\in H\), define
\[
  (Ku)(t)=\int_0^T e^{-st}u(s)\,ds,\qquad 0\le t\le T.
\]
Prove that \(K:H\to H\) is a compact operator. What are the consequences of this fact for
the utility of the Laplace transform as a numerical method?
\end{enumerate}