Math 6710 • Applied Linear Operators and Spectral Methods

Let \(a\) and \(b\) be non-zero (but possibly complex) elements in the complex Hilbert space
\(\ell^2\). Consider the bounded linear operator \(T : \ell^2 \to \ell^2\) defined by

\[T u = \langle u, b \rangle a.\]

\begin{enumerate}

\item
Use the Fredholm Alternative Theorem to find a simple condition on \(a\) and \(b\) guar-
anteeing that the equation

\[(T - I)u = v\]

has a unique solution \(u\) for all \(v \in \ell^2\).


\item
Find the Hilbert adjoint operator \(T^* : \ell^2 \to \ell^2\).


\item
Given \(T : X \to Y\) a bounded linear operator, recall that the adjoint operator
\(T^\times : Y'\to X'\) is defined

\[(T^\times g)(x) = g(Tx).\]

Find the adjoint operator \(T^\times : \ell^2 \to \ell^2\). Hint: Express \(g\) and \(T^\times g\) using the Riesz
Representation Theorem.


\item
Use the Fredholm Alternative Theorem to give conditions on \(v_0\) guaranteeing that

\[(T - I)u = v_0\]

has a unique solution \(u\). When it exists, find the solution.

\end{enumerate}