Math 6420 • PDE

Let \((\lambda_j)\) be a sequence real numbers with \(\lambda_j\ne1\) for all \(j\) and
\(\lambda_j\to1\). Consider the operator \(T:\ell^2\to\ell^2\) defined for
\((\xi_j)\in\ell^2\) by
\[
  T(\xi_j)=(\lambda_j\xi_j).
\]
\begin{enumerate}
\item[(a)] Find the spectrum \(\sigma(T)\), point spectrum \(\sigma_p(T)\), continuum
spectrum \(\sigma_c(T)\), residual spectrum \(\sigma_r(T)\) and resolvent set \(\rho(T)\).

\item[(b)] Give a condition on the \(\lambda_j\) for \(T\) to be invertible.
\end{enumerate}