Math 6420 • PDE

Let \(X\) be a Banach space, let \(z\in X\) with \(z\ne0\), and let \(f\in X'\) with
\(f\ne0\). Consider the linear operator \(T:X\to X\) defined for \(x\in X\) by
\[
  Tx=f(x)z.
\]
\begin{enumerate}
\item[(a)] Briefly explain why \(T\) is compact.

\item[(b)] Use \textbf{Neumann series} to give a condition on \(f\) and \(z\) that ensures the
equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).

\item[(c)] Use the \textbf{Fredholm alternative} to find a condition on \(f\) and \(z\) that
ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).
\end{enumerate}