Math 6420 • PDE

Let \(a\in\ell^2\setminus\{0\}\). Consider the bounded linear operator
\(T:\ell^2\to\ell^2\) defined by
\[
  Tu=\langle u,a\rangle a.
\]
\begin{enumerate}
\item[(a)] Show that the operator \(T\) is compact and self-adjoint.
\item[(b)] Use the Fredholm Alternative Theorem to find a simple condition on \(a\)
guaranteeing that the equation
\[
  (T-I)u=v
\]
has a unique solution \(u\) for all \(v\in\ell^2\). What is the solution?
\end{enumerate}